People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is.
Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.
Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.