TITLE: Applications of Morley's Categoricity Theorem
QUESTION [9 upvotes]: I just attended a lecture by Rami Grossberg and he mentioned that he is not aware of any applications of Morley's Categoricity Theorem. This is exactly my question.
Question: Do you know of any applications of Morley's Categoricity Theorem outside of Logic?
Morley's Categoricity Theorem If $T$ is a first-order theory in a countable vocabulary and $T$ is categorical in one uncountable cardinal, then it is categorical in all uncountable cardinals.

REPLY [3 votes]: If you want to apply the theorem without using the structure theory arising from the proof, then I claim that there cannot be an application. This is since the ordinary mathematician is not interested in comparing uncountable structures of different cardinality. In this sense Morley's Categoricity Theorem is a negative result, i.e. an uncountable structure (which is uncountably categorical) cannot be elementarily equivalent to some uncountable structure with exotic properties. Of course if you take the structure theory arising from the proof into account then the picture is completely different.