http://modelsofpa.info/api.php?action=feedcontributions&user=Rkossak&feedformat=atom
Peano's Parlour - User contributions [en]
2024-03-29T12:43:42Z
User contributions
MediaWiki 1.24.4
http://modelsofpa.info/index.php?title=Integer_parts_of_real_closed_fields&diff=277
Integer parts of real closed fields
2013-06-20T14:03:36Z
<p>Rkossak: </p>
<hr />
<div>== Tennenbaum property ==<br />
<br />
Je\v{z}\'abek and Ko{\l}odziejczyk show <cite> jezabekkolodziejczyk2013:real </cite> that if $A$ is a model of Open Induction and the real closure of $A$ is recursively saturated, then $+^A$ and $\leq^A$ cannot be both recursive, and they ask whether it is true that if every unbounded real closed field with<br />
an integer part satisfying an arithmetic theory $T$ is recursively saturated, then $T$ has no recursive<br />
nonstandard models.<br />
<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Integer_parts_of_real_closed_fields&diff=276
Integer parts of real closed fields
2013-06-20T14:00:09Z
<p>Rkossak: </p>
<hr />
<div>== Tennenbaum property ==<br />
<br />
Je\v{z}\'abek and Ko{\l}odziejczyk show <cite> jezabekkolodziejczyk2013:real </cite> that if $A$ is a model of Open Induction and the real closure of $A$ is recursively saturated, then $+^A$ and $\leq^A$ cannot be both recursive, and they ask whether it is true that if every unbounded real closed field with<br />
an integer part satisfying an arithmetic theory $T$ is recursively saturated, then $T$ has no recursive<br />
nonstandard models.</div>
Rkossak
http://modelsofpa.info/index.php?title=Integer_parts_of_real_closed_fields&diff=275
Integer parts of real closed fields
2013-06-20T13:58:45Z
<p>Rkossak: </p>
<hr />
<div>== Tennenbaum property ==<br />
<br />
Je\v{z}\'abek and Ko{\l}odziejczyk show <cite> jezabekkolodziejczyk2013:real <\cite> that if $A$ is a model of Open Induction and the real closure of $A$ is recursively saturated, then $+^A$ and $\leq^A$ cannot be both recursive, and they ask whether it is true that if every unbounded real closed field with<br />
an integer part satisfying an arithmetic theory $T$ is recursively saturated, then $T$ has no recursive<br />
nonstandard models.</div>
Rkossak
http://modelsofpa.info/index.php?title=Integer_parts_of_real_closed_fields&diff=274
Integer parts of real closed fields
2013-06-20T13:53:12Z
<p>Rkossak: </p>
<hr />
<div>== Tennenbaum property ==<br />
<br />
Je\v{z}\'abek and Ko{\l}odziejczyk show <\cite> jezabekkolodziejczyk2013:real <\cite></div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=273
Library
2013-06-20T13:52:03Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#engstrom2008:anote bibtex=@article {engstrom2008:anote,<br />
AUTHOR = {Engstr{\"o}m, Fredrik},<br />
TITLE = {A note on standard systems and ultrafilters},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {824--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C20 03C57)},<br />
MRNUMBER = {2444270 (2009i:03030)},<br />
DOI = {10.2178/jsl/1230396749},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396749},<br />
}<br />
<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gao2009:invariant bibtex=@book {gao2009:invariant,<br />
AUTHOR = {Gao, Su},<br />
TITLE = {Invariant descriptive set theory},<br />
SERIES = {Pure and Applied Mathematics (Boca Raton)},<br />
VOLUME = {293},<br />
PUBLISHER = {CRC Press},<br />
ADDRESS = {Boca Raton, FL},<br />
YEAR = {2009},<br />
PAGES = {xiv+383},<br />
ISBN = {978-1-58488-793-5},<br />
MRCLASS = {03-02 (03E15 22E05 28A05 28D05 37A20 54A35)},<br />
MRNUMBER = {2455198 (2011b:03001)},<br />
MRREVIEWER = {{\'E}tienne Matheron},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of Peano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#jezabekkolodziejczyk2013:real bibtex=@article {MR3017281,<br />
AUTHOR = {Je{\v{r}}{\'a}bek, Emil and Ko{\l}odziejczyk, Leszek<br />
Aleksander},<br />
TITLE = {Real closures of models of weak arithmetic},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {52},<br />
YEAR = {2013},<br />
NUMBER = {1-2},<br />
PAGES = {143--157},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03F30 (03C62)},<br />
MRNUMBER = {3017281},<br />
DOI = {10.1007/s00153-012-0311-x},<br />
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.1007/s00153-012-0311-x},<br />
}<br />
<br />
<br />
#kanovei1998:onstar bibtex=@incollection {kanovei1998:onstar,<br />
AUTHOR = {Kanovei, Vladimir},<br />
TITLE = {On "star'' schemata of Kossak and Paris},<br />
BOOKTITLE = {Logic Colloquium '96 (San Sebasti\'an)},<br />
SERIES = {Lecture Notes Logic},<br />
VOLUME = {12},<br />
PAGES = {101--114},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1998},<br />
MRCLASS = {03F35 (03C62)},<br />
MRNUMBER = {1674945 (2000f:03179)},<br />
MRREVIEWER = {Roman Kossak},<br />
} <br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1984:saturation bibtex=@article {kaufmannschmerl1984:saturation,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Saturation and simple extensions of models of Peano<br />
arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {27},<br />
YEAR = {1984},<br />
NUMBER = {2},<br />
PAGES = {109--136},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {763736 (85j:03051)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1016/0168-0072(84)90008-3},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(84)90008-3},<br />
}<br />
<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of Peano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A Galois correspondence for countable recursively saturated<br />
models of Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to NF},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the $\omega$-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
#kossak1991:exercises bibtex=@inproceedings{kossak1991:exercises,<br />
title={Exercises in ‘back-and-forth’},<br />
author={Kossak, R.},<br />
booktitle={Proceedings of the Nineth Easter Conference on Model Theory, Gosen},<br />
year={1991}<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#kossak2004:anote bibtex=@article {kossak2004:anote,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on a theorem of Kanovei},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {43},<br />
YEAR = {2004},<br />
NUMBER = {4},<br />
PAGES = {565--569},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2060400 (2005a:03073)},<br />
DOI = {10.1007/s00153-004-0218-2},<br />
URL = {http://dx.doi.org/10.1007/s00153-004-0218-2},<br />
}<br />
<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of PA},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakparis1992:subsets bibtex=@article {kossakparis1992:subsets,<br />
AUTHOR = {Kossak, Roman and Paris, Jeffrey B.},<br />
TITLE = {Subsets of models of arithmetic},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {32},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {65--73},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03H15 (03C62)},<br />
MRNUMBER = {1186468 (94e:03066)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1007/BF01270396},<br />
URL = {http://dx.doi.org/10.1007/BF01270396},<br />
}<br />
<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of Peano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in Mathematical Logic---London '70 (Bedford<br />
Coll., London, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the 1-3-1 lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of Peano<br />
arithmetic},<br />
BOOKTITLE = {Logic Year 1979--80 (Proc. Seminars and Conf. Math.<br />
Logic, Univ. Connecticut, Storrs, Conn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
#schmerl2002:automorphism bibtex=@article {schmerl2002:automorphism,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Automorphism groups of models of Peano arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {67},<br />
YEAR = {2002},<br />
NUMBER = {4},<br />
PAGES = {1249--1264},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (06A05 20A15)},<br />
MRNUMBER = {1955236 (2005b:03090)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1190150283},<br />
URL = {http://dx.doi.org/10.2178/jsl/1190150283},<br />
}<br />
<br />
<br />
#schmerl2004:diversity bibtex=@article {schmerl2004:diversity,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Diversity in substructures},<br />
BOOKTITLE = {Nonstandard models of arithmetic and set theory},<br />
SERIES = {Contemp. Math.},<br />
VOLUME = {361},<br />
PAGES = {145--161},<br />
PUBLISHER = {Amer. Math. Soc.},<br />
ADDRESS = {Providence, RI},<br />
YEAR = {2004},<br />
MRCLASS = {03H15 (03C62 05D10 06E05)},<br />
MRNUMBER = {2089281 (2005g:03117)},<br />
MRREVIEWER = {M. Yasuhara},<br />
DOI = {10.1090/conm/361/06593},<br />
URL = {http://dx.doi.org/10.1090/conm/361/06593},<br />
}<br />
<br />
#schmerl2008:nondiversity bibtex=@article {schmerl2008:nondiversity,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Nondiversity in substructures},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {1},<br />
PAGES = {193--211},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2387939 (2008k:03074)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1208358749},<br />
URL = {http://dx.doi.org/10.2178/jsl/1208358749},<br />
}<br />
<br />
<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of Peano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of Peano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#schmerl2012:theAutomorphism bibtex=@incollection {schmerl2012theAutomorphism,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {The automorphism group of a resplendent model},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {51},<br />
YEAR = {2012},<br />
NUMBER = {5-6},<br />
PAGES = {647--649},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03B25 (03C50 20F10)},<br />
MRNUMBER = {2945573},<br />
DOI = {10.1007/s00153-012-0288-5},<br />
URL = {http://dx.doi.org/10.1007/s00153-012-0288-5},<br />
}<br />
<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in Peano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Integer_parts_of_real_closed_fields&diff=272
Integer parts of real closed fields
2013-06-20T13:47:32Z
<p>Rkossak: Created page with " == Tennenbaum property == Je\v{z}\'abek and Ko{\l}odziejczyk show"</p>
<hr />
<div><br />
== Tennenbaum property ==<br />
<br />
Je\v{z}\'abek and Ko{\l}odziejczyk show</div>
Rkossak
http://modelsofpa.info/index.php?title=Model_theory_of_weak_fragments_of_arithmetic&diff=271
Model theory of weak fragments of arithmetic
2013-06-20T13:44:21Z
<p>Rkossak: Created page with "Integer parts of real closed fields."</p>
<hr />
<div>[[Integer parts of real closed fields]].</div>
Rkossak
http://modelsofpa.info/index.php?title=Errata&diff=270
Errata
2013-04-07T11:05:45Z
<p>Rkossak: </p>
<hr />
<div>Errata for Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006.<br />
<br />
<br />
== Corrections ==<br />
<br />
<br />
p. vii, l. -3: Should be: ``type $\omega+(\omega^{*}+\omega)\rho$..." <br />
<br />
p. 21, Exercise 1.14.5: $b\in M$ should be $c\in M$<br />
<br />
p. 22, l. -12: [70] should be [71] <br />
<br />
p. 23, l. 19: [209] refers to a paper of A. Wilkie, it should refer to George Wilmer's thesis which is missing in the references.<br />
<br />
p. 48, l. 3: the second instance of 2.3.2 should be 2.3.4. <br />
<br />
p. 57, Proof of Corollary 3.1.17. Should be ``Theorem 3.1.16..." To finish the argument one also needs to evoke Theorem 2.1.1.<br />
<br />
p. 85, Exercise 3.6.17: Delete the hint.<br />
<br />
p.98, l.-3: In the first line of the displayed formula, $x\leq x'$ should be replaced by $x < x'$ . The same change should be made in the second line.<br />
<br />
p.106, paragraph starting with l. -9: Replace the third sentence with: ``Let $A = M (D)$ be the set of meet-irreducibles." Then at the end of the paragraph replace the last part of the last sentence starting with ". . . containing $0_D$. . ." with ``. . . containing $1_D$ , which is $\{x \in M (D) : r \leq x\}$." <br />
<br />
