A bio of this mathematician can be found at http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html
"Goedel" is the standard ASCII-fication of "Gödel", but the name is commonly misspelled "Godel".
Click here to see related pages for Gödel's evil twin: http://c2.com/cgi/wiki?search=Godel
Gödel is most famous for "Gödel's Incompleteness Theory". This states that any attempt to describe Gödel's Incompleteness Theory will always be incomplete.
Actually it states that any complete formal system can not describe itself and therefore is actually incomplete.
The above gloss is a little loose. It is not difficult to create formal systems that do "describe themselves". A more precise version is (if my memory is working): Any formal system that includes integer arithmetic and is consistent, contains true statements that cannot be proved within that formal system.
There is a much better gloss on the GoedelsIncompletenessTheorem page. Follow the 'evil twin' link at the top of this page for other discussions on Goedel.SamuelDelany (TheEinsteinIntersection?) came up with this inspired gloss: "There are more things in heaven and earth than are dreamt of in your philosophy."
The quote is from Hamlet, in case anyone's wondering.
(And this sort of application of the quotation is not new. It was applied to the RussellParadox back when that was new.)
People often forget Kurt's Completeness theorem: All true statements in the lower predicate calculus can be proved within the lower predicate calculus.
GeneralSystemantics (non-fiction book by Gall) is solidly based on the incompleteness results: In any system there are unpredictable behaviors because prediction is a kind of proof and most systems are kind of logical and include counting.