Sets and Supersets

Toby Meadows

Research output: Contribution to journalArticle

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Abstract

It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe [Halmos, 1960]. In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.
Original languageEnglish
Pages (from-to)1875-1907
Number of pages33
JournalSynthese
Volume193
Issue number6
Early online date8 Jul 2015
DOIs
Publication statusPublished - Jun 2016

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Keywords

  • Philosophy of mathematics
  • Philosophy of set theory
  • Formal theories of truth
  • Absolute generality
  • Indefinite extensibility

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