From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4103 Path: news.gmane.org!not-for-mail From: Dana Scott Newsgroups: gmane.science.mathematics.categories Subject: Re: countable Heyting algebras Date: Mon, 26 Nov 2007 10:09:27 -0800 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v915) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019721 11722 80.91.229.2 (29 Apr 2009 15:42:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:42:01 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Nov 26 17:11:26 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Nov 2007 17:11:26 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iwl99-0006xa-3e for categories-list@mta.ca; Mon, 26 Nov 2007 17:05:52 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 58 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:4103 Archived-At: Woke up in the middle of the night and realized I said something obviously wrong. (Not the first time, and won't be the last, I'm afraid.) On Nov 25, 2007, at 10:03 PM, Dana Scott wrote: > Now, F can also be considered as the clopens of the Cantor space > 2^N. The cHa of all opens is fully non-atomic, but it is uncountable. > Call it C for Cantor. The not-not stable elements of C are the so- > called regular open sets. They form an uncountable cBa which is the > completion of F. But we want to ask if there are there interesting > countable subalgebras of C? We note that elements of C are determined > by the elements of F they contain. (In fact, the cHa C is isomorphic > to the lattice of ideals of F.) Well, it is true that the stable elements of C form the completion of F. And, every element of C is a sup of elements of F, so C is atomless in the sense of having no minimal non-zero elements. And, the stables of C form an atomless cBa. BUT -- and here is my oversight -- C does have gaps, and so the cHa is NOT fully non-atomic. Think of the Cantor set as a subspace T of the unit interval. There is a blank from 1/3 to 2/3, if we make the construction via the middle-third process. This means that [0,1/3) meet T is open in T, but [0,1/3] meet T = [0,2/3) meet T is both open and closed. This gives a gap between two opens in the cHa C. So C is not fully gapless. This gap also exists in the subalgebra A of arithmetically definable opens. Aarrgghh. Is there a fix? Can we take a quotient in the category of Ha's that closes the gaps? Maybe, and maybe not. In countable Ba's, dividing by the ideal generated by the atoms can result in a quotient algebra that still has atoms. Aarrgghh! I will have to think further. Rats!