Potentialism holds that certain objects are successively generated in an incompletable process. While it is natural to analyze this view modally, there are theorems that connect the resulting modal analysis of potentialism with the non-modal languages of ordinary mathematics. By extending this approach to plural languages, this article proves a far stronger result about definitional equivalence. This opens the door to a new and entirely non-modal explication of potentialism, using a restricted plural logic. Some advantages of this “demodalized” explication are discussed. It is certainly simpler and more user-friendly than the extant modal analysis, as illustrated by an application to potentialist set theory. More ambitiously, I suggest that the explication might also enable potentialists to sidestep the tricky question of which mathematical objects are actual.

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.

When properly arithmetized, Yablo’s paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but $\omega $-inconsistent. Adding either uniform disquotation or the $\omega $-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn’t arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by M. Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is $\omega $-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic (developed by one of the authors), the paradox strikes back—it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the $\omega $-rule.

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.