Realizability as a kind of truth-making Øystein Linnebo and Stewart Shapiro December 6, 2017 1 Background on potential infinity Although this paper is self-contained, it is part of a larger project concerning potentiality in mathematics. The first and simplest case is the traditional Aris- totelian notion of potential infinity (see [10]). An issue much like that of truth- making arises in our explication of one of the options for potential infinity, namely how to make sense of generalizations from that perspective. We use the traditional intuitionistic notion of realizability to resolve the issue, and to help settle the correct logic for one kind of potential infinity. We begin with some highlights of our account(s) of potential infinity. From Aristotle until the nineteenth century, the vast majority of major philosophers and mathematicians rejected the notion of the actual infinite. They argued that the only sensible notion is that of potential infinity—at least for scientific or, later, non-theological purposes. In Physics 3.6 (206a27-29), Aristotle wrote, “For generally the infinite is as follows: there is always another and another to be taken. And the thing taken will always be finite, but always different”(2o6a27-29). As Richard Sorabji [12] (322-3) puts it, for Aristotle, “infinity is an extended finitude” (see also [7], [8]). This orientation towards the infinite was endorsed by mainstream mathe- maticians as late as Gauss [5], who in 1831 wrote: “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking.” A definitive change in mathematicians’ orientation towards the infinite took place late in the nineteenth century, resulting in large part from pioneering work by Georg Cantor, who showed us how to make mathematical sense of completed infinite collections or sets, and how to assign a size or cardinal num- ber to such sets. Cantor’s theory of infinite sets and numbers proved so el- egant, insightful, and useful for mathematical purposes that it was quickly assimilated into mathematical practice, where it came to serve an important foundational role. From this point on, the only sustained opposition to the Cantorian concep- tion of the actual infinite came from intuitionists and constructive mathematics. 1