From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9474 Path: news.gmane.org!.POSTED!not-for-mail From: Neil Barton Newsgroups: gmane.science.mathematics.categories Subject: Re: How analogous are categorial and material set theories? Date: Tue, 12 Dec 2017 13:08:22 +0100 Message-ID: References: Reply-To: Neil Barton NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1513111952 31600 195.159.176.226 (12 Dec 2017 20:52:32 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 12 Dec 2017 20:52:32 +0000 (UTC) To: categories@mta.ca, Patrik Eklund Original-X-From: majordomo@mlist.mta.ca Tue Dec 12 21:52:23 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eOrXI-0007eh-Iz for gsmc-categories@m.gmane.org; Tue, 12 Dec 2017 21:52:20 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:41690) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eOrXv-0007iL-9s; Tue, 12 Dec 2017 16:52:59 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eOrWQ-0000Bq-U3 for categories-list@mlist.mta.ca; Tue, 12 Dec 2017 16:51:26 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9474 Archived-At: Thanks to everyone for their replies. I provide some quick clarifications to some of Patrik's comments below: ``You now introduce the word "core", and I would need to understand what that mathematically and logically really means, at least intuitively.'' I mean ``core'' in the trivial sense. The axioms defining what it is to be a category (of some kind) are just first-order. The same is true, of course, for ZFC. However, in the set theory context it seems at least sensible to try and provide a second-order (quasi) categoricity proof, despite the diversity of models for first-order ZFC (and indeed we have one by the work of Zermelo, Shepherdson, etc.). To attempt the same for category theory would seem to be an absurd strategy---the point of categories is that they can be instantiated in many non-isomorphic models. ``Yes, indeed, across diverse contexts, with "context" somehow related to "system" in Hidekazu Iwaki's reply involving "mathematical system".'' I cannot find Hidekazu Iwaki's reply (this is rather strange, I do not know why). However it seems like similar usage to me: I don't have a formal definition of `mathematical context', but the point is just that there's diverse non-isomorphic models where the same categorial properties pop up. e.g. (somewhat obviously for this list, but I found it striking learning some category theory) the notion of topos appears all over the place in diverse structures, and allows us to have a reasonable notion of internal logic. Your comments on the different notions of hierarchy are interesting, but I will have to think about this some more before I have anything reasonable to add. Best Wishes, Neil --=20 Dr. Neil Barton Postdoctoral Research Fellow Kurt G=C3=B6del Research Center for Mathematical Logic University of Vienna [For admin and other information see: http://www.mta.ca/~cat-dist/ ]