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Level A · Machine-checkable Hard Logic & formalisation P-vaught-conjecture

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, aleph_0 or 2^aleph_0.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-vaught-conjecture,
  title        = {Vaught conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/vaught-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

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 .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.VaughtConjecture.

theorem vaught_conjecture {L : FirstOrder.Language} (hL : Countable L.Symbols)
                          {T : L.Theory} (hT : T.IsComplete) :
  numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.