p. 109, Theorem 4.5.5: Definition of $\alpha^n$ is missing. Before the theorem insert: ``If $\alpha: L\rightarrow {\rm Eq}(A)$ and $\beta: L\rightarrow {\rm Eq}(B)$ are representations, the product $\gamma=\alpha\beta$ is a function $\gamma:L\rightarrow {\rm Eq}(A\times B)$ defined by $((a_1,b_1),(a_2,b_2))\in \gamma(r)$ iff $(a_1,a_2)\in\alpha(r)$ and $(b_1,b_2)\in \beta(r)$. Then $\alpha^1=\alpha$ and $\alpha^{n+1}=\alpha^n\alpha$."<br />
<br />
p.110/l.15: Should be: $A =\{1, 2, 3, 4\}$. <br />
<br />
p.110/l.18: Delete “$f (5) = 1,$”. <br />
<br />
p. 118, l. 12: Near end of last line of the Theorem, delete the extraneous “) ”<br />
<br />
p.128/Lemma 4.7.4: Interchange “$\alpha_1 : D_1\rightarrow {\rm Eq}(A_1 )$” and “$\alpha_2 : D_2\rightarrow {\rm Eq}(A_2 )$”. <br />
<br />
p.152/l.21: Expression at end of line should be: $F_n (x_0 , x_1 , \dots , x_{n-1} , u) = $. <br />
<br />
p. 159, l. -1. $f:{\mathbb Q}\rightarrow{\mathbb P}$<br />
<br />
p. 159. Lemma 6.2.5. Delete the last part on the last sentence starting with ``and if..."<br />
<br />
p. 177, l. -12: $f(t_{1})$ should be $f(t_{2})$ <br />
<br />
p. 179, l. -12: [132] should be [130] <br />
<br />
p. 179, l. -8: should [166] be J. Schmerl, Peano models with many generic classes. Pacific Journal of Mathematics 46, 523-536 (1973). This entry is missing in the references.<br />
<br />
p. 182, Proposition 7.1.3: $I$ is supposed to be just a cut, but it should be also assumed to be closed under addition and<br />
multiplication<br />
<br />
p. 209, Theorem 8.4.7: In the assumptions, $a$ is definable without parameters in $(M,I)$.<br />
<br />
p. 246. Lemma 9.4.3 (1). One has to assume that $I$ and $J$ are not infimum and supremum of the same gap.<br />
<br />
p. 292, Question 17 is garbled. It should say: Suppose $M$ is countable recursively saturated and $X\in{\mathcal P}(M)\setminus{\rm Def}(M)$ is such that ${\rm Th}(M,X)\in {\rm SSy}(M)$. Is there a countable recursively saturated $N$ such that $M\prec_{end} N$, and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv (M,X)$?<br />
<br />
<br />
== Typos ==<br />
<br />
<br />
p. vi, l. 7: fragment = fragment of<br />
<br />
p. vi, l. -17: While, = While <br />
<br />
p. vi, l, -9: proves = and proves<br />
<br />
p. vi, l. -2 f.b.: purpose = purpose of <br />
<br />
p. vii, l. -9: delete ``and"<br />
<br />
p. vii, l. -2 f: For every countable model $M$, the isomorphism type if its reducts...<br />
<br />
p. viii, l. -14: the Chapter 7 = Chapter 7 <br />
<br />
p. 1, l. 3: delete ``a'' <br />
<br />
p. 3, l. -11: is $B$ = $B$ is <br />
<br />
p. 3, l. -4: $y\in M$ = $y\in X$ <br />
<br />
p. 8, l. -5: $\neg \Theta(y)$ = $\neg \theta(y)$ <br />
<br />
p. 14, Definition 1.9.1: ``partial inductive satisfaction class''should be in italics <br />
<br />
p. 14, l. -9 and 7: instead of Con(Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$) one should read "Th($\frak{A},\bar{a})+\exists X\Psi(X,\bar{a})$ is consistent'' <br />
<br />
p. 19, l. 14: proof the = proof of the <br />
<br />
p. 24, l. 13: [69] . = [69]. <br />
<br />
p. 24, l. 17: Kotlarski and Kaye = Kaye and Kotlarski <br />
<br />
p. 25, l. 1: models arithmetic = models of arithmetic <br />
<br />
p.27, l.22: In Corollary 2.1.4, last word of first line should be “end”, not “and”. <br />
<br />
p. 32, l. 7: the contrary = to the contrary <br />
<br />
p. 33, l. -5: replaced = replaced by <br />
<br />
p. 33, l. 4 f.b.: single = a single <br />
<br />
p. 48, l. 15: Wikie = Wilkie<br />
<br />
p. 49, l. -6: each of which = each element of which<br />
<br />
p. 51, l. 7: $n_{i}$ should be $n$ (similarly for p. 51, l. 9) <br />
<br />
p. 51, l. 17: unboounded = unbounded <br />
<br />
p. 52, l. 21: infinte = infinite<br />
<br />
p. 53, l. 2: type = types <br />
<br />
p. 54, l. 6: do not look = do not look the same <br />
<br />
p. 55, l. 11: the another = another <br />
<br />
p. 57, l. 14: realizes = realize <br />
<br />
p. 57, l. -10: Should be ``Theorem 3.1.16..."<br />
<br />
p. 57, l. -7; Should be ``$a\in {\rm Scl}(b)$...<br />
<br />
p. 58, l. 10: since $p(x)$, is rare = , since $p(x)$ is rare <br />
<br />
p. 60, l. 13: gap(b)$\setminus$gap(b) = gap(b) <br />
<br />
p. 60, all $M$'s in the lines -8,-7, and -6 should be $M_0$'s.<br />
<br />
p. 61, l. 3 $s(x)$ should be $s'(x)$.<br />
<br />
p. 61, l. -5 : The are = There are <br />
<br />
p. 67, l. -18: types. = types, <br />
<br />
p. 67, l. -17: Then = then <br />
<br />
p. 80, l. 7: $F$ = $f$ <br />
<br />
p. 85, Exercise 3.6.25: are = There are<br />
<br />
p. 87, Definition after 3.6.38 should be $M\prec N$, not $M\prec_{end} N$.<br />
<br />
p. 100, l. 18: Should be $K=M_{1}(b)= M\star K$<br />
<br />
p. 107, l. 3: Theorem 4.5.32 = Corollary 4.5.32 <br />
<br />
p. 107, l. -16: representation <br />
<br />
p. 107, l. -3: $r$ should be $k$.<br />
<br />
p. 108, l. 23: be list all = be a list of all <br />
<br />
p. 109. l. 4: Should be ``The inclusion $M_{i\land j}\subseteq M_i\cap M_j$..."<br />
<br />
p. 122, l. -15: used get = used to get <br />
<br />
p. 124, l. 14: hold for = hold for<br />
<br />
p. 125, l. -17: the tree = on the tree<br />
<br />
p. 128, l. -9: representations<br />
<br />
p. 131, l. 3: (see Exercise 4.8.2)<br />
<br />
p. 133, l. 8: characterization <br />
<br />
p. 135, l. 7: introduced <br />
<br />
p. 142, l. 1: there some = there exists some <br />
<br />
p. 145, l. 2: instead = instead of ? <br />
<br />
p. 151, l. 17: is not be = is not <br />
<br />
p. 152, l. 17: off = of <br />
<br />
p. 156, l. -18: linearly set = linearly ordered set <br />
<br />
p. 157, l. 17: Theorems 5.3.4 = Theorem 5.3.4 <br />
<br />
p. 162, l. 10,12: Theorem 6.2.6 = Lemma 6.2.6 <br />
<br />
p. 166, l. -10: functions<br />
<br />
p. 168, l. -6: would no = would be no <br />
<br />
p. 169, second line of Theorem 6.4.3: $(M,X)_{X\in{\cal G}}$<br />
<br />
p. 171, l. 10: the proof = of the proof <br />
<br />
p. 173. First line of Theorem 6.4.8: $|M|\leq \kappa$.<br />
<br />
p. 177, l. -14: possibilities <br />
<br />
p. 177, l. -5: not = are not <br />
<br />
p. 179, l. -14: independently <br />
<br />
p. 184, l. -4: Propositional = Proposition <br />
<br />
p. 189, l. -11: Theorem 2.2.8 = Theorem 2.2.16<br />
<br />
p. 191, l.-8: Should be $b\in T^K$<br />
<br />
p. 192, l. 8: if = of <br />
<br />
p. 194, l. -6: Theorembut = Theorem but <br />
<br />
p. 197, l. 12: model IA = model of IA <br />
<br />
p. 200, l. -10: which = which is <br />
<br />
p. 201, l. -12: models = models of <br />
<br />
p. 203, l. 10: $ a_{\frak{A}} = \frak{a}_{\frak{A}}$ <br />
<br />
p. 206, l. 10: model = models <br />
<br />
p. 209, Fourth line of the proof of Corollary 8.4.6: $b_1,b_2=b_0,b_1$<br />
<br />
p. 214, l. -17: than least = than the least <br />
<br />
p. 221, l. -12: delete ``a''<br />
<br />
p. 222, l. 5: Corollary 3.2.4 = Lemma 3.2.4 <br />
<br />
p. 225, l. 7: maximal = a maximal <br />
<br />
p. 227, l. 12: Corollary 8.1.2 = Proposition 8.1.3. Delete the statement in parenthesis.<br />
<br />
p. 227, l. -10: Back-and-forth, = Back-and-forth <br />
<br />
p. 228, l. 20 f.b.: Frederike = Friederike <br />
<br />
p. 229, l. 9: proof the = proof of the <br />
<br />
p. 229, l. -14: the question mark appears ``upside down'' <br />
<br />
p. 233, l. 12: aid = and <br />
<br />
p. 234, l. 8: Proposition 9.1.3 = Lemma 9.1.3 <br />
<br />
p. 235, l. 2: models = model <br />
<br />
p. 239, l. 13: index if = index of <br />
<br />
p. 250, l. -6 f: realizes = realize <br />
<br />
p. 253, l. -7: of countable = of a countable <br />
<br />
p. 254, l. 14: definition <br />
<br />
p. 267, l. -12: devoted the = devoted to the <br />
<br />
p. 268, l. 6: delete ``an'' <br />
<br />
p. 276, l. -1: Use previous = Use the previous <br />
<br />
p. 280, l. 11: theory = theory of<br />
<br />
p. 284, l. 5: is get = is to get <br />
<br />
Reference [36]: G\"{o}tenborg = G\"{o}teborg <br />
<br />
Reference [43]: add ``{\it Mathematical Logic and Foundations of Set Theory} (Proc. Internat. Colloq., Jerusalem, 1968)'' <br />
<br />
Reference [54]: add ``Volume 1292 of {\it Lecture Notes in Mathematics}'' <br />
<br />
Reference [83]: of 619 = 619 of <br />
<br />
Reference [85]: delete ``(1983)''; this reference should appear after reference [88] <br />
<br />
Reference [109]: od = of <br />
<br />
Reference [113]: of pa = of ${\rm PA}$ <br />
<br />
Reference [120]: add `` in Automorphisms of first-order structures, R. Kaye, D. Macpherson (eds.)'' <br />
<br />
Reference [148]: {\bf CIII} = {\bf 103} <br />
<br />
Reference [153]: characterization <br />
<br />
Reference [164]: add ``Volume 859 of {\it Lecture Notes in Mathematics}'' <br />
<br />
Reference [167]: add ``Stud. Logic Found. Math., 120'' <br />
<br />
Reference [172]: add ``Lecture Notes Logic, 12'' <br />
<br />
Reference [188]: Unmglichkeit = Unm\"{o}glichkeit; vollstndigen = vollst\"{a}ndigen<br />
<br />
Reference [189]: abzhlbar = abz\"{a}hlbar; Fundam. = Fund. <br />
<br />
Reference [199]: add ``Volume 834 of {\it Lecture Notes in Mathematics}'' <br />
<br />
Reference [204]: poljak-r\"{o}dl = Poljak-R\"{o}dl <br />
<br />
Reference [206]: the name of the journal is usually abbreviated as ``Algebra Logic Appl.'' <br />
<br />
Reference [208]: complete = complete models<br />
<br />
Reference [212]: add ``Proceedings of the International Congress of Mathematicians''</div>
Rkossak
http://modelsofpa.info/index.php?title=Uncountable_models_with_interesting_second-order_properties&diff=268
Uncountable models with interesting second-order properties
2013-02-18T19:39:06Z
<p>Rkossak: /* Rigid models */</p>
<hr />
<div>== Friedman's 14th problem ==<br />
<br />
Let $T$ be a completion of PA. Let ${\rm Ot}(T)$ be the spectrum of order types of nonstandard models of $T$. <br />
In <cite> friedman1975:onehundred </cite> Harvey Friedman asked: Does ${\rm Ot(T)}$ depend on $T$?<br />
<br />
Shelah has recently announced that the answer is negative, but there is no paper available yet. See Sh[924] for an interesting effort in the opposite direction.<br />
<br />
== Rather classless models ==<br />
<br />
<br />
$X\subseteq M\models PA$ is a ''class'' if for all $a\in M$, $\{x\in X: x<a\}$ is definable (coded) in $M$.<br />
$M$ is ''rather classless'' if each class of $M$ is definable.<br />
<br />
Using the fact that every model of $PA$ has a conservative elementary end extension <cite> gaifman1976:models </cite>, Schmerl <cite> schmerl1981:recursively </cite> proved that for each cardinal $\kappa$ such that ${\rm cf}(\kappa)>\aleph_0$, there are $\kappa$-like rather classless models of $PA$. A model is ''$\kappa$-like'' if it is of cardinality $\kappa$ and each of its proper initial segments is of smaller cardinality.<br />
<br />
Kaufmann <cite> kaufmann1977:arather </cite>, assuming $\lozenge$, proves that there are recursively saturated $\aleph_1$-like rather classless models. Later Shelah showed that $\lozenge$ can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in <cite> hodges1985:building </cite>: Prove the existence of rather classless recursively saturated models of $PA$ in cardinality $\aleph_1$ without assuming diamond at any stage of the proof.<br />
<br />
== Jonsson models ==<br />
<br />
A model $\mathfrak B$ is ''Jonsson'' if $|{\mathfrak B}|>\aleph_0$ and for every ${\mathfrak A}\prec {\mathfrak B}$, if $|{\mathfrak A}|=|{\mathfrak B}|$, then ${\mathfrak A}={\mathfrak B}$.<br />
<br />
Gaifman <cite> gaifman1976:models </cite> and Knight <cite> knight1976:hanf </cite> independently showed that there are Jonsson models of $PA$. <br />
. <br />
Jonsson models $M$ of $PA$ of cardinality $\aleph_1$ are either $\aleph_1$-like or are ''short''. A model $M$ of $PA$ is short there is an $a\in M$ such that the Skolem closure of $a$ is cofinal in $M$. Each known Jonsson model realizes uncountably many complete types. <br />
<br />
Kossak has asked: Is there an $\aleph_1$-like Jonsson model $M\models PA$ such that $|\{{\rm tp}(a): a\in M\}|=\aleph_0$? <br />
