The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences

This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity.I begin Part I by rehearsing an argument--which is due to Linnebo and which I partially endorse--that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first-order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting.I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. I argue that there is an important range of potential infinities, which I call infinities with branching possibilities, for which S4 is the right modal logic. I further argue that the usual necessity operator is not sufficiently expressive for reasoning about branching potential infinities. A new operator is needed, which I call the inevitability operator. (In tense logic, the same operator has been called the strong future tense). I show that first-order S4 with the inevitability operator is unaxiomatizable. This makes it improbable that there can be a faithful translation of non-modal mathematical discourse into modal language. I suggest that this means standard mathematical theories of infinite sequences are thus committed to actual infinities.In Part II, I apply the modal account of potential infinity to develop a theory of free choice sequences. Free choice sequences are a concept from intuitionistic mathematics. They can be thought of as potentially infinite sequences of natural numbers whose values are freely chosen one at a time by an (idealized) mathematician. They figure prominently in results of intuitionistic mathematics that contradict classical mathematics. I propose an axiomatic theory of free choice sequences in a modal extension of classical second-order arithmetic, called MC, with the aim of providing modal analogues of key ideas and results from the intuitionistic analysis. In this theory I define the temporal-potential continuum, which serves as an ersatz intuitionistic continuum in my modal theory.I show that the temporal-potential continuum exhibits many characteristically intuitionistic properties. The main results are these: it is not the case that every real number is determinately rational or irrational; the natural order on real numbers is not linear; a bounded monotone sequence of rationals need not be Cauchy; if two disjoint sets A and B decompose the temporal-potential continuum, then both A and B are topologically open; finally, I introduce a notion of sharp discontinuity and show there is no function on the temporal-potential continuum which is sharply discontinuous.