Infinite Sets and Foundations (Joel David Hamkins) | Ep. 17

Joel David Hamkins is a Professor of Logic with appointments in Philosophy and Mathematics at Oxford University. His main interest is in set theory. We discuss the field of set theory: what it can say about infinite sets and which issues are unresolved, and the relation of set theory to philosophical issues concerning the foundations of mathematics. Joel's website: http://jdh.hamkins.org Joel's Youtube channel:    / @joeldavidhamkins5484   Joel's post summarizing ongoing work in set theory: https://math.stackexchange.com/a/2556... Joel's recent books: Proof and the Art of Mathematics https://amzn.to/3DsjmFR Lectures on the Philosophy of Mathematics https://amzn.to/3DlpPCJ (these are my affiliate links) Timestamps 0:00 Intro 2:11 Joel's background. Interaction between math and philosophy 9:04 Joel's work; infinite chess. 14:45 Infinite ordinals 22:27 The Cantor-Bendixson process 29:41 Uncountable ordinals 32:10 First order vs. second order theories 41:16 Non-standard analysis 46:57 The ZFC axioms and well-ordering of the reals 58:11 Showing independence of statements. Models and forcing. 1:04:38 Sets, classes, and categories 1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable? 1:30:20 What does set theory look like if certain axioms are rejected? 1:36:06 How to judge philosophical positions about math 1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials. 1:48:48 Goodstein sequences and the use of infinite ordinals 1:58:43 The state of set theory today 2:01:41 Joel's recent books Check out my discussion with Norman Wildberger on whether to admit infinite sets to which we refer several times:    • Math Debate: Real numbers and the infinite...   Playlist of full episodes of this podcast:    • Daniel Rubin Show, Full episodes   Also check out my Tricky Parts of Calculus series of lectures:    • Tricky Parts of Calculus