  Probably the most important result about first-order languages is the Gödel completeness theorem which of course says that the set of all valid formulas of any first-order language  L  can be generated from a simple set of axioms by means of a few straightforward rules of inference. A major consequence of this theorem is that, if the formulas of  L  are coded as natural numbers in some constructive way, then the set of (codes of) valid sentences is recursively enumerable. Thus, the completeness of a first-order language implies that the set of its valid sentences is definable in a particularly simple way. It would accordingly seem reasonable, given an arbitrary language L, to turn this implication around and suggest that, if the set of valid L-sentences is not definable in some simple fashion, then no meaningful completeness result can be established for L, or, as we shall say, that L is incomplete. In this section we are going to employ this suggestion in sketching a proof that “most” infinite quantifier languages are incomplete in this sense. 