application of Lowenheim-Skolem theorem

Question

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?

Answer 1

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.