From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4098 Path: news.gmane.org!not-for-mail From: Dana Scott Newsgroups: gmane.science.mathematics.categories Subject: Re: when are Lindenbaum-Tarski algebras complete? Date: Wed, 21 Nov 2007 13:30:37 -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; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019718 11694 80.91.229.2 (29 Apr 2009 15:41:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:58 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Nov 21 19:39:35 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Nov 2007 19:39:35 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iuz4v-0005wI-Jw for categories-list@mta.ca; Wed, 21 Nov 2007 19:33:57 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 52 Original-Lines: 37 Xref: news.gmane.org gmane.science.mathematics.categories:4098 Archived-At: On Nov 21, 2007, at 6:33 AM, Thomas Streicher wrote: > I also suspect that Sub(1) of the free topos with nno is not complete. > But countability does not suffice for refuting completeness (the > ordinal > \omega + 1 is an infinite countable cHa which nevertheless is > complete). >> > From Goedel's Theorem for HAH (higher order intuit. arithmetic) it > follows > that Sub(1) of the free topos with nno is not atomic. But that also > doesn't > suffice for refuting completeness. > > On p.169 of Freyd, Friedman and Scedrov's paper > "Lindenbaum algebras of intuitionistic theories and free > categories" (APAL 35) > they claim "Lindenbaum algebras are almost never complete" but don't > give > a proof. Ah, but if a Heyting algebra is complete, then so is the Boolean algebra of all not-not-stable elements. Familiar example: the regular open subsets of a topological space form a complete Boolean algebra. As remarked, the Sub(1) of the free topos with nno is not atomic, and with reference again to Godel's theorem via the not-not translation, the Boolean algebra of not-not-stable elements is also non atomic. But all countable, non-atomic Boolean algebras are isomorphic to the clopen subsets of the Cantor space (or the Lindenbaum algebra of classical propositional calculus, or the free Boolean algebra on countably many generators). That algebra is not complete -- as can be seen in many ways. Q.E.D.