Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!linus!philabs!cmcl2!floyd!vax135!cornell!uw-beaver!microsof!decvax!harpo!seismo!rochester!ee461 From: ee461@rochester.UUCP Newsgroups: net.politics Subject: Re: (Godel theorem revisited. FLAME !! NITPICKING !!!) Message-ID: <2215@rocheste.UUCP> Date: Sun, 10-Jul-83 17:56:44 EDT Article-I.D.: rocheste.2215 Posted: Sun Jul 10 17:56:44 1983 Date-Received: Tue, 19-Jul-83 09:45:01 EDT References: <254@isrnix.UUCP>, <237@cbscd5.UUCP> <2134@rochester.UUCP>, <975@uwvax.ARPA> Lines: 58 First, my apologies to all those interested in politics and not in metamathematics for submitting again... Second, a proposition to those interested in Godel's theorem: if there are any more doubts about it, let's move the discussion to net.math. I wouldn't like to appear as a nitpicker that wants to have the last word on the subject, but I can not agree with confusion being spread ... Jeff Myers has written recently: "... symbols in ANY formal system can be represented as integers! Hence, any relationship expressable withing a formal system is representable as a relationship between integers. (...) What the theorem says about formal systems dealing with non-negative integers is applicable to ALL formal systems.". The logic of this statement fully justifies its appearance in net.politics - it is an excellent example of twisting the real meaning and of backward reasoning. What you do with the REPRESENTATION of something has no effect on the subject!!! An easy example: if you write the representation of some system on a sheet of paper and tear it into pieces, the system will remain intact, don't you think so? I didn't believe Jeff and I read Chapter 1 of "Theory of Computation" by Brainerd and Landweber. It's clear enough: "for any consistent formal system whose axioms adequately define addition and multiplication of natural numbers, there are propositions which are true (...) but not derivable from the axioms using the rules of inference". And also: "essential incompleteness remained so long as the resulting system was adequate for describing the natural numbers". (pages 5-6) Stated simply: integers must be expressable using the tools provided by the formal system in question, not the other way around. Just to clear the picture: a formal system is charcterized by: - a set of objects - a set of axioms (basic theorems) - a set of inference rules describing the ways of combining theorems in order to create new theorems. There are examples of provably complete and consistent formal systems. Completeness and consistency may be particularly easy to demonstrate for systems dealing with finite number of objects. (Have a theorem? Check all situations where it applies!). And those systems still can be expressed in integers. For a neat treatment of Godel's theorem on not too complicated level, I'd recommend "Godel's proof" by E.Nagel and J.Newman (NY Univ.Press. 1960). Concluding remark, in order to justify the appearance of this article in net.politics: I can't think of a political system that agrees with the definition of the formal system in the sense explained above. You have the set of objects (people), the set of axioms (laws), but what about the derivation rules ??? (should they be like "Take two statements of two laws and interchange every second sentence. The result is a valid law." or what ??) As for political systems, I think that incompleteness is unavoidable because of non-deterministic human behaviour. Boring you again (hopefully for the last time) Krzysztof Kozminski. PS: In my previous article, Fermat's theorem was attributed to Cardano by a mistake. Sorry for that.