Morley's categoricity theorem

morley_categoricity_theorem

Submitter: A. M. Berns.

Notes: A complete theory in a countable language with only infinite models that is categorical in some uncountable cardinal is categorical in every uncountable cardinal.

Source: M. Ramsey, *Morley's Categoricity Theorem* (UChicago VIGRE REU 2010), Corollary 7.3. https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/RamseyN.pdf

Informal solution: Show that categoricity in an uncountable cardinal forces the theory to be ω-stable, and then use Vaughtian pairs to prove that any two models of the same uncountable size are isomorphic and categoricity transfers to all uncountable cardinals.

theorem declaration uses `sorry`morley_categoricity_theorem (L : FirstOrder.Language.{0, 0}) (hL : L.card ≤ ℵ₀)
    (T : L.Theory) (hT : T.IsComplete)
    (hInf : ∀ M : FirstOrder.Language.Theory.ModelType.{0, 0, 0} T, Infinite M)
    {κ : Cardinal.{0}} (hκ : ℵ₀ < κ) (hcat : κ.Categorical T)
    {μ : Cardinal.{0}} (hμ : ℵ₀ < μ) :
    μ.Categorical T := byL:FirstOrder.LanguagehL:L.card ≤ ℵ₀T:L.TheoryhT:T.IsCompletehInf:∀ (M : T.ModelType), Infinite ↑Mκ:Cardinal.{0}hκ:ℵ₀ < κhcat:κ.Categorical Tμ:Cardinal.{0}hμ:ℵ₀ < μ⊢ μ.Categorical T
  sorryAll goals completed! 🐙

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