<br />
If $M\models PA$ is $\aleph_1$-like and recursively saturated, then $|\{{\rm tp}(a): a\in M\}|=\aleph_0$, but $M$ is not Jonsson. Therefore, another related question is: Is there a ''weakly Jonsson model'' $M\models PA$, i.e. a recursively saturated model $M\models PA$ such that for every recursively saturated $K\prec M$, if $|K|=|M|$, then $K=M$? The problem was posed in <cite> kossak1995:four </cite>.<br />
<br />
== Rigid models ==<br />
<br />
There are $\aleph_1$-like rigid models of $PA$. Two different constructions are given in <cite> kossakschmerl1991:minimal </cite> and <cite> koskot1992:game </cite> <br />
<br />
Problem: Are there rigid recursively saturated $M\models PA$ such that ${\rm cf}(M)=\aleph_0$?<br />
<br />
== Lofty models ==<br />
<br />
A model $M$ of PA is '''lofty''' if there is $e \in M$ such that whenever $a \in M$ and <br />
$\Phi = \{\varphi_i(a,x) : i < \omega\}$ is a recursive set of formulas that is finitely satisfiable <br />
in $M$, then there is a definable $D \subseteq M$ such that $M \models |D| \leq e$ and <br />
$\Phi \cup \{x \in D\}$ is finitely satisfiable in $M$. This notion was defined and studied by <br />
Kaufmann and Schmerl in <cite> kaufmannschmerl1984:saturation </cite> (and also in <cite> kaufmannschmerl1987:remarks </cite>), <br />
where the following was proved: <br />
If $M$ is a countable model of PA, then $M$ is lofty iff $M$ has a simple elementary extension (i.e. generated by a single element) that is recursively saturated. The question was asked whether <br />
the modifier "countable" could be eliminated. Further progress was made in <br />
<cite> kaufmannschmerl1987:remarks </cite>, but the question remains open.<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automoprhism_groups&diff=254
Automoprhism groups
2013-02-06T22:34:12Z
<p>Rkossak: Created page with "Automorphism groups in general. Automorphisms groups of countable recursively saturated models."</p>
<hr />
<div>[[Automorphism groups in general]].<br />
<br />
[[Automorphisms groups of countable recursively saturated models]].</div>
Rkossak
http://modelsofpa.info/index.php?title=Peano%27s_Parlour&diff=253
Peano's Parlour
2013-02-06T22:31:28Z
<p>Rkossak: </p>
<hr />
<div>This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. <br />
<br />
[[Cuts in models of PA and independence results]].<br />
<br />
[[Recursively saturated, resplendent models, and saturated models]].<br />
<br />
[[Standard systems and the Scott set problem]].<br />
<br />
[[Lattices of elementary substructures]].<br />
<br />
[[Uncountable models with interesting second-order properties]].<br />
<br />
[[Nonstandard satisfaction classes]].<br />
<br />
[[Complexity and classification of countable models]].<br />
<br />
[[Automoprhism groups]].<br />
<br />
[[Model theory of strong fragments of arithmetic]].<br />
<br />
[[Model theory of weak fragments of arithmetic]].<br />
<br />
[[Model theoretic methods in proof theory]].<br />
<br />
The last chapter of ''The structure of models of Peano arithmetic'' by Kossak and Schmerl has a list of 20 questions with comments and references. Here is an [[errata]] for the book.<br />
<br />
== Contributing to Peano's Parlour ==<br />
<br />
All registered members can contribute to problems/solutions/corrections/remarks. To register, send an email to the webmaster Victoria Gitman at vgitman@nylogic.org. To contribute without registering, send an email to Victoria Gitman or to Roman Kossak at rkossak@nylogic.org.</div>
Rkossak
http://modelsofpa.info/index.php?title=Lattices_of_elementary_substructures&diff=245
Lattices of elementary substructures
2013-02-06T19:10:53Z
<p>Rkossak: </p>
<hr />
<div>For $M\models PA$ let ${\rm{Lt}(M)}=(\{K: K\prec M\},\prec)$ and for $M\prec N$, let ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$.<br />
<br />
In general, the lattice problem is: Which lattices can be represented as ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$, for some $M\prec N$?<br />
<br />
<br />
There is a vast literature on the problem and many special cases remain open. Here are basic references: <cite> gaifman1976:models, mills1979:substructure, paris1972:onmodels, paris1977:modelsof, schmerl1978:extending, wilkie1977:onmodels </cite>.<br />
Chapter 4 of <cite> kossakschmerl:modelsofpa </cite> is devoted to the lattice problem.<br />
<br />
<br />
== Finite lattices ==<br />
<br />
Is every finite lattice lattice a substructure lattice of a model of $PA$? <br />
<br />
By a result of Schmerl the answer is positive for all ${\bf M}_n$, where $n=q+1$ or $n=q+2$ and $q$ is a power of a prime. ${\bf M}_n$ is the lattice with a top element, bottom element, and $n$ incomparable elements in between.<br />
The simplest lattice for which the problem is open is ${\bf M}_{16}$.<br />
<br />
<br />
Every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M)$ is isomorphic to the pentagon lattice ${\bf N}_5$ <cite> wilkie1977:onmodels </cite>, but no $M\models PA$ at all has an elementary end extension such that ${\rm Lt}(N/M) \cong {\bf M}_3$ <cite> paris1977:modelsof </cite>.<br />
<br />
Schmerl has asked: What finite lattices $L$ are such that<br />
every $M\models PA$ has an elementary end extension $N$ such that ${\rm}Lt(N/M)<br />
\cong L$? What finite lattices $L$ are such that every countable $M\models PA$ has an elementary end extension $N$ such that<br />
${\rm Lt}(N/M) \cong L$?<br />
<br />
<br />
== First-order theory of ${\rm Lt}(N/M)$ ==<br />
<br />
Suppose $M_1\prec_{cof} N_1$, $M_2\prec_{cof} N_2$, and $(N_1,M_1)\equiv (N_2,M_2)$. Is ${\rm Lt}(N_1/M_1)$ elementarily equivalent to ${\rm Lt}(N_2/M_2)$? The problem is motivated by a result from <cite> kossakschmerl2012:oncofinal </cite> showing that the answer is positive for lattices of finitely generated interstructures.<br />
<br />
<br />
== Diversity and Nondiversity ==<br />
<br />
A model is '''diverse''' if no two distinct elementary substructures are isomorphic and is '''nondiverse''' otherwise.<br />
It was shown in <cite> schmerl2004:diversity </cite> that if $L \cong {\rm Lt}(M)$ is a finite <br />
lattice, then there is a diverse $N \equiv M$ such that ${\rm Lt}(N) \cong L$. Obtaining <br />
nondiverse models seems more difficult (<cite> schmerl2008:nondiversity </cite>). For example, the answer to the following <br />
question is, at present, unknown: Is there a nondiverse $M$ such that <br />
${\rm Lt}(M) \cong {\bf M}_3$ or ${\rm Lt}(M) \cong {\bf N}_5$?<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Cuts_in_models_of_PA_and_independence_results&diff=231
Cuts in models of PA and independence results
2013-02-01T22:33:59Z
<p>Rkossak: </p>
<hr />
<div>== Diversity in elementary cuts ==<br />
<br />
Let $M\models PA$ be countable and recursively saturated.<br />
<br />
Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.<br />
<br />
Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$.<br />
<br />
Problem: How many theories of pairs $(M,K)$ are there, such that $K$ is semiregular, but not regular in $M$.<br />
<br />
By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$.<br />
<br />
The problem was stated in <cite> kossakschmerl2012:oncofinal</cite> related results can be found in <cite>smorynski1982:anoteoninitial, smorynski1981:elementary, kossak1985:anoteon, kayetinlokwong2010:truth </cite>. <br />
<br />
<br />
== Stu Smith's question ==<br />
<br />
Question 6.6 of <cite> smith1989:extendible </cite>: Let $M\models PA$ be nonstandard and let $I$ be a cut of $M$ definable in $(M,\omega)$. Is $\omega$ the (arithmetic) cofinality of $I$ in $M$?<br />
<br />
<br />
== "Star" schemata ==<br />
<br />
Answering a question from <cite>kossakparis1992:subsets </cite> Kanovei <cite> kanovei1998:onstar </cite> gave a forcing construction to show that for every $n$, every countable recursively saturated $M\models PA$ has an elementary end extension $N$ such that for every $X\subseteq M$ which is coded in $N$, $(M,X)\models I\Sigma_n$<br />
and for some such $X$, $(M,X)\not\models I\Sigma_{n+2}$. Later in <cite> kossak2004:anote </cite> $n+2$ was reduced to $n+1$ by another method.<br />
<br />
<br />
Problem: The assumption that $M$ is recursively saturated is not necessary for $n=1,2$ <cite>kossakparis1992:subsets </cite>. Is it necessary for higher $n$?<br />
<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Cuts_in_models_of_PA_and_independence_results&diff=230
Cuts in models of PA and independence results
2013-02-01T22:32:22Z
<p>Rkossak: </p>
<hr />
<div>== Diversity in elementary cuts ==<br />
<br />
Let $M\models PA$ be countable and recursively saturated.<br />
<br />
Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.<br />
<br />
Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$.<br />
<br />
Problem: How many theories of pairs $(M,K)$ are there, such that $K$ is semiregular, but not regular in $M$.<br />
<br />
By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$.<br />
<br />
The problem was stated in <cite> kossakschmerl2012:oncofinal</cite> related results can be found in <cite>smorynski1982:anoteoninitial, smorynski1981:elementary, kossak1985:anoteon, kayetinlokwong2010:truth </cite>. <br />
<br />
<br />
== Stu Smith's question ==<br />
<br />
Question 6.6 of <cite> smith1989:extendible </cite>: Let $M\models PA$ be nonstandard and let $I$ be a cut of $M$ definable in $(M,\omega)$. Is $\omega$ the (arithmetic) cofinality of $I$ in $M$?<br />
<br />
<br />
== "Star" schemata ==<br />
<br />
Answering a question from <cite>kossakparis1992:subsets </cite> Kanovei <cite> kanovei1998:onstar </cite> gave a forcing construction to show that for every $n$, every countable recursively saturated $M\models PA$ has an elementary end extension $N$ such that for every $X\subseteq M$ which is coded in $N$, $(M,X)\models I\Sigma_n$<br />
and for some such $X$, $(M,X)\not\models I\Sigma_{n+2}$. Later in <cite> kossak2004:anote </cite> $n+2$ was reduced to $n+1$.<br />
<br />
<br />
Problem: The assumption that $M$ is recursively saturated is not necessary for $n=1,2$ <cite>kossakparis1992:subsets </cite>. Is it necessary for higher $n$?<br />
<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=229
Library
2013-02-01T22:29:45Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#engstrom2008:anote bibtex=@article {engstrom2008:anote,<br />
AUTHOR = {Engstr{\"o}m, Fredrik},<br />
TITLE = {A note on standard systems and ultrafilters},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {824--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C20 03C57)},<br />
MRNUMBER = {2444270 (2009i:03030)},<br />
DOI = {10.2178/jsl/1230396749},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396749},<br />
}<br />
<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kanovei1998:onstar bibtex=@incollection {kanovei1998:onstar,<br />
AUTHOR = {Kanovei, Vladimir},<br />
TITLE = {On ``star'' schemata of {K}ossak and {P}aris},<br />
BOOKTITLE = {Logic {C}olloquium '96 ({S}an {S}ebasti\'an)},<br />
SERIES = {Lecture Notes Logic},<br />
VOLUME = {12},<br />
PAGES = {101--114},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1998},<br />
MRCLASS = {03F35 (03C62)},<br />
MRNUMBER = {1674945 (2000f:03179)},<br />
MRREVIEWER = {Roman Kossak},<br />
} <br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
#kossak1991:exercises bibtex=@inproceedings{kossak1991:exercises,<br />
title={Exercises in ‘back-and-forth’},<br />
author={Kossak, R.},<br />
booktitle={Proceedings of the Nineth Easter Conference on Model Theory, Gosen},<br />
year={1991}<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#kossak2004:anote bibtex=@article {kossak2004:anote,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on a theorem of {K}anovei},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {43},<br />
YEAR = {2004},<br />
NUMBER = {4},<br />
PAGES = {565--569},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2060400 (2005a:03073)},<br />
