From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2633 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: arithmetical and geometric reals in (models of) SDG Date: Wed, 07 Apr 2004 15:00:23 +0200 Message-ID: <834qrw0wpk.fsf@haka.fmf.uni-lj.si> References: <200403261407.PAA29318@fb04209.mathematik.tu-darmstadt.de> Reply-To: Andrej Bauer NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018792 5075 80.91.229.2 (29 Apr 2009 15:26:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:32 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Apr 7 21:31:41 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Apr 2004 21:31:41 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BBNQ7-00012z-00 for categories-list@mta.ca; Wed, 07 Apr 2004 21:29:27 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 24 Xref: news.gmane.org gmane.science.mathematics.categories:2633 Archived-At: Thomas Streicher writes: > > Recently I was asking myself what is the relation between the arithmetical > (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms > (i.e. at least the Kock-Lawvere) axiom. Perhaps it is worth mentioning that in the ring of smooth reals R the sequence a_k = 2^(-k) is Cauchy but has many "limits" because every infinitesimal dx satisfies the condition "dx is the limit of a_k". This shows that R is not Cauchy complete, not because limits of Cauchy sequences are missing but because there are too many. I once thought the above observation implied there can be no isometric embedding of a Cauchy-complete field (e.g. the Dedekind reals) into R, but now I am not convinced anymore. Andrej Bauer Department of Mathematics and Physics University of Ljubljana http://andrej.com