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Synthese

, Volume 192, Issue 8, pp 2463–2488 | Cite as

Multiverse conceptions in set theory

  • Carolin Antos
  • Sy-David Friedman
  • Radek Honzik
  • Claudio Ternullo
Article

Abstract

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (Bull Symb Logic 19(1):77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory.

Keywords

Set theory Universe of sets Set-theoretic multiverse Hyperuniverse programme New axioms of set theory 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Carolin Antos
    • 1
  • Sy-David Friedman
    • 1
  • Radek Honzik
    • 1
    • 2
  • Claudio Ternullo
    • 1
  1. 1.KGRCViennaAustria
  2. 2.Charles UniversityPragueCzech Republic

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