From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2111 Path: news.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: Cauchy completeness of Cauchy reals Date: Wed, 22 Jan 2003 10:13:57 +0000 Message-ID: <15918.28389.594290.761117@acws-0054.cs.bham.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII X-Trace: ger.gmane.org 1241018418 2536 80.91.229.2 (29 Apr 2009 15:20:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:18 +0000 (UTC) To: CATEGORIES LIST Original-X-From: rrosebru@mta.ca Wed Jan 22 13:56:02 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 22 Jan 2003 13:56:02 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18bP24-0006uH-00 for categories-list@mta.ca; Wed, 22 Jan 2003 13:51:24 -0400 In-Reply-To: X-Mailer: VM 6.89 under 21.1 (patch 14) "Cuyahoga Valley" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 36 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:2111 Archived-At: Andrej Bauer writes: > find a topos in which Cauchy reals > are not Cauchy complete (i.e., not every Cauchy sequence of reals has > a limit). For extra credit, make it so that the Cauchy completion of > Cauchy reals is strictly smaller than the Dedekind reals. One small clarification: Regarding the extra credit, there doesn't seem to be a reasonable "absolute" way of defining the completion in question. E.g. the various, constructively different, notions of metric completion already assume the existence of some given complete reals. However, one can always embed the Cauchy reals into the Dedekind reals, which are always Cauchy complete, and then look at the smallest "subset" containing this which is closed under limits of Cauchy sequences (where Cauchy sequences are defined as in Andrej's message). We sometimes call this, somewhat verbosely, "the Cauchy completion of the Cauchy reals within the Dedekind reals". But notice that this is the same as what one gets starting from the rationals within the Dedekind reals and then closing under limits and hence could be called the "Cauchy completion of the rationals within the Dedekind reals". NB. Freyd characterized the Dedekind reals as a final coalgebra. Alex Simpson and I characterized "the Cauchy completion of the rationals within the Dedekind reals" as a free algebra (to be precise, we started from the algebras as a primitive notion and later found this construction of the free one). But this has already been discussed in postings in the past few years. Martin Escardo