[soulofmischief]

You're referring to Skolem's paradox. It just shows that first-order logic is incomplete.

Ernst Zermelo resolved this by stating that his axioms should be interpreted within second-order logic, and as such it doesn't contradict Cantor's theorem since the Löwenheim–Skolem theorem only applies in first-order logic.

[zozbot234]

The standard semantics for second-order logic are not very practical and arguably not even all that meaningful or logical (as argued e.g. by Willard Quine); you can use Henkin semantics (i.e. essentially a many-sorted first-order theory) to recover the model-theoretic properties of first-order logic, including Löwenheim-Skolem.