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So if minimal model of ZF exists, it is said that it is countable set by Lowenheim-Skolem. So, is Lowenheim-Skolem saying that for any countable theory with existence of infinite model there exists standard model, respecting normal element relation, that is countable infinite?

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No. The existence of standard models is strictly stronger.

It is consistent that there are no standard models, to see this note that the standard models are well-founded, in the sense that there is no infinite decreasing chain of standard models such that $M_{n+1}\in M_n$, simply because $\in$ itself is well-founded and standard models use the real $\in$ for their membership relation.

So there is a minimal standard model. But this model has the standard $\omega$ for its integers, so it cannot possible satisfy $\lnot\text{Con}(\mathsf{ZFC})$, so it must have a model of $\sf ZFC$ inside, but this model cannot be standard.

See also: Transitive ${\sf ZFC}$ model on Cantor's Attic.

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OK, then how can L be minimal and standard both? –  user61182 Feb 21 '13 at 12:45
    
@user81182: $L$ is minimal among class models that contain all the ordinals. The minimal standard model does not contain all the ordinals and is of the form $L_\alpha$ for some countable ordinal $\alpha$. –  Carl Mummert Feb 21 '13 at 12:48
    
@user61182: I'm not sure I understand your question. If there is a standard model then there is a minimal one. Since every model has an inner model which satisfies $V=L$, if $M$ is a minimal standard model then $L^M$ is a minimal standard model which satisfies $V=L$. Since $V=L$ is absolute we have that $L^M$ is actually $L_\eta$ (of the "real $L$") for some $\eta$, and this is the least transitive model in the sense of inclusion as well, not only in the sense of membership. –  Asaf Karagila Feb 21 '13 at 12:48
    
@Carl: You might want to post a more elaborated answer, I am just leaving for office hours and won't have time to do this for a couple of hours. :-) –  Asaf Karagila Feb 21 '13 at 12:50
    
@CarlMummert Oh got that part. I was mistaken. –  user61182 Feb 21 '13 at 12:58

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