From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2149 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Cauchy completeness of Cauchy reals Date: Wed, 5 Feb 2003 16:06:02 +0000 (GMT) Message-ID: References: <200302040047.QAA03714@coraki.Stanford.EDU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018449 2733 80.91.229.2 (29 Apr 2009 15:20:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:49 +0000 (UTC) To: CATEGORIES LIST Original-X-From: rrosebru@mta.ca Wed Feb 5 15:35:06 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 05 Feb 2003 15:35:06 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18gVI7-0007O9-00 for categories-list@mta.ca; Wed, 05 Feb 2003 15:33:03 -0400 In-Reply-To: <200302040047.QAA03714@coraki.Stanford.EDU> X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *18gS3m-00001q-00*TL.8jifFciY* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 46 Xref: news.gmane.org gmane.science.mathematics.categories:2149 Archived-At: On Mon, 3 Feb 2003, Vaughan Pratt wrote: > > 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). > No constructivist (of whatever stripe) would be happy talking about Q - (L U R), as in your second and third definitions, since he wouldn't want to assume that L and R were complemented as subsets of Q. Your first definition, if I understand it correctly, is equivalent to what most toposophers would call the MacNeille reals -- that is, the (Dedekind-MacNeille) order-completion of Q. If you "positivize" the second and third (which would appear to be equivalent, for any sensible notion of finiteness) by saying "Whenever p and q are rationals with p < q, then either p \in L or q \in R", you get the Dedekind reals, which are a proper subset of the MacNeille reals (though they coincide iff De Morgan's law holds) but have rather better algebraic properties. Peter Johnstone