Comment on Consistent extrapolated beliefs about math?

Tue, 09 Sep 2014 00:57:00 GMT

I think you're imagining an exit that isn't there. For example, ZFC can formalize Gentzen's proof of consistency of PA, and supports quite a bit of transfinite induction. Yet it still has to obey Gödel's theorem, like any other recursively axiomatizable theory.

(Also, all sets of axioms are countable, because they are subsets of the set of all finite strings, which is countable. I assume you meant to say something else, but I can't guess what.)

The amazing thing about Gödel's theorem is how general it is. I mentioned that in the post. Any Turing machine that prints theorems (regardless of the internal mechanism) must obey Gödelian limitations, as must any Turing machine that receives proofs and checks them for correctness by any method. The only way to escape these limits is by hypercomputation, but I wouldn't hold my breath.

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