https://en.wikipedia.org/wiki/Skolem%27s_paradox
In reading about Skolem's paradox on wikipedia, I encountered the claim that every consistent countable axiomatisation of set theory in first order logic has a countable model. That got me thinking that it's strange that set theory could have more than one axiomatisation at all, countably modeled or otherwise. The only set theory I typically hear about is ZFC, and then I recalled NBG set theory, but that is an extension of ZFC, not a separate axiomatisation. Or is NBG considered a separate axiomatisation?
Moreover, I had thought that set theory required second order logic, since it involves quantifying over both sets and relations between sets. How could it have an axiomatisation in first-order logic at all, let alone more than one?
Finally, if indeed set theory (and n.b. that they didn't clarify which set theory, so I would take it to mean ZFC) has more than one axiomatisation in first-order logic, does it have finitely many, denumerably many, or perhaps even non-denumerably many?
Thank you for any advice.
You can indeed get the same theory from a different set of axioms. A bit more precisely, for a set of sentences $\Gamma$, let the deductive closure of $\Gamma$ be the set $$Ded(\Gamma):=\{\varphi:\Gamma\vdash\varphi\}$$ of sentences which $\Gamma$ proves (I'm tacitly fixing a particular language to work in here, e.g. the language of set theory). Then we can have $Ded(\Gamma_1)=Ded(\Gamma_2)$ even if $\Gamma_1\not=\Gamma_2$.
For example, given $\Gamma$ consider $$\Gamma_{double}:=\{\varphi\wedge\varphi:\varphi\in\Gamma\}.$$ Obviously $Ded(\Gamma)=Ded(\Gamma_{double})$, but in general $\Gamma$ will be a different set of sentences than $\Gamma_{double}$. More interestingly, we can show the following:
Every finitely axiomatizable theory is axiomatizable by a single sentence. That is, if $\Gamma$ is finite, then there is some single sentence $\gamma$ such that $Ded(\Gamma)=Ded(\{\gamma\})$. There's actually less to this than meets the eye: just let $\gamma$ be the conjunction of the finitely many sentences in $\Gamma$.
Every computably enumerably axiomatizable theory is in fact computably axiomatizable, and indeed primitive recursively axiomatizable and more - this is Craig's trick, and is often useful in the context of incompleteness problems.
And getting back to $\mathsf{ZFC}$ in particular, it's easy to show that the Separation scheme is in fact redundant: we can prove each instance of Separation from the remaining axioms of $\mathsf{ZFC}$. Similarly, we can replace Choice with Zorn's Lemma, the Well-Ordering Principle, or various other statements. So there are lots of natural alternative axiomatizations of $\mathsf{ZFC}$.
(As to how many axiomatizations $\mathsf{ZFC}$ has, it's a good exercise to show that there are in fact continuum-many ways to axiomatize $\mathsf{ZFC}$. In fact, there are continuum-many ways to do this non-redundantly, but that takes more work. Even constructing a single non-redundant axiomatization is nontrivial - see here.)