Supertasks: Doing Infinitely Many Things — Lectures on Infinity (Lecture 2)

Lecture 2 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 What happens when you complete infinitely many steps? Joel David Hamkins walks through a series of supertasks that challenge ordinary intuition about infinity. He begins with Thompson's Lamp, a puzzle about toggling a light switch infinitely many times, then connects it to Zeno's paradox to show that supertasks may not be as exotic as they seem. The lecture builds toward deeper puzzles: a deal with the devil, balls placed into and removed from a sack where the final count defies expectation, and a stochastic version where probability-zero events become surprisingly relevant. Hamkins carefully distinguishes between what holds for every individual ball and what holds for all balls collectively, exposing a subtle logical gap. The lecture culminates in the chocolatier's game, where questions about finite versus infinite servings, memory, and the axiom of choice determine whether a glutton can guarantee tasting every flavor. Throughout, Hamkins shows how rigorous thinking about infinity upends common sense. 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 What Are Supertasks? 00:10 Thompson's Lamp: On or Off After Infinity? 04:00 Zeno's Paradox Makes Supertasks Real 05:03 Trivial Supertasks Hide in Plain Sight 06:22 The Deal with the Devil 12:03 Balls in a Sack: Empty After Infinity? 16:04 Controlling What Remains in the Sack 17:01 Random Removal and Vanishing Probability 21:18 Probability Zero Does Not Mean Impossible 23:40 Every Ball vs. All Balls: A Subtle Gap 25:00 Dartboards and Countable Additivity 30:12 The Chocolatier's Game: Finite Servings 35:31 Infinite Servings and the Zigzag Strategy 39:32 Can the Glutton Win Without Memory? 41:56 Countable vs. Uncountable Creativity 45:35 Wrapping Up Supertasks