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In class theories, the axiom of limitation of size says that for any class ''C'', ''C'' is a proper class, that is a class which is not a set (an element of other classes), if and only if it can be mapped onto the class ''V'' of all sets.〔This is roughly von Neumann's original formulation, see Fraenkel & al, p. 137.〕 : This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union〔showing directly that a set of ordinals has an upper bound, see A. Levy, " On von Neumann's axiom system for set theory ", Amer. Math. Monthly, 75 (1968), p. 762-763.〕 at one stroke. The axiom of limitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is a surjection from the ordinals to the universe, thus an injection from the universe to the ordinals, that is, the universe of sets is well-ordered. Together the axiom of replacement and the axiom of global choice (with the other axioms of von Neumann–Bernays–Gödel set theory) imply this axiom. This axiom is thus equivalent to the combination of replacement, global choice, specification and union in von Neumann–Bernays–Gödel or Morse–Kelley set theory. However, the axiom of replacement and the usual axiom of choice (with the other axioms of von Neumann–Bernays–Gödel set theory) do not imply von Neumann's axiom. In 1964, Easton used forcing to build a model that satisfies the axioms of von Neumann–Bernays–Gödel set theory with one exception: the axiom of global choice is replaced by the axiom of choice. In Easton's model, the axiom of limitation of size fails dramatically: the universe of sets cannot even be linearly ordered.〔Easton 1964.〕 It can be shown that a class is a proper class if and only if it is equinumerous to ''V'', but von Neumann's axiom does not capture all of the "limitation of size doctrine",〔Fraenkel & al, p. 137. A guiding principle for ZF to avoid set theoretical paradoxes is to restrict to instances of full (contradictory) comprehension scheme that do not give sets "too much bigger" than the ones they use; it is known as "limitation of size", Fraenkel & al call it "limitation of size doctrine", see p. 32.〕 because the axiom of power set is not a consequence of it. Later expositions of class theories (Bernays, Gödel, Kelley, ...) generally use replacement and a form of the axiom of choice rather than the axiom of limitation of size. ==History== Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z () as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments."〔''Historical Introduction'' in Bernays 1991, p. 31.〕 The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to support his proof of the well-ordering theorem and to avoid contradictory sets.〔"... we must, on the one hand, restrict these principles () sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation, p. 200). Gregory Moore analyzed Zermelo's reasons behind his axiomatization and concluded that "his axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem …" and "For Zermelo, … the paradoxes were an inessential obstacle to be circumvented with as little fuss as possible." (Moore 1982, p. 159–160).〕 In 1922, Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set where ''Z''0 is the set of natural numbers, and ''Z''''n''+1 is the power set of ''Z''''n''.〔Fraenkel 1922, p. 230–231; Skolem 1922 (English translation, p. 296–297).〕 They also introduced the axiom of replacement, which guarantees the existence of this set.〔Ferreirós 2007, p. 369. In 1917, Mirimanoff published a form of replacement based on cardinal equivalence (Mirimanoff 1917, p. 49).〕 However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor clarifies the difference between sets that are safe to use and collections that lead to contradictions. In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identifies the sets that are "too big" (now called proper classes) and that can lead to contradictions.〔He gave a detailed exposition of his set theory in two articles: von Neumann 1925 and von Neumann 1928.〕 Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is equivalent to the set of all things."〔Hallett 1984, p. 288.〕 He then restricted how these sets may be used: "… in order to avoid the paradoxes those () which are 'too big' are declared to be impermissible as ''elements''."〔Hallett 1984, p. 290.〕 By combining this restriction with his criterion, von Neumann obtained the axiom of limitation of size (which in the language of classes states): A class X is not an element of any class if and only if X is equivalent to the class of all sets.〔Hallett 1984, p. 290. Von Neumann later changed "equivalent to the class of all sets" to "can be mapped onto the class of all sets."〕 So von Neumann identified sets as classes that are not equivalent to the class of all sets. Von Neumann realized that, even with his new axiom, his set theory does not fully characterize sets.〔To be precise, von Neumann investigated whether his set theory is categorical; that is, whether it uniquely determines sets in the sense that any two of its models are isomorphic. He showed that it is not categorical because of a weakness in the axiom of regularity: this axiom only excludes descending ∈-sequences from existing in the model; descending sequences may still exist outside the model. A model having "external" descending sequences is not isomorphic to a model having no such sequences since this latter model lacks isomorphic images for the sets belonging to external descending sequences. This led von Neumann to conclude "that no categorical axiomatization of set theory seems to exist at all" (von Neumann 1925, p. 239; English translation: p. 412).〕 Gödel found von Neumann's axiom to be "of great interest": :"In particular I believe that his (Neumann's ) necessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles. The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view.〔For example, von Neumann's proof that his axiom implies the well-ordering theorem uses the Burali-Forte paradox (von Neumann 1925, p. 223; English translation: p. 398).〕 Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, will the basic problems of abstract set theory be solved."〔From a Nov. 8, 1957 letter Gödel wrote to Stanislaw Ulam (Kanamori 2003, p. 295).〕 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 limitation of size」の詳細全文を読む スポンサード リンク
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