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| '''Tarski's undefinability theorem''', stated and proved by [[Alfred Tarski]] in 1936, is an important limitative result in [[mathematical logic]], the [[foundations of mathematics]], and in formal [[semantics]]. Informally, the theorem states that ''arithmetical truth cannot be defined in arithmetic''.
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| The theorem applies more generally to any sufficiently strong [[formal system]], showing that truth in the standard model of the system cannot be defined within the system.
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| == History ==
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| In 1931, [[Kurt Gödel]] published his famous [[Gödel's incompleteness theorems|incompleteness theorems]], which he proved in part by showing how to represent ''syntax'' within [[first-order arithmetic]]. Each expression of the language of arithmetic is assigned a distinct number. This procedure is known variously as [[Gödel numbering]], ''coding'', and more generally, as arithmetization. | |
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| In particular, various ''sets'' of expressions are coded as sets of numbers. It turns out that for various syntactic properties (such as ''being a formula'', ''being a sentence'', etc.), these sets are [[computable set|computable]]. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences.
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| The undefinability theorem shows that this encoding cannot be done for [[semantic]]al concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any [[metalanguage]] capable of expressing the semantics of some [[object language]] must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.
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| The undefinability theorem is conventionally attributed to [[Alfred Tarski]]. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to [[John von Neumann]]. Tarski had obtained almost all results of his 1936 paper ''Der Wahrheitsbegriff in den formalisierten Sprachen'' between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result not obtained by him earlier. According to the footnote of the undefinability theorem (Satz I) of the 1936 paper, the theorem and the sketch of the proof were added to the paper only after the paper was sent to print. When he presented the paper to the Warsaw Academy of Science on March 21 1931, he wrote only some conjectures instead of the results after his own investigations and partly after Gödel's short report on the incompleteness theorems "Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit", Akd. der Wiss. in Wien, 1930.
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| == Statement of the theorem== | |
| We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski actually proved in 1936.
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| Let ''L'' be the language of [[first-order arithmetic]], and let ''N'' be the standard [[structure (mathematical logic)|structure]] for ''L''. Thus (''L'', ''N'') is the "interpreted first-order language of arithmetic." Each sentence ''x'' in ''L'' has a [[Gödel number]] ''g''(''x''). Let ''T'' denote the set of ''L''-sentences true in ''N'', and ''T''* the set of Gödel numbers of the sentences in ''T''. The following theorem answers the question: Can ''T''* be defined by a formula of first-order arithmetic?
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| ''Tarski's undefinability theorem'': There is no ''L''-formula ''[[Truth_predicate|True''(''n'')]] which defines ''T''*.
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| That is, there is no ''L''-formula ''True''(''n'') such that for every ''L''-formula ''A'', ''True''(''g''(''A'')) ↔ ''A'' holds.
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| Informally, the theorem says that given some formal arithmetic, the concept of truth in that arithmetic is not definable using the expressive means that arithmetic affords. This implies a major limitation on the scope of "self-representation." It ''is'' possible to define a formula ''True''(''n'') whose extension is ''T''*, but only by drawing on a [[metalanguage]] whose expressive power goes beyond that of ''L''. For example, a [[Truth_predicate|truth predicate]] for first-order arithmetic can be defined in [[second-order arithmetic]]. However, this formula would only be able to define a truth predicate for sentences in the original language ''L''. To define a truth predicate for the metalanguage would require a still higher "metametalanguage", and so on.
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| The theorem just stated is a corollary of [[Post's theorem]] about the [[arithmetical hierarchy]], proved some years after Tarski (1936). A semantic proof of Tarski's theorem from Post's theorem is obtained by [[reductio ad absurdum]] as follows. Assuming ''T''* is arithmetically definable, there is a natural number ''n'' such that ''T''* is definable by a formula at level <math>\Sigma^0_n</math> of the [[arithmetical hierarchy]]. However, ''T''* is <math>\Sigma^0_k</math>-hard for all ''k''. Thus the arithmetical hierarchy collapses at level ''n'', contradicting Post's theorem.
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| == General form of the theorem ==
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| Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting theorem applies to any formal language with [[negation]], and with sufficient capability for [[self-reference]] that the [[diagonal lemma]] holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more general formal systems.
