Does it matter whether there are infinite sets?

Abstract

This talk is mostly an exposition of a bit of philosophy and a bit of math due to JP Mayberry, included by but not necessarily co-limited to (1) the claim that yes, Virginia, you actually do want a foundation (2) that it’s set theory (3) that this has to be given in the naive Euclid-style sense of the axiomatic method (4) that what this foundation founds is, mainly, the modern structuralist sense of the axiomatic method (so that set theory and category theory are friends after all!) (5) that you’re supposed to actually believe the axioms in a traditional Euclid-style axiomatic system (6) that, actually, it’s not hard to give an explanation of set theory that leads to you actually believing all the axioms (7) EXCEPT the so-called axiom of infinity, which is profoundly non-obvious (8) but highly fruitful–so could we really get away without it? (9) a beginning of an anti-Cantorian set theory (so every set is finite) in which nonetheless you seem to have a good chance at doing modern math.
Well…Who cares? I suggest that you might care if you are (a) someone who programs, no doubt having noted that your data structures are actually always finite (b) someone who deals with large objects such as the category of “all” sets in your math. Mayberry’s anti-Cantorian set theory has a clearer treatment of how we ought to correctly approach “big” objects than any other treatment I know.