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

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.