I am interested in the following question just because it feels as if I am not able to figured it out on my own. It seems there is some mistake in my thinking.
Let $T$ be a theory in some first-order language and $\sigma$ a sentence in the same language. The Completeness Theorem says that $T \vdash \sigma$ if and only if $T \models \sigma$.
To me, it seems that the first statement depends only on logical axioms (how to deduce etc.) while the second statement depends on the existence or non-existence of certain models of $T$ which is (on first glance) a question of set-theoretical axioms.
I cannot believe that the existence of models of $T$ satisfying $\sigma$ is independent of the choice of set theory. This cannot be true.
Thank you for your help!
Edit making the question very concrete:
Suppose that a theory $T$ in a language (say the theory $\mathsf{ACF}$ of algebraically closed field together with one sentences $\sigma$, in the language of rings) has no models under the assumption of $\mathsf{ZFC}+ \mathsf{GCH}$ . Hence, by logical operation, one can infer a contradiction from $T$ (Incompleteness Theorem). But since there is a model of $\mathsf{ZFC}$ where natural numbers coincide with the ones from our original model of $\mathsf{ZFC}+ \mathsf{GCH}$, the proof of the contradiction can also be encoded in the ''smaller'' model, and this model therefore does not contain a model of $T$.
Is this correct?