From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8662 Path: news.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: Re: Current Issues in the Philosophy of Practice of Mathematics & Informatics Date: Wed, 29 Jul 2015 02:42:23 +0100 Message-ID: References: Reply-To: Martin Escardo NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1438262660 28080 80.91.229.3 (30 Jul 2015 13:24:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 30 Jul 2015 13:24:20 +0000 (UTC) To: Patrik Eklund , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Thu Jul 30 15:24:14 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.19]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZKnoi-0002Fh-QN for gsmc-categories@m.gmane.org; Thu, 30 Jul 2015 15:24:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56735) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ZKnnc-0007qA-Ng; Thu, 30 Jul 2015 10:23:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ZKnnd-0002LW-3T for categories-list@mlist.mta.ca; Thu, 30 Jul 2015 10:23:05 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8662 Archived-At: I am not sure why these questions are being asked in this list: On 26/07/15 16:33, Patrik Eklund wrote: > Why, for instance, is it so clear that G??del's Incompleteness Theorem is > a "theorem" and not a "paradox"? I am not sure what you mean by a paradox. But let me take this as a possible interpretation: A paradox is a statement P such that both P and not P are theorems (or, equivalently, such that P holds iff not P holds). As far as current mathematical knowledge goes, Goedel's Incompleteness Theorem is just a theorem, with significant further work needed to elevate it to the status of a paradox. > After all, it is nothing but a bit more subtle version of the Liar > paradox. I paradox means Fix it!, whereas a theorem means Don't > touch!. For comparison, in naive set theory, the set of all sets that don't belong to themselves does lead to a paradox, corresponding to the Liar Paradox: this set belongs to itself if and only it doesn't. Naive set theory is inconsistent (and hence deserves its name). In ZFC, however, the same argument *proves* that there is no set of all sets, and no set of sets that don't belong to themselves. It is important here that ZFC can actually formulate the question of whether there is a set of all sets. And the answer is no. Of course, in principle, the possibility is open that ZFC, too, has a some paradox. This involves exhibiting a statement P and two proofs, following strict rules of mathematical rigour, one of the statement P and another of the statement not P. Nobody has so far managed to exhibit such three things. This is so hard that probably deserves a Fields Medal (followed by immediate eviction from the mathematical community). M. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]