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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク History of the Church–Turing thesis ： ウィキペディア英語版 History of the Church–Turing thesis The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing. The debate and discovery of the meaning of "computation" and "recursion" has been long and contentious. This article provides detail of that debate and discovery from Peano's axioms in 1889 through recent discussion of the meaning of "axiom". ==Peano's nine axioms of arithmetic== In 1889, Giuseppe Peano presented his ''The principles of arithmetic, presented by a new method'', based on the work of Dedekind. Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although :"Well before the nineteenth century mathematicians used the principle of defining a function by induction. Dedekind 1888 proved, using accepted axioms, that such a definition defines a unique function, and he applied it to the definition of the functions m+n, m x n, and mn. Based on this work of Dedekind, Peano 1889 and 1891 wrote the familiar five () axioms for the positive integers. As a companion to his fifth () axiom, mathematical induction, Peano used definition by induction, which has been called ''primitive'' recursion (since Péter 1934 and Kleene 1936) ... ."〔Soare 1996:5〕 Observe that ''in fact'' Peano's axioms are ''9'' in number and axiom ''9'' is the recursion/induction axiom.〔cf: van Heijenoort 1976:94〕 :"Subsequently the 9 were reduced to 5 as "Axioms 2, 3, 4 and 5 which deal with identity, belong to the underlying logic. This leaves the five axioms that have become universally known as "the Peano axioms ... Peano acknowledges (1891b, p. 93) that his axioms come from Dedekind ... ."〔van Heijenoort 1976:83〕 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「History of the Church–Turing thesis」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.