From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/70 Path: news.gmane.org!not-for-mail From: Toby Kenney Newsgroups: gmane.science.mathematics.categories Subject: Re: "Kantor dust" Date: Wed, 11 Feb 2009 11:55:49 -0400 (AST) Message-ID: Reply-To: Toby Kenney NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1234405182 10105 80.91.229.12 (12 Feb 2009 02:19:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 12 Feb 2009 02:19:42 +0000 (UTC) To: Vaughan Pratt , Original-X-From: categories@mta.ca Thu Feb 12 03:20:57 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 1LXRB8-0006gF-U2 for gsmc-categories@m.gmane.org; Thu, 12 Feb 2009 03:19:51 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LXQWS-0004pf-Tj for categories-list@mta.ca; Wed, 11 Feb 2009 21:37:48 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:70 Archived-At: On Tue, 10 Feb 2009, Vaughan Pratt wrote: > As a (surely constructive!) witness to the surjectivity, indeed > bijectivity, of R, define the inverse S: [0,oo) --> N^N of R as > follows. (R converts sequences to Reals, which S turns back into > Sequences.) > > S(x)(0) = floor(x) > S(x)(n+1) = S(g(x mod 1))(n) > > where x mod 1 = x - floor(x) and g(x) = x/(1-x) : [0,1) --> [0,oo) is > the inverse of f(x) = x/(1+x) : [0,oo) --> [0,1) used in the definition > of R. > The trouble with this is that the floor function isn't constructive - the question "is x<2" is undecidable in the reals, but decidable in the natural numbers. The problem with the obvious definition: "Take the set of natural numbers below x, and take the join of this set." is that the natural numbers don't have K~-finite joins, only K-finite ones. Incidentally, has anyone looked at semilattices with K~-finite joins? (Or whatever your favourite notion of finiteness is.) Is there any use for something like the completion of N under K~-finite joins, other than allowing us to define the floor function? Toby