I found the NPR story ... search for
"Goedel" on npr.org and you'll get one hit. ("Gödel" doesn't get any matches,
sigh ...). There's a Real Audio file containing the segment.
The mathematician is Keith Devlin, the "math guy". He says, and I freely paraphrase ... Goedel is hovering behind the semantic web ... to do something like what Tim Berners-Lee is trying to do, you have to code in axioms and rules of deduction. When you do that, there are things that are true that users want to know that can't be deduced from the axioms. The Semantic Web will fail to do some of the things that the users want it to do, and there is no escape from this within the rules of logic.
Other fun facts about Gödel (courtesy Dr. Devlin)... he had to be restrained from explaining the logical inconsistencies in the US Constitution at the ceremony naturalizing him as a US citizen ... and starved to death because he feared that someone was poisoning him, which (quite logically!) led him to avoid food.
Based on my 20-year-old recollection of "Gödel, Escher, Bach" and "Gödel's Proof", and discussions thereof in a long-ago AI course ... I'd guess that Gödel's theorem would suggest that there are "truths" that a Google-like search engine (that gives credibility to assertions that are widely quoted) could find that a Semantic Web logical inference engine (that gives credibility to assertions that can be proven) couldn't find. Whethere there are *interesting* truths that would suffer this fate is another matter ...
The bigger problem with the SW, as many have noted, is that Gödel will never hava a chance to screw up the works, because he only talks about the incompleteness of CONSISTENT sets of axioms. Getting consistency in the SW's vast network of RDF metadata will be a monumental problem, and ANYTHING can be proven in with an inconsistent set of axioms (as my poor remaining neurons dimly recall my higher education).
So, at best the SW will have to employ some heuristics for finding useful axioms to feed into a logical inference engine. Whether this is worth the cost is another matter.