Gödel incompleteness theorems
Teoremas de incompletitud de Gödel
- Date
- 1931 · confidence high
- Place
- Vienna, Austria
- Credited to
- Kurt Gödel
- People
- 👤 Kurt Gödel1906–1978
- Field
- Knowledge & symbols
What was discovered
Every consistent formal system capable of expressing arithmetic contains true statements that cannot be proved, and it cannot prove its own consistency. It brought down Hilbert's programme and fixed the limits of the axiomatic method.
Gödel had previously proved (1929) the completeness of first-order logic, which does not contradict the incompleteness of arithmetical theories.
What it led to
- computability theory
- the Turing machine
- model theory
- the philosophy of mathematics
- independence results
Built on
- Peano axioms1889
- Cantor's diagonal argument1891
- Russell paradox and Principia Mathematica1901
- Gödel's completeness theorem1929
- Hilbert's programme
- Cantor's diagonal argument
Made possible
- Turing machine (universal computation)1936
- Turing machine, halting problem and Church-Turing thesis1936
- Goodstein's theorem1944
- Godel numbering (encoding programs as data)