Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 5/3/83; site rochester.UUCP Path: utzoo!linus!philabs!seismo!rochester!ee461 From: ee461@rochester.UUCP Newsgroups: net.politics Subject: Re: Limited Laws for All Time? (Godel theorem revisited) Message-ID: <2134@rochester.UUCP> Date: Sun, 3-Jul-83 20:04:50 EDT Article-I.D.: rocheste.2134 Posted: Sun Jul 3 20:04:50 1983 Date-Received: Mon, 4-Jul-83 16:27:15 EDT References: <254@isrnix.UUCP>, <237@cbscd5.UUCP> Organization: University of Rochester Lines: 54 Whoops .. I hope you are not reading this for the second time - when I submitted it for the first time, the first few lines got lost. I'm not sure if I managed to cancel this first posting... Now the full story: The followup posted by Larry Cipriani, where he tries to explain Godel's theorem to Tim Sevener, introduces only more misunderstanding over the subject. This news item will deal only with this. The subject is more suitable for net.math newsgroup, but Godel's theorem received the crippling treatment here, in net.politics... First, a quotation from Larry's article: > Godel's proof states that under a "certain set of circumstances" > there exists a problem which cannot be proven true using statements of > the same "form" as the problem. > A few examples will illustrate what I mean. > 1 ) The problem x * x + 1 = 0 can be solved only by using complex > numbers yet it is stated without using complex numbers. > 2 ) The problem x * x - 2 = 0 can be solved only by introducing > irrational numbers, yet they are not in the problem. The theorem, as it appears in the original Godel's work, is preceded by some 30 pages of definitions necessary to make the precise statement. The attempt to compress these to: "certain set of circumstances" and "form" appears to fail completely. Let me state the main result in an acceptably (I think) simplified version: Godel says that within each system of axioms that is consistent and powerful enough to express the arithmetic of integers there are THEOREMS that can not be proved WITHIN the system. (Theorem means here: a statement that is true). Note the difference: "theorems", NOT "problems"! Also, note that the proof is required to use only the fundamental axioms of the system and theorems derivable from these axioms (this is the meaning of: "within the system"). When Larry's examples are restated in the form of theorems, they are PROVABLY FALSE WITHIN THE RESPECTIVE SYSTEMS, hence they bear no relevance whatsoever to Godel's theorem. For example the following statement: "there exists a rational number x such that x*x - 2 = 0" can be disproved within the arithmetic of rationals. Note again: NOT "a number x", but "a rational number". Stay within the system! One of the most famous examples of assertions that appear to be true but nobody was able to prove them yet, is Cardano's theorem (a**n + b**n = c**n has no integer solutions in a,b,c for integer n>2). Another example: for any integer n there are two primes a,b such that a + b = 2*n. However, Larry is perfectly correct in that Godel's theorem was misused by Tim Sevener. Tim, unless you are discussing a moral or political system that contains the arithmetic of integers (wow! this would be something!) Godel's theorem does not apply. Krzysztof Kozminski