In his 1965 paper *Omitting Classes of Elements* (found in *The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley*, published by North-Holland Publ. Co., Amsterdam (1965)), Morley proved the following omitting types theorem:

For a complete countable theory $T$ and a 1-type $\Sigma$, if for every $\alpha<\omega_1$ there exists a model of $T$ with cardinality $>\beth_\alpha$ which omits $\Sigma$ then there is a model of $T$ omitting $\Sigma$ in every infinite cardinality.

Morley then claims without proof that this theorem still holds if $T$ is of cardinality $\lambda>\aleph_0$ by replacing $\omega_1$ in the statement by $(2^\lambda)^+$, but the proof given in the paper appears to fail for uncountable languages. Specifically, even after replacing $\omega_1$ with $(2^\lambda)^+$, the inductive step of the proof (Theorem 3.1 in the paper) fails at the limit cases.

So is there a way to prove that this statement indeed holds for uncountable languages? Or is the statement actually false when $T$ is uncountable?