Suppose you have a saturated model N of a complete theory T without finite models. How is it possibile to construct a proper saturated elementary substructure of N of the same cardinality of N ?
I tried to use a construction similar to the donward lowheneim-skolem theorem but I can not prove the saturation ...
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$\begingroup$ Try, instead, to give a proper elementary superstructure of the same cardinality which is also saturated. Then check why this is equivalent to what you want, using the fact that saturated models of the same cardinality must be isomorphic. $\endgroup$– Richard RastCommented Apr 9, 2015 at 16:11
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$\begingroup$ I understand what you mean but I think that the existence of satured superstructure of the same cardinality is related to that the cardinality of N is an inaccessible cardinal using $\endgroup$– eagleCommented Apr 9, 2015 at 16:35
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2$\begingroup$ You certainly don't need to use inaccessible cardinals. $\endgroup$– Richard RastCommented Apr 9, 2015 at 17:01
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5$\begingroup$ Take any elementary extension $M$ of $N$ of the same cardinaility. There is an elementary embedding of $M$ into $N$. Thus $N$ has an isomorphic proper elementary submodel, or, equivalently an isomorphic proper elementary extentsion. $\endgroup$– Dave MarkerCommented Apr 9, 2015 at 17:15
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1$\begingroup$ crossposted from math.stackexchange.com/questions/1227098/… $\endgroup$– Will JagyCommented Apr 9, 2015 at 18:48
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