Towards an axiomatization of the multiverse conception of set theory (original) (raw)

An axiomatic approach to the multiverse of sets

arXiv (Cornell University), 2022

Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a simultaneous consideration of all set theoretical universes and the relationships between them, eliminating the need for recourse 'outside the theory' when carrying out constructions like forcing etc. We also explore multiversal category theory, showing that we are finally free of questions about 'largeness' at each stage of the categorification process when working in this theory-the category of categories we consider for a given universe contains all large categories in that universe without taking recourse to a larger universe. We leverage this newfound freedom to define a category Force whose objects are universes and whose arrows are forcing extensions, a 2-category Verse whose objects are the categories of sets in each universe and whose component categories are given by functor categories between these categories of sets, and a tricategory Cat whose objects are the 2-categories of categories in each universe and whose component bicategories are given by pseudofunctor, pseudonatural transformation and modification bicategories between these 2-categories of categories in each universe.

Multiverse Conceptions in Set Theory

Synthese , 2015

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.

Multiverse Conceptions in Set Theory (with Sy-David Friedman, Radek Honzik, Claudio Ternullo)

Synthese

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Section 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 Section 2, we use this classification to review four major conceptions. Finally, in Section 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 Friedman-Arrigoni, \cite{friedman-arrigoni2013}. 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.

A Natural Model of the Multiverse Axioms

Notre Dame Journal of Formal Logic, 2010

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of [Hama].

A naturalistic justification of the generic multiverse with a core (slides)

2018

In this paper, I argue that a naturalist approach in philosophy of mathematics justies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic multiverse to be the best framework for mathematical practice. According to UNIFY, an adequate set theory should be foundational, in the sense that it should allow one to represent all the currently accepted mathematical theories. As for MAXIMIZE, this states that any adequate set theory should be as powerful as possible, allowing one to prove as many results and isomorphisms as possible. In a recent paper, Maddy (2017) has argued that this two principle justify ZF C as the best framework for mathematical practice. I argue that, pace Maddy, these two principles justify a multiverse conception of set theory, more precisely, the generic multiverse with a core (GM H). 1 The Multiverse The concept of multiverse was born following the discovery of the phenomenon of independence in set theory: sentences in the language of set theory, such as the Continuum Hypothesis (C H), that turned to be independent from the axioms of ZF C. In order to prove these independence results, 1

The Copernican Multiverse of Sets

2020

We develop an untyped semantic framework for the multiverse of set theory and show that its proof-theoretic commitments are mild. mathsfZF\mathsf{ZF}mathsfZF is extended with semantical axioms utilizing the new symbols mathsfM(mathcalU)\mathsf{M}(\mathcal{U})mathsfM(mathcalU) and mathsfMod(mathcalU,sigma)\mathsf{Mod}(\mathcal{U, \sigma})mathsfMod(mathcalU,sigma), expressing that mathcalU\mathcal{U}mathcalU is a universe and that sigma\sigmasigma is true in the universe mathcalU\mathcal{U}mathcalU, respectively. Here sigma\sigmasigma ranges over the augmented language, leading to liar-style phenomena that are analysed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation to the effect that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a {\em Copernican principle} that the background theory does not hold a privileged positi...

Reinterpreting the universe-multiverse debate in light of inter-model inconsistency in set theory

In this paper I apply the concept of inter-Model Inconsistency in Set Theory (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s Ultimate LLL, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the spectrum from inconsistency avoidance to inconsistency toleration. By doing so, I connect a debate in philosophy of set theory with a debate in philosophy of science about the role of inconsistencies in the natural sciences. While there are important differences, like the lack of threatening explosive inferences, I show how specific philosophical positions in the philosophy of set theory can be interpreted as reactions to a state of inconsistency similar to analogous reactions studied in the philosophy of science literature. My hope is that this transfer operation from philosophy of science to mathematics sheds a new light on the current discussion in philosophy of set theory; and that it can help to bring philosophy of mathematics and philosophy of science closer together.

A justification of the Generic Multiverse with a core

WIP Seminar, 2018

In this presentation I sketch an argument in favour of the multiverse conception of set theory. I also describe a particular kind of multiverse, the set generic multiverse with a core, as introduced by Steel (2014)

Abolishing Platonism in Multiverse Theories

Axiomathes, 2020

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 (CH), 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 address ontological multiversism as advocated, for instance, by Hamkins or Vaatanen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter.