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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
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	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.