FW: NPR, Godel, Semantic Web

["Bullard, Claude L (Len)" <clbullar@ingr.com>]

Forwarded by request

Len 
http://www.mp3.com/LenBullard

Ekam sat.h, Vipraah bahudhaa vadanti.
Daamyata. Datta. Dayadhvam.h

-----Original Message-----
From: Jay Zhang [mailto:jayzhangsj@hotmail.com]

I put in quite some time studying Math Logic 20 years ago.

When we discuss Goedel in the context of Semantic Web, we
need to understand it better than heresay but intuitive
enough to apply. It is not hard, only hindered by
exagerated claims vaguely based on it.

In a nutshell, it says: if a first-order logic (understand
it as a language design that makes statements) is
sophisticated enough to "paraphrase" number theory, there
can never be an axiom system that is complete for this
logic (language): i.e., answer true or false for any
statement expressed in this language.

Let's clarify a few points here before stretching it:

1. "First order" means we apply "for any" and "exists"
qualitatives on the variables in our statements.
Increasing "order" (apply qualitatives on predicates)
does not increase expressiveness or complexity.

2. "paraphrase" means the language does not have to talk
about number theory directly. As long as its syntax
constructs can be made complex enough.

3. "paraphrase Number Theory" essentially means that
math induction (true for 1, true after increament, then
true for any) is expressed/paraphrased.

The proof is that the number theory can be used to
paraphrase the paradox that is "This statement is
false" using properties of primes and powers. As Tim
Bray pointed out that this is hardly conclusion
we need to draw by Semantic Web.

However, that sending mafia guy to jail with tax evasion
charge does not mean he did not kill. This theorem kind
of pinpoint a point you cannot reach with any formal
axiom system. When attempting to build a "good enough"
axiom system, the achievable seems to be quite limited.
That is why mathematicians are so poorly paid:-)

Without a system of stereotypes ("for any" and "there
always exists") to help us draw conclusions, a logic
is only a brute force search algorithm on data. We
failed to find a magic.

The Semantic Web could hit the wall of Goedel if it
attempts to get meta-conclusions. Without
meta-conclusions to work on, are we looking at a
data search framework on the Web? In that case,
inefficiency of formal deduction is an issue.

Goldbach is not an example. Continum problem is more
relevant (is there a set denser than the set of
rational numbers but thinner than the set of real
numbers?), which is unprovable in layman's terms.
NP problem is also a potential candidate (but not
quite, with evidences so far).

Would you please repost this to the xml-dev mailing
list for me? I have problem posting these days.

Jay Zhang, Ph.D.
IntermicsTech, Inc.


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