I'm studying set theory which leads me to some confusions of model theory.
For the context, I know that the set $\mathbb{N}$ with the usual operations $(0,+,×)$ is a model of Peano arithmetic. It seems to me it is quite obvious (except for the axiom of mathematical induction, which I don't know how to prove) to verify that this structure satisfies the axioms of Peano. For example the 2 following axioms:
$0$ $\in$ $\mathbb{N}$ (I point at the number $0$ in the structure)
if $n \in \mathbb{N}$ then $n+1 \in \mathbb{N}$ (I point at 2 numbers $n$ and $n+1$ in the set $\mathbb{N}$ of the structure)
Concerning ZFC, I know that it has many models. My concern is how can we verify that a given model satisfies ZFC ? It seems to me that we need to verify if the given model satisfies all the axioms of ZFC or not (then it will automatically satisfies all the sentences of the theory). But I can't think of a way to actually verify if a structure (i.e. a model) satisfies or not an axiom of ZFC, for example :
Axiom of extension: Two sets are equal if and only if they have the same elements.
Axiom of specification: To every set A and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.
As for me, these axioms are so obvious, much more obvious than those of Peano. How can we prove that a given structure satisfies these obvious axioms ? Or in general, how can we prove that a given structure satisfies ZFC ?
Could you please enlighten me ? Thank you very much for your help!