From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2861 Path: news.gmane.org!not-for-mail From: wlawvere@buffalo.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Two topos questions Date: Thu, 03 Nov 2005 11:18:38 -0500 Message-ID: NNTP-Posting-Host: main.gmane.org Content-Type: text/plain Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241018948 6124 80.91.229.2 (29 Apr 2009 15:29:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Nov 4 07:25:42 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 04 Nov 2005 07:25:42 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXzYh-0001fW-9U for categories-list@mta.ca; Fri, 04 Nov 2005 07:16:35 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 31 Xref: news.gmane.org gmane.science.mathematics.categories:2861 Archived-At: What I was talking about 15 Jan 1997 was (not hoping for an axiom of infinity without assuming one, but) the fact that most of the mathematical uses of the rig N of natural numbers do not work in a topos, if one interprets that rig to mean the one characterized by Dedekind recursion. 1. starting with characteristic functions of subobjects, then adding and multiplying them for various combinatorial calculations 2. applying the least number principle 3. measuring the fiber dimension of a bundle of linear spaces all require the inf-completion of N, also known as the semicontinuous natural numbers. It contains the truth-value object omega and is contained in the semicontinuous reals (themselves indispensible for norming internal Banach spaces, and constructible simply as one-sided Dedekind cuts). Yet another way to picture these objects in the case of a Grothendieck topos E is to consider the sheaf of germs of continuous maps from E to the appropriate locale : the order topology (not the discrete one) on N, the order topology (not the interval topology) on nonnegative reals. Is any more known now as opposed to 9 years ago about the mathematical applications of finiteness to variable and cohesive sets ? The fact that K-finiteness is appropriate for some applications and that its theory resembles the classical theory for constant discrete sets should not distract us from the achievements of geometers in using coherence, Notherianness,etc., nor from the fact that our "logic" should serve to partly guide the learning of also those developments of thought.