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 $M$.
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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 $M$.
 
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 ==
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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.
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== "Star" schemata ==