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[XML-DEV Mailing List Archive Home] [By Thread] [By Date] [Recent Entries] [Reply To This Message] FW: NPR, Godel, Semantic Web
 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@h...] 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. _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com 
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