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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.

Martin Sleziak
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Ioannis Souldatos
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  • A good place to look would be examples of objects with uncountable cardinalities other than continuum: http://mathoverflow.net/questions/44705/cardinalities-larger-than-the-continuum-in-areas-besides-set-theory –  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. –  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
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    This question and answer seem relevant. – Alex Kruckman Oct 06 '16 at 13:17
  • @AlexKruckman: If I understand you correctly you are saying that all known applications of Morley's Theorem outside of Logic were already discovered before Morley. But if the step from "categoricity" to "a strongly minimal formula with a dimension function" was always so easy, then Morley's theorem wouldn't worth so much. There must be some examples where this step is not so easy (at least for the non-logicians). – Ioannis Souldatos Oct 13 '16 at 18:49
  • @IoannisSouldatos I agree, the jump from categoricity to a strongly minimal formula and a dimension is not easy. And given an uncountably categorical theory, it could definitely be useful to conclude that it has a notion of dimension and strongly minimal geometry. The issue is that it's hard to imagine how to prove categoricity for a theory found in the wild without already having access to a notion of dimension for models of that theory. Indeed, the content of Morley's theorem is that this is always the explanation for categoricity. – Alex Kruckman Oct 14 '16 at 01:26

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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.

TimZ
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