From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2112 Path: news.gmane.org!not-for-mail From: "Mamuka Jibladze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Cauchy completeness of Cauchy reals Date: Thu, 23 Jan 2003 13:50:00 +0400 Message-ID: <000b01c2c2c4$eba605e0$b1e493d9@alg1> References: <15918.28389.594290.761117@acws-0054.cs.bham.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018419 2541 80.91.229.2 (29 Apr 2009 15:20:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:19 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Thu Jan 23 13:27:04 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Jan 2003 13:27:04 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18bl4L-0004C9-00 for categories-list@mta.ca; Thu, 23 Jan 2003 13:23:13 -0400 X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4807.1700 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4807.1700 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 37 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:2112 Archived-At: > 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 Does one get any known versions of reals by performing the Cauchy or Dedekind construction starting with initial algebras I for non-decidable lifts L instead of the NNO? It would be then also natural to interpret Cauchy sequences and completeness using appropriate I-indexed families, of course. Even for "integers" Z one has at least three different options: taking I^op+1+I, taking the colimit of I->LI->LLI->..., each map being the unit, or taking the colimit of the corresponding I-indexed diagram. It would be strange if these turn out to be isomorphic. Is any of them an initial algebra for some simple functor? Similarly there are various possibilities for rationals - taking fractions, i.e. a quotient of ZxZ, or colimit of all multiplication maps Z->Z, or of the corresponding Z-indexed diagram. Actually I have not followed the ongoing research for a while, so maybe my questions are outdated. I would be grateful for any references to related work. Mamuka Jibladze