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Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 and ''c''. The theory was first suggested by Rudy Rucker in his ''Infinity and the Mind''.〔Rucker, Rudy, ''Infinity of the Mind'', Princeton UP, 1995, p.253.〕 The details set out in this entry are due to the American mathematician Randall M. Holmes. == Arguments supporting PST == There are at least two independent arguments in favor of a small set theory like PST. #One can get the impression from mathematical practice outside set theory that there are “only two infinite cardinals which demonstrably ‘occur in nature’ (the cardinality of the natural numbers and the cardinality of the continuum),”〔''Pocket Set Theory'', p.8.〕 therefore “set theory produces far more superstructure than is needed to support classical mathematics.”〔''Alternative Set Theories'', p.35.〕 Although it may be an exaggeration (one can get into a situation in which one has to talk about arbitrary sets of real numbers or real functions), with some technical tricks〔See ''Pocket Set Theory'', p.8. on encoding.〕 a considerable portion of mathematics can be reconstructed within PST; certainly enough for most of its practical applications. #A second argument arises from foundational considerations. Most of mathematics can be implemented in standard set theory or one of its large alternatives. Set theories, on the other hand, are introduced in terms of a logical system; in most cases it is first-order logic. The syntax and semantics of first-order logic, on the other hand, is built on set-theoretical grounds. Thus, there is a foundational circularity, which forces us to choose as weak a theory as possible for bootstrapping. This line of thought, again, leads to small set theories. Thus, there are reasons to think that Cantor's infinite hierarchy of the infinites is superfluous. Pocket set theory is a “minimalistic” set theory that allows for only two infinites: the cardinality of the (standard) natural numbers and the cardinality of the (standard) reals. 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「Pocket set theory」の詳細全文を読む スポンサード リンク
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