(July 11, 2019 at 9:30 am)Jehanne Wrote:(July 11, 2019 at 2:52 am)polymath257 Wrote: Well, I wouldn't expect to see much about finitism in such textbooks. I have, however, seen good discussions in introductory books on axiomatic set theory and the issues come up extensively in proof theory. Kunen's book on the Foundations of Mathemtics talks quite a bit about finitism, formalism, and Platonism in the context of introductory set theory. The point is that the meta-theory used to discuss proofs tends to be finitistic by nature.
Fact is that most professional mathematicians accept ZFC and the proofs of Cantor, which was the whole point of my OP. I don't think that it is an either/or proposition -- one can accept ZFC while having intelligent conversations about finitism; without having to reject the former while working within the confines of the latter.
Philosophically, I am a formalist. You tell me which axioms to work with, and I will work with them.
I *prefer* a set of axioms that has classes and a single choice function for all sets, but that's just me. I can live with ZFC.
That said, working in the hereditarily finite sets has advantages in some contexts.