The Logic of Potential Infinity

Abstract

Michael Dummett argues that acceptance of potentially infinite collections requires that we abandon classical logic and restrict ourselves to intuitionistic logic. In this paper we examine whether Dummett is correct. After developing two detailed accounts of what, exactly, it means for a concept to be potentially infinite (based on ideas due to Charles McCarty and Øystein Linnebo, respectively), we construct a Kripke structure that contains a natural number structure that satisfies both accounts. This model supports a logic much stronger than intuitionistic logic, demonstrating that Dummett was wrong. We conclude by briefly examining ways to extend the account(s) in question to indefinitely extensible concepts such as Cardinal, Ordinal, and Set.