What Does Finite Really Mean?

Defining what it means for a set to be finite seems straightforward, but the history of mathematics reveals a tangle of competing proposals, each with subtle strengths and surprising failures. In this lecture, Joel David Hamkins surveys the landscape of finiteness definitions, beginning with Aristotle's potentialist view of infinity and moving through Galileo's paradox, Dedekind infinite sets, and a range of alternatives including order infinite, discretely finite, Stöckel finite, and Tarski finite sets. He examines how these notions relate to one another and where they come apart, especially in the absence of the axiom of choice. Hamkins then turns to the numerical approach championed by Frege and Dedekind, showing how Dedekind's three axioms ground arithmetic and why this framework ultimately won out. The lecture concludes by revealing that expressing finiteness inherently requires second-order logic, explaining why the concept is far more delicate than it first appears.