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Diagonal lemma
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{{Short description|Statement in mathematical logic}} {{Other uses|Diagonal argument (disambiguation){{!}}Diagonal argument|Diagonalization (disambiguation)}}In [[mathematical logic]], the '''diagonal lemma''' (also known as '''diagonalization lemma''', '''self-reference lemma''' or '''fixed point theorem''') establishes the existence of [[self-referential]] sentences in certain formal theories. A particular instance of the diagonal lemma was used by [[Kurt Gödel]] in 1931 to construct his proof of the [[Gödel's incompleteness theorems|incompleteness theorems]] as well as in 1933 by [[Alfred Tarski|Tarski]] to prove his [[Tarski's undefinability theorem|undefinability theorem]]. In 1934, [[Rudolf Carnap|Carnap]] was the first to publish the diagonal lemma at some level of generality.<ref>See Smoryński 2022, Sec. 3.</ref> The diagonal lemma is named in reference to [[Cantor's diagonal argument]] in set and number theory. The diagonal lemma applies to any sufficiently strong theories capable of representing the diagonal function. Such theories include [[Peano axioms#First-order theory of arithmetic|first-order Peano arithmetic]] <math>\mathsf{PA}</math>, the weaker [[Robinson arithmetic]] <math>\mathsf{Q}</math> as well as any theory containing <math>\mathsf{Q}</math> (i.e. which interprets it).<ref>See Hájek and Pudlák 2016, Chap. III.</ref> A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all [[General recursive function|recursive functions]], but all the theories mentioned have that capacity, as well. == Background == === Gödel Numbering === The diagonal lemma also requires a [[Gödel numbering]] <math>\alpha</math>. We write <math>\alpha (\varphi)</math> for the code assigned to <math>\varphi</math> by the numbering. For <math>\overline{n}</math>, the standard numeral of <math>n</math> (i.e. <math>\overline{0} =_{df} \mathsf{0} </math> and <math>\overline{n+1} =_{df} \mathsf{S}(\overline{n}) </math>), let <math>\ulcorner \varphi \urcorner </math> be the standard numeral of the code of <math>\varphi</math> (i.e. <math>\ulcorner \varphi \urcorner </math> is <math>\overline{\alpha(\varphi)}</math>). We assume a [[Gödel's encoding|standard Gödel numbering]] === Representation Theorem === Let <math>\mathbb{N}</math> be the set of [[Natural number|natural numbers]]. A [[First-order logic|first-order]] [[Theory (mathematical logic)|theory]] <math>T</math> in the language of arithmetic containing <math>\mathsf{Q}</math> ''represents'' the <math>k</math>-ary recursive function <math>f: \mathbb{N}^k\rightarrow\mathbb{N}</math> if there is a [[First-order logic#Formulas|formula]] <math>\varphi_f(x_1, \dots, x_k, y)</math> in the language of <math>T</math> s.t. for all <math>m_1, \dots, m_k \in \mathbb{N} </math>, if <math>f(m_1, \dots, m_k) = n</math> then <math>T \vdash \forall y (\varphi_f (\overline{m_1}, \dots, \overline{m_k}, y) \leftrightarrow y = \overline{n} )</math>. The representation theorem is provable, i.e. every recursive function is representable in <math>T</math>.<ref>See Hinman 2005, Chap 4.6 for additional details and a proof of this theorem.</ref> == The Diagonal Lemma and its proof == <blockquote>'''Diagonal Lemma''': Let <math>T</math> be a first-order theory containing <math>\mathsf{Q}</math> ([[Robinson arithmetic]]) and let <math>\psi (x)</math> be any formula in the language of <math>T</math> with only <math>x</math> as free variable. Then there is a sentence <math>\varphi</math> in the language of <math>T</math> s.t. <math>T \vdash \varphi \leftrightarrow \psi (\ulcorner \varphi \urcorner)</math>. </blockquote> Intuitively, <math>\varphi</math> is a [[self-referential]] sentence which "says of itself that it has the property <math>\psi</math>." '''Proof''': Let <math>diag_T:\mathbb{N}\to\mathbb{N}</math> be the recursive