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?



How to extend Morley's omitting type theorem to uncountable languages?
The key step in the proof is to show that given a sequence $(a\_i : i < \beth\_{(2^{|T|})^+})$, there is a sequence $(b\_i : i < \omega)$ of indiscernibles such that for every $m$ there are $i\_0 < i\_1 < ... < i\_{m-1}$ with
$$tp(b\_0...b\_{m-1})=tp(a\_{i\_0}...a\_{i\_{m-1}}).$$


The proof of this assertion (in this generality) can be found in say Tent and Ziegler's book "A Course in Model Theory", or in Casanovas' book "Simple Theories and Hyperimaginaries".


The full proof of Morley's omitting types theorem can be found in the appendix of Baldwin's book Categoricity (I think it is even available for free from Baldwin's webpage.)


