Difference between revisions
hilbert-second
Status and infobox
- status:
—→ refuted - tier:
—→ V3 - kind:
—→ problem - year:
—→ 1900 - statement:
—→ Prove, by finitary means, that the axioms of arithmetic are consistent.
Source +21 −0
+---
+title: Hilbert's second problem: consistency of arithmetic
+kind: problem
+status: refuted
+tier: V3
+year: 1900
+authors: David Hilbert
+statement: Prove, by finitary means, that the axioms of arithmetic are consistent.
+space: logic-foundations
+---
+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)
+
+[[Space:logic-foundations|Logic and foundations]] · [[Continent:mathematics|Mathematics]]
Revision 48 as rendered
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.