As I learn the Compactness and Löwenheim-Skolem theorems of first-order logic and I begin to have a better understanding of what they really mean, something has me baffled.
As I have learned it, the Löwenheim-Skolem theorem says that if a set $S$ of $L$-sentences of a first-order language $L$ is satisfiable at all, then $S$ is satisfiable in a countable structure.
Now, mathematics proves the existence of uncountable sets. To be more specific, take the language $L$ of set theory, and let $\mathrm{ZFC}$ denote the set of axioms of set theory with the axiom of choice. Then, $\mathrm{ZFC}$ is a set of $L$-sentences and (if consistent) by the Löwenheim-Skolem it must be satisfiable in an $L$-structure $\mathcal{M}$ having a countable domain. On the other hand, the statement there is an uncountable set can be expressed as an $L$-sentence $\sigma$, and $S\vDash\sigma$, i.e. the existence of an uncountable set is a logical conclusion $\mathrm{ZFC}$.
Hence, the axioms which allow us to conclude that an uncountable set exists are themselves satisfied by a countable set. I am a novice, and my question is: Is there a particular meaning that professional logicians are giving to this situation? Am I suppose to interpret it in a particular way? It is quite remarkable!
This MSE question is related.