From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7108 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: __?__ Date: Wed, 14 Dec 2011 18:35:06 -0300 Message-ID: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1323961044 4861 80.91.229.12 (15 Dec 2011 14:57:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 15 Dec 2011 14:57:24 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Thu Dec 15 15:57:19 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpz.mta.ca ([138.73.1.187]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RbCkH-0007XW-M4 for gsmc-categories@m.gmane.org; Thu, 15 Dec 2011 15:57:17 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:52946) by smtpz.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1RbCiq-0000l0-4w; Thu, 15 Dec 2011 10:55:48 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RbCiv-00066h-Mj for categories-list@mlist.mta.ca; Thu, 15 Dec 2011 10:55:53 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7108 Archived-At: Is the following nonsense ? http://arxiv.org/abs/1112.2141 A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to G\"odel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms is computed from the solution to the corresponding system of polynomial equations. A reference by a formula to its own provability is shown to be a recurrence relation, which can be either interpreted as such to generate a discrete dynamical system, or interpreted in a static way to create an additional simultaneous equation. In this framework the truth values of logical formulas and other polynomial objectives have complex data structures: sets of elementary values, or dynamical systems that generate sets of infinite sequences of such solution-value sets. Besides the routine result that a formula has a definite elementary value, these data structures encode several exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or contingent. These exceptions represent several semantically different types of undecidability; none causes any fundamental problem for mathematics. It is simple to calculate that G\"odel's formula, which asserts that it cannot be proven, is exceptional in specific ways: interpreted statically, the formula defines an inconsistent system of equations (thus it is called unsatisfiable); interpreted dynamically, it defines a dynamical system that has a periodic orbit and no fixed point (thus it is called unsteady). These exceptions are not catastrophic failures of logic; they are accurate mathematical descriptions of G\"odel's self-referential construction. G\"odel's analysis does not reveal any essential incompleteness in formal reasoning systems, nor any barrier to proving the consistency of such systems by ordinary mathematical means. \\ ( http://arxiv.org/abs/1112.2141 , 60kb) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]