Born in Brünn (now Brno in the Czech Republic), Gödel initially studied physics at the University of Vienna, but his interest soon turned to mathematics and mathematical logic. He obtained his PhD in 1930 and the same year joined the faculty at Vienna. He became a member of the Institute for Advanced Study, Princeton, in 1938 and in 1940 emigrated to America. He was a professor at the Institute from 1953 to 1976, and received many scientific honors and awards including the National Medal of Science in 1975. He became a naturalized American citizen in 1948.

In 1930 Gödel published his doctoral dissertation, the proof that first-order logic is complete – that is to say that every sentence of the language of first-order logic is provable or its negation is provable. The completeness of logical systems was then a concept of central importance owing to the various attempts that had been made to reveal a logical axiomatic basis for mathematics. Completeness can be thought of as ensuring that all logically valid statements that a formal (logical) system can produce can be proved from the axioms of the system, and that every invalid statement is disprovable.

In 1931 Gödel presented his famous incompleteness proof for arithmetic. He showed that in any consistent formal system complicated enough to describe simple arithmetic there are propositions or statements that can neither be proved nor disproved on the basis of the axioms of the system – intuitively speaking, there are logical truths that cannot be proved within the system. Moreover, as a corollary Gödel showed (what is known as his second incompleteness theorem) that the consistency of any formal system including arithmetic cannot be proved by methods formalizable within that system; consistency can only be proved by using a stronger system – whose own consistency has to be assumed. This latter result showed the impossibility of carrying out Hilbert's program, at least in its original form.

Gödel's second great result concerned two important postulates of set theory, whose consistency mathematicians had been trying to prove since the turn of the century. Between 1938 and 1940 he showed that if the axioms of (restricted) set theory are consistent then they remain so upon the addition of the axiom of choice and the continuum hypothesis, and that these postulates cannot, therefore, be disproved by restricted set theory. (In 1963 Paul Cohen showed that they were independent of set theory.)

Gödel has also worked on the construction of alternative universes that are models of the general theory of relativity, and has produced a rotating-universe model.

Gödel apparently suffered from depression throughout much of his life. In 1936–37 he spent some time in an Austrian sanatorium being treated for the condition. He was also something of a hypochondriac. He retired from the institute in 1976 and when, soon after, his wife underwent major surgery, he seems to have stopped eating. Apparently he was convinced that he was being poisoned. In late 1977 he was admitted to hospital dehydrated and undernourished. He refused to eat and two weeks later died from “malnutrition and inanition caused by personality disturbance.”