Lectures on Infinity
Joel David Hamkins opens this course with an intriguing observation: infinity has bewildered thinkers for millennia, yet we have made genuine progress in understanding it. Beginning with Zeno's ancient paradoxes about motion and the impossibility of traversing infinitely many points, Hamkins shows how these puzzles force us to confront what it means to complete an infinite task. The mathematics of infinite sums provides one resolution, but deeper questions remain about whether infinity can ever be grasped as a completed whole.
The course traces the long debate between two conceptions of infinity. For Aristotle and most mathematicians until the nineteenth century, infinity was merely potential: you could always count one more, but never possess all the numbers at once. Hamkins explores how this view shaped ancient geometry and even Archimedes' calculation of areas, before turning to the revolutionary work of Georg Cantor, who insisted that infinite collections could be treated as completed totalities.
Through vivid thought experiments like Hilbert's Hotel, Hamkins reveals the strange arithmetic of the infinite. A hotel with infinitely many rooms can accommodate infinitely many new guests even when completely full. The natural numbers, the integers, and even the rational numbers all turn out to be the same size of infinity: they are countable. But Cantor's diagonal argument presents a different case: the real numbers cannot be counted. There are genuinely different sizes of infinity.
Along the way, Hamkins examines Galileo's paradox about squares and whole numbers, the impossibility of giants according to scaling laws, and the peculiar behavior of supertasks where infinitely many actions occur in finite time. These explorations illuminate how infinity defies ordinary intuition while remaining mathematically coherent.
The course concludes with one of mathematics' great unsolved mysteries: Cantor's continuum hypothesis, which asks whether any infinity lies between the countable infinity and the continuum. Hamkins explains how this question proved undecidable within standard mathematics, leaving a profound puzzle at the heart of set theory. No prior mathematical background is required, only curiosity about one of humanity's most fascinating ideas.
You can watch the lectures below, browse the chapter index, or watch on YouTube.
Joel David Hamkins
Joel David Hamkins is the John Cardinal O’Hara Professor of Logic at the University of Notre Dame. He previously held positions at the University of Oxford, where he was Professor of Logic and Sir Peter Strawson Fellow at University College, and at the City University of New York Graduate Center. His work spans mathematical logic, set theory, philosophy of mathematics, computability theory, and infinite games.
He is known for influential work on the set-theoretic multiverse, potentialism, large cardinals, modal logic, and infinite-time Turing machines. His research explores the nature of mathematical truth, infinity, and the foundations of mathematics, often bringing together technical set theory and philosophical questions about mathematical reality. He has also written extensively on infinite chess, computability, and the philosophy of the infinite.
He is the author of The Book of Infinity, Lectures on the Philosophy of Mathematics, Proof and the Art of Mathematics, and Proof and the Art of Mathematics: Examples and Extensions. He writes on mathematics and philosophy at infinitelymore.xyz.
