Exploring the Intricacies of Mathematical Systems and Philosophical Implications

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The chapter delves into the complexities of systems of axioms and consistency in mathematics, touching on topics like the continuum hypothesis, number theory, Gödel's incompleteness theorem, set theory, and large cardinal axioms. It also explores the concepts of mathematical realism, consistency, transfinite hierarchies, potentialism, infinity, and the interaction between mathematics, physics, and philosophy. The conversation showcases the intricate relationship between mathematics and philosophy, emphasizing the importance of both disciplines to advance in the field.

The philosophy of mathematics would be so much easier if it weren't for infinity. The concept seems natural, but taking it seriously opens the door to counterintuitive results. As mathematician and philosopher Joel David Hamkins says in this conversation, when we say that the natural numbers are "0, 1, 2, 3, and so on," that "and so on" is hopelessly vague. We talk about different ways to think about the puzzles of infinity, how they might be resolved, and implications for mathematical realism.

Blog post with transcript: https://www.preposterousuniverse.com/podcast/2024/07/15/282-joel-david-hamkins-on-puzzles-of-reality-and-infinity/

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Joel David Hamkins received his Ph.D. in mathematics from the University of California, Berkeley. He is currently the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He is a pioneer of the idea of the set theory multiverse. He is the top-rated user by reputation score on MathOverflow. He is currently working on The Book of Infinity, to be published by MIT Press.

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