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Concept
Diagonal lemma
Summary
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 of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem.
Let be the set of natural numbers. A first-order theory in the language of arithmetic represents the computable function if there exists a "graph" formula in the language of such that for each
Here is the numeral corresponding to the natural number , which is defined to be the th successor of presumed first numeral in .
The diagonal lemma also requires a systematic way of assigning to every formula a natural number (also written as ) called its Gödel number. Formulas can then be represented within by the numerals corresponding to their Gödel numbers. For example, is represented by
The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include first-order Peano arithmetic and the weaker Robinson arithmetic, and even to a much weaker theory known as R. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all computable functions, but all the theories mentioned have that capacity, as well.
Let be a first-order theory in the language of arithmetic and capable of representing all computable functions, and be a formula in with one free variable. Then there exists a sentence such that
Intuitively, is a self-referential sentence: says that has the property . The sentence can also be viewed as a fixed point of the operation assigning to each formula the sentence . The sentence constructed in the proof is not literally the same as , but is provably equivalent to it in the theory .
Official source
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