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Multiversism and Concepts of Set: How Much Relativism Is Acceptable?

In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing. pp. 189-209 (2016)

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  1. Set-theoretic pluralism and the Benacerraf problem.Justin Clarke-Doane - 2020 - Philosophical Studies 177 (7):2013-2030.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
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  • Steel’s Programme: Evidential Framework, the Core and Ultimate- L.Joan Bagaria & Claudio Ternullo - 2023 - Review of Symbolic Logic 16 (3):788-812.
    We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where (...)
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  • Hierarchical Multiverse of Sets.Ahmet Çevik - 2023 - Notre Dame Journal of Formal Logic 64 (4):545-570.
    In this article, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of “degrees of intentionality” of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favor of a hierarchical picture of the multiverse in which theories or models are thought (...)
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  • Mathematical Pluralism.Edward N. Zalta - 2023 - Noûs.
    Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand (...)
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  • Abolishing Platonism in Multiverse Theories.Stathis Livadas - 2020 - Axiomathes 32 (2):321-343.
    A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis, by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily (...)
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  • Can we resolve the continuum hypothesis?Shivaram Lingamneni - 2020 - Synthese 197 (2):599-622.
    I argue that contemporary set theory, as depicted in the 2011–2012 EFI lecture series, lacks a program that promises to decide, in a genuinely realist fashion, the continuum hypothesis (CH) and related questions about the “width” of the universe. We can distinguish three possible objectives for a realist completion of set theory: maximizing structures, maximizing sets, and maximizing interpretive power. However, none of these is allied to a program that can plausibly decide CH. I discuss the implications of this for (...)
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  • Maddy On The Multiverse.Claudio Ternullo - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78.
    Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...)
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