From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7049 Path: news.gmane.org!not-for-mail From: Andreas Blass Newsgroups: gmane.science.mathematics.categories Subject: Re: Symmetric models of ZF and GSet for large G Date: Fri, 11 Nov 2011 09:52:28 -0500 Message-ID: References: Reply-To: Andreas Blass NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1321054147 3637 80.91.229.12 (11 Nov 2011 23:29:07 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 11 Nov 2011 23:29:07 +0000 (UTC) Cc: "categories@mta.ca list" To: David Roberts Original-X-From: majordomo@mlist.mta.ca Sat Nov 12 00:29:03 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RP0Ws-0007tg-B2 for gsmc-categories@m.gmane.org; Sat, 12 Nov 2011 00:29:02 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:57703) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RP0VT-00023s-4b; Fri, 11 Nov 2011 19:27:35 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RP0VR-0004m3-NU for categories-list@mlist.mta.ca; Fri, 11 Nov 2011 19:27:33 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7049 Archived-At: There's a logical morphism from the topos of sets to the topos of = G-sets, so the internal logics of their ZF-parts (built by iterating the = power set operation starting from the empty set) look the same. As a = first step toward symmetric models, you want (whether or not the group G = is large) to form the topos of continuous G-sets for some group topology = on G (where the sets on which G acts are given the discrete topology). = The ZF part of the resulting topos is still logically equivalent to the = original universe of sets. But, at least in the case of small G the = topos may also contain objects A such that, if you iterate power set = starting with A (instead of the empty set), then new phenomena, = including violations of choice, can appear. (I haven't checked what = happens with large G; the problem is that A might then also be large and = thus not in your topos; whether this problem actually arises may depend = on the topology you put on G.) This power-set iteration over A produces = a model of the variant of ZF that allows atoms (=3D urelements), and the = violations of choice you get this way amount to those that set-theorists = get in Fraenkel-Mostoski-Specker style permutation models. To get to ZF (without atoms) and the symmetric models you asked about, a = further step is needed, namely to force over the topos of continuous = G-sets to produce an A-indexed family of pure sets (i.e., sets that = don't involve atoms from A). That forcing can move the choice-violating = features of A (and iterated power sets over it) into the ZF-part of the = topos (without atoms). The models you get this way are essentially the = same as symmetric submodels of forcing extensions, and that's covered in = considerable detail in my memoir with Andre Scedrov. =20 Andreas Blass On 10 Nov, 2011, at 9:14 PM, David Roberts wrote: > Hi, >=20 > this has been bugging me lately, and my copy of Blass and Scedrov's > AMS Memoir on order for interlibrary loan is being slow. > ... >=20 > Recall that the topos of G-sets for a large group G gives us a model > of ZF via its internal logic (use a universe if desired). >=20 > Has anyone written down the precise relation to symmetric models as > defined by set theorists? Roughly speaking, these > are submodels of generic extensions that are fixed pointwise by a > family of subgroups of the automorphism group of > the generic filter (if I have the basic idea correct). >=20 > Perhaps this is in Blass-Scedrov, and I just need to be patient... >=20 > Regards, >=20 > David Roberts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]