Hauser, Chomsky and Fitch (2002) have speculated that the language faculty narrowly construed (LFN) is comprised of the core mechanisms of recursion. This paper examines one of the founding results in recursion theory, Gödel's Incompleteness Theorem, in the context not of logical theories, but grammatical theories. It is demonstrated that when recursion is assumed to be a core property of natural language, an argument analogous to Gödel's exists which, given any theory of grammar G, establishes the existence of a grammatical sentence which G cannot show is either grammatical or ungrammatical.