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| ''Tarski's undefinability theorem (general form)'': Let (''L'',''N'') be any interpreted formal language which includes negation and has a Gödel numbering ''g''(''x'') such that for every ''L''-formula ''A''(''x'') there is a formula ''B'' such that ''B'' ↔ ''A''(''g''(''B'')) holds. Let ''T''* be the set of Gödel numbers of ''L''-sentences true in ''N''. Then there is no ''L''-formula ''True''(''n'') which defines ''T''*. That is, there is no ''L''-formula ''True''(''n'') such that for every ''L''-formula ''A'', ''True''(''g''(''A'')) ↔ ''A'' holds.
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| The proof of Tarski's undefinability theorem in this form is again by [[reductio ad absurdum]]. Suppose that an ''L''- formula ''True''(''n'') defines ''T''*. In particular, if ''A'' is a sentence of arithmetic then ''True''(''g''(''A'')) holds in ''N'' if and only if ''A'' is true in ''N''. Hence for all ''A'', the Tarski ''T''-sentence ''True''(''g''(''A'')) ↔ ''A'' is true in ''N''. But the diagonal lemma yields a counterexample to this equivalence, by giving a "Liar" sentence ''S'' such that ''S'' ↔ ¬''True''(''g''(''S'')) holds. Thus no ''L''-formula ''True''(''n'') can define ''T''*. QED.
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| The formal machinery of this proof is wholly elementary except for the [[diagonalization]] that the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for example, it does not invoke [[recursion#Functional recursion|recursive function]]s in any way. The proof does assume that every ''L''-formula has a [[Gödel number]], but the specifics of a coding method are not required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated [[Gödel's incompleteness theorems|theorems of Gödel]] about the metamathematical properties of [[Peano arithmetic|first-order arithmetic]]. | |
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| ==Discussion==
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| [[Raymond Smullyan|Smullyan]] (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by [[Gödel's incompleteness theorem]]s. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., [[John Lucas (philosopher)|Lucas]] 1961) is less than evident. Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough [[self-reference]] for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more strikingly evident.
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| An interpreted language is ''strongly-semantically-self-representational'' exactly when the language contains [[Predicate (grammar)|predicates]] and [[function symbol]]s defining all the [[semantic]] concepts specific to the language. Hence the required functions include the "semantic valuation function" mapping a formula ''A'' to its [[truth value]] ||''A''||, and the "semantic denotation function" mapping a term ''t'' to the object it denotes. Tarski's theorem then generalizes as follows: ''No sufficiently powerful language is strongly-semantically-self-representational''.
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| The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For example, the set of (codes for) formulas of first-order [[Peano arithmetic]] that are true in ''N'' is definable by a formula in [[second order arithmetic]]. Similarly, the set of true formulas of the standard model of second order arithmetic (or ''n''-th order arithmetic for any ''n'') can be defined by a formula in first-order [[Zermelo%E2%80%93Fraenkel_set_theory|ZFC]].
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| == References ==
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| * J.L. Bell, and M. Machover, 1977. ''A Course in Mathematical Logic''. North-Holland.
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| * [[George Boolos|G. Boolos]], [[John P. Burgess|J. Burgess]], and [[Richard Jeffrey|R. Jeffrey]], 2002. ''Computability and Logic'', 4th ed. Cambridge University Press.
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| * [[John Lucas (philosopher)|J.R. Lucas]], 1961. "[http://users.ox.ac.uk/~jrlucas/Godel/mmg.html Mind, Machines, and Gödel]". Philosophy 36: 112-27.
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| * R. Murawski, 1998. [http://www.staff.amu.edu.pl/~rmur/hpl1.ps Undefinability of truth. The problem of the priority: Tarski vs. Gödel]. History and Philosophy of Logic 19, 153-160
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| * [[Raymond Smullyan|R. Smullyan]], 1991. ''Godel's Incompleteness Theorems''. Oxford Univ. Press.
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| * [[Raymond Smullyan|R. Smullyan]], 2001. "Gödel’s Incompleteness Theorems". In L. Goble, ed., ''The Blackwell Guide to Philosophical Logic'', Blackwell, 72-89.
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| * {{cite journal| author=[[Alfred Tarski|A. Tarski]]| title=Der Wahrheitsbegriff in den formalisierten Sprachen| journal=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}}
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| * [[Alfred Tarski|A. 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.
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| {{Theories of Truth}}
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| [[Category:Mathematical logic]]
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| [[Category:Theories of truth]]
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| [[Category:Metatheorems]]
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| [[Category:Philosophy of logic]]
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