From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2052 Path: news.gmane.org!not-for-mail From: S Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Two constructivity questions Date: Sun, 09 Dec 2001 10:35:26 +0000 Message-ID: <3.0.5.32.20011209103526.00854aa0@TESLA.open.ac.uk> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241018370 2210 80.91.229.2 (29 Apr 2009 15:19:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:19:30 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Dec 9 19:23:45 2001 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Dec 2001 19:23:45 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 3.33 #2) id 16DD2s-0003xF-00 for categories-list@mta.ca; Sun, 09 Dec 2001 19:07:42 -0400 X-Sender: sjv22@TESLA.open.ac.uk X-Mailer: QUALCOMM Windows Eudora Light Version 3.0.5 (32) In-Reply-To: Original-References: Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 25 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:2052 Archived-At: As before, let S be the Stone locale of square roots of the generic complex number. The question is, In what sense can S be considered finite? Here is one idea that occurs to me. If a set is acted on transitively by a finite group, then classically it must be finite (and I dare say some constructive statement of this is also true). S is acted on by the discrete group {+1, -1} (by multiplication in C). Hence if that action can be considered transitive in some way, that would be a finiteness property of S (or, rather, finiteness _structure_ on S). If a: S x {+1, -1} -> S is the action, then I believe I can prove (by techniques involving the upper powerlocale) that : S x {+1, -1} -> S x S is a proper surjection. This would seem to be a natural way to capture transitivity of a and hence a finiteness property of S. More generally, if an action on a locale by a finite group has only finitely many orbits (using the above idea to specify transitivity on the orbits), then that would be a finiteness property of the locale. One might ask whether, by Galois theory, this can be applied to arbitrary polynomials over C. Steve Vickers.