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Kurt Godel. Any logical system up to the level of specificity to make it worthwhile to use will have truths that are incapable of proof within it.

Of course, that misses a predicate. The predicate is "Why bother?" -- to state in its Eeyore form.

Why do people even care what is proved or not? Why bother?

The answer is we do care. And where does that sense of caring come from? What does man's very being infer?

It sounds to me that you are (mis)using Godel to essentially state that we know God exists because we can't prove he exists. That's a cop out. Furthermore, you aren't answering any of my specific questions. Fine. Whatever. I don't care anymore.

Why do we care? Because evolution has primed us to be the most intelligent and inquisitive of animals. Death is the ultimate mystery; one that we are frightened and fascinated by. We fear that ultimately we have no more meaning than that of a cooling corpse and we reject and rebel against that idea, because our ego, developed over the eons to ensure our survival and reproductive success, was designed by evolution to reject it.

Man's very being infers nothing more than man exists now, and is the product of countless years of evolution.

This neither proves nor disproves nothing of the supernatural. For that, you must ask, "What does my faith tell me?"

Any logical system up to the level of specificity to make it worthwhile to use will have truths that are incapable of proof within it.

Not really. Euclidean geometry is catagorical. All theorems are provable. The same is true for Pressberger arithmetic (ordinary arithmetic without multiplication; repeated addition up to any number is allowed.)

I don't know what your term "level of specificity" is supposed to mean, it does not appear in any of the works I have read on Gödel's theorem. I do know what the hypotheses of Gödel's theorem actually is though.