|
|
(6 intermediate revisions by one other user not shown) |
Line 3: |
Line 3: |
| Let $M\models PA$ be countable and recursively saturated. | | Let $M\models PA$ be countable and recursively saturated. |
| | | |
− | Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.
| + | Smorynski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$. |
| | | |
| Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$. | | Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$. |
Line 11: |
Line 11: |
| By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$. | | By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$. |
| | | |
| + | 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>. |
| | | |
| + | == Stu Smith's question == |
| | | |
− | References:
| + | 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 $M$? |
| | | |
| | | |
− | Smoryński, C. ''Elementary extensions of recursively saturated models of arithmetic''. Notre Dame J. Formal Logic 22 (1981), no. 3, 193–203.
| + | == "Star" schemata == |
| | | |
− | Smoryński, C. ''A note on initial segment constructions in recursively saturated models of arithmetic''. Notre Dame J. Formal Logic 23 (1982), no. 4, 393–408
| + | 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 $N$, $(M,X)\models I\Sigma_n$ |
| + | 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. |
| | | |
− | Kossak, Roman ''A note on satisfaction classes''. Notre Dame J. Formal Logic 26 (1985), no. 1, 1–8.
| |
| | | |
− | Kaye, Richard; ''Wong, Tin Lok Truth in generic cuts''. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005.
| + | Problem: The assumption that $M$ is recursively saturated is not necessary for $n=1,2$ <cite>kossakparis1992:subsets </cite>. Is it necessary for higher $n$? |
| + | |
| + | |
| + | |
| + | |
| + | {{References}} |
Latest revision as of 12:26, 16 February 2013
Diversity in elementary cuts
Let $M\models PA$ be countable and recursively saturated.
Smorynski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.
Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$.
Problem: How many theories of pairs $(M,K)$ are there, such that $K$ is semiregular, but not regular in $M$.
By recent results of Kaye and Tin Lok Wong, every countable recursively saturated $M$ has $K$ and $L$ as in the question, such that $(M,K)\not\cong (M,L)$.
The problem was stated in [1] related results can be found in [2, 3, 4, 5].
Stu Smith's question
Question 6.6 of [6]: 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 $M$?
"Star" schemata
Answering a question from [7] Kanovei [8] 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 $N$, $(M,X)\models I\Sigma_n$
and for some such $X$, $(M,X)\not\models I\Sigma_{n+2}$. Later in [9] $n+2$ was reduced to $n+1$ by another method.
Problem: The assumption that $M$ is recursively saturated is not necessary for $n=1,2$ [7]. Is it necessary for higher $n$?
References
- Roman Kossak and James H. Schmerl. On cofinal extensions and elementary interstices. Notre Dame J. Formal Logic 53(3):267--287, 2012. www bibtex
@article {kossakschmerl2012:oncofinal,
AUTHOR = {Kossak, Roman and Schmerl, James H.},
TITLE = {On cofinal extensions and elementary interstices},
JOURNAL = {Notre Dame J. Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {53},
YEAR = {2012},
NUMBER = {3},
PAGES = {267--287},
ISSN = {0029-4527},
CODEN = {NDJFAM},
MRCLASS = {03C62 (03C62)},
MRNUMBER = {},
MRREVIEWER = {},
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1348524112},
}
- C. Smoryński. A note on initial segment constructions in recursively saturated models of arithmetic. Notre Dame J. Formal Logic 23(4):393--408, 1982. www MR bibtex
@article {smorynski1982:anoteoninitial,
AUTHOR = {Smory{\\'n}ski, C.},
TITLE = {A note on initial segment constructions in recursively saturated models of arithmetic},
JOURNAL = {Notre Dame J. Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {23},
YEAR = {1982},
NUMBER = {4},
PAGES = {393--408},
ISSN = {0029-4527},
CODEN = {NDJFAM},
MRCLASS = {03C62 (03C57 03F30 03H15)},
MRNUMBER = {669146 (83j:03058)},
