Vaught conjecture
Vaught conjecture
Inferred from the community field; finer classification has not been reviewed.
Problem statement
vaught_conjecture
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, or .
Lean 4
Formal statement
Lean 4/-
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-/
import FormalConjecturesUtil
/-!
# Vaught conjecture
*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Vaught_conjecture)
-/
namespace VaughtConjecture
open FirstOrder.Language
/-- The 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
:= by sorry
end VaughtConjecture
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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