Re: Challenge from Harvey Friedman

[categories <cat-dist@mta.ca>]

Date: Fri, 23 Jan 1998 21:22:39 -0500 (EST)
From: Michael Barr <barr@triples.math.mcgill.ca>

I guess simplicity is in the eye of the beholder.  For example, I do not
consider the categorical version of either choice (epis split) or
well-pointed (1 is a generator) to be translations of set theory, but
perfectly natural categorical axioms.  The point is that Harvey is a
set-theorist, so he thinks comprehension and all that stuff (which at
least 95% of all mathematicians could not state properly if their lives
depended on it) is perfectly natural and I don't.  

But my actual criticism of ZF(C) is much simpler.  I have taught these
courses in set theory and we spend a lot of time developing these epsilon
trees and then totally ignore the structure.  In other words, the epsilon
tree structure of sets is totally irrelevant to what you do with them.
There are a number of definitions of pairs, but they are irrelevant.  The
only thing we need to know about pairs (and the only thing a categoriest
does know) is when two pairs are equal.  All the defintions of pairs have
that property of course, but they also have irrelevant properties.

Mike