From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4310 Path: news.gmane.org!not-for-mail From: "Michael Shulman" Newsgroups: gmane.science.mathematics.categories Subject: Re: categorical formulations of Replacement Date: Fri, 14 Mar 2008 11:02:39 -0500 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019864 12705 80.91.229.2 (29 Apr 2009 15:44:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:24 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Mar 14 20:25:55 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 14 Mar 2008 20:25:55 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JaJ8y-0007HE-To for categories-list@mta.ca; Fri, 14 Mar 2008 20:16:56 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 74 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:4310 Archived-At: On Thu, Mar 13, 2008 at 9:34 PM, Thomas Streicher wrote: > But even in this framework it is > too strong due to its requirement that every external family arises from an > internal one. So it fails for example for the model of ETCS arising from > a countable model of ZFC because there are only countably many internal > families over N whereas there uncountable many external families indexed > by (the global elements of) N. I do not understand what is meant by "external" here. What Colin and Osius's replacement scheme states is that every *definable* family of sets is internal. This is the same as the ZF replacement axiom: every *definable* function defined on a set is a set. Since there are only countably many logical formulas, there are only countably many definable families for them to define, so there is no problem with countable models. > A defect of the work from the 70ies (Cole, Osius at.al.) is that it just proves > equiconsistency of ETCS and bZ (bounded Zermelo set theory) and not an > equivalence between models of ETCS and bZ. In Osius' paper "Categorical set theory" he does prove exactly this sort of equivalence, by adding a couple weaker extra axioms to ETCS and bZ relating to the existence of transitive closures and collapses. An account can also be found in Johnstone's "Topos Theory", chapter 9. Mike