Galileo's Paradox of Infinity — Lectures on Infinity (Lecture 6)

Lecture 6 of "Lectures on Infinity." Watch the full lecture series:    • Lectures on Infinity   Presented by Ergo, a nonprofit that publishes structured philosophical lectures online, free to anyone. https://ergo.org About this lecture How can a part be the same size as the whole? Joel David Hamkins walks through Galileo's famous paradox, which reveals a deep tension between two seemingly obvious principles: that sets matched one-to-one must be the same size, and that the whole must be greater than any proper part. Beginning with a simple dinner party analogy for one-to-one correspondence, Hamkins traces the history of this puzzle through Aristotle's wheel paradox, Bolzano's geometric observations about arcs and line segments, and Galileo's original 1638 dialogue about natural numbers and perfect squares. Along the way, he shows how finite line segments can be matched point-for-point with infinite lines, and how circles of different sizes contain equally many points. The lecture concludes with the modern resolution: the Cantor-Hume principle and the Cantor-Schröder-Bernstein theorem, which together provide a coherent framework for comparing the sizes of infinite sets. About this course In Lectures on Infinity, Joel David Hamkins guides viewers through the great paradoxes and discoveries about infinity, from Zeno's ancient puzzles to Cantor's revolutionary proof that some infinities are larger than others. The course traces the long debate between potential and actual infinity, exploring thought experiments like Hilbert's Hotel and Galileo's paradox. It concludes with Cantor's continuum hypothesis, a profound puzzle at the heart of set theory. No prior mathematical background is required, only curiosity. About 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. His research addresses the set-theoretic multiverse, potentialism, large cardinals, modal logic, infinite-time Turing machines, and the foundations of mathematics. He has also written 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. Chapters 00:00 The Puzzle of Infinite Set Sizes 00:24 A Dinner Party and One-to-One Correspondence 01:07 Defining Equinumerosity 02:29 The Cantor-Hume Principle Explained 04:02 The Paradox of Aristotle's Wheel 08:16 Hexagonal Wheels and Vanishing Gaps 12:21 Bolzano on Arcs, Circles, and Bijections 14:06 Matching Points on Unequal Line Segments 15:35 A Finite Segment Equals the Infinite Line 17:32 Natural Numbers vs. Perfect Squares 20:36 Reading the Original 1638 Dialogue 23:31 How Modern Math Resolves the Paradox 25:38 Comparing Sizes of Infinite Sets 28:48 The Cantor-Schröder-Bernstein Theorem