Re: Challenge from Harvey Friedman

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

Date: Mon, 26 Jan 1998 14:21:40 +0000 (GMT)
From: Ronnie Brown <r.brown@bangor.ac.uk>


How does Harvey Friedman know that the formulation of real analysis as 
carried out in set theory will do all that real analysis *should* do? 

My favourite example is that of partial functions. Most teachers of real 
analysis (calculus) rightly impress on students the importance of the domain 
of a function, 
and the domain of f+g, etc. So a student might think that the algebra and 
analysis of partial functions would occupy a good part of the literature. 
Solutions of ODEs (such as dy/dx=exp(-y) ) are often given by partial 
functions 
with domain involving a parameter, and the solution (including its 
domain) seems to vary smoothly with this parameter. In fact there is very 
little in the literature on such matters. I had a small go with 
29. (with A.M. ABD-ALLAH), ``A compact-open topology on partial maps with 
open domain'', {\em J. London Math Soc.} (2) 21 (1980) 480-486. 

It is not clear that the most general case of 
the functional analysis of partial functions with domain neither open nor 
closed can be successfully handled within classical set theory. There is 
a chance it can be handled within topos theory. (Try functions defined on 
the leaves of foliations. Any answers?) 

Another point of topos theory is to handle categories such as that of 
directed graphs in a similar manner to the category of sets, and to make 
comparisons between such categories. (Bill Lawvere has of course written 
a lot on this.) 

Ronnie Brown