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| Let $M\models PA$ be countable and recursively saturated. | | Let $M\models PA$ be countable and recursively saturated. |
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− | Smorybski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$.
| + | Smorynski proved that here are $2^{\aleph_0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$. |
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| 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 $(M,K)\not\cong (M,L)$. |
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| + | 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>. |
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| + | == Stu Smith's question == |
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− | == References ==
| + | 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 $M$? |
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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.
| + | == "Star" schemata == |
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