# Difference between revisions of "Cuts in models of PA and independence results"

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Let $M\models PA$ be countable and recursively saturated. | 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 $M$. | Kossak proved that there are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$. | ||

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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 == | ||

− | + | 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 $M$? | |

− | + | == "Star" schemata == | |

− | + | 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. | ||

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− | + | Problem: The assumption that $M$ is recursively saturated is not necessary for $n=1,2$ <cite>kossakparis1992:subsets </cite>. Is it necessary for higher $n$? | |

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