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Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

For the earlier theory about the correspondence between truth and provability, see Gödel's completeness theorem.

If the system is ω-consistent, it can prove neither p nor its negation, and so p is undecidable.

If the system is consistent, it may have the same situation, or it may prove the negation of p. In the later case, we have a statement ("not p") which is false but provable, and the system is not ω-consistent.

Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", , v. 38 n. 1, pp. 173–198. doi:10.1007/BF01700692

Monatshefte für Mathematik und Physik

—, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in , ed., 1986. Kurt Gödel Collected works, Vol. I. Oxford University Press, pp. 144–195. ISBN 978-0195147209. The original German with a facing English translation, preceded by an introductory note by Stephen Cole Kleene.

Solomon Feferman

—, 1951, "Some basic theorems on the foundations of mathematics and their implications", in , ed., 1995. Kurt Gödel Collected works, Vol. III, Oxford University Press, pp. 304–323. ISBN 978-0195147223.

Solomon Feferman

on In Our Time at the BBC

Godel's Incompleteness Theorems

entry by Juliette Kennedy in the Stanford Encyclopedia of Philosophy, July 5, 2011.

"Kurt Gödel"

entry by Panu Raatikainen in the Stanford Encyclopedia of Philosophy, November 11, 2013.

"Gödel's Incompleteness Theorems"

entry in the Stanford Encyclopedia of Philosophy.

Paraconsistent Logic § Arithmetic and Gödel's Theorem

Kurt Gödel.

by Karlis Podnieks. An online free book.

What is Mathematics:Gödel's Theorem and Around

using a printing machine as an example.

World's shortest explanation of Gödel's theorem

about/including Gödel's Incompleteness theorem

October 2011 RadioLab episode

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Gödel incompleteness theorem"

by Natalie Wolchover, Quanta Magazine, July 14, 2020.

How Gödel's Proof Works

and [2] Gödel's incompleteness theorems formalised in Isabelle/HOL

[1]

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