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 $(M,K)\not\cong (M,L)$.
  
 
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Several results related to the problem can be found in <cite>kossak1985:anoteon, kayetinlokwong2010:truth </cite>,
 
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References:
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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.

Revision as of 18:03, 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],

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.