
Contents
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7.1 THE NAME OF THE GAME 7.1 THE NAME OF THE GAME
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7.2 THE QUASI-SET THEORY 𝔔 7.2 THE QUASI-SET THEORY 𝔔
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7.2.1 Relations and Quasi-Functions 7.2.1 Relations and Quasi-Functions
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7.2.2 Quasi-Cardinals 7.2.2 Quasi-Cardinals
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7.2.3 ‘Weak’ Extensionality 7.2.3 ‘Weak’ Extensionality
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7.2.4 Replacement Axioms 7.2.4 Replacement Axioms
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7.2.5 The Strong Singleton 7.2.5 The Strong Singleton
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7.2.6 Permutations are not Observable 7.2.6 Permutations are not Observable
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7.2.7 The Axiom of Choice 7.2.7 The Axiom of Choice
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7.2.8 Remark on the Existence of Atoms: The Theory 𝔔m 7.2.8 Remark on the Existence of Atoms: The Theory 𝔔m
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7.3 RELATIVE CONSISTENCY 7.3 RELATIVE CONSISTENCY
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7.4 QUASET IDEAS WITHIN QUASI-SET THEORY 7.4 QUASET IDEAS WITHIN QUASI-SET THEORY
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7.5 CHANGES IN TIME: THE THEORY 𝔔t 7.5 CHANGES IN TIME: THE THEORY 𝔔t
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7.6 QUANTUM STATISTICS WITHIN 𝔔 7.6 QUANTUM STATISTICS WITHIN 𝔔
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7.7 ON JUSTIFYING QUASI-SET THEORY 7.7 ON JUSTIFYING QUASI-SET THEORY
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7.7.1 Quasi-Sets and Quasets: A New Look 7.7.1 Quasi-Sets and Quasets: A New Look
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7 The Mathematics of Non-Individuality
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Published:June 2006
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Abstract
This chapter presents the theory of quasi-sets, and argues that this offers an appropriate set-theoretic formalism for non-individual quantum objects. The basic idea is that in a quasi-set, there may exist elements for which the traditional concept of identity does not apply — these elements are called the ‘m-atoms’. Thus, a quasi-set may have a cardinal but not an associated ordinal. The other elements can be regarded as standard elements of a set. Hence, the theory encompasses a copy of Zermelo-Frankel set theory with Urelement. A number of alternatives are also presented with a view to possible physical applications, such as allowing the cardinality of a quasi-set to vary in time and thereby accommodating aspects of relativistic quantum theory. The formal framework is also applied to the development of quantum statistics, which can be naturally captured by the theory.
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