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 PAofcardinality\aleph_1areeither\aleph_1-likeorare″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 16:37, 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.