# Applications of Morley's Categoricity Theorem

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.

• A good place to look would be examples of objects with uncountable cardinalities other than continuum: mathoverflow.net/questions/44705/… – Matt F. Sep 30 '16 at 14:42
• @MattF. I have to look Charles Staat's answer under the question you linked. I fail to see why structures with size continuum (or less if CH fails) are not good examples for Morley's Theorem. – Ioannis Souldatos Sep 30 '16 at 15:41
• The theorem needs two uncountable cardinalities. It is not a requirement that one be larger than the continuum -- but an example where one cardinality is continuum and one is provably less than continuum would be interesting enough to show a contradiction in ZFC. – Matt F. Sep 30 '16 at 15:56
• @MattF. "to show a contradiction in ZFC" wait, what? Can you explain what you mean? – Noah Schweber Sep 30 '16 at 21:33
• This question and answer seem relevant. – Alex Kruckman Oct 6 '16 at 13:17