Set-theoretic foundations
Penelope Maddy · Contemporary mathematics - American Mathematical Society · 2017
Set-theoretic Foundations 1It's more or less standard orthodoxy these days that set theory --ZFC, extended by large cardinals --provides a foundation for classical mathematics.Oddly enough, it's less clear what 'providing a foundation' comes to.Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can't do it at all, and that category theory can.There are also those insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called 'multiverse' of set-theoretic universes --while retaining its foundational role.I won't pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1 It's an honor to be included in this 60 th birthday tribute to Hugh Woodin, who's done so much to further, and often enough to re-orient, research on the fundamentals of contemporary set theory.I'm grateful to the organizers for this opportunity, and especially, to Professor Woodin for his many contributions.