¿countably natural models?

Question

¿ A set theory can have countably models models that can it be natural models, where ∈ relation in the internal model , it is, can be the real ∈ in the external model?

Answer 1

I believe the question you're asking is: "If $V$ is some uncountable background model of set theory, can we have a countable $M\subseteq V$ such that the substructure $(M, \in)$ is a model of set theory?" That is, can we find countable models of set theory with the "right" notion of "element of"?

If this is what you're asking, then the answer is "yes" (assuming there are models of set theory at all): the Lowenheim-Skolem theorem gives us such submodels (which are elementary submodels as well as just being models of set theory). Moreover, we can even have such models which are transitive - $x\in M, y\in x\implies y\in M$ (though in this case we have to give up elementarity - e.g., such an $M$ will never compute $\mathbb{R}$ correctly).

Does this answer your question? (If there are language issues, you might want to write your question in your native language, and then someone can translate it accurately for you.)