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Richard,
The context of Gödel's first paragraph I believe is in response to Hilbert's formalization program where he (David Hilbert) set out to rescue mathematics (specifically Cantor's set theory) from the criticisms of the constructivists and intuitionists. See:
http://www.math.psu.edu/simpson/papers/hilbert/node2.html
I admit there is some controversy about the meaning of Gödel's first and second theorem, but after reviewing the first theorem in that web link address I gave you, I don't see how one can say that he was against theorems being derived from their axioms. He did say they couldn't be decided if they were true, but that means something different than being derived (for example, Goldbach's conjecture is derivable but so not decided as true).
Perhaps I misunderstand Gödel?
Warm regards, Harv |
