不完全性定理 | ゲーデルの証明、数学的論理

Incompleteness theorem | Gödel’s Proof, Mathematical Logic, Undecidability | Britannica Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos incompleteness theorem Introduction References & Edit History Related Topics Contents CITE verified Cite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style MLA APA Chicago Manual of Style Copy Citation Share Share Share to social media Facebook X URL https://www.britannica.com/topic/incompleteness-theorem Feedback External Websites Feedback Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). 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External Websites Massachusetts Institute of Technology - GödelÂ’s First Incompleteness Theorem (PDF) The University of Chicago - Department of Mathematics - G�del's Completeness and Incompleteness Theorem University of Toronto - Department of Mathematics - Godel�s Incompleteness Theorems University of Hawaii - Godel's Incompleteness Theorem Stanford University - Mathematics School of Humanities And Sciences - The nature and significance of Godel's Incompleteness Theorems (PDF) University of Hamburg - Department of Mathematics - Godel�s Incompleteness Theorem CORE - On the Philosophical Relevance of GödelÂ’s Incompleteness Theorems (PDF) University of Basel - Department of Mathematics and Computer Science - G�del's Incompleteness Theorem in its Historical Context Seminar (PDF) New Mexico State University - Department of Computer Science - G�odel�s Incompleteness Theorems Stanford Encyclopedia of Philosophy - G�del�s Incompleteness Theorems BBC Sounds - In Our Time - Godel's Incompleteness Theorems Scholarship at Claremont - On GödelÂ’s Incompleteness Theorem (PDF) University of South Carolina - My Computer Science and Engineering Department - Godel�s Incompleteness Theorem for Computer Users Simon Fraser University - Incompleteness Theorem American Mathematical Society - The Incompleteness Theorem Mathematics LibreTexts - The Incompleteness Theorems The Guardian - Can you solve it? G�del�s incompleteness theorem incompleteness theorem logic Ask Anything Homework Help Written by William L. Hosch William L. Hosch was an editor at Encyclopædia Britannica. William L. Hosch Fact-checked by Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... Britannica Editors Last updated Apr. 7, 2026 • History Britannica AI Ask Anything Table of Contents Table of Contents Ask Anything incompleteness theorem , in foundations of mathematics , either of two theorems proved by the Austrian-born American logician Kurt Gödel . In 1931 Gödel published his first incompleteness theorem , “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally Undecidable Propositions of Principia Mathematica and Related Systems”), which stands as a major turning point of 20th-century logic . This theorem established that it is impossible to use the axiomatic method to construct a formal system for any branch of mathematics containing arithmetic that will entail all of its truths. In other words, no finite set of axioms can be devised that will produce all possible true mathematical statements, so no mechanical (or computer-like) approach will ever be able to exhaust the depths of mathematics. It is important to realize that if some particular statement is undecidable within a given formal system, it may be incorporated in another formal system as an axiom or be derived from the addition of other axioms. For example, German mathematician Georg Cantor ’s continuum hypothesis is undecidable in the standard axioms, or postulates, of set theory but could be added as an axiom. Related Topics: foundations of mathematics axiomatic method Gödel’s first incompleteness theorem Gödel’s second incompleteness theorem (Show more) On the Web: Mathematics LibreTexts - The Incompleteness Theorems (Apr. 07, 2026) (Show more) See all related content The second incompleteness theorem follows as an immediate consequence, or corollary , from Gödel’s paper. Although it was not stated explicitly in the paper, Gödel was aware of it, and other mathematicians, such as the Hungarian-born American mathematician John von Neumann , realized immediately that it followed as a corollary. The second incompleteness theorem shows that a formal system containing arithmetic cannot prove its own consistency. In other words, there is no way to show that any useful formal system is free of false statements. The loss of certainty following the dissemination of Gödel’s incompleteness theorems continues to have a profound effect on the philosophy of mathematics . William L. Hosch