From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/614 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Challenge from Harvey Friedman Date: Sat, 24 Jan 1998 13:34:43 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017072 26515 80.91.229.2 (29 Apr 2009 14:57:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:52 +0000 (UTC) To: categories Original-X-From: cat-dist Sat Jan 24 13:34:45 1998 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id NAA19677; Sat, 24 Jan 1998 13:34:43 -0400 (AST) Original-Lines: 21 Xref: news.gmane.org gmane.science.mathematics.categories:614 Archived-At: Date: Sat, 24 Jan 1998 15:59:46 +0000 (GMT) From: Dr. P.T. Johnstone > But the axioms of elementary topoi are already incomparably more > complicated than the axioms for set theory presented here. What on earth does Friedman mean by complicated? As Peter Freyd pointed out a long time ago, the axioms for an elementary topos are essentially algebraic -- that is, they live at a very low level of logical complexity. The very first axiom in anyone's (including Friedman's) axiomatization of set theory is the axiom of extensionality, which is not expressible even in coherent logic (at least, not unless you take not-membership as a primitive predicate, on the same level as membership). Unless Friedman can put forward an objective measure of complexity (as opposed to "unfamiliarity to H. Friedman") which justifies the above quote, then his challenge is not worth considering. Peter Johnstone