"We must know. We shall know." So said David Hilbert, one of the leading mathematicians at the turn of the century. Hilbert was gung ho about the future of mathematics. No-go areas should not exist, he believed, and he even had the outlines of a program to prove it. Yet within a few years, Hilbert's dream lay in ruins—a young logician named Kurt Goedel had proved that some mathematical questions simply don't have answers. What Goedel showed in 1930 was that any logical system rich enough to model mathematics will always have insoluble problems. For instance, it is impossible to prove that mathematics contains no logical inconsistencies. Of course, you can deal with any particular insoluble problem by adding a new mathematical rule, but a new insoluble problem will always appear in the patched-up system.
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