Paris of the year 1900 left two landmarks: theTour Eiffel,and David Hilbert's celebrated list of twenty-four mathematical problems presented at a conference opening the new century. Kurt Gödel, a logical icon of that time, showed Hilbert's ideal of complete axiomatization of mathematics to be unattainable. The result, of 1931, is called Gödel'sincompleteness theorem. Gödel then went on to attack Hilbert's first and second Paris problems, namely Cantor'scontinuum problemabout the type of infinity of the real numbers, and the freedom from contradiction of the theory of real numbers. By 1963,...