From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2051 Path: news.gmane.org!not-for-mail From: "Dr. P.T. Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Constructive finiteness Date: Sun, 9 Dec 2001 15:20:47 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018369 2208 80.91.229.2 (29 Apr 2009 15:19:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:19:29 +0000 (UTC) To: Categories mailing list Original-X-From: rrosebru@mta.ca Sun Dec 9 19:23:42 2001 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 09 Dec 2001 19:23:42 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 3.33 #2) id 16DD3b-0001Yu-00 for categories-list@mta.ca; Sun, 09 Dec 2001 19:08:27 -0400 X-Scanner: exiscan *16D5l3-0003yp-00*2PWaFQC3HiI* http://duncanthrax.net/exiscan/ Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 21 Xref: news.gmane.org gmane.science.mathematics.categories:2051 Archived-At: It is not constructively true, as I conjectured yesterday, that the set of roots of a polynomial over C is Russell-finite. Let X be the subspace of C consisting of 0 and all points whose argument is a rational multiple of \pi, and consider the sheaf of solutions of z^2 - f = 0, where f: X --> C is the inclusion map. It is easy to see that the stalk of this sheaf at 0 is uncountably infinite. Since R-finiteness is preserved by inverse image functors, this yields a counterexample. This doesn't, of course, answer Steve Vickers' original question whether there is a sense in which the *locale* of roots of a polynomial can be said to be finite. But it does indicate that the appropriate notion of finiteness, if it exists, must be a rather delicate one. Peter Johnstone