On a Relationship between Godel's Second Incompleteness Theorem and Hilbert's Program

概要

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We can find in several places an assertion that Godel's second incompleteness theorem defeated Hilbert's program. But, (as M. Detlefsen argued in his book) in order to establish this assertion, we need to address additional issues. First we formulate Hilbert's program. Second we reconstruct a standard argument for the claim that Godel's second incompleteness theorem defeated Hilbert's program. In doing so, we formulate a critical, and problematic assumption which we call "DCT" (Derivability Conditions Thesis). Finally we examine three arguments whose aims are to justify DCT. We show that the first and the second argument are not valid, and discuss the third argument, which is based on Kreisel's idea. We identify a difficulty in this argument as well. After examining the difficulty, we conclude that we cannot claim that Godel's second incompleteness theorem defeats Hilbert's program. Moreover we clarify what is essentially needed for such an argument to succeed.
2009-03-31