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Article 7365 of comp.ai.philosophy:
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>From: g89p4455@giraffe.ru.ac.za (MR L PEDERSEN)
Newsgroups: comp.ai,comp.ai.philosophy
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <g89p4455.16.719764005@giraffe.ru.ac.za>
Date: 22 Oct 92 14:26:45 GMT
Article-I.D.: giraffe.g89p4455.16.719764005
References: <1992Oct16.180352.13326@oracorp.com> <1992Oct18.171354.6029@wixer.cactus.org>
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In article <1992Oct18.171354.6029@wixer.cactus.org> sparky@wixer.cactus.org (Timothy Sheridan) writes:
>From: sparky@wixer.cactus.org (Timothy Sheridan)
>Subject: Re: Human intelligence vs. Machine intelligence
>Date: Sun, 18 Oct 92 17:13:54 GMT
>In article <1992Oct16.180352.13326@oracorp.com> daryl@oracorp.com (Daryl McCull
>ough) writes:
>
>....   Is it the case however that the godel formulation addresses
>statements like the following:
>
>This statement is false.
>
>The "deep structure" of the above seems to also directly say:
>
>if X is 'true' then X is 'false'.
>
>This directly states:
>
>There exists a statement that is true and false. (undecidable)
>
>However It also seems to imply that:
>
>Truth does not always imply falsity.
>
>If this statement is taken as an axiom then we have just invented a tautology
>of demanding that we use a mathamatics that allows a contradiction. And then
>we show that if we accept that (a truth can be a falsity) then all else holds.
>
>The result is: If you choose a system that acepts contradictions then there
>will be undecidables.  If one can formalise such statements as contradictions
>then the undecidables become nothing more than falsitys.

Not so.

As I understand it, the Godel number does NOT say "This statement is 
false".  Rather, if expressed in a formal system X, it sais:
     
     This statement in unprovable in system X.

Unprovable is not the same as false.  Clearly the above statement is true, 
since if it were not, then it could be proven and so would be true.  The 
consequence of this is that any sufficiently powerful system (ie one that 
can embody number theory) will always be incomplete, that is it will contain 
such unprovable statements.

Even worse, it was shown that there is no way of enumerating all the 
undecidable propositions.  They form an uncountable set and cannot all be 
specified by a finite set of rules (or meta rules that to specify the rules, 
or meta meta rules .... and so on).

This is known as Godels incompleteness theorem and applies only to 
consistent (contradiction free) formal systems.  In any formal system that 
accepts a contradiction EVERY possible statement in that system CAN BE 
PROVEN.

To see this, suppose Q is any statement and P is a statement such that 
the contradiction P ^ ~P holds.  But A ==> B is always true if A is false, 
hence (P ^ ~P) ==> Q is true.  But as (P ^ ~P) is true, then so is Q.

Therefore, an inconsistent formal system cannot be incomplete.

I look forward to hearing anyones comments.
Liam


