From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8725 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: not(CH) and cardinal inequality in the absence of LEM Date: Wed, 28 Oct 2015 21:55:24 +1030 Message-ID: Reply-To: David Roberts NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1446075482 8454 80.91.229.3 (28 Oct 2015 23:38:02 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 28 Oct 2015 23:38:02 +0000 (UTC) To: "categories@mta.ca list" Original-X-From: majordomo@mlist.mta.ca Thu Oct 29 00:37:51 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZraHu-0000Wc-US for gsmc-categories@m.gmane.org; Thu, 29 Oct 2015 00:37:51 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:41749) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ZraHH-0003TU-F6; Wed, 28 Oct 2015 20:37:11 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ZraHJ-0006mh-8E for categories-list@mlist.mta.ca; Wed, 28 Oct 2015 20:37:13 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8725 Archived-At: Dear all, as an obvious-in-hindsight followup in my part 2), the idea that Epi(Y,X) =3D 0 gives strictness of a cardinal inequality clearly loses its meaning in the absence of the Cantor-Bernstein-Schr=C3=B6der Theorem. What should (perhaps) be considered is the subobject Iso(Y,X) of Epi(Y,X), and if Y injects to X and Iso(Y,X) =3D 0, then one can unambiguously say Y < X. Or at least less ambiguously! There are of course variants where one says instead that (X surjects onto Y) xor (Y=3D0), or Y is a subquotient of X. One gets of course variants, for fixed X, analogous to the case X=3D|N where different notions of finiteness arise (Kuratowski finite, subfinite etc) Best regards, David On 28 October 2015 at 18:10, David Roberts wrote: > Dear all, > > 1) not(CH) > > I'm reacquainting myself with the proof using toposes that not(CH) is > relatively consistent. There's one step that is not quite as I'd like > it, and that is as follows. > > Let P be a poset with the countable chain condition and the double > negation topology (or indeed any site where all covering sieves are > generated by families that are at most countable), then for objects > X,Y in the base topos S that are infinite (in the sense that N x X =3D > X), having Epi(Y,X) =3D 0 in S implies that Epi(Y^,X^) =3D 0 in Sh(P). > > It seems to me that the proof, as recounted in Mac Lane--Moerdijk or > Johnstone's Baby Elephant for instance, uses LEM, by assuming that > Epi(Y^,X^) =3D/=3D 0 and showing that Epi(Y,X) =3D/=3D 0. > > I haven't thought this through, but it seems like it might be possible > to rework things so that the proof in fact shows that from an > isomorphism 0--> Epi(Y,X) one can show that any two maps Epi(Y^,X^) > --> \Omega are equal, and hence since 0 --> Epi(Y^,X^) is a subobject, > it must be an isomorphism. > > Has anyone thought about this before? Of course, the previous > paragraph may be nonsense, I admit... > > 2) Cardinal inequality in the absence of LEM > > This does raise the related question as to what the definition or > definitions of <, strict inequality of cardinalities (i.e. just sets), > could be in the absence of LEM and still be useful. If one were to > adopt the following: > > Definition: B < A if and only if there a mono B >--> A and a > factorisation B >--> A' >--> A such that Epi(B,A') =3D 0. > > then---modulo the issue in part 1)---the existing proof of the > preservation of inequalities of sets seems to work without the > assumption the base topos is boolean (which is needed in the existing > proof to find a retract A -->> A' ). > > In the presence of LEM the definition of < reduces to the usual one, > but I don't see a good reason or philosophy to single out this > particular definition over other potential definitions except that it > is what the proof uses. > > Thoughts? > > Best regards, > > David --=20 Dr David Roberts http://ncatlab.org/nlab/show/David+Roberts Visiting Fellow School of Mathematical Sciences University of Adelaide SA 5005 AUSTRALIA "When I consider what people generally want in calculating, I found that it always is a number." -- al-Khw=C4=81rizm=C4=AB CRICOS Provider Number 00123M IMPORTANT: This message may contain confidential or legally privileged information. If you think it was sent to you by mistake, please delete al= l copies and advise the sender. For the purposes of the SPAM Act 2003, this email is authorised by The University of Adelaide. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]