Existing in Mathlib · Logic and foundations
Łoś's Theorem
Build an ultraproduct from a family of nonempty structures and an ultrafilter on their indices. A first-order sentence is true in the resulting ultraproduct exactly when the set of indices whose structures satisfy that sentence belongs to the ultrafilter.
- first-order language
- ultrafilter
- ultraproduct
- sentence satisfaction
- model-theoretic transfer
Exact theorem
Exact Mathlib statement
theorem FirstOrder.Language.Ultraproduct.sentence_realize {α : Type*} {M : α → Type*} {u : Ultrafilter α} {L : FirstOrder.Language.{u, v}} [∀ a, L.Structure (M a)] [∀ a : α, Nonempty (M a)] (φ : L.Sentence) : (u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φThe theorem at a glance
Łoś's theorem at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Łoś's theorem at a glance
Detailed visual description
The poster keeps the exact sentence-level endpoint prominent. Every displayed factor feeds the construction, a gold contour distinguishes an ultrafilter-large satisfying region without implying a vote, and a balanced two-way mark communicates the if-and-only-if truth transfer.
Statement structure
Statement and scope
Read the exact Mathlib declaration
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
Łoś’s Theorem — scientific diagram
Detailed visual description
The left column shows seven differently shaped first-order structure glyphs followed by continuation dots. A strand from every displayed factor enters the product braid, passes through a single quotient ring, and reaches one ultraproduct. Separately, the satisfying-index subset is a card labeled Sφ, and the membership symbol places that entire card in a stacked collection of subsets labeled u. The drawing represents ultrafilter membership structurally; no count, fraction, area, probability, measure, or numerical majority encodes largeness.
Why it matters
A mathematical landmark
Łoś's theorem is the central transfer principle for ultraproducts: first-order truth in the quotient structure is governed exactly by ultrafilter-large truth among the factors. It is foundational to compactness arguments, nonstandard models, and modern model theory.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This target indexes Mathlib's sentence-level form of Łoś's theorem. For an ultrafilter u and a family M of nonempty L-structures, the ultraproduct satisfies a sentence φ exactly when M a satisfies φ for u-almost every a. “u-almost every” means membership in the ultrafilter; it is not numerical majority, probability, or measure. The selected endpoint does not itself display parameters for formulas with free variables.
- Proof Atlas did not originate Łoś's theorem or Mathlib's declaration.
- The theorem does not require every factor to satisfy the sentence; it requires the satisfying indices to belong to the ultrafilter.
- The selected endpoint is for sentences and does not directly state the more general formula-with-parameters result used in its proof.
- The theorem does not identify ultrafilter membership with finite majority, probability, or measure.
- The generated explanation and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
FirstOrder.Language.Ultraproduct.sentence_realizein mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.los-theorem.v001