DOI = {10.1007/s00153-004-0218-2},<br />
URL = {http://dx.doi.org/10.1007/s00153-004-0218-2},<br />
}<br />
<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakparis1992:subsets bibtex=@article {kossakparis1992:subsets,<br />
AUTHOR = {Kossak, Roman and Paris, Jeffrey B.},<br />
TITLE = {Subsets of models of arithmetic},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {32},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {65--73},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03H15 (03C62)},<br />
MRNUMBER = {1186468 (94e:03066)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1007/BF01270396},<br />
URL = {http://dx.doi.org/10.1007/BF01270396},<br />
}<br />
<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=228
Library
2013-02-01T22:20:14Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#engstrom2008:anote bibtex=@article {engstrom2008:anote,<br />
AUTHOR = {Engstr{\"o}m, Fredrik},<br />
TITLE = {A note on standard systems and ultrafilters},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {824--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C20 03C57)},<br />
MRNUMBER = {2444270 (2009i:03030)},<br />
DOI = {10.2178/jsl/1230396749},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396749},<br />
}<br />
<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kanovei1998:onstar bibtex=@incollection {kanovei1998:onstar,<br />
AUTHOR = {Kanovei, Vladimir},<br />
TITLE = {On ``star'' schemata of {K}ossak and {P}aris},<br />
BOOKTITLE = {Logic {C}olloquium '96 ({S}an {S}ebasti\'an)},<br />
SERIES = {Lecture Notes Logic},<br />
VOLUME = {12},<br />
PAGES = {101--114},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1998},<br />
MRCLASS = {03F35 (03C62)},<br />
MRNUMBER = {1674945 (2000f:03179)},<br />
MRREVIEWER = {Roman Kossak},<br />
} <br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
#kossak1991:exercises bibtex=@inproceedings{kossak1991:exercises,<br />
title={Exercises in ‘back-and-forth’},<br />
author={Kossak, R.},<br />
booktitle={Proceedings of the Nineth Easter Conference on Model Theory, Gosen},<br />
year={1991}<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakparis1992:subsets bibtex=@article {kossakparis1992:subsets,<br />
AUTHOR = {Kossak, Roman and Paris, Jeffrey B.},<br />
TITLE = {Subsets of models of arithmetic},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {32},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {65--73},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03H15 (03C62)},<br />
MRNUMBER = {1186468 (94e:03066)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1007/BF01270396},<br />
URL = {http://dx.doi.org/10.1007/BF01270396},<br />
}<br />
<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Standard_systems_and_the_Scott_set_problem&diff=227
Standard systems and the Scott set problem
2013-02-01T15:11:31Z
<p>Rkossak: </p>
<hr />
<div>== The Scott set problem ==<br />
<br />
A Scott set is a set of sets of natural numbers $\mathfrak X$ such that $(\omega, {\mathfrak X})\models {\sf WKL}_0$, Assume $\lnot{\sf CH}$. Is every Scott set the standard system of a nonstandard model of PA?<br />
<br />
The origins of the problem goes back the paper of Scott <cite> scott1962:algebras </cite>, where it is shown that the answer is positive for countable sets $\mathfrak X$. <br />
<br />
Knight and Nadel <cite> knightnadel1982:models </cite>, and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$. <br />
<br />
Jim Schmerl <cite> schmerl1973:peano </cite> gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality. <br />
<br />
For more recent attempts employing forcing axioms see <cite> engstrom2008:anote </cite> and <cite> gitman2008:scott </cite>.<br />
<br />
<br />
== Kanovei's question ==<br />
<br />
Is there a Borel model $M\models PA$ whose standard system is the power set of $\omega$?<br />
<br />
<br />
== Woodin's question ==<br />
<br />
If $\mathfrak X$ is a Borel Scott set, is there a Borel model $M\models PA$ whose standard system is $\mathfrak X$?<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=226
Library
2013-02-01T15:10:03Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#engstrom2008:anote bibtex=@article {engstrom2008:anote,<br />
AUTHOR = {Engstr{\"o}m, Fredrik},<br />
TITLE = {A note on standard systems and ultrafilters},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {824--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C20 03C57)},<br />
MRNUMBER = {2444270 (2009i:03030)},<br />
DOI = {10.2178/jsl/1230396749},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396749},<br />
}<br />
<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
#kossak1991:exercises bibtex=@inproceedings{kossak1991:exercises,<br />
title={Exercises in ‘back-and-forth’},<br />
author={Kossak, R.},<br />
booktitle={Proceedings of the Nineth Easter Conference on Model Theory, Gosen},<br />
year={1991}<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_of_countable_recursively_saturated_models&diff=223
Automorphisms of countable recursively saturated models
2013-01-29T03:19:25Z
<p>Rkossak: </p>
<hr />
<div>== Extending automorphisms ==<br />
<br />
<br />
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$ <cite> kossak1995:four </cite>?<br />
<br />
<br />
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated <cite> kossak1995:four </cite>?<br />
<br />
<br />
== Bumps ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $I$ be a strong elementary cut of $M$. An automorphism $f$ of $M$ is an ''$I$-bump'' if for all $x\in I$, $f(x)=x$ and for all $x>I$, $f(x)>x$. Bumps do exist <cite> kossak1991:exercises </cite>.<br />
<br />
<br />
Problem: Under the assumptions above, let $f$ and $g$ be $I$-bumps. Are $f$ and $g$ cojugate, i.e. is there an automorphism $h$ such that $f=h^{-1}gh$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_of_countable_recursively_saturated_models&diff=222
Automorphisms of countable recursively saturated models
2013-01-29T03:19:03Z
<p>Rkossak: </p>
<hr />
<div>== Extending automorphisms ==<br />
<br />
<br />
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$ <cite> kossak1995:four </cite>?<br />
<br />
<br />
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated <cite> kossak1995:four </cite>?<br />
<br />
<br />
== Bumps ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $I$ be a strong elementary cut of $M$. An automorphism $f$ of $M$ is an ''$I$-bump'' if for all $x\in I$, $f(x)=x$ and for all $x>I$, $f(x)>x$. Bumps do exist <cite> kossak1991:exercises </cite>.<br />
<br />
<br />
Problem: Under the assumptions above, let $f$ and $g$ be $I$-bumps. Are $f$ and $g$ cojugate, i.e. is there an automprphism $h$ such that $f=h^{-1}gh$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_of_countable_recursively_saturated_models&diff=221
Automorphisms of countable recursively saturated models
2013-01-29T03:18:16Z
<p>Rkossak: </p>
<hr />
<div>== Extending automorphisms ==<br />
<br />
<br />
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$ <cite> kossak1995:four </cite>?<br />
<br />
<br />
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated <cite> kossak1995:four </cite>?<br />
<br />
<br />
== Bumps ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $I$ be a strong elementary cut of $M$. An automorphism $f$ of $M$ is an ''$I$-bump'' if for all $x\in I$, $f(x)=x$ and for all $x>I$, $f(x)>x$. Bumps do exist <cite> kossak1991:exercises </cite>.<br />
<br />
<br />
Problem: Under the assumptions above, let $f$ and $g$ be $I$-bumps. Are $f$ and $g$ cojugate, i.e.~is there an automprphism $h$ such that $f=h^{-1}gh$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_of_countable_recursively_saturated_models&diff=220
Automorphisms of countable recursively saturated models
2013-01-29T03:17:30Z
<p>Rkossak: </p>
<hr />
<div>== Extending automorphisms ==<br />
<br />
<br />
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$ <cite> kossak1995:four </cite>?<br />
<br />
<br />
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated <cite> kossak1995:four </cite>?<br />
<br />
<br />
== Bumps ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $I$ be a strong elementary cut of $M$. An automorphism $f$ of $M$ is an ''$I$-bump'' if for all $x\in I$, $f(x)=x$ and for all $x>I$, $f(x)>x$. Bumps do exist <cite> kossak1991:exercises </cite>.<br />
<br />
<br />
Problem: Under the assumptions above, let $f$ and $g$ be $I$-bumps. Are $f$ and $g$ cojugate, i.e.~is there an automprphism $h$ such that $f=h^-{1}gh$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=219
Library
2013-01-29T03:11:46Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
#kossak1991:exercises bibtex=@inproceedings{kossak1991:exercises,<br />
title={Exercises in ‘back-and-forth’},<br />
author={Kossak, R.},<br />
booktitle={Proceedings of the Nineth Easter Conference on Model Theory, Gosen},<br />
year={1991}<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_of_countable_recursively_saturated_models&diff=218
Automorphisms of countable recursively saturated models
2013-01-29T03:10:02Z
<p>Rkossak: </p>
<hr />
<div>== Extending automorphisms ==<br />
<br />
<br />
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$ <cite> kossak1995:four </cite>?<br />
<br />
<br />
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is recursively, but not arithmetically, saturated <cite> kossak1995:four </cite>?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_groups_of_countable_recursively_saturated_models&diff=217
Automorphisms groups of countable recursively saturated models
2013-01-29T03:00:45Z
<p>Rkossak: </p>
<hr />
<div>__FORCETOC__<br />
== Generic automorphisms and the small index property ==<br />
<br />
Lascar <cite> lascar1994:thesmall </cite> showed that countable arithmetically saturated models of $PA$ have generic<br />
automorphisms and used them to show that automorphism groups of these models have the small index property. <br />
<br />
Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property?<br />
<br />
It was shown in <cite> koskotsch1993:onmaximal </cite> that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?<br />
<br />
<br />
== How many automorphism groups are there? ==<br />
<br />
By a result of <cite> kossakschmerl1995:theautomorphism </cite>, if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.<br />
<br />
Nurkhaidarov <cite> nurkhaidarov2006:automorphism </cite> proved that there is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=4$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, and $M\models T$ and $M'\models T'$ are countable and arithmetically saturated, then ${\rm Aut}(M)\not\cong{\rm Aut}(M')$. Surely, there must be more than four such completions. How many are there?<br />
<br />
<br />
== Normal subgroups ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$. By a theorem of Kaye <cite> kaye1994:agalois </cite> every closed normal subgroup of $G$ is $G_{(I)}$, the pointwise stabilizer of an invariant initial segment $I$ of $M$. If $I$ is an inariant initial segment of $M$, then the group $G_{(>I)}$ of those automorphisms that fix pointwise an initial segment $J>I$ is also a normal subgroup, whose closure in $G$ is $G_{(I)}$.<br />
<br />
<br />
Problem: Does $G$ have any other normal subgroups?<br />
<br />
<br />
== Recursively saturated automorphisms ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$ and let ${\rm RS}(M)$ be the subgroup of $G$ generated by $f\in G$ such that $(M,f)$ is recursively saturated.<br />
${\rm RS}(M)$ is a normal subgroup of $G$, and every $f\in {\rm RS}(M)$ fixes pointwise a nonstandard initial segment of $M$ <cite> kkk1991:automorphisms </cite>.<br />
<br />
<br />
Problem: Is ${\rm RS}(M)$ the group of those $f\in G$ that fix pointwise a nonstandard initial segment of $M$? <br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_groups_of_countable_recursively_saturated_models&diff=216
Automorphisms groups of countable recursively saturated models
