From pbs@MATH.AMS.ORG Thu Mar 6 09:44:07 1997 Received: from math.ams.org (MATH.AMS.ORG [130.44.210.14]) by mail1.its.rpi.edu (8.8.5/8.8.5) with SMTP id JAA48990 for ; Thu, 6 Mar 1997 09:44:06 -0500 Received: from axp14.ams.org by math.ams.org via smtpd (for mail1.its.rpi.edu [128.113.100.7]) with SMTP; 6 Mar 1997 14:44:05 UT Received: from achilles.mr.ams.org by AXP14.AMS.ORG (PMDF V5.1-8 #16534) with SMTP id <01IG6EU3DGJK0001ZC@AXP14.AMS.ORG> for brings@rpi.edu; Thu, 6 Mar 1997 09:44:04 EST Received: by achilles.mr.ams.org (5.57/Ultrix3.0-C) id AA27526; Thu, 06 Mar 1997 09:43:44 -0500 Received: from poseidon.mr.ams.org by mr4.mr.ams.org; (5.65v3.2/1.1.8.2/28Sep94-0231PM) id AA08512; Thu, 06 Mar 1997 09:43:22 -0500 Date: Thu, 06 Mar 1997 14:43:49 +0000 From: pbs@MATH.AMS.ORG (Perry B. Smith) Subject: Re: history of proof X-Sender: pbs@mr4.mr.ams.org To: Selmer Bringsjord Cc: pbs@MATH.AMS.ORG Message-id: MIME-version: 1.0 Content-type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by mail1.its.rpi.edu id JAA48990 X-UIDL: 4c0c8fb674e44e40ca0371fda384ad3f Status: R I'm looking at Gregory H. Moore's book entitled Zermelo's Axiom of Choice: Its Origins, Development, and Influence (Springer-Verlag, 1982), which has a fair amount to say about the axiom (it is in the index under the heading Axiom of Foundation rather than under either Regularity or Foundation). On page 165 it is stated that "Zermelo had initially taken as a postulate that A is not an element of A, but had abandoned it" and also that Schoenflies in 1911 had proposed postulates for set theory including one that there exist no sets A and B such that A is an element of B and B is an element of A, although Shoenflies's postulates were too weak to be an adequate foundation for set theory. On page 269, Moore notes that Zermelo's new axiom system of 1930 contained the Axiom of Foundation, "probably adopted from von Neumann but perhaps stated independently", in two formulations, (1) "there is no infinite descending epsilon-sequence" (A contains B contains C etc.) and (2) every nonempty set A contains an epsilon-minimal element, i.e. an element disjoint from A. Zermelo stated without proof that these two formulations were equivalent. The natural way to prove that (1) implies (2) is to suppose that some set A has no epsilon-minimal element and then to construct an infinite descending epsilon-sequence of elements of A; specifically, if A has no epsilon-minimal element then the axiom of choice gives us a function f with domain A that assigns to each element x of A an element in the intersection of x and A, and now for any element a of A we have an infinite descending epsilon-sequence a, f(a), f(f(a)), etc. The proof that (2) implies (1) is easier: the set of terms of an infinite descending epsilon-sequence would have no epsilon-minimal element. So Zermelo presumably had worked out both of these proofs. Moore states that Mendelson proved in 1958 that (2) cannot be proved from (1) without using the axiom of choice. The proof appeared in his article "The axiom of Fundierung and the axiom of choice", Archiv für mathematische Logik und Grundlagenforschung, volume 4, pp. 67--70. Either (1) or (2) readily implies that a set cannot belong to itself, two sets cannot belong to each other, etc. The proof from (1) is more obvious, since if A contains A then A, A, A, ... is an infinite descending epsilon-sequence, and if A and B contain each other then A, B, A, B, ... is an infinite descending epsilon-sequence. Jane Kister already gave the proof from (2). There is some discussion of descending epsilon-sequences on page 261 of Moore's book, which I mention because this topic is not in the index. This page repeats that Schoenflies had an axiom excluding the possibility of two sets belonging to each other, and adds that Schoenflies did not say anything about three-cycles (A contains B contains C contains A). In the index of Moore's book, under Axiom of Foundation, page 266 should have been cited since it mentions von Neumann's introduction of this axiom, although the axiom is not named but just called "an axiom which prohibited infinite descending epsilon-sequences". From page 269 it seems that the axiom simply said "there are no infinite descending epsilon-sequences", which is Zermelo's form (1) given above. Von Neumann's paper is "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, vol. 27 (1928), pp. 669-752. Maybe it is translated in From Frege to Gödel. I don't know whether the paper states explicitly that this axiom rules out finite epsilon-cycles, though that would have been obvious to von Neumann. That's all the information I have at present. You might find relevant papers in Volume 5 of the Omega-Bibliography of Mathematical Logic, starting on page 252, but especially after 1930 which starts on page 254. The set theory textbooks written by Fraenkel might discuss the Axiom of Regularity also. >Perry, hi. Do you know if there are any early English publications >on the Axiom of Regularity and its blockage of "pathological" circles >or cycles? Yrs, //Selmer > >=============================================================== > Selmer Bringsjord * selmer@rpi.edu * Associate Professor > Dept. of Philosophy, Psychology & Cognitive Science > Department of Computer Science > ----------------------------- > http://www.rpi.edu/~brings > ----------------------------- > Rensselaer Polytechnic Institute Troy New York 12180 >===============================================================