From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6394 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology of locally small categories without replacement Date: Thu, 02 Dec 2010 06:01:29 -0800 Message-ID: References: Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1291343998 15362 80.91.229.12 (3 Dec 2010 02:39:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Dec 2010 02:39:58 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Dec 03 03:39:54 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1POLYw-0007gK-5V for gsmc-categories@m.gmane.org; Fri, 03 Dec 2010 03:39:54 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:54216) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POLYW-0002Ky-EO; Thu, 02 Dec 2010 22:39:28 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POLYS-0002fc-JJ for categories-list@mlist.mta.ca; Thu, 02 Dec 2010 22:39:24 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6394 Archived-At: On 12/1/2010 2:00 PM, Colin McLarty wrote: > These two [weak and strong notions of locally small] are not > equivalent in the absence of the axiom scheme of > replacement. There the second is much stronger, but it remains > important. Is there a good term for it? Sure: "Locally small." In the absence of Replacement it would make more sense to call the weaker concept "weakly locally small" than the stronger one "strongly locally small" since it is presumably the strong one that is more often intended. As you say, Replacement identifies the concepts, and one then defines the common concept with whichever definition is shorter or simpler, namely the weak one. A downside of allowing multiple set theories is the proliferation of a menagerie of definitions. Considerations like the above can help manage the menagerie, though the benefit of the menagerie in the first place would seem to accrue more to logic than to mathematics. The role of logic in mathematics should be to understand the latter, not to complicate it. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]