Cuts in models of PA and independence results

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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

  1. Roman Kossak and James H. Schmerl. On cofinal extensions and elementary interstices. Notre Dame J. Formal Logic 53(3):267--287, 2012. www   bibtex
  2. 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
  3. C. Smoryński. Elementary extensions of recursively saturated models of arithmetic. Notre Dame J. Formal Logic 22(3):193--203, 1981. www   MR   bibtex
  4. Roman Kossak. A note on satisfaction classes. Notre Dame J. Formal Logic 26(1):1--8, 1985. www   DOI   MR   bibtex
  5. Richard Kaye and Tin Lok Wong. Truth in generic cuts. Ann. Pure Appl. Logic 161(8):987--1005, 2010. www   DOI   MR   bibtex
  6. Stuart T. Smith. Extendible sets in Peano arithmetic. Trans. Amer. Math. Soc. 316(1):337--367, 1989. www   DOI   MR   bibtex
  7. Roman Kossak and Jeffrey B. Paris. Subsets of models of arithmetic. Arch. Math. Logic 32(1):65--73, 1992. www   DOI   MR   bibtex
  8. Vladimir Kanovei. On "star" schemata of Kossak and Paris. Logic Colloquium '96 (San Sebastián)12:101--114, Berlin, 1998. MR   bibtex
  9. Roman Kossak. A note on a theorem of Kanovei. Arch. Math. Logic 43(4):565--569, 2004. www   DOI   MR   bibtex
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