Gödel

Kurt Gödel (1906–1978) was an Austrian-American logician and mathematician whose work laid the foundations of modern logic. He is best known for two landmark results: the completeness theorem for first-order logic, which shows that every logically valid sentence is provable in a suitable deductive system; and the incompleteness theorems, demonstrating that any consistent, sufficiently strong formal system cannot prove all true arithmetic statements and cannot prove its own consistency.

Gödel introduced arithmetization of syntax, Gödel numbering, and the diagonal lemma, enabling formal self-reference. These ideas underlie results in logic, the theory of computation, and the philosophy of mathematics. His work showed inherent limitations of formal axiomatic systems and influenced later developments in computer science, including undecidability results by Church and Turing.

Life and career: Gödel was born in 1906 in Brünn, then part of Austria-Hungary (present-day Brno, Czech Republic). He studied at the University of Vienna, where he developed his foundational ideas in logic. In 1940 he emigrated to the United States, joining the Institute for Advanced Study in Princeton, and later became a U.S. citizen. He remained at IAS for most of his career and formed a notable intellectual rapport with Albert Einstein. Gödel died in Princeton in 1978, leaving a lasting influence on mathematics, logic, and the philosophy of mathematics.