Logic and foundations · existing Mathlib declaration

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

The theorem at a glance

Łoś's theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory and forest-green theorem poster states that a sentence is true in the ultraproduct exactly when it is true in ultrafilter-many factors, above a family of structures feeding one quotient construction and a two-way truth seam. — Sentence truth in an ultraproduct is equivalent to truth on an ultrafilter-large set of factors.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

At original pixels, all five labels are character-complete: NONEMPTY L-STRUCTURES M a; (u : Filter α).Product M; ULTRAPRODUCT ⊨ φ; ⇔; ∀ᶠ a in u, M a ⊨ φ. Both satisfaction symbols have the required double horizontal stroke, α and φ are distinct Greek glyphs, and the f in ∀ᶠ is visibly raised. Six unnumbered structure boxes each connect through a right-pointing construction arrow to the complete product bundle and central product/quotient chamber. Rust and gold box borders distinguish factors outside and inside the ultrafilter-large satisfying region without implying a count-based vote. The only two-way symbol is the centered truth equivalence; connectors do not overlap labels, nothing is cropped, and the Scientific journal-figure layout is precise and attractive. — The full family forms the ultraproduct, while sentence truth is detected by the ultrafilter-large set of satisfying factors.

Complementary intuition

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

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 ⊨ φ

Selected evidence declaration text

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 ⊨ φ

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

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.

Provenance

Origin and evidence stay separate

Existing declaration: FirstOrder.Language.Ultraproduct.sentence_realize in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.los-theorem.v001
Source handling: Verified upstream reference; no mirrored source package