Kurt Gödel proved that mathematics can never fully prove itself
At 25, Kurt Gödel wrote a formula that effectively says: I cannot be proved. If it could be proved it would be false, so any consistent system rich enough for arithmetic must contain truths it cannot reach. That short argument ended a half-century quest to put all of mathematics on a complete, self-certified footing.
Gödel came into the world on April 28, 1906, in the Moravian city of Brünn, today's Brno, then part of Austria-Hungary. His family spoke German, and his father ran a large textile firm. His relatives called him Mr Why for his endless questions. A bout of rheumatic fever at six or seven left him, though fully recovered, convinced his heart was damaged, and poor health dogged him all his life. He collected nationalities without moving much: Czechoslovak at 12, Austrian by choice in 1929, German after the 1938 annexation, and American in 1948.
At 18 he joined his brother at the University of Vienna, already fluent in university mathematics. A seminar on Bertrand Russell's philosophy of mathematics, run by Moritz Schlick of the Vienna Circle, turned him to logic, and David Hilbert had posed the question he took up: are the axioms of a formal system enough to derive everything true in all its models? His 1929 dissertation, written under Hans Hahn, showed that for first-order logic they are. At a conference in Königsberg in 1930 he hinted that the result would not stretch further.
The incompleteness theorems followed in 1931. To make mathematics talk about itself, he invented Gödel numbering, turning formulas into numbers. The consequences were stark: such a system cannot settle every statement about whole numbers, and it cannot demonstrate its own consistency. Later he proved that standard set theory, if consistent, cannot rule out either the axiom of choice or the continuum hypothesis, which freed mathematicians to use the axiom of choice.
He met Einstein on his first American trip in 1933, and they became close friends. The 1936 murder of Schlick by a former student threw Gödel into a severe crisis, with fears of being poisoned and months in a sanitarium. In 1938 he married Adele Nimbursky, a former dancer his parents had disapproved of, and in 1939 he left for the United States. His fear of poisoned food returned late in life; refusing to eat, he died on January 14, 1978.
Source: Kurt Gödel
