Gödel's Incompleteness Theorems

Any consistent arithmetic rich enough for whole numbers contains true statements it cannot prove, and cannot prove its own consistency.

Specifications & Metrics

proved

Kurt Gödel, 1931

Core Principles & Analysis

• Gödel announced the result at a 1930 Konigsberg conference; only von Neumann grasped it at once.
• The proof encodes the liar paradox arithmetically: a sentence saying 'I am not provable'.
• It demolished Hilbert's program of grounding all mathematics in a finitistic consistency proof.