Theory
Gödel’s incompleteness theorems
Two theorems showing that any consistent formal system rich enough for arithmetic contains true statements it cannot prove.
Updated Oct 18, 2026 · 1 min read · 4 connections
- Published
- 1931
- Author
- Kurt Gödel
- Field
- Mathematical logic
Statement
The first theorem states that a consistent formal system capable of expressing basic arithmetic is incomplete: some statements in its language can be neither proved nor disproved within it. The second states that such a system cannot prove its own consistency.
Influence
The theorems ended the hope of a complete and provably consistent foundation for mathematics, reshaping logic. Alan Turing reached a related limit with the turing machine, and the results are still debated in the philosophy of mind.