From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2133 Path: news.gmane.org!not-for-mail From: Alex Simpson Newsgroups: gmane.science.mathematics.categories Subject: Re: Cauchy completeness of Cauchy reals Date: Mon, 27 Jan 2003 03:57:45 +0000 (GMT) Message-ID: <1043639865.3e34ae39d965f@mail.inf.ed.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1241018434 2645 80.91.229.2 (29 Apr 2009 15:20:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:34 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Jan 27 12:33:44 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 27 Jan 2003 12:33:44 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18dC75-0000g0-00 for categories-list@mta.ca; Mon, 27 Jan 2003 12:27:59 -0400 X-Authentication-Warning: topper.inf.ed.ac.uk: apache set sender to als@localhost using -f In-Reply-To: User-Agent: IMP/PHP IMAP webmail program 2.2.8 X-Originating-IP: 130.54.16.90 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 58 Original-Lines: 26 Xref: news.gmane.org gmane.science.mathematics.categories:2133 Archived-At: Peter Johnstone writes: > I'm sorry, but this won't do. In a topos, equality is equality; you > can't > treat it "lazily". So a Cauchy real has to be an equivalence class of > Cauchy sequences, and there is in general no way of choosing a > canonical representative for it. Markov's principle would, I think > (I haven't checked the details), suffice to give a canonical > representation as a binary expansion with no infinite sequence of > 1's, In fact not. Markov's principle holds in the effective topos, and there, unless I'm much mistaken, it is not even true that the map from binary representations to Cauchy (= Dedekind) reals in [0,1] is epi, let alone split epi. Alex Simpson Alex Simpson, LFCS, Division of Informatics, Univ. of Edinburgh Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://www.dcs.ed.ac.uk/home/als Fax: +44 (0)131 667 7209