From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/618 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Challenge from Harvey Friedman Date: Mon, 26 Jan 1998 15:01:05 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017075 26533 80.91.229.2 (29 Apr 2009 14:57:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:55 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Jan 26 15:01:07 1998 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id PAA26123; Mon, 26 Jan 1998 15:01:06 -0400 (AST) Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:618 Archived-At: Date: Mon, 26 Jan 1998 14:21:40 +0000 (GMT) From: Ronnie Brown 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