Exploring Axioms, Models, and Theorems in Mathematics

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The chapter delves into the complexities of mathematics, exploring the addition of axioms, different models within theories like geometry and arithmetic, and various theorems such as Dedekin's Categoricity Theorem and Gödel's incompleteness theorems. The conversation challenges the idea of a unique intended model of arithmetic, discusses the halting problem and computability theory with Turing machines, and explores the concept of well-formed statements in a system that are true but unprovable if the system is consistent. The chapter concludes with discussions on the density of provable versus unprovable true statements and the undecidability of the halting problem, showcasing the intricacies and challenges within mathematical theories.

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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