X-Message-Number: 21702
From: 
Date: Thu, 1 May 2003 20:17:58 EDT
Subject: =?ISO-8859-1?Q?G=D6del=20and=20Physics?=

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New Scientist magazine (5 April 2003) has an article by Michael Brooks based 
on a  talk by Stephen Hawking recently in California. Below is what I sent 
New Scientist, and following that a little more.
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G del's work and its alleged implications are widely misunderstood by many, 
including your Michael Brooks and perhaps Stephen Hawking too.

G del merely showed that, in a broad class of formal systems, it is possible 
to frame a statement that is unprovable within the system. We already knew 
that, or should have; it can be shown quickly and easily, without G del's 
cumbersome approach. Formal systems have other, much more important 
limitations also--e.g. the axioms may not correspond to anything in reality; 
and the system has syntax only, without semantics.

Brooks' article suggests that "unprovable" equates with "unknowable." For a 
formal system, this is NOT true, obviously, since we (outside the system) can 
see that the G del sentence is true, although not provable within the system.

Finally, there simply is no "analogy" between the incompleteness theorem and 
physics. There may be unknowables in the real world, but you can't prove it 
by G del.
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Now, here's one simple way of proving incompleteness with no abstruse math:

We assume a sufficiently rich system which includes propositional logic and 
arithmetic. Then we assign a unique number (positive integer) to each 
possible finite string of symbols, there being also a finite number of 
symbols. This can be done in many ways, and it doesn't matter how we do it, 
so long as the number 1 is assigned to the sentence,

"Sentence number 1 is unprovable."

This obviously does the trick.

We can also generate an infinite number of unprovable (and undecidable) 
sentences in the following way: To the sentence, "Sentence number 1 is 
unprovable" we assign the number 1; to the sentence, "Sentence number 2 is 
unprovable" we assign the number two; and so on. (This would use up all the 
integers and leave all the other types of sentences without numbers if we 
stopped there, but of course we have other recourse, such as starting over 
and using labels a1, a2, .......... for the other types of sentences.

Robert Ettinger

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