In this book I advocate a formulation of potentialist set theory in terms of (a generalization of) the logical possibility operator. I show that, working in this framework, we can justify mathematicians' use of the ZFC axioms from general modal principles which (unlike those used in prior potentialist justifications for use of the ZFC axioms) all seem clearly true. This provides an appealing answer to classic questions about how anyone (realist or potentialist) can satisfyingly justify use of the axiom of replacement.
Looking beyond set theory, I argue that philosophical analyses using this generalized logical possibility operator can illuminate topics like: grounding, neo-Carnapian theories of ontological knowledge by convention, varieties of (post) Quinean indispensability arguments, and the heterogeneity of applied mathematics. I'll also argue that the above logical-potentialist approach to set theory fits naturally into a more general set-theoretic paradox driven, modality-first, modestly neo-Carnapian approach to mathematics as a whole.