# Gödel's Incompleteness Theorems

**Type:** Theorem  
**Domain:** Named Results · Foundations  
**Proved:** 1931, Kurt Gödel  
**Codex URL:** /mathematics/named-results/godels-incompleteness-theorems/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Any consistent formal system powerful enough for arithmetic contains true statements it cannot prove (First Theorem) and cannot prove its own consistency (Second Theorem). Ended Hilbert's Programme.

## First Incompleteness Theorem
Any consistent F (encoding Peano arithmetic) is incomplete: ∃ statement G true but unprovable in F.

## Construction — Gödel numbering
Assigns unique numbers to symbols/formulas/proofs, encoding "G is not provable in F" as an arithmetic statement — a rigorous Liar's Paradox.

## Second Incompleteness Theorem
No consistent F can prove its own consistency.

## Related results
- Turing's Halting Problem (1936) — similar diagonalization
- Rosser (1936) — weakened hypothesis to simple consistency
- Gentzen (1936) — proved PA consistency using methods beyond PA itself

## Sources
### Tier 1
- Gödel, K. (1931). Über formal unentscheidbare Sätze. *Monatshefte für Mathematik und Physik*, 38, 173-198.
- Smullyan, R.M. (1992). *Gödel's Incompleteness Theorems*. Oxford University Press.
### Tier 2
- Nagel, E. and Newman, J.R. (2001). *Gödel's Proof*. Revised ed.
### Tier 3
- Hofstadter, D.R. (1979). *Gödel, Escher, Bach*. Basic Books.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*