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This Berkeley Seminar features Kevin Carlson exploring the philosophical and mathematical question "Does it matter whether there are infinite sets?" Delve into an exposition of JP Mayberry's ideas covering fundamental concepts about mathematical foundations, including why set theory serves as a proper foundation, how it relates to the axiomatic method in both naive Euclidean and modern structuralist senses, and why set theory and category theory can be viewed as complementary rather than opposing frameworks. Examine the axioms of set theory with particular focus on the non-obvious but fruitful axiom of infinity, and explore the possibilities of an anti-Cantorian set theory where every set is finite yet still capable of supporting modern mathematics. Discover why this perspective matters particularly for programmers working with inherently finite data structures and mathematicians dealing with "large" objects like the category of "all" sets, offering a clearer approach to handling such big objects than conventional treatments.