2013-01-29T02:59:27Z
<p>Rkossak: </p>
<hr />
<div>__FORCETOC__<br />
== Generic automorphisms and the small index property ==<br />
<br />
Lascar <cite> lascar1994:thesmall </cite> showed that countable arithmetically saturated models of $PA$ have generic<br />
automorphisms and used them to show that automorphism groups of these models have the small index property. <br />
<br />
Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property.<br />
<br />
It was shown in <cite> koskotsch1993:onmaximal </cite> that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?<br />
<br />
<br />
== How many automorphism groups are there? ==<br />
<br />
By a result of <cite> kossakschmerl1995:theautomorphism </cite>, if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.<br />
<br />
Nurkhaidarov <cite> nurkhaidarov2006:automorphism </cite> proved that there is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=4$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, and $M\models T$ and $M'\models T'$ are countable and arithmetically saturated, then ${\rm Aut}(M)\not\cong{\rm Aut}(M')$. Surely, there must be more than four such completions. How many are there?<br />
<br />
<br />
== Normal subgroups ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$. By a theorem of Kaye <cite> kaye1994:agalois </cite> every closed normal subgroup of $G$ is $G_{(I)}$, the pointwise stabilizer of an invariant initial segment $I$ of $M$. If $I$ is an inariant initial segment of $M$, then the group $G_{(>I)}$ of those automorphisms that fix pointwise an initial segment $J>I$ is also a normal subgroup, whose closure in $G$ is $G_{(I)}$.<br />
<br />
<br />
Problem: Does $G$ have any other normal subgroups?<br />
<br />
<br />
== Recursively saturated automorphisms ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$ and let ${\rm RS}(M)$ be the subgroup of $G$ generated by $f\in G$ such that $(M,f)$ is recursively saturated.<br />
${\rm RS}(M)$ is a normal subgroup of $G$, and every $f\in {\rm RS}(M)$ fixes pointwise a nonstandard initial segment of $M$ <cite> kkk1991:automorphisms </cite>.<br />
<br />
Problem: Is ${\rm RS}(M)$ the group of those $f\in G$ that fix pointwise a nonstandard initial segment of $M$? <br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=215
Library
2013-01-29T02:58:32Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Automorphisms_groups_of_countable_recursively_saturated_models&diff=214
Automorphisms groups of countable recursively saturated models
2013-01-29T02:54:17Z
<p>Rkossak: </p>
<hr />
<div>__FORCETOC__<br />
== Generic automorphisms and the small index property ==<br />
<br />
Lascar <cite> lascar1994:thesmall </cite> showed that countable arithmetically saturated models of $PA$ have generic<br />
automorphisms and used them to show that automorphism groups of these models have the small index property. <br />
<br />
Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property.<br />
<br />
It was shown in <cite> koskotsch1993:onmaximal </cite> that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?<br />
<br />
<br />
== How many automorphism groups are there? ==<br />
<br />
By a result of <cite> kossakschmerl1995:theautomorphism </cite>, if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.<br />
<br />
Nurkhaidarov <cite> nurkhaidarov2006:automorphism </cite> proved that there is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=4$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, and $M\models T$ and $M'\models T'$ are countable and arithmetically saturated, then ${\rm Aut}(M)\not\cong{\rm Aut}(M')$. Surely, there must be more than four such completions. How many are there?<br />
<br />
<br />
== Normal subgroups ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$. By a theorem of Kaye <cite> kaye1994:agalois </cite> every closed normal subgroup of $G$ is $G_{(I)}$, the pointwise stabilizer of an invariant initial segment $I$ of $M$. If $I$ is an inariant initial segment of $M$, then the group $G_{(>I)}$ of those automorphisms that fix pointwise an initial segment $J>I$ is also a normal subgroup, whose closure in $G$ is $G_{(I)}$.<br />
<br />
<br />
Problem: Does $G$ have any other normal subgroups?<br />
<br />
<br />
== Recursively saturated automorphisms ==<br />
<br />
Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$ and let ${\rm RS}(M)$ be the subgroup of $G$ generated by $f\in G$ such that $(M,f)$ is recursively saturated.<br />
${\rm RS}(M)$ is a normal subgroup of $G$, and every $f\in {\rm RS}(M)$ fixes pointwise a nonstandard initial segment of $M$.<br />
<br />
Problem: Is ${\rm RS}(M)$ the group of those $f\in G$ that fix pointwise a nonstandard initial segment of $M$? <br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=213
Library
2013-01-29T02:35:18Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
#kaye1994:agalois bibtex=@incollection {kaye1994:agalois,<br />
AUTHOR = {Kaye, Richard},<br />
TITLE = {A {G}alois correspondence for countable recursively saturated<br />
models of {P}eano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {293--312},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62 20B27)},<br />
MRNUMBER = {1325480},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=212
Library
2013-01-24T18:13:35Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1985:recursively bibtex=@article {schmerl1985:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated models generated by indiscernibles},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {2},<br />
PAGES = {99--105},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {783590 (87g:03038)},<br />
MRREVIEWER = {Alain Lewis},<br />
DOI = {10.1305/ndjfl/1093870818},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870818},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Resplendent_models&diff=211
Resplendent models
2013-01-24T18:04:47Z
<p>Rkossak: </p>
<hr />
<div>== Chronically resplendent models ==<br />
<br />
This problem, and the nest, are not specifically about models of $PA$, but both questions are interesting<br />
in the context of $PA$.<br />
<br />
Every countable resplendent model is ''chronically resplendent'', which means that the expansions given by replendency can be also made resplendent.<br />
<br />
<br />
Problem <cite> schlipf1977:aguide </cite>: Is every resplendent model chronically resplendent?<br />
<br />
<br />
== Totally resplendent models ==<br />
<br />
A model $M$ is ''totally resplendent'' if there are countably many relations $R_0, R_1,\dots$ on $M$ such that each expansion $(M,R_0,R_1,\dots, R_{n-1})$ is resplendent and moreover if $(M,R_0,R_1,\dots)\models\exists R\ \varphi (\bar a, R)$, then $(M,R_0,R_1,\dots)\models \varphi (\bar a, R)$ for some $R$ parametrically definable in $(M,R_0,R_1,\dots)$ <cite> schmerl1989:large </cite>.<br />
<br />
Every countable resplendent model is totally resplendent.<br />
<br />
<br />
Problem: Is every resplendent model totally resplendent?<br />
<br />
<br />
<br />
== A converse to Schmerl's theorem? ==<br />
<br />
By a theorem of Schmerl <cite> schmerl1985:recursively </cite>, every countable recursively saturated model of $PA$ is generated by a set of indiscernibles of any coutnable order type without a last element.<br />
<br />
<br />
Problem: Suppose $M$ is a countable, tall model of $PA$, and suppose $M$ is generated by sets of indiscernibles of two different order types. Is $M$ recursively saturated?<br />
<br />
<br />
== Gentle expansions ==<br />
This problem is not directly about models of PA, but is motivated by results<br />
concerning maximal automorphisms, first proved in the context of arithmetically <br />
saturated models of $PA$ <cite> kkk1991:automorphisms </cite>. For results on maximal automorphisms in general setting see <cite> korner1998:automorphisms, duby2003:automorphisms </cite> <br />
<br />
An expansion $M^+$ of a structure $M$ is ''gentle'' if all algebraic elements of $M^+$ are<br />
already algebraic in $M$.<br />
<br />
<br />
Problem: Let $M$ be a countable resplendent model. Can $M$ always be gently expanded to a linearly ordered<br />
resplendent $M^+$?<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Resplendent_models&diff=210
Resplendent models
2013-01-24T18:04:05Z
<p>Rkossak: </p>
<hr />
<div>== Chronically resplendent models ==<br />
<br />
This problem, and the nest, are not specifically about models of $PA$, but both questions are interesting<br />
in the context of $PA$.<br />
<br />
Every countable resplendent model is ''chronically resplendent'', which means that the expansions given by replendency can be also made resplendent.<br />
<br />
<br />
Problem <cite> schlipf1977:aguide </cite>: Is every resplendent model chronically resplendent?<br />
<br />
<br />
== Totally resplendent models ==<br />
<br />
A model $M$ is ''totally resplendent'' if there are countably many relations $R_0, R_1,\dots$ on $M$ such that each expansion $(M,R_0,R_1,\dots, R_{n-1})$ is resplendent and moreover if $(M,R_0,R_1,\dots)\models\exists R\ \varphi (\bar a, R)$, then $(M,R_0,R_1,\dots)\models \varphi (\bar a, R)$ for some $R$ parametrically definable in $(M,R_0,R_1,\dots)$ <cite> schmerl1989:large </cite>.<br />
<br />
Every countable resplendent model is totally resplendent.<br />
<br />
<br />
Problem: Is every resplendent model totally resplendent?<br />
<br />
<br />
<br />
== A converse to Schmerl's theorem? ==<br />
<br />
By a theorem of Schmerl <cite> schmerl1985:recursively </cite>, every countable recursively saturated model of $PA$ is generated by a set of indiscernibles of any coutnable order type without a last element.<br />
<br />
<br />
Problem: Suppose $M$ is a countable, tall model of $\PA$, and suppose $M$ is generated by sets of indiscernibles of two different order types. Is $M$ recursively saturated?<br />
<br />
<br />
== Gentle expansions ==<br />
This problem is not directly about models of PA, but is motivated by results<br />
concerning maximal automorphisms, first proved in the context of arithmetically <br />
saturated models of $PA$ <cite> kkk1991:automorphisms </cite>. For results on maximal automorphisms in general setting see <cite> korner1998:automorphisms, duby2003:automorphisms </cite> <br />
<br />
An expansion $M^+$ of a structure $M$ is ''gentle'' if all algebraic elements of $M^+$ are<br />
already algebraic in $M$.<br />
<br />
<br />
Problem: Let $M$ be a countable resplendent model. Can $M$ always be gently expanded to a linearly ordered<br />
resplendent $M^+$?<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=209
Library
2013-01-24T17:29:08Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl1989:large bibtex=@article {schmerl1989,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Large resplendent models generated by indiscernibles},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {4},<br />
PAGES = {1382--1388},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C57},<br />
MRNUMBER = {1026604 (91b:03063)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.2307/2274820},<br />
URL = {http://dx.doi.org/10.2307/2274820},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Resplendent_models&diff=208
Resplendent models
2013-01-24T17:25:45Z
<p>Rkossak: </p>
<hr />
<div><br />
== Chronically resplendent models ==<br />
<br />
This problem, and the nest, are not specifically about models of $PA$, but both questions are interesting<br />
in the context of $PA$.<br />
<br />
Every countable resplendent model is chronically resplendent, which means that the expansions given by replendency can be also made resplendent.<br />
<br />
<br />
Problem <cite> schlipf1977:aguide </cite>: Is every resplendent model chronically resplendent?<br />
<br />
<br />
<br />
<br />
<br />
== Gentle expansions ==<br />
This problem is not directly about models of PA, but is motivated by results<br />
concerning maximal automorphisms, first proved in the context of arithmetically <br />