function which associates the code of each formula <math>\varphi (x)</math> with only one free variable <math>x</math> in the language of <math>T</math> with the code of the closed formula <math>\varphi (\ulcorner \varphi \urcorner )</math> (i.e. the substitution of <math>\ulcorner \varphi \urcorner </math> into <math>\varphi</math> for <math>x</math>) and <math>0</math> for other arguments. (The fact that <math>diag_T</math> is recursive depends on the choice of the Gödel numbering, here the [[Gödel's encoding|standard one]].) By the representation theorem, <math>T</math> represents every recursive function. Thus, there is a formula <math>\delta(x,y)</math> be the formula representing <math>diag_T</math>, in particular, for each <math>\varphi (x)</math>, <math>T \vdash \delta(\ulcorner \varphi \urcorner , y) \leftrightarrow y = \ulcorner \varphi (\ulcorner \varphi \urcorner) \urcorner </math>. Let <math>\psi(x)</math> be an arbitrary formula with only <math>x</math> as free variable. We now define <math>\chi (x)</math> as <math>\exists y (\delta(x,y) \land \psi(y))</math>, and let <math>\varphi </math> be <math>\chi (\ulcorner \chi \urcorner)</math>. Then the following equivalences are provable in <math>T</math>: <math>\varphi \leftrightarrow \chi(\ulcorner \chi \urcorner) \leftrightarrow \exists y (\delta(\ulcorner \chi \urcorner,y) \land \psi(y)) \leftrightarrow \exists y (y = \ulcorner \chi (\ulcorner \chi \urcorner) \urcorner \land \psi(y)) \leftrightarrow \exists y (y = \ulcorner \varphi \urcorner \land \psi(y)) \leftrightarrow \psi (\ulcorner \varphi \urcorner) </math>. == Some Generalizations == There are various generalizations of the Diagonal Lemma. We present only three of them; in particular, combinations of the below generalizations yield new generalizations.<ref>See Smoryński 2022, Sec. 3 or Hájek and Pudlák 2016, III.2.a</ref> Let <math>T</math> be a first-order theory containing <math>\mathsf{Q}</math> ([[Robinson arithmetic]]). === Diagonal Lemma with Parameters === Let <math>\psi (x, y_1, \dots , y_n) </math> be any formula with free variables <math>x, y_1, \dots , y_n</math>. Then there is a formula <math>\varphi (y_1, \dots y_n)</math> with free variables <math>y_1, \dots , y_n </math> s.t. <math>T \vdash \varphi (y_1 , \dots , y_n) \leftrightarrow \psi (\ulcorner \varphi (y_1 , \dots , y_n) \urcorner, y_1 , \dots , y_n)</math>. === Uniform Diagonal Lemma === Let <math>\psi (x, y_1, \dots , y_n) </math> be any formula with free variables <math>x, y_1, \dots , y_n</math>. Then there is a formula <math>\varphi (y_1, \dots y_n)</math> with free variables <math>y_1, \dots , y_n </math> s.t. for all <math>m_1 , \dots , m_n \in \mathbb{N} </math>, <math>T \vdash \varphi (\overline{m_1} , \dots , \overline{m_n}) \leftrightarrow \psi (\ulcorner \varphi (\overline{m_1} , \dots , \overline{m_n}) \urcorner, \overline{m_1} , \dots , \overline{m_n}) </math>. === Simultaneous Diagonal Lemma === Let <math>\psi_1 (x_1 , x_2)</math> and <math>\psi_2 (x_1 , x_2) </math> be formulae with free variable <math>x_1</math> and <math>x_2</math>. Then there are sentence <math>\varphi_1</math> and <math>\varphi_2 </math> s.t. <math>T \vdash \varphi_1 \leftrightarrow \psi_1(\ulcorner \varphi_1 \urcorner, \ulcorner \varphi_2 \urcorner)</math> and <math>T \vdash \varphi_2 \leftrightarrow \psi_2(\ulcorner \varphi_1 \urcorner, \ulcorner \varphi_2 \urcorner)</math>. The case with <math>n</math> many formulae is similar. ==History== The lemma is called "diagonal" because it bears some resemblance to [[Cantor's diagonal argument]].<ref>See, for example, Gaifman (2006).