The modal logic of arithmetic potentialism and the universal algorithm

Abstract

I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I shall consider the potentialist conceptions arising from a model-theoretic view of the models of arithmetic as possible arithmetic realms of feasibility, considering them under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more, which in effect express distinct potentialist ideas. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The main philosophical point is that fundamentally different potentialist conceptions -- linear inevitability, convergent potentialism and radical branching possibility -- are revealed by the precise modal validities of the corresponding potentialist systems in which those attitudes are expressed, and so it is important to discover them.