The term "omega-consistent" was introduced by Gödel, presumably because he wanted to formulate the conditions on the theory T in terms of "consistency". It is a misleading term in that there doesn't have to be anything at all wrong with an omega-inconsistent theory. For example, let T be the theory of the real field, where the quantifiers range over the real numbers. In this theory it is provable for each natural number n that n² is not equal to 2, while "for every x, x² is not equal to 2" is disprovable in the theory. This is just as it should be, since the intended interpretation of the quantifier "for every x" is "for every real number x".