Gödel's incompleteness theorems
Any consistent formal system rich enough for arithmetic contains truths it cannot prove.
- When
- Known to
- ± 1 year
- Era
- Industrial, Holocene
Gödel shows in 1931 that Hilbert's programme of establishing all mathematics on a complete and provably consistent axiomatic foundation cannot succeed. The method is a formal version of the liar paradox: he constructs a statement that asserts its own unprovability. It is one of the deepest results in logic, and Turing's halting problem five years later is closely related.