A set of sentences is ω‎-complete if whenever it deductively yields every instance of a universal generalization, it also yields the generalization itself. It is ω‎-consistent if when it yields every instance of such a generalization, it does not also yield that there is an instance not satisfying it. The notions are particularly important in connection with systems generalizing over all the natural numbers, and form a key element in the proof of Gödel’s theorem.