Beginning with Aristotle, almost every major philosopher and mathematician before the nineteenth century rejected the notion of the actual infinite. They all argued that the only sensible notion is that of potential infinity. The list includes some of the greatest mathematical minds ever. Due to Georg Cantor’s influence, the situation is almost the opposite nowadays (with some intuitionists as notable exceptions). The received view is that the notion of a merely potential infinity is dubious: it can only be understood if there is an actual infinity that underlies it.

After a sketch of some of the history, our aim is to analyze the notion of potential infinity, in modal terms, and to assess its scientific merits. This leads to a number of more specific questions. Perhaps the most pressing of these is whether the conception of potential infinity can be explicated in a way that is both interesting and substantially different from the now-dominant conception of actual infinity. One might suspect that, when metaphors and loose talk give way to precise definitions, the apparent differences will evaporate.

As we will explain, however, a number of differences still remain. Some of the most interesting and surprising differences concern consequences that one’s conception of infinity has for higher-order logic. Another important question concerns the relation between potential infinity and mathematical intuitionism. We show that potential infinity is not inextricably tied to intuitionistic logic. There are interesting explications of potential infinity that underwrite classical logic, while still differing in important ways from actual infinity. However, we also find that on some more stringent explications, potential infinity does indeed lead to intuitionistic logic.