saturated models of $PA$ <cite> kkk1991:automorphisms </cite>. For results on maximal automorphisms in general setting see <cite> korner1998:automorphisms, duby2003:automorphisms </cite> <br />
<br />
An expansion $M^+$ of a structure $M$ is ''gentle'' if all algebraic elements of $M^+$ are<br />
already algebraic in $M$.<br />
<br />
<br />
Problem: Let $M$ be a countable resplendent model. Can $M$ always be gently expanded to a linearly ordered<br />
resplendent $M^+$?<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=207
Library
2013-01-24T17:18:21Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#schlipf1977:aguide bibtex=@article{schlipf1977:aguide,<br />
author={Schlipf, John S.},<br />
title={A guide to the identification of admissible sets above structures},<br />
journal={Ann. Math. Logic},<br />
volume={12},<br />
date={1977},<br />
number={2},<br />
pages={151--192},<br />
issn={0168-0072},<br />
MRNUMBER={0485330 (58 \#5177)},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=Cuts_in_models_of_PA_and_independence_results&diff=206
Cuts in models of PA and independence results
2013-01-24T17:09:46Z
<p>Rkossak: </p>
<hr />
<div>== Diversity in elementary cuts ==<br />
<br />
Let $M\models PA$ be countable and recursively saturated.<br />
<br />
Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.<br />
<br />
Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$.<br />
<br />
Problem: How many theories of pairs $(M,K)$ are there, such that $K$ is semiregular, but not regular in $M$.<br />
<br />
By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$.<br />
<br />
The problem was stated in <cite> kossakschmerl2012:oncofinal</cite> related results can be found in <cite>smorynski1982:anoteoninitial, smorynski1981:elementary, kossak1985:anoteon, kayetinlokwong2010:truth </cite>. <br />
<br />
<br />
== Stu Smith's question ==<br />
<br />
Question 6.6 of <cite> smith1989:extendible </cite>: Let $M\models PA$ be nonstandard and let $I$ be a cut of $M$ definable in $(M,\omega)$. Is $\omega$ the (arithmetic) cofinality of $I$ in $M$?<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Library&diff=205
Library
2013-01-24T17:05:17Z
<p>Rkossak: </p>
<hr />
<div>{{DISPLAYTITLE: The Peano's parlour library}}<br />
<br />
<br />
Welcome to the library, our central repository for references cited here on Peano's parlour. <br />
<br />
== Library holdings ==<br />
<br />
<!-- <br />
Please add new entries below in alphabetical order by first author. <br />
- Follow our format for IDs with the form AuthornameYEAR:Titlefragment, e.g. GitmanHamkins2027:ZFCisInconsistent.<br />
- Please endeavor to include an url field, doi field and eprint field, since these will become linkable, which makes the entry far more useful.<br />
- You must not put spaces around the = sign in bibtex=@article{etc}<br />
- Author field must have form: Lastname, First name. <br />
- Titles may not have commas!<br />
//--><br />
<br />
<biblio force=true><br />
<br />
#coskeykossak2010:thecomplexity bibtex=@article {coskeykossak2010:thecomplexity,<br />
AUTHOR = {Coskey, Samuel and Kossak, Roman},<br />
TITLE = {The complexity of classification problems for models of<br />
arithmetic},<br />
JOURNAL = {Bull. Symbolic Logic},<br />
FJOURNAL = {Bulletin of Symbolic Logic},<br />
VOLUME = {16},<br />
YEAR = {2010},<br />
NUMBER = {3},<br />
PAGES = {345--358},<br />
ISSN = {1079-8986},<br />
MRCLASS = {03C62 (03E15 03H15)},<br />
MRNUMBER = {2731248 (2011k:03071)},<br />
MRREVIEWER = {Thomas McLaughlin},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.bsl/1286284557},<br />
}<br />
<br />
#duby2003:automorphisms bibtex=@article {duby2003:automorphisms,<br />
AUTHOR = {Duby, Gr{\'e}gory},<br />
TITLE = {Automorphisms with only infinite orbits on non-algebraic<br />
elements},<br />
JOURNAL = {Arch. Math. Logic},<br />
FJOURNAL = {Archive for Mathematical Logic},<br />
VOLUME = {42},<br />
YEAR = {2003},<br />
NUMBER = {5},<br />
PAGES = {435--447},<br />
ISSN = {0933-5846},<br />
CODEN = {AMLOEH},<br />
MRCLASS = {03C50 (03C57)},<br />
MRNUMBER = {1990734 (2004e:03068)},<br />
MRREVIEWER = {H. Dugald Macpherson},<br />
DOI = {10.1007/s00153-002-0146-y},<br />
URL = {http://dx.doi.org/10.1007/s00153-002-0146-y},<br />
}<br />
<br />
#friedman1975:onehundred bibtex=@article {friedman1975:onehundred,<br />
AUTHOR = {Friedman, Harvey},<br />
TITLE = {One hundred and two problems in mathematical logic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {40},<br />
YEAR = {1975},<br />
PAGES = {113--129},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02-02},<br />
MRNUMBER = {0369018 (51 \#5254)},<br />
MRREVIEWER = {Andreas Blass},<br />
}<br />
<br />
<br />
#gaifman1976:models bibtex=@article {gaifman1976:models,<br />
AUTHOR = {Gaifman, Haim},<br />
TITLE = {Models and types of {P}eano's arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {9},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {223--306},<br />
ISSN = {0168-0072},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0406791 (53 \#10577)},<br />
MRREVIEWER = {S. R. Kogalovskii},<br />
}<br />
<br />
<br />
#gitman2008:scott bibtex=@article {gitman2008:scott,<br />
AUTHOR = {Gitman, Victoria},<br />
TITLE = {Scott's problem for proper Scott sets},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {73},<br />
YEAR = {2008},<br />
NUMBER = {3},<br />
PAGES = {845--860},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03E65)},<br />
MRNUMBER = {2444272 (2009f:03047)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1230396751},<br />
URL = {http://dx.doi.org/10.2178/jsl/1230396751},<br />
}<br />
<br />
#hodges1985:building bibtex=@book {hodges1985:building,<br />
AUTHOR = {Hodges, Wilfrid},<br />
TITLE = {Building models by games},<br />
SERIES = {London Mathematical Society Student Texts},<br />
VOLUME = {2},<br />
PUBLISHER = {Cambridge University Press},<br />
ADDRESS = {Cambridge},<br />
YEAR = {1985},<br />
PAGES = {vi+311},<br />
ISBN = {0-521-26897-4; 0-521-31716-9},<br />
MRCLASS = {03C30 (03-01 03Cxx)},<br />
MRNUMBER = {812274 (87h:03045)},<br />
MRREVIEWER = {John T. Baldwin},<br />
}<br />
<br />
#kaufmann1977:arather bibtex=@article {kaufmann1977:arather,<br />
AUTHOR = {Kaufmann, Matt},<br />
TITLE = {A rather classless model},<br />
JOURNAL = {Proc. Amer. Math. Soc.},<br />
FJOURNAL = {Proceedings of the American Mathematical Society},<br />
VOLUME = {62},<br />
YEAR = {1977},<br />
NUMBER = {2},<br />
PAGES = {330--333},<br />
ISSN = {0002-9939},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0476498 (57 \#16058)},<br />
MRREVIEWER = {James H. Schmerl},<br />
}<br />
<br />
#kaufmannschmerl1987:remarks bibtex=@article {kaufmannschmerl1987:remarks,<br />
AUTHOR = {Kaufmann, Matt and Schmerl, James H.},<br />
TITLE = {Remarks on weak notions of saturation in models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {52},<br />
YEAR = {1987},<br />
NUMBER = {1},<br />
PAGES = {129--148},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {877860 (88g:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2273867},<br />
URL = {http://dx.doi.org/10.2307/2273867},<br />
}<br />
<br />
<br />
#kayetinlokwong2010:truth bibtex=@article {kayetinlokwong2010,<br />
AUTHOR = {Kaye, Richard and Wong, Tin Lok},<br />
TITLE = {Truth in generic cuts},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {161},<br />
YEAR = {2010},<br />
NUMBER = {8},<br />
PAGES = {987--1005},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {2629502 (2011f:03048)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1016/j.apal.2009.11.001},<br />
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},<br />
}<br />
<br />
#knight1976:hanf bibtex=@article {knight1976:hanf,<br />
AUTHOR = {Knight, Julia F.},<br />
TITLE = {Hanf numbers for omitting types over particular theories},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {41},<br />
YEAR = {1976},<br />
NUMBER = {3},<br />
PAGES = {583--588},<br />
ISSN = {0022-4812},<br />
MRCLASS = {02H05},<br />
MRNUMBER = {0485323 (58 \#5170)},<br />
}<br />
<br />
#knightnadel1982:models bibtex=@article {knightnadel1982:models,<br />
AUTHOR = {Knight, Julia and Nadel, Mark},<br />
TITLE = {Models of arithmetic and closed ideals},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {47},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {833--840 (1983)},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03D30)},<br />
MRNUMBER = {683158 (85d:03072)},<br />
MRREVIEWER = {S. S. Goncharov},<br />
DOI = {10.2307/2273102},<br />
URL = {http://dx.doi.org/10.2307/2273102},<br />
}<br />
<br />
#korner1998:automorphisms bibtex=@article {korner1998:automorphisms,<br />
AUTHOR = {K{\"o}rner, Friederike},<br />
TITLE = {Automorphisms moving all non-algebraic points and an<br />
application to {NF}},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {63},<br />
YEAR = {1998},<br />
NUMBER = {3},<br />
PAGES = {815--830},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C50 (03C57 03E70)},<br />
MRNUMBER = {1649063 (2000c:03030)},<br />
MRREVIEWER = {M. Randall Holmes},<br />
DOI = {10.2307/2586714},<br />
URL = {http://dx.doi.org/10.2307/2586714},<br />
}<br />
<br />
#kossak1985:anoteon bibtex=@article {kossak1985:anoteon,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {A note on satisfaction classes},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {26},<br />
YEAR = {1985},<br />
NUMBER = {1},<br />
PAGES = {1--8},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03H15 (03C50 03C62)},<br />
MRNUMBER = {766663 (86c:03055)},<br />
MRREVIEWER = {Bernd Dahn},<br />
DOI = {10.1305/ndjfl/1093870757},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},<br />
}<br />
<br />
#kossak1989:models bibtex=@article {kossak1989:models,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Models with the {$\omega$}-property},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {The Journal of Symbolic Logic},<br />
VOLUME = {54},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {177--189},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {987330 (90a:03053)},<br />
MRREVIEWER = {A. J. Wilkie},<br />
DOI = {10.2307/2275023},<br />
URL = {http://dx.doi.org/10.2307/2275023},<br />
}<br />
<br />
<br />
#kossak1995:four bibtex=@article {kossak1995:four,<br />
AUTHOR = {Kossak, Roman},<br />
TITLE = {Four problems concerning recursively saturated models of<br />
arithmetic},<br />
NOTE = {Special Issue: Models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {36},<br />
YEAR = {1995},<br />
NUMBER = {4},<br />
PAGES = {519--530},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1368464 (98a:03056)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1305/ndjfl/1040136913},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},<br />
}<br />
<br />
#koskot1992:game bibtex=@article {koskot1992:game,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Game approximations of satisfaction classes and the problem of<br />
rather classless models},<br />
JOURNAL = {Z. Math. Logik Grundlag. Math.},<br />
FJOURNAL = {Zeitschrift f\"ur Mathematische Logik und Grundlagen der<br />
Mathematik},<br />
VOLUME = {38},<br />
YEAR = {1992},<br />
NUMBER = {1},<br />
PAGES = {21--26},<br />
ISSN = {0044-3050},<br />
CODEN = {ZMLGAQ},<br />
MRCLASS = {03C62 (03C57)},<br />
MRNUMBER = {1253748 (95c:03089)},<br />
MRREVIEWER = {David E. Marker},<br />
DOI = {10.1002/malq.19920380103},<br />
URL = {http://dx.doi.org/10.1002/malq.19920380103},<br />
}<br />
<br />
<br />
#koskot2008:moreon bibtex=@article {koskot2008:moreon,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {More on extending automorphisms of models of Peano<br />
arithmetic},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Fundamenta Mathematicae},<br />
VOLUME = {200},<br />
YEAR = {2008},<br />
NUMBER = {2},<br />
PAGES = {133--143},<br />
ISSN = {0016-2736},<br />
MRCLASS = {03C62 (03C50)},<br />
MRNUMBER = {2426391 (2009d:03084)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.4064/fm200-2-2},<br />
