# Varieties of Potentialism

Potentialism in mathematics is, roughly, the view that a mathematical universe is never fully completed because there always could be more objects of the relevant kind to consider. In particular, set-theoretic potentialism has it that the set-theoretic universe is never fully completed, since for any circumscribed totality of sets it seems there could have been further sets, and thus a larger totality of sets to take into account.

Contemporary exploration of potentialism has revealed that there are several important theoretical choices that potentialists face when formulating their view. This gives rise to different varieties of potentialism. This mini-workshop will explore issues and disagreements that could arise between them. In particular, we are interested in the following issues:

*-The question of the correct modal logic for the potentialist (S4, S4.2 or S4.3 are common candidates). Also, should the logic be classical, intuitionistic, or something else?*

*-branching vs. non-branching possibility (either for set-theoretic hierarchies or other mathematical structures)
-whether translations between the modal, potentialist language and the non-modal, actualist language are desirable at all
-in what way answers to the previous issues/questions depend on how the relevant modality is understood*

**Program (time displayed in CEST): (**click on title for slides)

12.00-12.10: Welcome and introduction

12:10-13.00: Sam Roberts (University of Konstanz) "Potentialism and ultimate V"

13.10-14.00: Joel David Hamkins (University of Oxford) "Modal Model Theory as Mathematical Potentialism"

14.00-14.30: Break

14.30-15.20: Hans Robin Solberg (University of Oxford) "Radical Potentialism"

15.30-16.20: Ethan Brauer (Lingnan University) "Intuitionism and Potentialism about Real Numbers"

16.20-16.50: Break

16.50-17.40: Stewart Shapiro (Ohio State University) and Øystein Linnebo (University of Oslo) "Choice sequences: a modal and classical analysis"

17.50-18.40: Laura Crosilla and Øystein Linnebo (University of Oslo) "Two kinds of potential domains: some logical and historical remarks"

**Registration**

The workshop will be held over Zoom. Non-speaking participants can register by emailing hans.solberg@philosophy.ox.ac.uk for a Zoom-invite.

**Abstracts**

**Title: Potentialism and ultimate V" **

**Speaker: Sam Roberts (University of Konstanz)**

Abstract: Potentialism is the view that the universe of mathematics is inherently potential. It comes in two main flavours: height-potentialism and width-potentialism. It is often thought that these are two aspects of a broader phenomenon: that the universe of sets is potential in both ways. In this talk, I show that this thought is mistaken: height-potentialism and width-potentialism are inconsistent with one another. In particular, I will argue that height-potentialism implies the existence of an ultimate background universe of sets—an ultimate V—to which no new sets can be added by forcing and in which every set-theoretic statement is either determinately true or determinately false. This directly contradicts the core claim of width-potentialism that there are no such universes.

**Title: Modal model theory as mathematical potentialism**

**Speaker: Joel David Hamkins, Oxford**

Abstract: I shall introduce and describe the subject of modal model theory, in which one studies a mathematical structure within a class of similar structures under an extension concept, giving rise to mathematically natural notions of possibility and necessity, a form of mathematical potentialism. We study the class of all graphs, or all groups, all fields, all orders, or what have you; a natural case is the class $\text{Mod}(T)$ of all models of a fixed first-order theory $T$. In this talk, I shall describe some of the resulting elementary theory, such as the fact that the $\mathcal{L}$ theory of a structure determines a robust fragment of its modal theory, but not all of it. The class of graphs illustrates the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_\omega$, $\beth_\omega$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. When augmented with the actuality operator @, modal graph theory becomes fully bi-interpretable with truth in the set-theoretic universe. This is joint work with Wojciech Wołoszyn.

**Title: Intuitionism and Potentialism about Real Numbers**

**Speaker: Ethan Brauer, Lingnan**

In intuitionistic analysis, real numbers are apprehended via *real number generators*, which are potentially infinite sequences of rational numbers satisfying the Cauchy convergence criterion. Using a modal theory of free choice sequences I will present a modal theory of real number generators with a classical background theory. Many concepts from intuitionism are nicely captured in this setting, and characteristic results from intuitionistic analysis have analogues in this modal theory. For instance, it is not the case that every real number is determinately rational or irrational; the natural order on real numbers is not linear; and there exists a bounded monotone sequence of rationals which is not necessarily Cauchy. Finally, I introduce a notion of sharp discontinuity and show there is no function on the reals that is sharply discontinuous.

**Title: Two kinds of potential domains: some logical and historical remarks**

**Speakers: Laura Crosilla and Øystein Linnebo, Oslo**

Abstract: Potentialists defend a distinction between actual and merely potential domains. We defend the philosophical and mathematical importance of a less familiar distinction that applies to the merely potential domains: those that are predetermined and those that are not. This is a question of whether we have clearly circumscribed the possibilities that we wish to consider. For example, a potentialist about the natural numbers can take the possibilities for generating natural numbers to be predetermined. The transition from Weyl 1918 to 1921 will be used to illustrate the distinction. We also make some remarks about how generalizations over the two different kinds of merely potential domains can be understood.