Existing in Mathlib · Logic and foundations

Compactness Theorem for First-Order Logic

Take any collection of first-order sentences, possibly infinite. If every finite selection from that collection can be satisfied in some nonempty structure, then there is one nonempty structure satisfying the entire collection at once.

first-order theory
finite subtheory
satisfiability
models
compactness

Exact theorem

Exact Mathlib statement

theorem FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable {L : FirstOrder.Language.{u, v}} {T : L.Theory} : T.IsSatisfiable ↔ T.IsFinitelySatisfiable

The theorem at a glance

First-order compactness at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory logic poster states that a theory is satisfiable exactly when every finite subtheory is satisfiable, with separate finite models feeding one whole-theory model. — Every finite part having some model is exactly enough for the complete first-order theory to have a model.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Many finite selections of sentence tiles each have their own model, and a coherent gold passage leads to one model satisfying the entire infinite theory. — Satisfiability of every finite subtheory is equivalent to satisfiability of the whole first-order theory.

Why it matters

A mathematical landmark

Logical compactness turns consistent finite behavior into a global model and underlies many of model theory's central transfer and existence arguments.

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

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 semantic compactness theorem for arbitrary first-order languages. Here finite satisfiability means that every finite set of sentences contained in the theory has some nonempty model; those finite models need not be the same. The endpoint is not a theorem about compact topological spaces.

Proof Atlas did not originate the Compactness Theorem or Mathlib's declaration.
The theorem does not say that one fixed finite-stage model already satisfies the whole theory.
The selected endpoint is semantic first-order compactness, not topological compactness.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.first-order-compactness-theorem.v001
Source: Open the pinned upstream reference