https://www.ellipticcurve.info/Category:Incompleteness_theorems/history?feed=atom 2026-06-20T09:19:15Z Revision history for this page on the wiki MediaWiki 1.42.3 https://www.ellipticcurve.info/index.php?title=Category:Incompleteness_theorems&diff=98&oldid=prev 2024-12-27T10:53:50Z

<p>categorize incompleteness</p> <p><b>New page</b></p><div>Kurt Gödel dealt a death blow to David Hilbert’s program of formalizing all of mathematics. That was the equivalent then, a century earlier, of Clay Mathematics Institute’s Millennium Problems today.<br /> <br /> == Gödel’s first incompleteness theorem ==<br /> Gödel first proved that&lt;blockquote&gt;In any consistent formal system of logic, there are statements which are true, but unprovable.&lt;/blockquote&gt;When we say “consistent,” we mean simply that the system is not self-contradictory. However Gödel used a slightly different notion of ω-consistency for technical reasons.<br /> <br /> == Gödel’s second incompleteness theorem ==<br /> The second incompleteness theorem is a considerable sharpening of the first.&lt;blockquote&gt;No formal system of logic, proving its own consistency, is consistent.&lt;/blockquote&gt;<br /> <br /> == Tarski’s theorem on the undefinability of truth ==<br /> Alfred Tarski proved that&lt;blockquote&gt;No well-formed formula exists within any consistent formal system of logic, that assigns a truth value to all well-formed formulæ with no free variables.&lt;/blockquote&gt;<br /> <br /> == Löb’s theorem ==<br /> Martin Hugo Löb proved that&lt;blockquote&gt;If any mathematical statement is provable assuming &#039;&#039;a priori&#039;&#039; that a valid proof for it exists already, then this assumption may be formally discharged, and the statement is provable &#039;&#039;prima facie&#039;&#039; without any assumption of a pre-existing proof.&lt;/blockquote&gt;<br /> <br /> == More ==<br /> Kleene’s recursion theorems, the &lt;big&gt;&lt;math&gt;s^m_n&lt;/math&gt;&lt;/big&gt; theorem, Rogers’ fixed-point theorem, and the diagonalization lemma also pertain to this category.</div>

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