From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6728 Path: news.gmane.org!not-for-mail From: Paul Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: ordinal dependent choice Date: Thu, 30 Jun 2011 17:07:10 +0100 Message-ID: References: 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 1309464284 19873 80.91.229.12 (30 Jun 2011 20:04:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 30 Jun 2011 20:04:44 +0000 (UTC) To: categories list Original-X-From: majordomo@mlist.mta.ca Thu Jun 30 22:04:36 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 1QcNTY-0005M4-Du for gsmc-categories@m.gmane.org; Thu, 30 Jun 2011 22:04:36 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:60314) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QcNRL-00011G-M0; Thu, 30 Jun 2011 17:02:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QcNRK-0007U2-Uf for categories-list@mlist.mta.ca; Thu, 30 Jun 2011 17:02:18 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6728 Archived-At: Thanks to all the people who sent me counterexamples, references and Aronszajn trees. I found the following document helpful: http://math.berkeley.edu/~gbergman/papers/unpub/emptylim.pdf It seems that Higman and Stone's result mentioned there is the same as Aronszajn's. Paul PS Of course I should have stated that alpha is a positive limit ordinal, for otherwise alpha-dependent choice holds trivially. On 28 Jun 2011, at 14:12, 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). > > 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/ ] -- 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/ ]