From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9129 Path: news.gmane.org!.POSTED!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Does equality between sets contradict the philosophy behind structural set theory? Date: Sun, 26 Feb 2017 00:23:43 -0500 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1488119717 29268 195.159.176.226 (26 Feb 2017 14:35:17 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 26 Feb 2017 14:35:17 +0000 (UTC) To: Leopold Schlicht , Original-X-From: majordomo@mlist.mta.ca Sun Feb 26 15:35:07 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 1chzuj-0006TZ-4D for gsmc-categories@m.gmane.org; Sun, 26 Feb 2017 15:35:05 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:48129) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1chzuK-0004N5-9N; Sun, 26 Feb 2017 10:34:40 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1chztj-000793-F7 for categories-list@mlist.mta.ca; Sun, 26 Feb 2017 10:34:03 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9129 Archived-At: Leopold Schlicht quotes, in part: > Notation 1.1: The arrow notation f : A -> B just means the domain of f = is A > and the codomain of f is B, and we write dom(f) =3D A and cod(f) =3D B.= and continues: > Here, the authors talk about the equality of two sets (dom(f) =3D A). No. The symbolatry dom(f) =3D A is NOT expressing the equality of two sets, dom(f), and A, it is an abbreviation for the assertion that the map= f has domain A, which is itself a gloss on (a part of) the significance = we may attach to the arrow notation f: A -> B. The remaining part of that= significance is expressed in the symbolatry cod(f) =3D B. There again it'= s NOT an expression of the equality of two sets, cod(f), and B. Rather, it's as if we split f: A -> B into two chunks, f: A -> _ (abbreviated dom(f) =3D A) and f: _ -> B (abbreviated cod(f) =3D B). (In the above, please treat the simple underscore _ as just more white sp= ace: my usa.net mail service will not permit a string of 3 or 4 space charact= ers to pass through without condensing them into a single space character.) Taking a fresh breath of air, now, and asking about objects of a category= : does one not have a right to suppose that, should you and I each focus ou= r respective attentions on one object in a category, you on A, say, and I o= n B, it should be possible to determine when we have chosen to focus on ONE AN= D THE SAME object, i.e., to determine when your A is my B, i.e., when A =3D= B ? No question on elementhood or membership enters into that question. And we NEVER actually say that TWO sets are equal -- only that ONE set is= = equal to itself. If you and I have focused on sets A and B, respectively,= and if it happens that A =3D B, it means exactly that we were NOT focused= on TWO distinct sets, but only on ONE. = Cheers, -- Fred Linton [For admin and other information see: http://www.mta.ca/~cat-dist/ ]