From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/73 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: "Kantor dust" Date: Wed, 11 Feb 2009 09:53:44 -0800 Message-ID: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1234405417 10641 80.91.229.12 (12 Feb 2009 02:23:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Feb 2009 02:23:37 +0000 (UTC) To: categories list Original-X-From: categories@mta.ca Thu Feb 12 03:24:53 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LXRG1-0008EA-55 for gsmc-categories@m.gmane.org; Thu, 12 Feb 2009 03:24:53 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LXQiR-0005ZQ-Ux for categories-list@mta.ca; Wed, 11 Feb 2009 21:50:11 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:73 Archived-At: Vaughan Pratt wrote: > Similar reasoning should also rehabilitate the constructivity of binary > fractions, where the final coalgebra surely deletes the open set > separating 0111... from 1000... Perhaps ASD has something to say about > this---Paul? Unfortunately the reasoning is not similar enough to make this work. The crucial difference is that in the representation of [0,oo) ~ [0,1)[1,2)[2,3)... as N^N, there is no largest element of [0,1) whence the Scott topology omits the open set separating [0,1) from [1,0). With binary fractions however we have two points 0111... and 1000... with nothing between them and the inequality 0111... < 1000..., which the Scott topology must respect by preserving the open set separating them. There is (as far as I'm aware) no such localic alternative of the kind I was envisaging to either omitting 0111... or identifying it with 1000..., both of which are intrinsically spatial solutions to the problem of converting Cantor space 2^N to the continuum. In contrast the Alexandroff-to-Scott conversion of N^N to the continuum is localic in character, in that it operates on open sets instead of points. The crucial difference between classical and sheaf-theoretic toposes is that localic procedures are meaningless in the former. The latter permit the finer distinctions to be drawn that are needed to explicate constructivity, starting with the distinction between not-not and identity. (Unlike some other buggy posts of mine that I've been able to retract before Bob forwarded them to the list, this one went through too promptly for me to catch it in time. I made this mistake by incorrectly visualizing the gap between 0111... and 1000... as though it were the gap between .0 < .01 < .011 < .0111 < ... and ... < .1000 < .100 < .10 < .1 This makes no sense (a) because the finite sequences on the right should all be identified and (b) there are no finite sequences to begin with, the binary fractions are properly understood as infinite sequences, namely maps N --> 2. The paragraph was an afterthought I tacked on with insufficient consideration before posting.) Vaughan Pratt