Kurt Gödel

Overview

Gödel proved in 1929, in his doctoral work, that first-order logic is complete: every statement true in all interpretations has a proof. Two years later he proved the opposite kind of result for arithmetic. By coding statements and proofs as numbers, he built a sentence that says of itself that it has no proof, and showed that if the system is consistent then that sentence is true and unprovable in it; a second theorem follows, that no such system can prove its own consistency. In 1938 he showed that the axiom of choice and the continuum hypothesis cannot be disproved from the standard axioms of set theory, and Cohen completed the picture in 1963 by showing they cannot be proved from them either. He left Austria in 1940 for Princeton, where he became a close friend of Einstein. Start with a guided exposition, such as Nagel and Newman's, rather than the 1931 paper.

Where to start reading

• On Formally Undecidable Propositions of Principia Mathematica and Related Systems 1931

  The incompleteness paper. Contains the coding of syntax into arithmetic and both theorems.

• The Consistency of the Continuum Hypothesis 1940

  Shows that the continuum hypothesis and the axiom of choice cannot be refuted from the standard axioms of set theory.

Part of

• Formalism and the Hilbert programme
• Set theory

Sources

1. Kurt Gödel MacTutor History of Mathematics
2. Gödel's Incompleteness Theorems Stanford Encyclopedia of Philosophy
3. Kurt Gödel Wikipedia