Difference between revisions of "Jonsson models"
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Gaifman and Knight independently showed in 1976 that there are Jonsson models of $PA$. | Gaifman and Knight independently showed in 1976 that there are Jonsson models of $PA$. | ||
| − | 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. | + | 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. |
Kossak has asked: Is there an $\aleph_1$-like Jonsson model $M\models\PA$ such that $|\{{\rm tp}(a): a\in M\}|=\aleph_0$? | 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 $K=M$? 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. | 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. | ||
Revision as of 15:37, 17 January 2013
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 and Knight independently showed in 1976 that there are Jonsson models of $PA$.
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.
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 $K=M$? 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.
