VaughtConjecture.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Vaught conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

def VaughtConjecture.numberOfCountableModels The number of countable models of some L-Theory T up to isomorphism line 33 source
Source documentation
The number of countable models of some L-Theory T up to isomorphism

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: FirstOrder.Language, FirstOrder.Language.Theory, Cardinal
Proof uses: Cardinal.mk, Quotient, Subtype, FirstOrder.Language.Theory.ModelType, Countable, Setoid.comap, CategoryTheory.Bundled, FirstOrder.Language.Structure, CategoryTheory.Bundled.mk
4 instances, coercions and auxiliary constants left out.
```
def numberOfCountableModels {L : FirstOrder.Language} (T : L.Theory) : Cardinal :=
…
```
theorem VaughtConjecture.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 sorry line 44 source
Source documentation
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 $2^{\aleph_0}$.

depends on sorry what this means
Statement uses: FirstOrder.Language, Countable, FirstOrder.Language.Symbols, FirstOrder.Language.Theory, FirstOrder.Language.Theory.IsComplete, Or, LE.le, Cardinal, VaughtConjecture.numberOfCountableModels, Cardinal.aleph0, Eq, Cardinal.continuum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: VaughtConjecture.numberOfCountableModels