Multiverse Conceptions in Set Theory

Synthese 192 (8):2463-2488 (2015)
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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 (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.

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Author Profiles

Claudio Ternullo
Universitat de Barcelona
Carolin Antos
Universität Konstanz

Citations of this work

Universism and extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - 2021 - Review of Symbolic Logic 14 (1):112-154.
Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
Are Large Cardinal Axioms Restrictive?Neil Barton - 2023 - Philosophia Mathematica 31 (3):372-407.
Independence Proofs in Non-Classical Set Theories.Sourav Tarafder & Giorgio Venturi - 2023 - Review of Symbolic Logic 16 (4):979-1010.

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Naturalism in mathematics.Penelope Maddy - 1997 - New York: Oxford University Press.
Platonism and anti-Platonism in mathematics.Mark Balaguer - 1998 - New York: Oxford University Press.
Models and reality.Hilary Putnam - 1980 - Journal of Symbolic Logic 45 (3):464-482.

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