URL = {http://dx.doi.org/10.4064/fm200-2-2},<br />
}<br />
<br />
#kkk1991:automorphisms bibtex=@article {kkk1991:automorphisms,<br />
AUTHOR = {Kaye, Richard and Kossak, Roman and Kotlarski, Henryk},<br />
TITLE = {Automorphisms of recursively saturated models of arithmetic},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {55},<br />
YEAR = {1991},<br />
NUMBER = {1},<br />
PAGES = {67--99},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1134917 (93c:03047)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1016/0168-0072(91)90098-7},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(91)90098-7},<br />
}<br />
<br />
#koskotsch1993:onmaximal bibtex=@article {koskotsch1993,<br />
AUTHOR = {Kossak, Roman and Kotlarski, Henryk and Schmerl, James H.},<br />
TITLE = {On maximal subgroups of the automorphism group of a countable<br />
recursively saturated model of {PA}},<br />
JOURNAL = {Ann. Pure Appl. Logic},<br />
FJOURNAL = {Annals of Pure and Applied Logic},<br />
VOLUME = {65},<br />
YEAR = {1993},<br />
NUMBER = {2},<br />
PAGES = {125--148},<br />
ISSN = {0168-0072},<br />
CODEN = {APALD7},<br />
MRCLASS = {03C62 (03C57 20B27)},<br />
MRNUMBER = {1257467 (94m:03061)},<br />
MRREVIEWER = {Fuxing Shen},<br />
DOI = {10.1016/0168-0072(93)90035-C},<br />
URL = {http://dx.doi.org/10.1016/0168-0072(93)90035-C},<br />
}<br />
<br />
#kossakschmerl1991:minimal bibtex=@article {kossakschmerl1991,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {Minimal satisfaction classes with an application to rigid<br />
models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {32},<br />
YEAR = {1991},<br />
NUMBER = {3},<br />
PAGES = {392--398},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {1124592 (92h:03051)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.1305/ndjfl/1093635835},<br />
URL = {http://dx.doi.org/10.1305/ndjfl/1093635835},<br />
}<br />
<br />
#kossakschmerl1995:theautomorphism bibtex=@article {kossakschmerl1995:theautomorphism,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The automorphism group of an arithmetically saturated model of<br />
Peano arithmetic},<br />
JOURNAL = {J. London Math. Soc. (2)},<br />
FJOURNAL = {Journal of the London Mathematical Society. Second Series},<br />
VOLUME = {52},<br />
YEAR = {1995},<br />
NUMBER = {2},<br />
PAGES = {235--244},<br />
ISSN = {0024-6107},<br />
CODEN = {JLMSAK},<br />
MRCLASS = {03C62 (20B27)},<br />
MRNUMBER = {1356139 (97a:03048)},<br />
MRREVIEWER = {Ali Enayat},<br />
DOI = {10.1112/jlms/52.2.235},<br />
URL = {http://dx.doi.org/10.1112/jlms/52.2.235},<br />
}<br />
#kossakschmerl:modelsofpa bibtex=@book {kossakschmerl:modelsofpa,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {The structure of models of Peano arithmetic},<br />
SERIES = {Oxford Logic Guides},<br />
VOLUME = {50},<br />
NOTE = {Oxford Science Publications},<br />
PUBLISHER = {The Clarendon Press Oxford University Press},<br />
ADDRESS = {Oxford},<br />
YEAR = {2006},<br />
PAGES = {xiv+311},<br />
ISBN = {978-0-19-856827-8; 0-19-856827-4},<br />
MRCLASS = {03-02 (03C62 03F30 03H15)},<br />
MRNUMBER = {2250469 (2008b:03001)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.1093/acprof:oso/9780198568278.001.0001},<br />
URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},<br />
}<br />
<br />
#kossakschmerl2012:oncofinal bibtex=@article{kossakschmerl2012:oncofinal,<br />
AUTHOR = {Kossak, Roman and Schmerl, James H.},<br />
TITLE = {On cofinal extensions and elementary interstices},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {3},<br />
PAGES = {267--287},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C62)},<br />
MRNUMBER = {},<br />
MRREVIEWER = {},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},<br />
}<br />
<br />
#lascar1994:thesmall bibtex=@incollection {lascar1994:thesmall,<br />
AUTHOR = {Lascar, Daniel},<br />
TITLE = {The small index property and recursively saturated models of<br />
Peano arithmetic},<br />
BOOKTITLE = {Automorphisms of first-order structures},<br />
SERIES = {Oxford Sci. Publ.},<br />
PAGES = {281--292},<br />
PUBLISHER = {Oxford Univ. Press},<br />
ADDRESS = {New York},<br />
YEAR = {1994},<br />
MRCLASS = {03C57 (03C62)},<br />
MRNUMBER = {1325479},<br />
}<br />
<br />
#mills1979:substructure bibtex=@article {mills1979:substructure,<br />
AUTHOR = {Mills, George},<br />
TITLE = {Substructure lattices of models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {16},<br />
YEAR = {1979},<br />
NUMBER = {2},<br />
PAGES = {145--180},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03H15},<br />
MRNUMBER = {537207 (81i:03105)},<br />
DOI = {10.1016/0003-4843(79)90007-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(79)90007-X},<br />
}<br />
<br />
#nurkhaidarov2006:automorphism bibtex=@article {nurkhaidarov2006:automorphism,<br />
AUTHOR = {Nurkhaidarov, Ermek S.},<br />
TITLE = {Automorphism groups of arithmetically saturated models},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {71},<br />
YEAR = {2006},<br />
NUMBER = {1},<br />
PAGES = {203--216},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03C62 (03C50 03F30)},<br />
MRNUMBER = {2210062 (2006k:03066)},<br />
MRREVIEWER = {Roman Kossak},<br />
DOI = {10.2178/jsl/1140641169},<br />
URL = {http://dx.doi.org/10.2178/jsl/1140641169},<br />
}<br />
<br />
#paris1972:onmodels bibtex=@incollection {paris1972:onmodels,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {On models of arithmetic},<br />
BOOKTITLE = {Conference in {M}athematical {L}ogic---{L}ondon '70 ({B}edford<br />
{C}oll., {L}ondon, 1970)},<br />
PAGES = {251--280. Lecture Notes in Math., Vol. 255},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1972},<br />
MRCLASS = {02H05 (02H13)},<br />
MRNUMBER = {0392552 (52 \#13369)},<br />
MRREVIEWER = {W. Marek},<br />
}<br />
<br />
#paris1977:modelsof bibtex=@article {paris1977:modelsof,<br />
AUTHOR = {Paris, J. B.},<br />
TITLE = {Models of arithmetic and the {$1-3-1$} lattice},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {195--199},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446953 (56 \#5270)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
#scott1962:algebras bibtex=@incollection {scott1962:algebras,<br />
AUTHOR = {Scott, Dana},<br />
TITLE = {Algebras of sets binumerable in complete extensions of<br />
arithmetic},<br />
BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},<br />
PAGES = {117--121},<br />
PUBLISHER = {American Mathematical Society},<br />
ADDRESS = {Providence, R.I.},<br />
YEAR = {1962},<br />
MRCLASS = {02.72},<br />
MRNUMBER = {0141595 (25 \#4993)},<br />
MRREVIEWER = {H. Ribeiro},<br />
}<br />
<br />
<br />
#schmerl1973:peano bibtex=@article {schmerl1973:peano,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Peano models with many generic classes},<br />
JOURNAL = {Pacific J. Math.},<br />
FJOURNAL = {Pacific Journal of Mathematics},<br />
VOLUME = {46},<br />
YEAR = {1973},<br />
PAGES = {523--536},<br />
ISSN = {0030-8730},<br />
MRCLASS = {02H20},<br />
MRNUMBER = {0354351 (50 \#6831)},<br />
MRREVIEWER = {M. Boffa},<br />
}<br />
<br />
#schmerl1978:extending bibtex=@article {schmerl1978:extending,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Extending models of arithmetic},<br />
JOURNAL = {Ann. Math. Logic},<br />
FJOURNAL = {Annals of Mathematical Logic},<br />
VOLUME = {14},<br />
YEAR = {1978},<br />
PAGES = {89--109},<br />
ISSN = {0003-4843},<br />
CODEN = {AMLOAD},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {506527 (80f:03036)},<br />
MRREVIEWER = {Andreas Blass},<br />
DOI = {10.1016/0003-4843(78)90013-X},<br />
URL = {http://dx.doi.org/10.1016/0003-4843(78)90013-X},<br />
}<br />
<br />
#schmerl1981:recursively bibtex=@incollection {schmerl1981:recursively,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Recursively saturated, rather classless models of {P}eano<br />
arithmetic},<br />
BOOKTITLE = {Logic {Y}ear 1979--80 ({P}roc. {S}eminars and {C}onf. {M}ath.<br />
{L}ogic, {U}niv. {C}onnecticut, {S}torrs, {C}onn., 1979/80)},<br />
SERIES = {Lecture Notes in Math.},<br />
VOLUME = {859},<br />
PAGES = {268--282},<br />
PUBLISHER = {Springer},<br />
ADDRESS = {Berlin},<br />
YEAR = {1981},<br />
MRCLASS = {03C62 (03C50 03H15)},<br />
MRNUMBER = {619874 (83b:03039)},<br />
MRREVIEWER = {Matt Kaufmann},<br />
}<br />
<br />
#schmerl2012:elementary bibtex=@article {schmerl2012:elementary,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Elementary cuts in saturated models of {P}eano arithmetic},<br />
JOURNAL = {Notre Dame J. Form. Log.},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {53},<br />
YEAR = {2012},<br />
NUMBER = {1},<br />
PAGES = {1--13},<br />
ISSN = {0029-4527},<br />
MRCLASS = {03C62},<br />
MRNUMBER = {2925265},<br />
DOI = {10.1215/00294527-1626491},<br />
URL = {http://dx.doi.org/10.1215/00294527-1626491},<br />
}<br />
<br />
#schmerl2010:infinite bibtex=@article {schmerl2010:infinite,<br />
AUTHOR = {Schmerl, James H.},<br />
TITLE = {Infinite substructure lattices of models of {P}eano<br />
arithmetic},<br />
JOURNAL = {J. Symbolic Logic},<br />
FJOURNAL = {Journal of Symbolic Logic},<br />
VOLUME = {75},<br />
YEAR = {2010},<br />
NUMBER = {4},<br />
PAGES = {1366--1382},<br />
ISSN = {0022-4812},<br />
CODEN = {JSYLA6},<br />
MRCLASS = {03F30 (03C62 06B05)},<br />
MRNUMBER = {2767974 (2012a:03162)},<br />
MRREVIEWER = {Constantine Dimitracopoulos},<br />
DOI = {10.2178/jsl/1286198152},<br />
URL = {http://dx.doi.org/10.2178/jsl/1286198152},<br />
}<br />
<br />
#smith1989:extendible bibtex=@article {smith1989:extendible,<br />
AUTHOR = {Smith, Stuart T.},<br />
TITLE = {Extendible sets in {P}eano arithmetic},<br />
JOURNAL = {Trans. Amer. Math. Soc.},<br />
FJOURNAL = {Transactions of the American Mathematical Society},<br />
VOLUME = {316},<br />
YEAR = {1989},<br />
NUMBER = {1},<br />
PAGES = {337--367},<br />
ISSN = {0002-9947},<br />
CODEN = {TAMTAM},<br />
MRCLASS = {03C62 (03H15)},<br />
MRNUMBER = {946223 (90b:03049)},<br />
MRREVIEWER = {Athanassios Tzouvaras},<br />
DOI = {10.2307/2001288},<br />
URL = {http://dx.doi.org/10.2307/2001288},<br />
}<br />
<br />
<br />
<br />
#smorynski1981:elementary bibtex=@article{smorynski1981:elementary,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {Elementary extensions of recursively saturated models of<br />
arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {22},<br />
YEAR = {1981},<br />
NUMBER = {3},<br />
PAGES = {193--203},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03H15)},<br />
MRNUMBER = {614117 (82g:03064)},<br />
MRREVIEWER = {Klaus Potthoff},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},<br />
}<br />
<br />
#smorynski1982:anoteoninitial bibtex=@article{smorynski1982:anoteoninitial,<br />
AUTHOR = {Smoryński, C.},<br />
TITLE = {A note on initial segment constructions in recursively<br />
saturated models of arithmetic},<br />
JOURNAL = {Notre Dame J. Formal Logic},<br />
FJOURNAL = {Notre Dame Journal of Formal Logic},<br />
VOLUME = {23},<br />
YEAR = {1982},<br />
NUMBER = {4},<br />
PAGES = {393--408},<br />
ISSN = {0029-4527},<br />
CODEN = {NDJFAM},<br />
MRCLASS = {03C62 (03C57 03F30 03H15)},<br />
MRNUMBER = {669146 (83j:03058)},<br />
MRREVIEWER = {Roman Murawski},<br />
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},<br />
}<br />
<br />
#wilkie1977:onmodels bibtex=@article {wilkie1977:onmodels,<br />
AUTHOR = {Wilkie, A. J.},<br />
TITLE = {On models of arithmetic having non-modular substructure<br />
lattices},<br />
JOURNAL = {Fund. Math.},<br />
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},<br />
VOLUME = {95},<br />
YEAR = {1977},<br />
NUMBER = {3},<br />
PAGES = {223--237},<br />
ISSN = {0016-2736},<br />
MRCLASS = {02H05 (02H20)},<br />
MRNUMBER = {0446952 (56 \#5269)},<br />
MRREVIEWER = {P. Hajek},<br />
}<br />
<br />
<br />
</biblio><br />
<br />
== User instructions == <br />
<br />
