Is that the kind of statement Godel was talking about? Seems to me to be somehow different. It's not really true or false, it's an arbitrary choice, so it's just something you either decide is part of your system or not, not a true statement that's unprovable.

Maybe I'm missing something?

Axiom of choice isn't really relevant to Godel's work, except for the fact that it's part of axiomatic systems. But in response to your comment, I was pointing out that there's lots of interesting math that comes from the results of his work; namely under different axiomatic systems you can find different and interesting results.

Axioms are, by definition, a choice.