Theorems · Theorem · logic and foundations
FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable
- 1000+ list: Compactness theorem
∀ {L : FirstOrder.Language} {T : L.Theory}, T.IsSatisfiable ↔ T.IsFinitelySatisfiableThe Compactness Theorem of first-order logic: A theory is satisfiable if and only if it is finitely satisfiable.
- Defined in
- Mathlib.ModelTheory.Satisfiability
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetproof · cited by 13,712
- SetLike.coeproof · cited by 8,199
- Set.Elemproof · cited by 7,166
- Filter.atTopproof · cited by 2,405
- FirstOrder.Languagestatement and proof · cited by 1,084
- Finset.mapproof · cited by 747
- Nonempty.someproof · cited by 340
- Ultrafilter.toFilterproof · cited by 172
- FirstOrder.Language.Theorystatement and proof · cited by 154
- Function.Embedding.subtypeproof · cited by 128
- FirstOrder.Language.Sentenceproof · cited by 127
- Finset.coe_mapproof · cited by 114
Cited by3
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Theory.models_iff_finset_modelsproof · cited by 1
- FirstOrder.Language.Theory.isSatisfiable_directed_union_iffproof · cited by 1