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Cardinality without Enumeration

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Abstract

We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.

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Correspondence to Athanassios Tzouvaras.

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Tzouvaras, A. Cardinality without Enumeration. Stud Logica 80, 121–141 (2005). https://doi.org/10.1007/s11225-005-6780-8

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Keywords

  • Cardinality notion
  • choice
  • wellordering
  • generic class
  • separation class
  • nonstandard model of arithmetic
  • quantum mechanics