MRREVIEWER = {Roman Murawski},
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093870152},
}
- C. Smoryński. Elementary extensions of recursively saturated models of arithmetic. Notre Dame J. Formal Logic 22(3):193--203, 1981. www MR bibtex
@article {smorynski1981:elementary,
AUTHOR = {Smory{\\'n}ski, C.},
TITLE = {Elementary extensions of recursively saturated models of arithmetic},
JOURNAL = {Notre Dame J. Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {22},
YEAR = {1981},
NUMBER = {3},
PAGES = {193--203},
ISSN = {0029-4527},
CODEN = {NDJFAM},
MRCLASS = {03C62 (03C57 03H15)},
MRNUMBER = {614117 (82g:03064)},
MRREVIEWER = {Klaus Potthoff},
URL = {http://projecteuclid.org/getRecord?id=euclid.ndjfl/1093883454},
}
- Roman Kossak. A note on satisfaction classes. Notre Dame J. Formal Logic 26(1):1--8, 1985. www DOI MR bibtex
@article {kossak1985:anoteon,
AUTHOR = {Kossak, Roman},
TITLE = {A note on satisfaction classes},
JOURNAL = {Notre Dame J. Formal Logic},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {26},
YEAR = {1985},
NUMBER = {1},
PAGES = {1--8},
ISSN = {0029-4527},
CODEN = {NDJFAM},
MRCLASS = {03H15 (03C50 03C62)},
MRNUMBER = {766663 (86c:03055)},
MRREVIEWER = {Bernd Dahn},
DOI = {10.1305/ndjfl/1093870757},
URL = {http://dx.doi.org/10.1305/ndjfl/1093870757},
}
- Richard Kaye and Tin Lok Wong. Truth in generic cuts. Ann. Pure Appl. Logic 161(8):987--1005, 2010. www DOI MR bibtex
@article {kayetinlokwong2010:truth,
AUTHOR = {Kaye, Richard and Wong, Tin Lok},
TITLE = {Truth in generic cuts},
JOURNAL = {Ann. Pure Appl. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {161},
YEAR = {2010},
NUMBER = {8},
PAGES = {987--1005},
ISSN = {0168-0072},
CODEN = {APALD7},
MRCLASS = {03C62 (03H15)},
MRNUMBER = {2629502 (2011f:03048)},
MRREVIEWER = {Constantine Dimitracopoulos},
DOI = {10.1016/j.apal.2009.11.001},
URL = {http://dx.doi.org/10.1016/j.apal.2009.11.001},
}
- Stuart T. Smith. Extendible sets in Peano arithmetic. Trans. Amer. Math. Soc. 316(1):337--367, 1989. www DOI MR bibtex
@article {smith1989:extendible,
AUTHOR = {Smith, Stuart T.},
TITLE = {Extendible sets in Peano arithmetic},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {316},
YEAR = {1989},
NUMBER = {1},
PAGES = {337--367},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {03C62 (03H15)},
MRNUMBER = {946223 (90b:03049)},
MRREVIEWER = {Athanassios Tzouvaras},
DOI = {10.2307/2001288},
URL = {http://dx.doi.org/10.2307/2001288},
}
- Roman Kossak and Jeffrey B. Paris. Subsets of models of arithmetic. Arch. Math. Logic 32(1):65--73, 1992. www DOI MR bibtex
@article {kossakparis1992:subsets,
AUTHOR = {Kossak, Roman and Paris, Jeffrey B.},
TITLE = {Subsets of models of arithmetic},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {32},
YEAR = {1992},
NUMBER = {1},
PAGES = {65--73},
ISSN = {0933-5846},
CODEN = {AMLOEH},
MRCLASS = {03H15 (03C62)},
MRNUMBER = {1186468 (94e:03066)},
MRREVIEWER = {Ali Enayat},
DOI = {10.1007/BF01270396},
URL = {http://dx.doi.org/10.1007/BF01270396},
}
- Vladimir Kanovei. On "star" schemata of Kossak and Paris. Logic Colloquium '96 (San Sebastián)12:101--114, Berlin, 1998. MR bibtex
@incollection {kanovei1998:onstar,
AUTHOR = {Kanovei, Vladimir},
TITLE = {On "star\'\' schemata of Kossak and Paris},
BOOKTITLE = {Logic Colloquium \'96 (San Sebasti{\\'a}n)},
SERIES = {Lecture Notes Logic},
VOLUME = {12},
PAGES = {101--114},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1998},
MRCLASS = {03F35 (03C62)},
MRNUMBER = {1674945 (2000f:03179)},
MRREVIEWER = {Roman Kossak},
}
- Roman Kossak. A note on a theorem of Kanovei. Arch. Math. Logic 43(4):565--569, 2004. www DOI MR bibtex
@article {kossak2004:anote,
AUTHOR = {Kossak, Roman},
TITLE = {A note on a theorem of Kanovei},
JOURNAL = {Arch. Math. Logic},
FJOURNAL = {Archive for Mathematical Logic},
VOLUME = {43},
YEAR = {2004},
NUMBER = {4},
PAGES = {565--569},
ISSN = {0933-5846},
CODEN = {AMLOEH},
MRCLASS = {03C62},
MRNUMBER = {2060400 (2005a:03073)},
DOI = {10.1007/s00153-004-0218-2},
URL = {http://dx.doi.org/10.1007/s00153-004-0218-2},
}
Main library