From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2147 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Cauchy completeness of Cauchy reals Date: Mon, 03 Feb 2003 16:47:58 -0800 Message-ID: <200302040047.QAA03714@coraki.Stanford.EDU> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018448 2730 80.91.229.2 (29 Apr 2009 15:20:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:48 +0000 (UTC) To: CATEGORIES LIST Original-X-From: rrosebru@mta.ca Wed Feb 5 10:49:48 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Feb 2003 10:49:48 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18gQko-0005Fp-00 for categories-list@mta.ca; Wed, 05 Feb 2003 10:42:22 -0400 X-Mailer: exmh version 2.1.1 10/15/1999 In-Reply-To: Message from Vaughan Pratt of "Wed, 22 Jan 2003 22:29:51 PST." Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 2 Original-Lines: 31 Xref: news.gmane.org gmane.science.mathematics.categories:2147 Archived-At: While I'm comfortable with coalgebraic presentations of the continuum, as well as such algebraic presentations as the field P/I (P being a ring of certain polynomials, I the ideal of P generated by 1-2x) that I mentioned a week or so ago, I'm afraid I'm no judge of constructive approaches to formulating Dedekind cuts. Would a toposopher (or a constructivist of any other stripe) view the following variants as all more or less equally constructive, for example? 1. Define a (closed) interval in the reals as a disjoint pair (L,R) consisting of an order ideal L and an order filter R, both in the rationals standardly ordered, both lacking endpoints. Order intervals by pairwise inclusion: (L,R) <= (L',R') when L is a subset of L' and R is a subset of R'. Define the reals to be the maximal elements in this order. Define an irrational to be a real for which (L,R) partitions Q. 2. Ditto but with the reals defined instead to be intervals for which Q - (L U R) is a finite set. ("Finite set" rather than just "finite" to avoid the other meaning of "finite interval." The order plays no role in this definition, maximality of reals in the order being instead a theorem.) 3. As for 2 but with "finite" replaced by "cardinality at most 1". The predicate "rational" is identified with the cardinality of Q - (L U R). Vaughan