Jonsson models

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 ${\mathfrak A}={\mathfrak B}$.

Gaifman (1976) and Knight (1976) independently showed that there are J\'onsson models of $PA$.

J\'onsson 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$. Each known J\'onsson model realizes uncountably many complete types. Kossak has asked: Is there an \olike\ J\'onsson model$M\models\PA$such that$|\{\tp(a): a\in M\}|=\aleph_0$? If$M\models\PA$is \olike\ and \rs\ then$|\{\tp(a): a\in M\}|=\aleph_0$, but$M$is not J\'onsson. Therefore, another related question is: Is there a '' weakly J\'onsson model''$M\models \PA$, i.e. a recursively saturated model$M$such that for every recursively saturated$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.