## Free cuts

A cut $I$ in a model $M\models PA$ is *free* if for all $a, b\in I$ if $(M,a)\equiv (M,b)$,
then $(M,I,a)\equiv (M,I,b)$. There are free elementary cuts in every countable recursively saturated
model of $PA$, and generic cuts of Kaye and Tin Lok Wong are free.

Problem: Let $M\models PA$ be countable and recursively saturated. Does $M$ have a free elementary cut $I$ such that the pair $(M,I)$ is recursively saturated?

The problem was posed in [1], freeness of elementary generic cuts is discussed in [2].

## Omitting theories of subsets

Suppose $M$ is countable recursively saturated and
$X$ is an undefinable subset of $M$. Is there a countable recursively saturated $N$ such that $N$ is an elementary end extension of $M$,
and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv
(M,X)$?

The answer if `yes' is either $(M,X)\not\models PA^*$ or ${\rm Th}(M,X)\notin {\rm SSy}(M)$.

This problem is listed in [3], but, unfortunately, with many typos.

## Elementarily equivalent nonisomorphic pairs?

Let $M\models PA$ be countable and recursively saturated and let $K$ and $K'$ be short elementary cuts of $M$ such that $(M,K)\equiv (M,K')$. Are $(M,K)$ and $(M,K')$ isomorphic?

The problem was posed in [4].

## References

- Roman Kossak.
*Four problems concerning recursively saturated models of arithmetic.* Notre Dame J. Formal Logic 36(4):519--530, 1995. (Special Issue: Models of arithmetic) www DOI MR bibtex@article {kossak1995:four,

AUTHOR = {Kossak, Roman},

TITLE = {Four problems concerning recursively saturated models of arithmetic},

NOTE = {Special Issue: Models of arithmetic},

JOURNAL = {Notre Dame J. Formal Logic},

FJOURNAL = {Notre Dame Journal of Formal Logic},

VOLUME = {36},

YEAR = {1995},

NUMBER = {4},

PAGES = {519--530},

ISSN = {0029-4527},

CODEN = {NDJFAM},

MRCLASS = {03C62 (03C57)},

MRNUMBER = {1368464 (98a:03056)},

MRREVIEWER = {Fuxing Shen},

DOI = {10.1305/ndjfl/1040136913},

URL = {http://dx.doi.org/10.1305/ndjfl/1040136913},

}

- 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},

}

- Roman Kossak and James H. Schmerl.
*The structure of models of Peano arithmetic.* Vol. 50, The Clarendon Press Oxford University Press, Oxford, 2006. (Oxford Science Publications) www DOI MR bibtex@book {kossakschmerl:modelsofpa,

AUTHOR = {Kossak, Roman and Schmerl, James H.},

TITLE = {The structure of models of Peano arithmetic},

SERIES = {Oxford Logic Guides},

VOLUME = {50},

NOTE = {Oxford Science Publications},

PUBLISHER = {The Clarendon Press Oxford University Press},

ADDRESS = {Oxford},

YEAR = {2006},

PAGES = {xiv+311},

ISBN = {978-0-19-856827-8; 0-19-856827-4},

MRCLASS = {03-02 (03C62 03F30 03H15)},

MRNUMBER = {2250469 (2008b:03001)},

MRREVIEWER = {Constantine Dimitracopoulos},

DOI = {10.1093/acprof:oso/9780198568278.001.0001},

URL = {http://dx.doi.org/10.1093/acprof:oso/9780198568278.001.0001},

}

- 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},

}

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