Difference between revisions of "Jonsson models"

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-A model $\mathfrak B$ is ''Jonsson'' if $|{\mathfrak B}|>\aleph_0$ and for every ${\mathfrak A}\prec {\mathfrak B}$, if $|{\mathfrak A}|=|{\mathfrak B}|$, then ${\mathfrak A}={\mathfrak B}$.
-Gaifman <cite> gaifman1976:models </cite> and Knight <cite> knight1976:hanf </cite> independently showed  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''.  A model $M$ of $PA$ is short  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.
-Kossak has asked: Is there an $\aleph_1$-like Jonsson model $M\models PA$ such that $|\{{\rm tp}(a): a\in M\}|=\aleph_0$?
-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 $K=M$? The problem was posed in <cite> kossak1995:four </cite>.
-{{References}}

Latest revision as of 12:45, 23 January 2013