Divergent potentialism: A modal analysis with an application to choice sequences Ethan Brauer, Øystein Linnebo, and Stewart Shapiro Abstract Modal logic has been used to analyze potential infinity and poten- tialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of free choice sequences, we motivate the need for a modal analysis of diver- gent potentialism and explain the challenges this involves. Then, using Beth-Kripke semantics for intuitionistic logic, we overcome those chal- lenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms. 1 Potentialism In mathematics, we are often interested in potential existence, namely in what can be constructed or generated. Here are some examples that de- rive from Aristotle and that remained highly influential right up until the Cantorian revolution of the late nineteenth century: (1) Necessarily, for any number m, possibly there is a successor ✷∀m✸∃n Succ(m, n) (2) Necessarily, for any line segment l, possibly l has bisects l 1 and l 2 Such potential existence claims are particularly important in connection with potential infinity. Each claim can be combined with the rejection of the corresponding actual infinity, for example: (3) For any number m, there is a successor ∀m∃n Succ(m, n) Potentialism is the view that potential existence, and modality more generally, have a role to play in mathematics, either explicitly or implicitly. 1