Set theory in the usual first order axiomatization, as far as I understand, is powerful enough to construct structures that satisfy the axioms of the second order axiomatization for naturals, integers, rationals and reals...
It has been proven that the second order axiomatizations presented above, are all categorical. Thus, for any formula $\phi$: $\Phi_2\models\phi$ or $\Phi_2\models\neg\phi$, where $\Phi_2$ is the set of second order axioms.
Gödel showed that first order logic is semantically complete, that is: $\Phi_1\models \phi$ implies $\Phi_1\vdash\phi$.
Gödel also proved the incompleteness result for first order logic. That is, if the set of axioms is strong enough to prove certain results about natural numbers, then there exists a formula such that: $\Phi_1\nvdash\phi$, $\Phi_1\nvdash\neg\phi$. From the result it's also possible to prove that in second order logic with sufficient axioms, using any proof system, there exists a formula such that if $\Phi_2\models\phi$, then $\Phi_2\nvdash\phi$ (ie. second order logic is never semantically complete). So no matter what, there is a formula that can't be proven. But as far as I understand this all only applies to theories about numbers, not sets.
However, I believe it's common knowledge that the incompleteness result applies to numbers constructed in set theory as well. But as discussed above, set theory has first order axiomatization, that is able to construct structures equivalent to the second order structures... so where/how exactly does it break down, where is the unprovable formula?