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Godel doesn't talk about space, time, human experience, truths that exist independently of human experience. I fail to see how he's relevant.

He does. Learn more about his biography and his interest in mysticism and philosophy: http://www.amazon.com/G%C3%B6del-Logic-John-L-Casti/dp/07382...

Perhaps he does, but I don't think he does so in the proofs of his incompleteness theorems.