• Korean Journal of Logic
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• v.8 no.2
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• pp.3-32
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• 2005

Hilbert's rational ambition to establish consistency in Number theory and mathematics in general was frustrated by the fact that the statement itself claiming consistency is undecidable within its formal system by $G\ddot{o}del's$ second theorem. Hilbert's optimism that a mathematician should not say "Ignorabimus" ("We don't know") in any mathematical problem also collapses, due to the presence of a undecidable statement that is neither provable nor refutable. The failure of his program receives more shock, because his system excludes any ambiguity and is based on only mechanical operations concerning signs and strings of signs. Above all, $G\ddot{o}del's$ theorem demonstrates the limits of formalization. Now, the notion of provability in the dimension of syntax comes to have priority over that of semantic truth in mathematics. In spite of his failure, the notion of algorithm(mechanical processe) made a direct contribution to the emergence of programming languages. Consequently, we believe that his program is failure, but a great one.