From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6787 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: size_question_encore Date: Thu, 14 Jul 2011 23:03:55 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1310722854 4804 80.91.229.12 (15 Jul 2011 09:40:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 15 Jul 2011 09:40:54 +0000 (UTC) Cc: Categories To: Toby Bartels Original-X-From: majordomo@mlist.mta.ca Fri Jul 15 11:40:50 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Qhet6-0000G2-Bg for gsmc-categories@m.gmane.org; Fri, 15 Jul 2011 11:40:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45200) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QheqW-0000br-L0; Fri, 15 Jul 2011 06:38:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QheqW-0006Vo-0e for categories-list@mlist.mta.ca; Fri, 15 Jul 2011 06:38:08 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6787 Archived-At: On Wed, Jul 13, 2011 at 9:10 PM, Toby Bartels wrote: >>the axiom of collection, which implies that we can find some *set* of >>objects containing *at least one* limit for every finite diagram in >>the original small subcategory; and then we can iterate countably many >>times to obtain a small category which contains at least one limit for >>any finite diagram therein. > > The axiom of collection guarantees only *some* appropriate set of objects= , > so you need to choose one. =A0To iterate this countably many times, > you might need dependent choice. That's a good point. However, I think we can get around it as follows. We can make finitely many choices without any axiom of choice. Thus, for any natural number n, by applying collection n times, we can find *some* n^th iterate of the "construction". (Formally, we prove this by induction on n.) Applying the axiom of collection again over the natural numbers, we obtain a set which contains at least one n^th iterate of the "construction" for every natural number n. Taking the union of this set, we should obtain a set of objects whose corresponding full subcategory contains at least one limit of every finite diagram therein. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]