David Hilbert, the great German mathematician (he died in 1943), had a stupendous, dazzling vision. He hoped and believed that some day mathematicians would construct one vast formal deductive system with axioms so powerful that every possible theorem in all of mathematics could be proved true or false. Such a system would have to be both consistent and complete. Consistent means it is impossible to prove both a statement and its negation. Complete means that every statement in the system can be proved true or false.

In 1931, to the astonishment of mathematicians, a shy, reclusive Austrian, Kurt Gödel, aged twenty-five, shattered Hilbert’s magnificent dream. Gödel showed that any formal system rich enough to include arithmetic and elementary logic could not be both consistent and complete. If complete, it would contain an infinity of true statements that could not be proved by the system’s axioms. What is worse, even the...

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