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    griffMay 5, 2009 at 10:41 pm

    Several of the Hilbert problems have been resolved (or arguably resolved) in ways that would have been profoundly surprising, and even disturbing, to Hilbert himself. Following Frege and Russell, Hilbert sought to place mathematics on a sound logical foundation using the method of formal systems, i.e., finitistic proofs from an agreed upon set of axioms. One of the main goals of Hilbert’s program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.

    However, Gödel’s second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Gödel’s theorem, but he does not seem to have written any formal response to Godel’s work. But there is no doubt that the significance of Godel’s work to mathematics as a whole (and not just the field of formal logic) was amply and dramatically illustrated by its applicability to one of Hilbert’s problems.

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