From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1098 Path: news.gmane.org!not-for-mail From: Mike Oliver Newsgroups: gmane.science.mathematics.categories Subject: Re: Is Zermelo-Fraenkel set theory inconsistent? Date: Thu, 01 Apr 1999 07:52:01 -0800 Organization: UCLA Message-ID: <37039621.7387D2A7@math.ucla.edu> References: <199904011210.NAA21705@wax.dcs.qmw.ac.uk> Reply-To: oliver@math.ucla.edu NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017566 29502 80.91.229.2 (29 Apr 2009 15:06:06 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:06 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Fri Apr 2 06:29:44 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id FAA28262 for categories-list; Fri, 2 Apr 1999 05:36:24 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.5 [en] (Win95; U) X-Accept-Language: it Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:1098 Archived-At: Paul Taylor wrote: > Now let L(infinity) be the union of L(n) over n:N. > > If L(infinity) |- false then L(n) |- false for some n. > > But L(infinity) |- ``L(n) is consistent,'' > > so L(infinity) proves its OWN consistency, > contradicting Godel's theorem. How do you conclude, from the fact that L(infinity) |- "L(n) is consistent", that L(n) is in fact consistent? Generally, if T1 |- "T2 is consistent", then to conclude "T2 is consistent", we use the following argument: Suppose T2 is inconsistent. Then there is some proof by which T2 |- false. Assuming T1 is strong enough to formalize the deductive system being used, then it follows that T1 |- "T2 is inconsistent". But by hypothesis, T1 |- "T2 is consistent", therefore T1 is inconsistent. But this is not a contradiction unless we were already *assuming* the consistency of T1 ! I.e. it follows from T1+Con(T1) that if T1 |- Con(T2), then Con(T2), but it does *not* in general follow from T1 alone. So the step from L(infinity) |- "L(n) is consistent" to L(n) is consistent cannot be formalized in any obvious way in L(infinity), and therefore you cannot (again in any obvious way) conclude L(infinity) |- "L(infinity) is consistent." -- Disclaimer: I could be wrong -- but I'm not. (Eagles, "Victim of Love") Finger for PGP public key, or visit http://www.math.ucla.edu/~oliver. 1500 bits, fingerprint AE AE 4F F8 EA EA A6 FB E9 36 5F 9E EA D0 F8 B9