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 lowheneimskolem theorem but I can not prove the saturation ...
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closed as offtopic by Will Jagy, Emil Jeřábek, Stefan Kohl, Dima Pasechnik, Alex Degtyarev Apr 9 '15 at 19:50
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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 Rast Apr 9 '15 at 16:11

$\begingroup$ 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 $\endgroup$ – eagle Apr 9 '15 at 16:35

2$\begingroup$ You certainly don't need to use inaccessible cardinals. $\endgroup$ – Richard Rast Apr 9 '15 at 17:01

4$\begingroup$ 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 Marker Apr 9 '15 at 17:15

1$\begingroup$ crossposted from math.stackexchange.com/questions/1227098/… $\endgroup$ – Will Jagy Apr 9 '15 at 18:48