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ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY
http://hdl.handle.net/2237/00027943
http://hdl.handle.net/2237/00027943b2782b37-ee7c-4fea-9154-e572ad5e681c
名前 / ファイル | ライセンス | アクション |
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MO170514_ppt (171.7 kB)
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Item type | 学術雑誌論文 / Journal Article(1) | |||||
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公開日 | 2018-05-08 | |||||
タイトル | ||||||
タイトル | ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY | |||||
言語 | en | |||||
著者 |
OZAWA, MASANAO
× OZAWA, MASANAO |
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アクセス権 | ||||||
アクセス権 | open access | |||||
アクセス権URI | http://purl.org/coar/access_right/c_abf2 | |||||
権利 | ||||||
言語 | en | |||||
権利情報 | This article has been published in a revised form in [The Review of Symbolic Logic] [https://doi.org/10.1017/S1755020317000120]. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2017 Cambridge University Press. | |||||
抄録 | ||||||
内容記述 | In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory. | |||||
言語 | en | |||||
内容記述タイプ | Abstract | |||||
出版者 | ||||||
言語 | en | |||||
出版者 | Cambridge University Press | |||||
言語 | ||||||
言語 | eng | |||||
資源タイプ | ||||||
資源タイプresource | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | journal article | |||||
出版タイプ | ||||||
出版タイプ | AM | |||||
出版タイプResource | http://purl.org/coar/version/c_ab4af688f83e57aa | |||||
DOI | ||||||
関連タイプ | isVersionOf | |||||
識別子タイプ | DOI | |||||
関連識別子 | https://doi.org/10.1017/S1755020317000120 | |||||
ISSN | ||||||
収録物識別子タイプ | PISSN | |||||
収録物識別子 | 1755-0203 | |||||
書誌情報 |
en : The Review of Symbolic Logic 巻 10, 号 4, p. 782-807, 発行日 2017-12 |
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著者版フラグ | ||||||
値 | author |