Peano's parlour users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit this page to make your contribution.</div>
Rkossak
http://modelsofpa.info/index.php?title=End_extensions,_cofinal_extensions&diff=204
End extensions, cofinal extensions
2013-01-24T15:11:19Z
<p>Rkossak: </p>
<hr />
<div>== Conservative cofinal extensions? ==<br />
<br />
<br />
For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$. <br />
<br />
Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is ''conservative'' if, for each $b\in N\setminus M$ there is $a\in M$ such that<br />
$b\cap M_b=a\cap M_b.$ <br />
<br />
Problem: Do conservative cofinal extensions exist?<br />
<br />
<br />
== The description property ==<br />
<br />
The extension $M\prec_{cof} N$ has the ''description property'' if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that <br />
<br />
1. $N\models a\in A_i$ for all $i<\omega$;<br />
<br />
2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.<br />
<br />
<br />
For each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?<br />
<br />
== Isolated gaps ==<br />
For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.<br />
<br />
If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is ''non-isolated'' if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is ''isolated''.<br />
<br />
If $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. Moreover, if the extension has the description property, then all new gaps are non-isolated <cite> koskot2008:moreon </cite>. <br />
<br />
<br />
Problem: Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?<br />
<br />
<br />
== Extending automorphisms to cofinal extensions ==<br />
<br />
If $M$ and $N$ are countable recursively saturated models of $PA$, $M\prec_{cof} N$ and $f$ is an automorphism of $M$, then $f$ is ''potentially extendable'' if for every bounded $A\subseteq M$, $A$ is coded in $N$ iff $f(A)$ is.<br />
<br />
If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ has the description property, then every potentially extendable automorphism of $M$ has an extension to an automorphism of $N$ <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
Problem: Find a sufficient and necessary condition on the extension $M\prec_{cof} N$, where $M$ and $N$ are countable recursively saturated models of $PA$, under which every potentially extendable automorphism of $M$ extends to an automorphism of $N$.<br />
<br />
<br />
== Strong cofinal extensions? ==<br />
<br />
Find a notion of strongness for cofinal extensions and prove that if If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ is strong than there is an automorphism $f$ of $N$ whose fixed point set is $M$ <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=End_extensions,_cofinal_extensions&diff=203
End extensions, cofinal extensions
2013-01-24T15:10:05Z
<p>Rkossak: </p>
<hr />
<div>== Conservative cofinal extensions? ==<br />
<br />
<br />
For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$. <br />
<br />
Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is ''conservative'' if, for each $b\in N\setminus M$ there is $a\in M$ such that<br />
$b\cap M_b=a\cap M_b.$ <br />
<br />
Problem: Do conservative cofinal extensions exist?<br />
<br />
<br />
== The description property ==<br />
<br />
The extension $M\prec_{cof} N$ has the ''description property'' if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that <br />
<br />
1. $N\models a\in A_i$ for all $i<\omega$;<br />
<br />
2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.<br />
<br />
<br />
For each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?<br />
<br />
== Isolated gaps ==<br />
For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.<br />
<br />
If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is ''non-isolated'' if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is ''isolated''.<br />
<br />
If $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. Moreover, if the extension has the description property, then all new gaps are non-isolated <cite> koskot2008:moreon </cite>. <br />
<br />
Problem: Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?<br />
<br />
<br />
== Extending automorphisms to cofinal extensions ==<br />
<br />
If $M$ and $N$ are countable recursively saturated models of $PA$, $M\prec_{cof} N$ and $f$ is an automorphism of $M$, then $f$ is ''potentially extendable'' if for every bounded $A\subseteq M$, $A$ is coded in $N$ iff $f(A)$ is.<br />
<br />
If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ has the description property, then every potentially extendable automorphism of $M$ has an extension to an automorphism of $N$ <cite> koskot2008:moreon </cite>.<br />
<br />
Problem: Find a sufficient and necessary condition on the extension $M\prec_{cof} N$, where $M$ and $N$ are countable recursively saturated models of $PA$, under which every potentially extendable automorphism of $M$ extends to an automorphism of $N$.<br />
<br />
<br />
== Strong cofinal extensions? ==<br />
<br />
Find a notion of strongness for cofinal extensions and prove that if If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ is strong than there is an automorphism $f$ of $N$ whose fixed point set is $M$ <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=End_extensions,_cofinal_extensions&diff=202
End extensions, cofinal extensions
2013-01-24T14:52:16Z
<p>Rkossak: </p>
<hr />
<div><br />
== Conservative cofinal extensions? ==<br />
<br />
<br />
For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$. <br />
<br />
Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is ''conservative'' if, for each $b\in N\setminus M$ there is $a\in M$ such that<br />
$b\cap M_b=a\cap M_b.$ <br />
<br />
Problem: Do conservative cofinal extensions exist?<br />
<br />
<br />
== The description property ==<br />
<br />
The extension $M\prec_{cof} N$ has the ''description property'' if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that <br />
<br />
1. $N\models a\in A_i$ for all $i<\omega$;<br />
<br />
2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.<br />
<br />
<br />
For each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated <cite> koskot2008:moreon </cite>.<br />
<br />
<br />
Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?<br />
<br />
== Isolated gaps ==<br />
For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.<br />
<br />
If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is ''non-isolated'' if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is ''isolated''.<br />
<br />
If $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. Moreover, if the extension has the [[Media:description property]], then all new gaps are non-isolated <cite> koskot2008:moreon </cite>. <br />
<br />
<br />
Problem: Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?<br />
<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=End_extensions,_cofinal_extensions&diff=201
End extensions, cofinal extensions
2013-01-24T14:43:05Z
<p>Rkossak: </p>
<hr />
<div>== Isolated gaps ==<br />
For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.<br />
<br />
If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is ''non-isolated'' if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is ''isolated''.<br />
<br />
<br />
It is known that if $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. <br />
<br />
<br />
Problem <cite> koskot2008:moreon </cite> Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?<br />
<br />
<br />
<br />
== Conservative cofinal extensions? ==<br />
<br />
<br />
For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$. <br />
<br />
Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is ''conservative'' if, for each $b\in N\setminus M$ there is $a\in M$ such that<br />
$b\cap M_b=a\cap M_b.$ <br />
<br />
Problem: Do conservative cofinal extensions exist?<br />
<br />
<br />
== The description property ==<br />
<br />
The extension $M\prec_{cof} N$ has the ''description property'' if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that <br />
<br />
1. $N\models a\in A_i$ for all $i<\omega$;<br />
<br />
2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.<br />
<br />
It is shown in <cite> koskot2008:moreon </cite> that for each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated.<br />
<br />
Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=End_extensions,_cofinal_extensions&diff=200
End extensions, cofinal extensions
2013-01-24T14:41:12Z
<p>Rkossak: </p>
<hr />
<div>== Isolated gaps ==<br />
For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.<br />
<br />
If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is ''non-isolated'' if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is ''isolated''.<br />
<br />
<br />
It is known that if $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. <br />
<br />
<br />
Problem <cite> koskot2008:moreon </cite> Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?<br />
<br />
<br />
<br />
== Conservative cofinal extensions? ==<br />
<br />
<br />
For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$. <br />
<br />
Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is ''conservative'' if, for each $b\in N\setminus M$ there is $a\in M$ such that<br />
$b\cap M_b=a\cap M_b.$ <br />
<br />
Problem: Do conservative cofinal extensions exist?<br />
<br />
<br />
== The description property ==<br />
<br />
The extension $M\prec_{cof} N$ has the ''description property'' if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that <br />
<br />
1. $N\models a\in A_i$ for all $i<\om$;<br />
<br />
2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.<br />
<br />
It is shown in <cite> koskot2008:moreon </cite> that for each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated.<br />
}<br />
Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?<br />
<br />
<br />
<br />
{{References}}</div>
Rkossak
http://modelsofpa.info/index.php?title=Peano%27s_Parlour&diff=199
Peano's Parlour
2013-01-23T19:49:44Z
<p>Rkossak: </p>
<hr />
<div>This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. <br />
<br />
[[Cuts in models of PA and independence results]].<br />
<br />
[[Recursively saturated, resplendent models, and saturated models]].<br />
<br />
[[Standard systems and the Scott set problem]].<br />
<br />
[[Lattices of elementary substructures]].<br />
<br />
[[Uncountable models with interesting second-order properties]].<br />
<br />
[[Nonstandard satisfaction classes]].<br />
<br />
[[Complexity and classification of countable models]].<br />
<br />
[[Model theory of strong fragments of arithmetic]].<br />
<br />
[[Model theory of weak fragments of arithmetic]].<br />
<br />
[[Model theoretic methods in proof theory]].<br />
<br />
The last chapter of ''The structure of models of Peano arithmetic'' by Kossak and Schmerl has a list of 20 questions with comments and references. Here is an [[errata]] for the book.</div>
Rkossak
http://modelsofpa.info/index.php?title=Uncountable_models_with_interesting_second-order_properties.&diff=198
Uncountable models with interesting second-order properties.
2013-01-23T19:46:38Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Rigid_models&diff=197
Rigid models
2013-01-23T19:46:22Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Jonsson_models&diff=196
Jonsson models
2013-01-23T19:45:58Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Rather_classless_models.&diff=195
Rather classless models.
2013-01-23T19:45:37Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Friedman%27s_14-th_problem.&diff=194
Friedman's 14-th problem.
2013-01-23T19:45:06Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Standard_systems_and_the_Scott_set_problem.&diff=193
Standard systems and the Scott set problem.
2013-01-23T19:43:51Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak
http://modelsofpa.info/index.php?title=Kanovei%27s_question&diff=192
Kanovei's question
2013-01-23T19:43:29Z
<p>Rkossak: Blanked the page</p>
<hr />
<div></div>
Rkossak