From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2249 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Kripke semantics and Alexandrov topology Date: Thu, 24 Apr 2003 15:54:04 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241018525 3241 80.91.229.2 (29 Apr 2009 15:22:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Apr 26 15:19:30 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 26 Apr 2003 15:19:30 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 199UDU-0002kT-00 for categories-list@mta.ca; Sat, 26 Apr 2003 15:16:04 -0300 X-Mailer: exmh version 2.1.1 10/15/1999 Original-Mime-Version: 1.0 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 20 Original-Lines: 47 Xref: news.gmane.org gmane.science.mathematics.categories:2249 Archived-At: YES, because it doesn't matter. Bill's question is not well framed (besides the potential for ambiguity in the answer arising from "Am I wrong to say that"). Is the question is about intuitionistic logic/Heyting algebra itself, or its domain of discourse, ostensibly posetal Kripke structures? If the former, then the canonical models obtained by dualizing Heyting algebras are Stone(-Priestley(-Heyting)) spaces, whose topology is compact, the logic being finitary. If the latter, and if Stone topology is more bother than discrete topology (surely the case from a pedagogical standpoint), then the Alexandrov topology is preferable; this in general is not compact. But variations between topologies having the same specialization order describe only what points are approached in the (infinite) limit. Since intuitionistic logic is entirely finitary and does not concern itself with limiting processes, it doesn't matter what topology Kripke semantics is based on. Hence YES, Bill is wrong to imply that it does matter (other than for the secondary reasons discussed above). Best, Vaughan >YES, because, given a poset X, the category of sheaves over X equipped = >with the Alexandrov topology is equivalent to the category of = >*covariant* functors X --> Set. A very important, basic fact, which = >is at the foundation of a lot of things in topos theory and computer = science. > >Cheers, >Fran=E7ois >> >> I have been re-reading the chapter on intuitionism >> in Goldblatt's book, specifically the section on >> Kripke semantics. Am I wrong to say that Kripke >> semantics is based on the Alexandrov topology >> generated by the underlying poset? >> >> Regards, Bill