Incompleteness theorem | Gödel’s Proof, Mathematical Logic, Undecidability

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class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/topic/incompleteness-theorem">incompleteness theorem</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <span class="toc-extra-link selected link-gray-900 mt-15"> Introduction </span> <a class="toc-extra-link link-gray-900" href="https://www.britannica.com/topic/incompleteness-theorem/additional-info">References & Edit History</a> <a class="toc-extra-link link-gray-900" href="/facts/incompleteness-theorem">Related Topics</a> </div> <div class="mb-30 tlr-read-next"> <div class="text-gray-900 p-5 font-serif font-14 font-weight-bold mx-10 mb-10"> Read Next </div> <div class="imagelink-with-image-on-the-side card card-horizontal tlr-img-with-side-link ml-15 link-gray-900 mb-10" > <div class="position-relative card-media" style="flex: 0;"> <a class="ilf-image position-relative" href="/story/unusual-counting-systems"> <img loading="lazy" src="https://cdn.britannica.com/41/191041-131-7ECE668A/bunch-numbers.jpg?w=200&h=200&c=crop" alt="bunch of numbers" width="200" height="200" /> </a> </div> <div class="card-body ilf-content"> <a class="font-weight-semi-bold d-block mb-5 font-16 ilf-title" href="/story/unusual-counting-systems" >Unusual Counting Systems</a> </div> </div> <div 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Gödel’s incompleteness theorem</a></li> </ul> </div> </div> </div> </div> </div> <div class="toc-header-marker"></div> <button class="ai-ask-button btn border-2 js-header-ai-ask-button d-none btn-sm btn-outline-red-400 border-red-400 mr-0 mr-lg-10 ml-5 ml-sm-10 ml-lg-0 p-10"> Ask the Chatbot a Question </button> <div class="md-byline module-spacing "> <div class="font-serif font-12"> <span class="written-by text-gray-700"> Written by </span> <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup gtm-byline font-12 byline-contributor" href="/editor/William-L-Hosch/6481" > <div class="editor-title font-16 font-weight-bold">William L. Hosch</div> <div class="editor-description font-12 font-serif mt-5 clamp-description text-black">William L. Hosch was an editor at Encyclopædia Britannica.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> William L. Hosch</span></div> <div class="font-serif font-12 text-gray-700"> <span class="qa-fact-checked-by">Fact-checked by</span> <div class="editor-popover popover p-0"> <a class="d-block p-20 qa-editor-popup font-12" href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419" > <div class="editor-title font-16 font-weight-bold">The Editors of Encyclopaedia Britannica</div> <div class="editor-description font-12 font-serif mt-5 text-black">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. They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link font-12 "> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <span class="text-gray-700"> Last Updated: <time datetime="2025-01-17T00:00:00CST" >Jan 17, 2025</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/topic/incompleteness-theorem/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn border-2 ai-ask-button btn border-2 module-spacing btn-sm js-inline-ai-ask-button btn-outline-red-400 border-red-400 p-10 ml-5"> Ask the Chatbot a Question </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/foundations-of-mathematics" topicid="369221">foundations of mathematics</a></dd> <dd><a href="/science/axiomatic-method" topicid="46255">axiomatic method</a></dd> <dd><a href="/topic/Godels-first-incompleteness-theorem" topicid="236794">Gödel’s first incompleteness theorem</a></dd> <dd><a href="/topic/Godels-second-incompleteness-theorem" topicid="236802">Gödel’s second incompleteness theorem</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/incompleteness-theorem"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref1025365"></span>incompleteness theorem</strong>, in <a href="https://www.britannica.com/science/foundations-of-mathematics" class="md-crosslink " data-show-preview="true">foundations of mathematics</a>, either of two theorems proved by the Austrian-born American logician <span id="ref1107611"></span><a href="https://www.britannica.com/biography/Kurt-Godel" class="md-crosslink " data-show-preview="true">Kurt Gödel</a>.</p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">In 1931 Gödel published his <span id="ref1107613"></span><a href="https://www.britannica.com/topic/Godels-first-incompleteness-theorem" class="md-crosslink ">first incompleteness theorem</a>, “Über formal unentscheidbare Sätze der <em>Principia Mathematica</em> und verwandter Systeme” (“On Formally Undecidable Propositions of <em>Principia Mathematica</em> and Related Systems”), which stands as a major turning point of 20th-century <a href="https://www.britannica.com/topic/logic" class="md-crosslink " data-show-preview="true">logic</a>. This <a href="https://www.britannica.com/topic/theorem" class="md-crosslink autoxref " data-show-preview="true">theorem</a> established that it is impossible to use the <a href="https://www.britannica.com/science/axiomatic-method" class="md-crosslink " data-show-preview="true">axiomatic method</a> to construct a <a href="https://www.britannica.com/topic/formal-system" class="md-crosslink " data-show-preview="true">formal system</a> for any branch of <a href="https://www.britannica.com/science/mathematics" class="md-crosslink " data-show-preview="true">mathematics</a> containing <a href="https://www.britannica.com/science/arithmetic" class="md-crosslink " data-show-preview="true">arithmetic</a> that will entail all of its truths. In other words, no finite set of <a href="https://www.britannica.com/topic/axiom" class="md-crosslink " data-show-preview="true">axioms</a> 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 <a href="https://www.britannica.com/topic/axiom" class="md-crosslink autoxref " data-show-preview="true">axiom</a> or be derived from the addition of other axioms. For example, German mathematician <a href="https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor" class="md-crosslink " data-show-preview="true">Georg Cantor</a>’s <a href="https://www.britannica.com/science/continuum-hypothesis" class="md-crosslink " data-show-preview="true">continuum hypothesis</a> is undecidable in the standard axioms, or postulates, of <a href="https://www.britannica.com/science/set-theory" class="md-crosslink " data-show-preview="true">set theory</a> but could be added as an axiom.</p><div class="one-good-fact-module"> </div><span class="marker MOD2 mod-inline"></span><span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">The <span id="ref1107614"></span><a href="https://www.britannica.com/topic/Godels-second-incompleteness-theorem" class="md-crosslink ">second incompleteness theorem</a> follows as an immediate consequence, or <a class="md-dictionary-link md-dictionary-tt-off mw" data-term="corollary" href="https://www.merriam-webster.com/dictionary/corollary" data-type="MW">corollary</a>, 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 <span id="ref1107615"></span><a href="https://www.britannica.com/biography/John-von-Neumann" class="md-crosslink " data-show-preview="true">John von Neumann</a>, 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 <a href="https://www.britannica.com/science/philosophy-of-mathematics" class="md-crosslink " data-show-preview="true">philosophy of mathematics</a>.</p><div class="module-spacing"> </div><span class="marker MOD3 mod-inline"></span></section> <span class="md-signature"><a href="/editor/William-L-Hosch/6481">William L. 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