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| − | A model $\mathfrak B$ is ''J\'onsson'' if $|{\mathfrak B}|>\aleph_0$ and for every ${\mathfrak A}\prec {\mathfrak B}$, if $|{\mathfrak A}|=|{\mathfrak B}|$, then ${\mathfrak A}={\mathfrak B}$.
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| − | Gaifman and Knight independently showed in 1976 that there are Jonsson models of $PA$.
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| − | Jonsson models $M$ of $PA of cardinality $\aleph_1$ are either $\aleph_1-like$ or are ''short'' i.e. there is an $a\in M$ such that the Skolem closure of $a$ is cofinal in $M$. Each known Jonsson model realizes uncountably many complete types.
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| − | Kossak has asked: Is there an $\aleph_1$-like Jonsson model $M\models\PA$ such that $|\{{\rm tp}(a): a\in M\}|=\aleph_0$?
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| − | If $M\models\PA$ is $\aleph_1$-like and recursively saturated, then $|\{{\rm tp}(a): a\in M\}|=\aleph_0$, but $M$ is not Jonsson. Therefore, another related question is: Is there a ''weakly Jonsson model'' $M\models \PA$, i.e. a recursively saturated model $M\models PA$ such that for every recursively saturated $K\prec M$, if $|K|=|M|$, then $K=M$? Some results related to this question are in Kossak, Roman, ''Four problems concerning recursively saturated models of arithmetic''. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519–530.
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