From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6726 Path: news.gmane.org!not-for-mail From: N.Bowler@dpmms.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: ordinal dependent choice Date: 29 Jun 2011 09:32:15 +0100 Message-ID: References: Reply-To: N.Bowler@dpmms.cam.ac.uk NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 X-Trace: dough.gmane.org 1309353942 26107 80.91.229.12 (29 Jun 2011 13:25:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 29 Jun 2011 13:25:42 +0000 (UTC) Cc: categories list To: Paul Levy Original-X-From: majordomo@mlist.mta.ca Wed Jun 29 15:25:38 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 1Qbult-0002Hv-Or for gsmc-categories@m.gmane.org; Wed, 29 Jun 2011 15:25:37 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44792) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QbukW-00033C-9W; Wed, 29 Jun 2011 10:24:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QbukV-0002fY-IG for categories-list@mlist.mta.ca; Wed, 29 Jun 2011 10:24:11 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6726 Archived-At: On Jun 28 2011, Paul Levy wrote: >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). In my last email, I showed that omega_1-dependent choice is false. In fact, there is a simple argument showing that if alpha has cofinality greater than omega then alpha-dependent choice is false. Let A_i be the set of all finite increasing sequences s of ordinals less than alpha such that only the final term of s is greater than or equal to i. Let the map A_i,j: A_i ---> A_j send a sequence s from A_i to the unique initial segment of s lying in A_j. An element of the limit of A would be a cofinal sequence for alpha of length at most omega, so if alpha has cofinality greater than omega then this limit is empty. Nathan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]