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 n^{2} is not equal to 2, while "for
every x, x^{2} 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".