Cuts in recursively saturated models

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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

  1. 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
  2. Richard Kaye and Tin Lok Wong. Truth in generic cuts. Ann. Pure Appl. Logic 161(8):987--1005, 2010. www   DOI   MR   bibtex
  3. 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
  4. Roman Kossak and James H. Schmerl. On cofinal extensions and elementary interstices. Notre Dame J. Formal Logic 53(3):267--287, 2012. www   bibtex
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