We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

Free choice sequences play a key role in the intuitionistic theory of the continuum and especially in the theorems of intuitionistic analysis that conflict with classical analysis, leading many classical mathematicians to reject the concept of a free choice sequence. By treating free choice sequences as potentially infinite objects, however, they can be comfortably situated alongside classical analysis, allowing a rapprochement of these two mathematical traditions. Building on recent work on the modal analysis of potential infinity, I formulate a modal theory of the free choice sequences known as lawless sequences. Intrinsically well-motivated axioms for lawless sequences are added to a background theory of classical second-order arithmetic, leading to a theory I call $MC_{LS}$. This theory interprets the standard intuitionistic theory of lawless sequences and is conservative over the classical background theory.

Modal potentialism as proposed by Barbara Vetter (2015) is the view that every possibility is grounded in something having a potentiality. Drawing from work by Jessica Leech (2017), Samuel Kimpton-Nye (2021) argues that potentialists can have an S5 modal logic. I present a novel argument to the conclusion that the most straightforward way of spelling out modal potentialism cannot validate an S5 modal logic. Then I will propose a slightly tweaked version of modal potentialism that can validate an S5 modal logic and still does justice to the core claim of potentialism.

Potentialists think that the concept of set is importantly modal. Using tensed language as a heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: ‘Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets’. Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with Level Theory, as developed in Part 1.

We introduce the $\Sigma _1$ -definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma _1$ -definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if M is a countable model of set theory in which the sequence is s and t is any finite extension of s in this model, then there is an end-extension of M to a model in which the sequence is t. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work of both these elements can be done by a single natural generalization of the logical possibility operator.

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta $ ), Grothendieck–Zermelo potentialism (true in all larger $V_\kappa $ for inaccessible cardinals $\kappa $ ), transitive-set potentialism (true in all larger transitive sets), forcing potentialism (true in all forcing extensions), countable-transitive-model potentialism (true in all larger countable transitive models of ZFC), countable-model potentialism (true in all larger countable models of ZFC), and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.