Tarski's indefinability theorem
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Tarski's indefinability 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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[edit] History
In 1931, Kurt Gödel published his famous 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.
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. 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.
Can the same be done for semantical concepts such as truth? In 1936, Tarski stated and proved the Indefinability Theorem bearing his name, implying that in many interesting cases the answer is no, because a sufficiently rich interpreted language cannot 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.
Gödel discovered the Indefinability Theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 publication of Tarski's work. While Gödel never published anything bearing on his independent discovery of indefinability, he did describe it in a 1931 letter to John von Neumann. But Tarski largely derived his Indefinability Theorem and spoke about it to Polish audiences during 1929-31, all before the appearance of Gödel's 1931 landmark paper. Hence Tarski discovered the theorem independently of Gödel, and it rightly bears Tarski's name.
[edit] 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. Let L be the language of first-order arithmetic, and let N be the standard structure for L. Thus (L, N) is the "interpreted first-order language of arithmetic." Let T denote the set of L-sentences true in N, and T* the set of code numbers of the sentences in T. The following theorem answers the question: Can T* be defined by a formula of first-order arithmetic?
Tarski's Indefinability Theorem: There is no L-formula True(x) which defines T*. That is, there is no L formula True(x) such that for every L-formula x, True(x) ↔ x comes out true.
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(x) whose extension is T*, but only by drawing on a metalanguage whose expressive power goes beyond that of L, second-order arithmetic for example.
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 
of the arithmetical hierarchy. However, T* is 
complete for all k. Thus the arithmetical hierarchy collapses at level n, contradicting Post's theorem.
[edit] General form of the theorem
Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method applicable to formal systems far more general than first-order arithmetic. The resulting theorem applies to any formal languages with negation and sufficient capability for self-reference that Gödel's Diagonal Lemma holds. First-order arithmetic satisfies these preconditions, of course.
Proof of Tarski's Indefinability Theorem in its most general form. By contradiction. Suppose that an L- formula True(x) defines T*. In particular, if A is a sentence of arithmetic then True("A") is true in N if and only if A is true in N. Hence for all A, the Tarski T-sentence True("A") ↔ A is true in N. But Gödel's diagonal lemma yields a counterexample to this equivalence: the "Liar" sentence S such that S ↔ ¬True("S") holds. Thus no L-formula True(x) can define T*. QED.
The formal machinery of this proof is wholly elementary except for the diagonalization encapsulated in the diagonal lemma. The proof of that Lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any way. The proof does assume that every L-formula has a Gödel number, but the specifics of the coding method are not required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated theorems of Gödel concerning the metamathematical properties of first-order arithmetic.
[edit] Discussion
Smullyan (1991, 2001) has argued forcefully that Tarski's Indefinability Theorem deserves much of the attention garnered by Gödel's incompleteness theorems. That the latter theorems have much to say about all of mathematics and more controversially, about a range of philosophical issues (e.g., 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 any 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.
Although some have claimed otherwise, Tarski's Theorem is not restricted to bivalent classical logic. For example, it can be generalized to interpreted languages based on many-valued logic, such as fuzzy logic, and to dialetheism, paraconsistent logic, etc. An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates and function symbols 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.
[edit] References
- Bell, J.L., and Machover, M., 1977. A Course in Mathematical Logic. North-Holland.
- George Boolos, John Burgess, and Richard Jeffrey, 2002. Computability and Logic, 4th ed. Cambridge University Press.
- Lucas, J. R., 1961, "Mind, Machines, and Gödel," Philosophy 36: 112-27.
- Raymond Smullyan, 1991. Godel's Incompleteness Theorems. Oxford Univ. Press.
- --------, 2001, "Gödel’s Incompleteness Theorems" in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell: 72-89.
- Alfred Tarski, 1983, "The Concept of Truth in Formalized Languages" in Corcoran, J., ed., Logic, Semantics and Metamathematics. Indianapolis: Hackett. The English translation of Tarski's 1936 Der Wahrheitsbegriff in den formalisierten Sprachen.
