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IMI/Publicaţii/CSJM/Ediţii/CSJM v.23, n.3 (69), 2015/

Extensionality, Proper Classes, and Quantum Non-Individuality

Authors: William J. Greenberg

Abstract

In this paper I address two questions: (1) What distinguishes proper classes from sets? (2) Are proper classes and quantum particles individuals? Against the familiar response to (1) that proper classes are "too big" to be sets, I propose that it is not a difference in size that distinguishes such collections but a difference in individuation. The linchpin of my proposal and centerpiece of an NBG-like fragment of class and set theory (``NBG'': von Neumann-Bernays-G\"odel), is an Axiom of Restricted Extensionality according to which sets are individuated by their members but proper classes are not. This setting (I call it NBG$^{-}$) I show to be equi-consistent with its NBG counterpart. I answer (2) by exhibiting a parallelism in NBG$^{-} between proper classes and quantum particles, the former unindividuated by their members and the latter unindividuated by their relational properties. Since both violate the (weak) principle of the identity of indiscernibles as well as the principle of reflexive identity, in NBG$^{-}$ neither proper classes nor quantum particles are individuals.

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