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import FormalConjecturesUtilVaught conjecture
namespace VaughtConjecture
open FirstOrder.LanguageThe number of countable models of some L-Theory T up to isomorphism
def numberOfCountableModels {L : FirstOrder.Language} (T : L.Theory) : Cardinal :=
Cardinal.mk (Quotient (Setoid.comap
(fun (model : {mt : T.ModelType // Countable mt.Carrier}) ↦
CategoryTheory.Bundled.mk model.val.Carrier model.val.struc)
equivSetoid))
The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, $\aleph_0$ or $2^{\aleph_0}$.
@[category research open, AMS 3]
theorem vaught_conjecture {L : FirstOrder.Language} (hL : Countable L.Symbols)
{T : L.Theory} (hT : T.IsComplete) :
numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum
:= L:FirstOrder.LanguagehL:Countable L.SymbolsT:L.TheoryhT:T.IsComplete⊢ numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum All goals completed! 🐙
end VaughtConjecture