One thing I've always wondered is if there's any _interesting_ statements that can't be proved or if they're all kind of self-referential nonsense statements.
On the one hand, I've often understood that it would be nice if we could assemble a list of "all the axioms" and prove it's consistent - at least that's how it was presented to me and roughly my interpretation of what eg. Russell and Whitehead were trying to do before being derailed by Godel proving it's impossible. On more recent reflection, it seems a little silly to want a proof in system X of the consistency of system X, even before we get to Godel, because if system X is inconsistent then it can definitely provide a proof that it is consistent by the principle of explosion.
There's whole _fields_ of inquiry that either take or leave the axiom of choice; so yes, absolutely.
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.
Sure there are, just choose your axioms that are interesting…