From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6733 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: size_question Date: Sun, 3 Jul 2011 11:01:54 -0400 Message-ID: References: , Reply-To: "F. William Lawvere" 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 1309777477 450 80.91.229.12 (4 Jul 2011 11:04:37 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 4 Jul 2011 11:04:37 +0000 (UTC) Cc: , categories To: THOMAS STREICHER , Original-X-From: majordomo@mlist.mta.ca Mon Jul 04 13:04:32 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 1Qdgx5-0005FE-Ng for gsmc-categories@m.gmane.org; Mon, 04 Jul 2011 13:04:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:35827) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QdguB-0005AA-Ae; Mon, 04 Jul 2011 08:01:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QdguA-0008Qe-Ia for categories-list@mlist.mta.ca; Mon, 04 Jul 2011 08:01:30 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6733 Archived-At: In the absence of AC=2C we need to specify which notions of"finite" we are = using. Certainly K-S (=3D locally quotient of standard numeral) while impor= tant is by no means the endof the story. We categorists do not seem to yet = have a way of dealing elegantly with the locally Noetherian=2C coherent=2C= etcsheaves in terms of internal finiteness notions. The SUBQUOTIENTS of standard numerals would surely be important. I recall t= he existence of intuitionist literature on this=2C but it seemed to assume = that subquots of subquots etc would be an infinite sequence. Of course in a= category with reasonable pullbacks and pushouts this relation is already t= ransitive (indeed an important subcategory of spans). But the original question actually had to do with the observation that for = any qualitative definition of finite set=2C there are probably as many as t= here things in the ambient universe. We need an axiom of infinity to say th= at there is a category object that represents that metacategory UP TO EQUIVA= LENCE OF COURSE.=20 The uniqueness is only relative to the ambient universe . The Incompletenes= s Theorem would seem to imply that there are an infinite number of non-eleme= ntarily-equivalent ambient universesand hence of these little metacategorie= s in particular. We really should overcome the ritual belief that such things must be define= d by iteration (as opposed to being partially investigated via iteration). A= lready Peano misrepresented Grassmann's views on this. Dedekind proposed that a set A is finite if any idempotent whose fixed par= t is isomorphic to all of A is itself an automorphism. This seems difficult= to relate to operations such as product. However note that it is an element= ary axiom to require that all objects of a given topos satisfy it. It would = see to propagate to any finite (in the sense of Artin) topos over such. Do = basictheorems=2C such as the essentialness of all geometric morphisms=2C e= xtend to this axiomatic setting ? A different elementary axiom on a topos is the requirement that every objec= t A is fixed by the monad obtained by composing 3^( ) with its adjoint from = the related topos of left actions of 3^3. Are these two theories equivalent ? Such a finitely-axiomatized T defines "= the" category of finite sets as any one that represents up to equivalence t= he submetacategory of the ambient universe consisting of all discrete catego= ries that are finite in the stated sense.( But then of course there are many= models of T not equivalent to that) I proposed the study of such an "Objective Number Theory" in Braga four yea= rs ago at a European computer science meeting without much response. It is = known since Craig-Vaught that Peano's arithmetic can be embedded in a finit= ely axiomatizable theory=2Cbut it seems reasonable to try to find such an e= xpansion with an objective content.If E is such a category=2C the Paris-Har= rington theory seems to be about the topos E^Iwhere I is the arrow category= =3B but don't the properties of that follow from those of E ? Bill PS Of course this ONT is not the same as (though related to) my ongoing= work with Schanuel and Menni. > Date: Sat=2C 2 Jul 2011 22:05:05 +0200 > From: streicher@mathematik.tu-darmstadt.de > To: andrej.bauer@andrej.com > CC: jlipton@wesleyan.edu=3B categories@mta.ca > Subject: categories: Re: size_question >=20 >> I should think that the hereditarily finite sets do not depend all >> that much on the background setting. After all=2C there are not very >> many of them and they are quite concrete. Can they really be hugely >> different depending on whether we work in ZF=2C ZFC=2C IZF=2C CZF etc? >=20 > If we are not working classically subsets of finite sets need not be > finite but the sets in V_\omega are closed under subsets. >=20 > Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]