Hello. I'm Joel David Hamkins, and welcome to this series of lectures on infinity.
I want to show you my favorite conundrums and paradoxes of infinity. We are going to see Zeno's paradox, the paradox of supertasks, and Galileo's paradox.
The distinction between the potentially infinite and the actually infinite is one of the oldest and most fundamental in the philosophy of mathematics. The potentially infinite refers to a process that goes on without end, a procedure that can always be continued one more step, but which at no point is completed. The actually infinite, by contrast, refers to a completed totality, a collection that is infinite all at once, as a finished whole.
Aristotle famously endorsed the potentially infinite while rejecting the actually infinite. For Aristotle, it made sense to say that the natural numbers are potentially infinite, you can always count one more, but it did not make sense to speak of the collection of all natural numbers as a completed, existing totality. That idea, he thought, was incoherent.
This Aristotelian position was the dominant view for centuries. Even many mathematicians who worked freely with infinite processes were reluctant to commit themselves to actual infinities as genuine mathematical objects. The resistance was not merely philosophical squeamishness; there were real conceptual puzzles about how a completed infinite collection could behave consistently.
What changed everything was Cantor. Georg Cantor developed a systematic, rigorous theory of actually infinite sets in the late nineteenth century, and in doing so he transformed the mathematical landscape entirely. Cantor insisted that we can treat infinite collections as completed wholes, that we can compare their sizes, and that, remarkably, not all infinities are the same size.
The distinction between potential and actual infinity remains philosophically alive today. A strict finitist rejects both. An intuitionist or constructivist may accept the potentially infinite while remaining suspicious of the actually infinite. And a classical mathematician working in the tradition of Cantor and Zermelo–Fraenkel set theory embraces the actually infinite without reservation, treating infinite sets as fully legitimate mathematical objects on a par with finite ones.
We will eventually arrive at the distinction between the countably infinite and the uncountably infinite, a profound idea due to Cantor.
We will start with Zeno and the classical ideas of Aristotle and Archimedes, before moving through Galileo and then on to Frege, Dedekind, and Cantor, and finally to the twentieth-century thinkers: Gödel, Tarski, Russell, and others, arriving at last at the contemporary ideas.
The study of infinity is, to my way of thinking, a case where there has been genuine progress. We have come to understand these incredibly confusing, mystifying ideas to a point where we can now say that we understand them far better than before. Concepts that were formerly bewildering are ones about which we now have real insight.
Please join me in these lectures and follow along.