From d_deramo@csc32.ENET.dec.com Thu Feb 15 20:55:26 1996 Date: Thu, 15 Feb 96 17:48:07 PST From: "Dan D'Eramo, Customer Support Center 15-Feb-1996 1847" To: rose@garnet.acns.fsu.edu Subject: Re: query on set theory axioms Status: RO Kermit, The axioms of what is usually called Zermelo-Fraenkel set theory, abbreviated ZFC, can be listed as 1) extensionality 2) foundation 3) empty set 4) pair set 5) union or sum set 6) subset (axiom scheme) 7) power set 8) infinity 9) replacement (axiom scheme) 10) choice Dan Dan D'Eramo d_deramo@csc32.enet.dec.com From owner-math-history-list@ENTERPRISE.MAA.ORG Fri Nov 14 07:43 EST 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.7/8.7.6) with ESMTP id HAA00584 for ; Fri, 14 Nov 1997 07:43:45 -0500 (EST) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.F79D3F00@enterprise.maa.org>; Fri, 14 Nov 1997 7:50:25 -0500 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 6619 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Fri, 14 Nov 1997 07:50:24 -0500 Received: from whorfin.sjca.edu by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.F72799D0@enterprise.maa.org>; Fri, 14 Nov 1997 7:50:24 -0500 Received: from [199.89.180.242] (slip-3.sjca.edu [199.89.180.242]) by whorfin.sjca.edu (8.8.7/8.8.5) with SMTP id HAA03537; Fri, 14 Nov 1997 07:43:11 -0500 (EST) Mime-Version: 1.0 Approved-By: "Samuel S. Kutler" Message-ID: Date: Fri, 14 Nov 1997 07:43:04 -0500 Reply-To: Discussion List on the History of Mathematics , "Samuel S. Kutler" From: "Samuel S. Kutler" Subject: Re: Sets Comments: To: Ralph Gainey To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG Content-Type: text/plain; charset="us-ascii" Content-Length: 1675 Status: RO Ralph Gainey: You wrote . . .when we don't even know what a set is. Georg Cantor thought that he DID know. He had three explanations: Here is the first: Any many (Viele) which can be thought of as one [Eines}, that is, every totality of definite elements which can be united to a whole through a law. By this I believe I have defined something related to the Platonic *eidos* or *idea*. [1883] The ancient Greek word *idea* does not mean what our thought has, but, rather, it means something that has our thought! Here is the second: Every collection to a whole M of definite, well-differentiated objects m of our intuition or our thought. [1895] Third: When . . . the totality of elements of a multiplicity can be thought without contradiction as 'being together', so that their collection into 'one thing' is posssible. [1932 letter] The translations are from Cantorian Set Theory & Limitation of Size by Michael Hallett. Best wishes, Sam Kutler > "It's important in mathematics that we be able > to argue clearly and correctly about the members > of the empty set, because in mathematics we often > discuss sets before we even know whether they are > empty or not." > John Conway > > >Clarity also seems to suffer when we discus sets before we have removed the >ambiguities from the definition of set, and thus it may be equally >alleged that we are discussing sets when we don't even know what a set is. > >My very best regards, >Ralph Gainey >[who is just a little boy] From owner-math-history-list@ENTERPRISE.MAA.ORG Fri Nov 14 09:40 EST 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.7/8.7.6) with ESMTP id JAA12713 for ; Fri, 14 Nov 1997 09:40:23 -0500 (EST) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.257A04C0@enterprise.maa.org>; Fri, 14 Nov 1997 9:46:14 -0500 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 6666 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Fri, 14 Nov 1997 09:46:13 -0500 Received: from VMS.HUJI.AC.IL (128.139.4.12) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.20AD8890@enterprise.maa.org>; Fri, 14 Nov 1997 9:46:06 -0500 Received: by HUJIVMS (HUyMail-V7c); Fri, 14 Nov 97 16:39:10 +0300 Received: by HUJIVMS (HUyMail-V7c); Fri, 14 Nov 97 16:38:13 +0300 Approved-By: MANN@VMS.HUJI.AC.IL Message-ID: <14110097163813@HUJIVMS> Date: Fri, 14 Nov 1997 16:38:00 +0300 Reply-To: Discussion List on the History of Mathematics , MANN@vms.huji.ac.il From: Avinoam Mann Subject: Empty sets To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG Content-Type: text Content-Length: 673 Status: RO "It's important in mathematics that we be able to argue clearly and correctly about the members of the empty set, because in mathematics we often discuss sets before we even know whether they are empty or not." John Conway Which reminds me of the following interesting proof by a fellow student that the empty set exists. He was just beginning his studies, while I was at my second year. The argument ran as follows: if the empty set exists, fine. Otherwise, the set of all empty sets is empty. Avinoam Mann From owner-math-history-list@ENTERPRISE.MAA.ORG Fri Nov 14 10:30 EST 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.7/8.7.6) with ESMTP id KAA20134 for ; Fri, 14 Nov 1997 10:30:06 -0500 (EST) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.1B892430@enterprise.maa.org>; Fri, 14 Nov 1997 10:36:03 -0500 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 6705 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Fri, 14 Nov 1997 10:35:57 -0500 Received: from dns1.mcn.org (204.189.12.26) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.179D4400@enterprise.maa.org>; Fri, 14 Nov 1997 10:35:57 -0500 Received: from pm25-3-men-a19.mcn.org (pm25-4-men-a07.mcn.org [204.189.12.231]) by dns1.mcn.org (8.8.7/8.8.7) with SMTP id HAA10780 for ; Fri, 14 Nov 1997 07:28:57 -0800 (PST) X-Sender: sesame@mail.mcn.org X-Mailer: Windows Eudora Pro Version 2.1.2 Mime-Version: 1.0 Approved-By: Ralph Gainey Message-ID: <199711141528.HAA10780@dns1.mcn.org> Date: Fri, 14 Nov 1997 07:28:57 -0800 Reply-To: Discussion List on the History of Mathematics , Ralph Gainey From: Ralph Gainey Subject: Re: Empty sets To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG Content-Type: text/plain; charset="us-ascii" Content-Length: 976 Status: RO "It's important in mathematics that we be able to argue clearly and correctly about the members of the empty set, because in mathematics we often discuss sets before we even know whether they are empty or not." John Conway Which reminds me of the following interesting proof by a fellow student that the empty set exists. He was just beginning his studies, while I was at my second year. The argument ran as follows: if the empty set exists, fine. Otherwise, the set of all empty sets is empty. Avinoam Mann Who says matheticians don't have any fun!!! This makes me laugh out loud ... because it is indeed, such a CLASSIC ... I wish I had heard this years ago. I think you should get a bumper sticker made and sell it. Regardless, it is destined to live forever ... of such is "greatness" made ... IMHO :)) rlg