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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.

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  • $\begingroup$ A good place to look would be examples of objects with uncountable cardinalities other than continuum: mathoverflow.net/questions/44705/… $\endgroup$ – Matt F. Sep 30 '16 at 14:42
  • $\begingroup$ @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. $\endgroup$ – Ioannis Souldatos Sep 30 '16 at 15:41
  • $\begingroup$ 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. $\endgroup$ – Matt F. Sep 30 '16 at 15:56
  • $\begingroup$ @MattF. "to show a contradiction in ZFC" wait, what? Can you explain what you mean? $\endgroup$ – Noah Schweber Sep 30 '16 at 21:33
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    $\begingroup$ This question and answer seem relevant. $\endgroup$ – Alex Kruckman Oct 6 '16 at 13:17
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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.

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