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In mathematical logic, the diagonal 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.〔See Boolos and Jeffrey (2002, sec. 15) and Mendelson (1997, Prop. 3.37 and Cor. 3.44 ).〕 == Background == Let N be the set of natural numbers. A theory ''T'' represents the computable function ''f'' : N→N if there exists a "graph" predicate Γf(''x'',''y'') in the language of ''T'' such that for each ''x'' in ''N'', ''T'' proves :(∀''y'') (= ''y'' ↔ Γf(°''x'',''y'') ).〔For details on representability, see Hinman 2005, p. 316〕 Here ''°x'' is the numeral corresponding to the natural number ''x'', which is defined to be the closed term 1+ ··· +1 (''x'' ones), and °''f''(''x'') is the numeral corresponding to ''f''(''x''). The diagonal lemma also requires that there be a systematic way of assigning to every formula θ a natural number #(θ) called its Gödel number. Formulas can then be represented within the theory by the numerals corresponding to their Gödel numbers. The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include Peano arithmetic and the weaker Robinson arithmetic. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all computable functions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「diagonal lemma」の詳細全文を読む スポンサード リンク
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