From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5255 Path: news.gmane.org!not-for-mail From: Andre.Rodin@ens.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: categorical foundations Date: Fri, 13 Nov 2009 01:42:16 +0100 Message-ID: References: Reply-To: Andre.Rodin@ens.fr 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: ger.gmane.org 1258079564 20345 80.91.229.12 (13 Nov 2009 02:32:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 13 Nov 2009 02:32:44 +0000 (UTC) To: Colin McLarty , categories@mta.ca Original-X-From: categories@mta.ca Fri Nov 13 03:32:37 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1N8lxk-00052R-Hc for gsmc-categories@m.gmane.org; Fri, 13 Nov 2009 03:32:36 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1N8la0-0006fA-DP for categories-list@mta.ca; Thu, 12 Nov 2009 22:08:04 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5255 Archived-At: > > The basic notions are in fact not very articulate in themselves, and > throughout the history of mathematics it has taken further ideas to > articulate them. Bill saw how to articulate these and many more, > quite directly, in categorical terms not assuming any prior set > theory. That articulation works even if you do not take it as > foundational. But it gets a natural foundational character in the > framework of the category of categories -- thus CCAF, the axiomatic > theory of the category of categories as a foundation. > I agree with you about generalities concerning pre-formal and formal concepts. A reason why I say CCAF is not a satisfactory categorical found= ation is different. ETC is the formal basis of CCAF and ETC relies on a pre-for= mal notion of set or collection just like ZF or any other axiomatic theory bu= ilt with Hilbert-Tarski axiomatic method. Elements of a new properly categori= cal method of theory-building are present in the "basic theory" (BC) that fol= lows ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A= , etc. The standard definition of functor given earlier in ETC never reappears i= n BC.) However in CCAF these new features are not yet developed into an autonomo= us axiomatic method - or into a new way of formalisation of pre-formal conce= pts, if you like. In my understanding, such a method should meake part of categorical foundations deserving the name. CCAF remains in this sense eclectic, it is a half-way to categorical foundations. best, andrei [For admin and other information see: http://www.mta.ca/~cat-dist/ ]