From: mccolm@darwin.math.usf.edu. (Gregory McColm)
Newsgroups: comp.ai.philosophy,sci.logic
Subject: Re: Self-Reference and Paradox (was Re: Human intelligence...)
Date: 9 Dec 92 19:35:23 GMT
Organization: Univ. of South Florida, Math Department

In article <1992Dec1.054427.5632@trilogy.com> eichen@trilogy.com (John Eichenseer) writes:
>In article <1992Nov25.154623.12002@th-ilmenau.RZ.TH-Ilmenau.DE> huebner@PrakInf.TH-Ilmenau.DE (Huebner) writes:
>>Anyway, I think there are some interesting extensions of classical logic 
>>we should discuss here (not only derived from these pathological cases above).
>
>Hasn't anybody here heard of Kurt Godel? 
>
>His insight into logical systems ties into every thread running through 
>comp.ai.philosophy, and somebody with a solid background could easily clean
>house with a little Godel theory.       
>
>The way I understand it, Godel proved that any closed system of axioms will be
>incomplete in that you can always find well-formed statements within the 
>system which are impossible to evaluate. Essentially, there will always be
>irreconcilable paradoxes that can only be understood by utilizing insight
>that is EXTERNAL to the system. 
>

Strictly speaking, given a recursively enumerable (r.e.) set of 
sentences (ie, a (possibly infinite) set of sentences that can 
be listed one after another by a computer) that is consistent, 
there exists a sentence that can be neither proven nor disproven 
from that r.e. set.  The assumption that the set of sentences is 
r.e. is necessary:  the (complete) set of all true sentences on 
the integers is an example of a non-r.e. set of sentences that 
will prove or disprove every sentence (written in the language 
of the integers).

>Therefore, you can only explain the liar paradox by introducing external
>insight or "extensions" to the formal system in question. And, once you
>[...]

While the diagonalization of The Cretan Liar underlies the proof 
of the Incompleteness Theorem, the relation is more tenuous than 
this.  I don't think that you can talk your way out of "This 
sentence is false" by appealing to even higher mathematics.  The 
paradox remains a paradox.

>
>I apologize for my lack of formality (it's been a while), but the concept should
>be clear. Roger Penrose offers a very lucid discussion of Godel's ideas and
>their extraordinary implications in his book, "The Emperor's New Mind." I would
>recommend this text to almost anyone with an interest in the philosophical
>issues surrounding logic, physics, and especially computer science. 
>

Being lazy, I skipped over the logic parts of Penrose's book.  
Two logicians I know claim that the logic part is a mess, and 
some of the scientific reviews are very negative.  While I 
can't comment on the physics (I suspect that he knows what he 
is doing there, but I am completely non-expert), I can say that 
his discussion on ethology (animal psychology) is uninformed 
and decades out of date.

-----Greg McColm
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