# Hilbert's second problem: consistency of arithmetic

- kind: problem
- status: refuted (evidence tier V3)
- year: 1900
- authors: David Hilbert

**Statement.** Prove, by finitary means, that the axioms of arithmetic are consistent.

Gödel showed that no consistent system strong enough for arithmetic can prove its own consistency, so Hilbert's program as stated cannot succeed. Gentzen proved consistency in 1936 using induction up to the ordinal ε₀, which trades one assumption for another. The problem is marked refuted because the kind of proof Hilbert asked for is impossible.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1900, David Hilbert. Closed 1931, Kurt Gödel's second incompleteness theorem.

## Sources

- [Wikipedia: Hilbert's second problem](https://en.wikipedia.org/wiki/Hilbert%27s_second_problem)

[Logic and foundations](/wiki/Space:logic-foundations.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [liar-paradox](/wiki/Problem:liar-paradox.md) — The original curation links this problem to Hilbert's second problem: consistency of arithmetic; the argument for the link has not been written yet.

---
Revision 48, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:hilbert-second?action=history. Edit: POST /api/wiki/page/Problem:hilbert-second {source, summary, base_rev: 48}.