From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6719 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: ordinal dependent choice Date: Tue, 28 Jun 2011 14:12:22 +0100 Message-ID: Reply-To: Paul Levy NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1309292619 31469 80.91.229.12 (28 Jun 2011 20:23:39 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 28 Jun 2011 20:23:39 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Tue Jun 28 22:23:35 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qbeol-0004tE-Iy for gsmc-categories@m.gmane.org; Tue, 28 Jun 2011 22:23:31 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:34198) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QbemH-00049a-TJ; Tue, 28 Jun 2011 17:20:57 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QbemH-0007q1-45 for categories-list@mlist.mta.ca; Tue, 28 Jun 2011 17:20:57 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6719 Archived-At: Dear all, Let alpha be an ordinal. Let $alpha be the totally ordered set of ordinals below alpha. "Alpha-dependent choice" is the following statement: for any functor A : $alpha ^ op ---> Set, if A_i is nonempty for all i < alpha, and A_i,j : A_j ---> A_i is surjective for all i <= j < alpha, then the limit of A is nonempty. If alpha has a cofinal omega-sequence (i.e. an omega-sequence of ordinals < alpha whose supremum is alpha), then alpha-dependent choice follows from dependent choice. I would think that, if alpha doesn't have a cofinal omega-sequence, then alpha-dependent choice is false. Is there a known counterexample? E.g. in the case alpha = omega_1 (the least uncountable ordinal). Thanks, Paul -- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]