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The Axiom of Reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the Axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory. ==History== With Russell's discovery (1901, 1902)〔According to van Heijenoort 1967:124, Russell discovered the paradox in June 1901. van Heijenoort in turn references Bertrand Russell (1944) "My mental development" in ''The philosophy of Bertrand Russell'', edited by Paul Arthur Schilpp (Tudor, New York), page 13. But Russell did not report it to Frege until his letter to Frege dated 16 June 1902. Livio 2009:186 reports the same date. Livio 2009:191 writes that Zermelo discovered the paradox as early as 1900, but does not give his source for this (Ewald 1996?). Indeed, Zermelo makes this claim in a footnote 9 to his 1908 ''A new proof of the possibility of a well-ordering'' in van Heijenoort 1967:191.〕 of a paradox in Gottlob Frege's 1879 ''Begriffsschrift'' and Frege's acknowledgment of the same (1902), Russell tentatively introduced his solution as "Appendix B: Doctrine of Types" in his 1903 ''Principles of Mathematics''.〔cf Introductory remarks by W. V. Quine preceding Bertrand Russell (1908a) reprinted in van Heijenoort 1967:150.〕 This contradiction can be stated as "the class of all classes that do not contain themselves as elements".〔cf Introductory remarks by W. V. Quine preceding Bertrand Russell (1908a) reprinted in van Heijenoort 1967:150.〕 At the end of this appendix Russell asserts that his "doctrine" would solve the immediate problem posed by Frege, but "there is at least one closely analogous contradiction which is probably not soluble by this doctrine. The totality of all logical objects, or of all propositions, involves, it would seem a fundamental logical difficulty. What the complete solution of the difficulty may be, I have not succeeded in discovering; but as it affects the very foundations of reasoning..."〔Russell 1903:528〕 By the time of his 1908 ''Mathematical logic as based on the theory of types''〔reprinted in van Heijenoort 150–182〕 Russell had studied "the contradictions" (among them the Epimenides paradox, the Burali-Forti paradox, and Richard's paradox) and concluded that "In all the contradictions there is a common characteristic, which we may describe as self-reference or reflexiveness".〔Russell 1908:154. The exact wording appears in Whitehead and Russell 1913 reprinted to *53 1962:60〕 In 1903, Russell defined ''predicative'' functions as those whose order is one more than the highest order function occurring in the expression of the function. While these were fine for the situation, ''impredicative'' functions had to be disallowed: He repeats this definition in a slightly different way later in the paper (together with a subtle prohibition that they would express more clearly in 1913): This usage carries over to Alfred North Whitehead and Russell's 1913 ''Principia Mathematica'' wherein the authors devote an entire subsection of their Chapter II: "The Theory of Logical Types" to subchapter I. ''The Vicious-Circle Principle'': "We will define a function of one variable as ''predicative'' when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. . . A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument."〔Whitehead and Russell 1913 reprinted to *53 1962:53〕 They again propose the definition of a ''predicative function'' as one that does not violate The Theory of Logical Types. Indeed the authors assert such violations are "incapable (achieve )" and "impossible": The authors stress the word ''impossible'': 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「Axiom of reducibility」の詳細全文を読む スポンサード リンク
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