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18min chapter
282 | Joel David Hamkins on Puzzles of Reality and Infinity
Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas
Exploring Axioms, Models, and Theorems in Mathematics
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.
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