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Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics.〔Concerning the origin of the term ''naive set theory'', Jeff Miller says, "''Naïve set theory'' (contrasting with axiomatic set theory) was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyl's review of P. A. Schilpp (ed) ''The Philosophy of Bertrand Russell'' in the ''American Mathematical Monthly'', 53., No. 4. (1946), p. 210 and Laszlo Kalmar's review of ''The Paradox of Kleene and Rosser'' in ''Journal of Symbolic Logic'', 11, No. 4. (1946), p. 136. (JSTOR)." () The term was later popularized by Paul Halmos' book, ''Naive Set Theory'' (1960).〕 Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday usage of set theory concepts in contemporary mathematics.〔. "The working mathematicians usually thought in terms of a naive set theory (probably one more or less equivalent to ZF) ... a practical requirement (any new foundational system ) could be that this system could be used "naively" by mathematicians not sophisticated in foundational research" ((p. 236 )).〕 Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory can be seen as a stepping-stone to more formal treatments, and suffices for many purposes. ==Method== A naive theory is considered to be a non-formalized theory, that is, a theory that uses a natural language to describe sets and operations on sets. The words and, or, if ... then, not, for some, for every are not here subject to rigorous definition. It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them. Furthermore, a firm grasp of set theory's concepts from a naive standpoint is a step to understanding the motivation for the formal axioms of set theory. As a matter of convenience, usage of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. Sets are defined informally and a few of their properties are investigated. Links to specific axioms of set theory describe some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis. The first development of set theory was a naive set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets and developed by Gottlob Frege in his ''Begriffsschrift''. Naive set theory may refer to several very distinct notions. It may refer to * Informal presentation of an axiomatic set theory, e.g. as in ''Naive Set Theory'' by Paul Halmos. * Early or later versions of Georg Cantor's theory and other informal systems. * Decidedly inconsistent theories (whether axiomatic or not), like a theory of Gottlob Frege〔 In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.〕 that yielded Russell's paradox, and theories of Giuseppe Peano〔 Axiom 52. chap. IV produces antinomies.〕 and Richard Dedekind. 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「naive set theory」の詳細全文を読む スポンサード リンク
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