Lewis is off at a philosophy convention right now. Filling in for his comments is a quote from Gödel's Theorem: An Incomplete Guide To Its Use and Abuse, by Torkel Franzén.

This idea that Gödel's Theorem does not have an ordinary mathematical proof seems to be based on a specific misunderstanding of Gödel's original proof of the first incompleteness theorem: the mistaken belief that Gödel's proof, which shows a certain arithmetical statement G depending on S to be unprovable in S if S is consistent, also shows G to be true.