In his Mathematical Analysis I, Zorich says the following after introducing the reals axiomatically:
In relation to any abstract system of axioms, at least two questions arise immedi- ately. First, are these axioms consistent? That is, does there exist a set satisfying all the conditions just listed? This is the problem of consistency of the axioms. Second, does the given system of axioms determine the mathematical object uniquely? That is, as the logicians would say, is the axiom system categorical? Here uniqueness must be understood as follows...
Now I am no logician or philosopher so I certainly don't expect, want, or need the full treatment, but I am wondering if it is at all "controversial" to assume that consistency implies existence in the sense which seems to be tacit in Zorich passing from
First, are these axioms consistent?
to using "that is" in
That is, does there exist a set satisfying all the conditions just listed?
Godel's perhaps-unfortunately-named completeness theorem gives a precise positive answer: if $T$ is a set of (first-order) sentences and every model of $T$ satisfies the (first-order) sentence $\varphi$, then in fact there is a proof of $\varphi$ from $T$. More snappily, we have $$T\vdash\varphi\iff T\models\varphi$$ (technically completeness is the right-to-left direction, with the left-to-right direction being soundness, but soundness is so trivial it's often subsumed by completeness).
In particular, let $T$ be "any abstract system of axioms" (as long as they're first-order) and let $\varphi$ be $\perp$, the always-false sentence. Then "$T\models\varphi$" means "Every model of $T$ satisfies the always-false sentence," which is another way of saying "$T$ is unsatisfiable;" meanwhile, "$T\vdash\varphi$" is just another way of saying "$T$ is consistent." Contrapositing, we get that if $T$ is consistent then $T$ is satisfiable (= has a model).
(Note that "consistent iff satisfiable" is equivalent to the completeness theorem as stated above, since we can shift from "$T\not\vdash\varphi$" to "$T\cup\{\neg\varphi\}$ is consistent" and similarly for $\models$. However, this trick doesn't work in logics without negation, so in general there is a real difference here. But this is a side issue.)
There are two crucial caveats here:
It is absolutely essential that we stick to first-order sentences. In general, logics admitting a completeness theorem are quite rare.
It is also absolutely general that we look at all models of $T$ in the definition of $T\models\varphi$. The common phrasing of Godel's incompleteness theorem as "There are true statements of arithmetic which aren't provable" may appear to contradict the completeness theorem, but of course it doesn't: the issue is that "true" here refers to truth in one particular model, and that's not something that the completeness theorem has bearing on.