Young Researchers in
University of Edinburgh,
19 June 2013.
Abstract At the heart of mathematical culture there is a paradox. Mathematicians use set-theoretic language all the time, and we are informed that the classical axioms of set theory, ZFC, are the "foundation of mathematics". Yet most of us sail through life neither knowing nor much caring what the ZFC axioms are; and if we do stop to look at them, they seem curiously remote from what we as mathematicians actually do.
I will present a radical solution, due to Lawvere. His axioms are 10 totally mundane properties of sets, used every day by ordinary mathematicians. They are of comparable strength to ZFC. So we can, if we wish, forget about ZFC entirely and adopt these as our axioms instead.
Slides In this pdf file.
References Reprint of Lawvere's paper on which this talk is based
My own paper corresponding to this talk (only eight pages, but with more detail than the talk)
Blog post and resulting discussion.