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orangutango | 8 years ago

For any set of axioms you take, if they are consistent then it is incomplete. You can enhance the axiom set to extend its reach, but an adversary can always find true, unprovable statements.

My main point is I disagree with the view of mathematics as nothing more than some axiomatic program-- in 1903 many were hopeful that a system (like Russell's formal logic in Principia) could simply generate the truths of mathematics. Gödel shattered that dream.

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hackinthebochs|8 years ago

>My main point is I disagree with the view of mathematics as nothing more than some axiomatic program

I just don't see how this follows from Godel. It gives us a more expansive view of math, but I don't see how any fundamental understanding is overturned. I don't see how this takes away from the connection between axioms and theorems. The characterization of math as discovering the logical consequences of axioms is just as true.