</ref> The terms "diagonal lemma" or "fixed point" do not appear in [[Kurt Gödel]]'s [[On Formally Undecidable Propositions of Principia Mathematica and Related Systems|1931 article]] or in [[Alfred Tarski]]'s [[The Concept of Truth in Formalized Languages|1936 article]]. In 1934, [[Rudolf Carnap]] was the first to publish the diagonal lemma in some level of generality, which says that for any formula <math>\psi (x)</math> with <math>x</math> as free variable (in a sufficiently expressive language), then there exists a sentence <math>\varphi</math> such that <math>\varphi \leftrightarrow \psi(\ulcorner \varphi \urcorner)</math> is true (in some standard model).<ref>See Carnap, 1934, and Gödel, 1986, p. 363, fn 23.</ref> Carnap's work was phrased in terms of [[Semantic theory of truth|truth]] rather than [[Proof calculus|provability]] (i.e. semantically rather than syntactically).<ref>See Smoryński 2022, Sec. 3.</ref> Remark also that the concept of [[General recursive function|recursive functions]] was not yet developed in 1934. The diagonal lemma is closely related to [[Kleene's recursion theorem]] in [[computability theory]], and their respective proofs are similar.<ref>See Gaifman, 2006 or Smoryński 2022, Sec. 3.</ref> In 1952, [[Leon Henkin|Léon Henkin]] asked whether sentences that state their own provability are provable. His question led to more general analyses of the diagonal lemma, especially with [[Löb's theorem]] and [[provability logic]].<ref>See Smoryński 2022, Sec. 3.</ref> ==See also== *[[Indirect self-reference]] *[[List of fixed point theorems]] *[[Self-reference]] *[[Self-referential paradoxes]] ==Notes== {{Reflist}} ==References== * [[George Boolos]] and [[Richard Jeffrey]], 1989. ''Computability and Logic'', 3rd ed. Cambridge University Press. {{ISBN|0-521-38026-X}} {{ISBN|0-521-38923-2}} * [[Rudolf Carnap]], 1934. ''Logische Syntax der Sprache''. (English translation: 2003. ''The Logical Syntax of Language''. Open Court Publishing.) * [[Haim Gaifman]], 2006. '[https://haimgaifman.files.wordpress.com/2016/07/22odel-to-kleene.pdf Naming and Diagonalization: From Cantor to Gödel to Kleene]'. ''Logic Journal of the IGPL'', 14: 709–728. * Petr Hájek & Pavel Pudlák, 2016 (first edition 1998). ''Metamathematics of First-Order Arithmetic.'' Springer Verlag. * Peter Hinman, 2005. ''Fundamentals of Mathematical Logic''. A K Peters. {{ISBN|1-56881-262-0}} * [[Elliott Mendelson|Mendelson, Elliott]], 1997. ''Introduction to Mathematical Logic,'' 4th ed. Chapman & Hall. * Panu Raatikainen, 2015a. [http://plato.stanford.edu/entries/goedel-incompleteness/sup2.html The Diagonalization Lemma]. In [[Stanford Encyclopedia of Philosophy]], ed. Zalta. * Panu Raatikainen, 2015b. [http://plato.stanford.edu/entries/goedel-incompleteness/index.html Gödel's Incompleteness Theorems]. In [[Stanford Encyclopedia of Philosophy]], ed. Zalta. * [[Raymond Smullyan]], 1991. ''Gödel's Incompleteness Theorems''. Oxford Univ. Press. * Raymond Smullyan, 1994. ''[https://philpapers.org/rec/SMUDAS Diagonalization and Self-Reference]''. Oxford Univ. Press. * Craig Smoryński, 2023. 'The early history of formal diagonalization'. ''Logic Journal of the IGPL'', 31.6: 1203–1224. * {{cite journal| author=Alfred Tarski| author-link=Alfred Tarski| title=Der Wahrheitsbegriff in den formalisierten Sprachen| journal=[[Studia Philosophica (Poland)|Studia Philosophica]]| year=1936| volume=1| pages=261–405| url=http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf| accessdate=26 June 2013| url-status=dead| archiveurl=https://web.archive.org/web/20140109135345/http://www.ifispan.waw.pl/studialogica/s-p-f/volumina_i-iv/I-07-Tarski-small.pdf| archivedate=9 January 2014}} * [[Alfred Tarski]], tr. J. H. Woodger, 1983. 'The Concept of Truth in Formalized Languages'. [http://www.thatmarcusfamily.org/philosophy/Course_Websites/Readings/Tarski%20-%20The%20Concept%20of%20Truth%20in%20Formalized%20Languages.pdf English translation of Tarski's 1936 article]. In A. Tarski, ed. J. Corcoran, 1983, ''Logic, Semantics, Metamathematics'', Hackett. {{DEFAULTSORT:Diagonal Lemma}} [[Category:Mathematical logic]] [[Category:Lemmas]] [[Category:Articles containing proofs]]
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