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S is an axiomatic set theory set out by George Boolos in his article, Boolos (1989). S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed S to embody his understanding of the “iterative conception of set“ and the associated iterative hierarchy. S has the important property that all axioms of Zermelo set theory ''Z'', except the axiom of Extensionality and the axiom of Choice, are theorems of S or a slight modification thereof. ==Ontology== Any grouping together of mathematical, abstract, or concrete objects, however formed, is a ''collection'', a synonym for what other set theories refer to as a class. The things that make up a collection are called elements or members. A common instance of a collection is the domain of discourse of a first order theory. All sets are collections, but there are collections that are not sets. A synonym for collections that are not sets is proper class. An essential task of axiomatic set theory is to distinguish sets from proper classes, if only because mathematics is grounded in sets, with proper classes relegated to a purely descriptive role. The Von Neumann universe implements the “iterative conception of set” by stratifying the universe of sets into a series of “stages,” with the sets at a given stage being possible members of the sets formed at all higher stages. The notion of stage goes as follows. Each stage is assigned an ordinal number. The lowest stage, stage 0, consists of all entities having no members. We assume that the only entity at stage 0 is the empty set, although this stage would include any urelements we would choose to admit. Stage ''n'', ''n''>0, consists of all possible sets formed from elements to be found in any stage whose number is less than ''n''. Every set formed at stage ''n'' can also be formed at every stage greater than ''n''.〔Boolos (1998:88).〕 Hence the stages form a nested and well-ordered sequence, and would form a hierarchy if set membership were transitive. The iterative conception has gradually become more accepted, despite an imperfect understanding of its historical origins. The iterative conception of set steers clear, in a well-motivated way, of the well-known paradoxes of Russell, Burali-Forti, and Cantor. These paradoxes all result from the unrestricted use of the principle of comprehension of naive set theory. Collections such as “the class of all sets” or “the class of all ordinals” include sets from all stages of the iterative hierarchy. Hence such collections cannot be formed at any given stage, and thus cannot be sets. 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「S (set theory)」の詳細全文を読む スポンサード リンク
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