Kurt Gödel (1906-1978) was probably the most strikingly original and important logician of the twentieth century. He proved the incompleteness of axioms for arithmetic (his most famous result), as well as the relative consistency of the axiom of choice and continuum hypothesis with the other axioms of set theory.
Gödel's work was the surprising culmination of a long search for foundations. Throughout the nineteenth century, mathematicians had tried to establish the foundations of calculus. First Cauchy gave the modern definition of limits; later Weierstrass and Dedekind gave rigorous definitions of the real numbers. By the end of the century, the foundations of calculus rested on integers and their arithmetic. This left the problem of putting the integers themselves on a sound logical basis, which Frege appeared to solve by defining the positive integers in terms of sets. But it soon became clear that naive use of sets could lead to contradictions (such as the set of all sets that aren't members of themselves). Set theory itself would have to be axiomatized. In their massive 3-volume Principia Mathematica, Russell and Whitehead built the foundations of mathematics on a set of axioms for set theory; they needed hundreds of preliminary results before proving that 1 + 1 = 2.
There remained the problem of analyzing the axioms of set theory. Mathematicians hoped that their axioms could be proved consistent (free from contradictions) and complete (strong enough to provide proofs of all true statements). Gödel showed these hopes were overly naive. He proved that any consistent formal system strong enough to axiomatize arithmetic must be incomplete; that is, there are statements that are true but not provable. Also, one can't hope to prove the consistency of such a system using the axioms themselves. The basic idea of Gödel's proof, indirect self-reference, is strikingly simple, but tricky to grasp. A book-long explanation for the general reader is offered in Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.
Gödel was born in Brno, which was then part of the Austria-Hungary. In 1924 he matriculated at the University of Vienna. He became interested in logic and was influenced by Hahn, who was to be his thesis advisor. From 1926-28 he participated in the Vienna Circle that was to become associated with Rudolf Carnap and logical positivism (though Gödel disagreed with most of Carnap's views). He completed his dissertation (on the completeness of first-order logic) in 1929. The next year he had already proved his incompleteness theorem, and it was published in 1931. (It is ironic that Gödel's first two major results were a completeness theorem and an incompleteness theorem. The two are not contradictory, but together they do show that no first-order axiomatization can capture all the truths of arithmetic). Gödel submitted his incompleteness paper to the University of Vienna as his Habilitationsschrift (probationary essay), and in 1933 he was confirmed as a Privatdozent: this was not a salaried position, but a certificate that gave him the right to lecture and collect fees from students. He taught his first course in the summer of 1933, and that fall he began a year-long appointment at the newly formed Institute for Advanced Study (IAS) in Princeton, New Jersey.
Upon his return to Austria the next year, Gödel had the first of several breakdowns; he spent several months in a sanatorium recovering from depression. In 1935 he proved the (relative) consistency of the axiom of choice with the other axioms of set theory. ("Relative" in this case means that if the axioms other than the axiom of choice are consistent, then so are these axioms together with the axiom of choice. As noted above, one can't hope to prove the consistency of the axioms from themselves.) A second visit to the IAS was cut short by a relapse of depression, and Gödel remained incapacitated until spring 1937. Later that year he proved the consistency of the generalized continuum hypothesis with the axioms of set theory, and he lectured on his set-theoretic results at the IAS in 1938-39. By now Austria had been incorporated into Hitler's Germany, and when he returned home he faced liability for military service. Though he was not Jewish, Gödel's academic associations put him in a precarious position. After protracted negotiations he received a U. S. visa late in 1939: in the early months of the Second World War he and his wife travelled to the U. S. via the Soviet Union and Japan. He was given a one-year appointment to the IAS upon his arrival in Princeton; this was renewed yearly until 1946, when he was appointed a permanent member.
In 1942 Gödel attempted to prove that the axiom of choice and continuum hypothesis are independent of (not implied by) the axioms of set theory. He did not succeed, and the problem remained open until 1963. (In that year, Paul Cohen proved that the axiom of choice is independent of the axioms of set theory, and that the continuum hypothesis is independent of both.) Gödel did little original work in logic after this, though he did publish a remarkable paper in 1949 on general relativity: he discovered a universe consistent Einstein's equations in which there were "closed timelike lines"--in such a universe, one could visit one's own past!
Gödel struck most people as eccentric. His political views were often surprising: for instance, while he condemned Truman for fomenting war hysteria and creating the climate for McCarthyism, he was a great admirer of Eisenhower. While studying for his U. S. citizenship examination in 1948, he became convinced he had found an inconsistency in the Constitution. (Fortunately, this did not disrupt Gödel's citizenship interview, as the judge brushed aside the point when Gödel tried to bring it up.) Gödel became increasingly reclusive in his later years. He was always somewhat prone to paranoia, was distrustful of doctors, and tended to feed himself poorly. When his wife was incapacitated with illness, these factors combined to cause his death from self-starvation.
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