From newshub.ccs.yorku.ca!torn!cs.utexas.edu!milano!cactus.org!wixer!sparky Mon Oct 19 16:59:46 EDT 1992
Article 7326 of comp.ai.philosophy:
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>From: sparky@wixer.cactus.org (Timothy Sheridan)
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <1992Oct18.171354.6029@wixer.cactus.org>
Organization: Real/Time Communications
References: <1992Oct16.180352.13326@oracorp.com>
Date: Sun, 18 Oct 92 17:13:54 GMT
Lines: 66

In article <1992Oct16.180352.13326@oracorp.com> daryl@oracorp.com (Daryl McCull
ough) writes:
>In article <Bvz82n.A7w@cs.bham.ac.uk>, axs@cs.bham.ac.uk (Aaron
>Sloman) writes:
>
>>My argument, summarised very briefly in an earlier posting, is that
>>even if F is consistent, Penrose is deluded in thinking that he can
>>use Godel's theorem as a basis for discovering that the Godel
>>sentence G(F) is true. Many of his critics share the same delusion.
>>
>>The full argument requires very detailed analysis of what the Godel
>>formula actually says: I claim it merely says that a very
>>complicated arithmetical property belongs to a certain very large
>>number. It gets subtly misinterpreted as saying something about a
>>certain formula and its relationship to a formal system.
>
>You are right that the Godel formula is a statement of pure
>arithmetic, but you are completely wrong to say that it is not also a
>statement about formal systems.
>
>Godel's arguments showed that the notion of consistency of a theory
>can be "arithmetized": he gives a construction which applied to any
>formal system S (expressive enough to interpret Peano arithmetic)
>yields a formula G of pure arithmetic such that (1) if S is
>consistent, then G is a true statement of arithmetic, and (2) if S is
>consistent, then S does not prove G.
>
>Of course, since G is a statement of pure arithmetic, you can ignore
>all of the metamathematical implications of G and simply consider it
>as a statement about numbers. However, as a statement about numbers,
>there is no reason to regard it as very interesting. The whole point
>of talking about G is its metamathematical implications.
>
>Daryl McCullough
>ORA Corp.
>Ithaca, NY


Well said.   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.

--your thoughts?

Tim.


