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

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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 $(M,K)\not\cong (M,L)$. | ||

− | Several results related to the problem can be found in <cite>kossak1985:anoteon, kayetinlokwong2010:truth </cite>, | + | Several results related to the problem can be found in <cite>smorynski1982:anoteoninitial, smorynski1981:elementary, kossak1985:anoteon, kayetinlokwong2010:truth </cite>, |

Smoryński, C. ''Elementary extensions of recursively saturated models of arithmetic''. Notre Dame J. Formal Logic 22 (1981), no. 3, 193–203. | Smoryński, C. ''Elementary extensions of recursively saturated models of arithmetic''. Notre Dame J. Formal Logic 22 (1981), no. 3, 193–203. | ||

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Kaye, Richard; ''Wong, Tin Lok Truth in generic cuts''. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005. | Kaye, Richard; ''Wong, Tin Lok Truth in generic cuts''. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005. | ||

+ | |||

+ | {{References}} |

## Revision as of 17:05, 20 January 2013

## Diversity in elementary cuts

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 $M$.

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 $(M,K)\not\cong (M,L)$.

Several results related to the problem can be found in [1, 2, 3, 4],

Smoryński, C. *Elementary extensions of recursively saturated models of arithmetic*. Notre Dame J. Formal Logic 22 (1981), no. 3, 193–203.

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

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.

## References

- 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