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Gödel's incompleteness theorems

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For everyone

Gödel proved that in every consistent formal system strong enough to include arithmetic there are statements that are true but cannot be proved within the system. Such a system also cannot prove its own consistency. The result changed the hope of placing all of mathematics on a complete formal foundation (Hilbert's programme).

For specialists

First theorem: every recursively axiomatisable, ω-consistent (after Rosser, merely consistent) theory T that contains Robinson arithmetic is incomplete. There is a sentence G_T such that neither it nor its negation is provable in T. Second theorem: if such a T is consistent, Con(T) is not provable in T. The proof rests on Gödel numbering and the diagonal lemma, constructing a sentence that says "this sentence is not provable".

Claims

Sufficiently strong consistent formal systems are incomplete and cannot prove their own consistency.^[1]

Evidence

Supporting

Mathematical proof

The proof in the 1931 paper has been examined many times in the decades since and has also been formally verified with proof assistants.^[1][2]

Challenging

—

Positions

The Penrose–Lucas argument: The theorems show that the human mind cannot be reduced to an algorithm.^[3]
Critics of that argument: The argument assumes that humans know their own consistency; the theorems do not support that conclusion.^[2]

Connections

Parent

Mathematics

Influenced

The Turing machine and computability

Consciousness (open question)Frege and the birth of modern logic Cantor and the sizes of infinity

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Sources

ArticleGödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198DOI 10.1007/BF01700692 archive
EncyclopediaRaatikainen, P.Gödel's Incompleteness Theorems. Stanford Encyclopedia of Philosophyplato.stanford.edu archive
BookPenrose, R. (1989). The Emperor's New Mind. Oxford University Press

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Cite this page

APA

Athena contributors. (2026). Gödel's incompleteness theorems. Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/godel/

BibTeX

@misc{athena_godel,
  title        = {Gödel's incompleteness theorems},
  author       = {{Athena contributors}},
  year         = {2026},
  howpublished = {\url{https://latifkedi.github.io/athena/en/d/godel/}},
  note         = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}

Added: September 27, 2026 · Last changed: September 27, 2026 · Page history

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