{ "artifactId": "artifact.library.mathlib.first-order-compactness-theorem.v001", "candidateOnly": false, "category": "Logic and foundations", "claimBoundary": "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 models of finite subtheories need not be the same. The endpoint is not a theorem about compact topological spaces.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable {L : FirstOrder.Language.{u, v}} {T : L.Theory} : T.IsSatisfiable ↔ T.IsFinitelySatisfiable", "family": "Models and compactness", "id": "library.mathlib.first-order-compactness-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/first-order-compactness-theorem/", "entryData": "/data/library-theorems/first-order-compactness-theorem.json", "evidence": "/proofs/artifact.library.mathlib.first-order-compactness-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.first-order-compactness-theorem.v001.evidence.json", "source": "/sources/upstream/first-order-compactness-theorem/" }, "landmark": { "completeFormalType": "theorem FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable {L : FirstOrder.Language.{u, v}} {T : L.Theory} : T.IsSatisfiable ↔ T.IsFinitelySatisfiable", "concepts": [ "first-order theory", "finite subtheory", "satisfiability", "models", "compactness" ], "plainLanguage": "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.", "poster": { "alt": "An ivory logic poster states that a theory is satisfiable exactly when every finite subtheory is satisfiable, with separate models of finite subtheories feeding one whole-theory model.", "byteSize": 2766914, "caption": "Every finite part having some model is exactly enough for the complete first-order theory to have a model.", "description": "The poster distinguishes an arbitrary theory, its many finite subtheories, their potentially different models, and the single model obtained for the whole theory. A restrained proof-route strip evokes the ultraproduct construction without turning it into evidence.", "derivatives": [ { "byteSize": 124648, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:61faffbef84c3499915f623f83fac6c16a96498606ac69be7d6a3140565da729", "width": 640 }, { "byteSize": 280968, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:aa5fcc05a5f4e3969b20b03c59c14c7b4ce5b04da4be2b6e528df07f21b8160b", "width": 1024 } ], "generatedAt": "2026-07-25T06:08:46.083Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-poster-v1.png", "sha256": "sha256:7ae3d447f82f26265f33b1b03f51b18ffd53ef90eeb64b030b473e8c89a67fd1", "title": "First-order compactness at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "LOGIC · MODELS AND FINITE SUBTHEORIES\nCOMPACTNESS THEOREM FOR FIRST-ORDER LOGIC\nT is satisfiable\nif and only if\nevery finite subtheory of T is satisfiable\nEACH FINITE SUBTHEORY MAY USE A DIFFERENT MODEL\nHOW THE CHECKED ROUTE MOVES\n1 · Give each finite subtheory a model\n2 · Organize the finite subtheories by inclusion\n3 · Form a filtered ultraproduct\n4 · Obtain one model of all sentences in T\nEXACT SCOPE\nSemantic first-order compactness with nonempty models; not topological compactness." }, "visual": { "alt": "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.", "byteSize": 2613392, "caption": "Satisfiability of every finite subtheory is equivalent to satisfiability of the whole first-order theory.", "description": "An unbounded field of sentence tiles represents an arbitrary theory. Several differently shaped finite windows select finite subtheories and connect to distinct model structures; a final coherent structure receives the whole field, without suggesting that the finite-stage models were identical.", "derivatives": [ { "byteSize": 41496, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:9c8cb1aad6618f26541764b27eb74e6952a323ae672761ecf01c783930c0b0f3", "width": 640 }, { "byteSize": 110870, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:4308480ffbe9075bb254dd9b170583856d837bdb65ab78554afbb32ead7e5727", "width": 1200 } ], "generatedAt": "2026-07-25T06:08:46.083Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/first-order-compactness-theorem/theorem-schematic-v1.png", "sha256": "sha256:0f5d19a5e0eec2681a3ef060689bbd88c5e92afd0c55924f8a8534cfb6cbdbbf", "title": "First-order compactness theorem schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "Logical compactness turns consistent finite behavior into a global model and underlies many of model theory's central transfer and existence arguments." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.first-order-compactness-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "A first-order theory has a model if and only if every finite subtheory has a model.", "targetId": "target.library.mathlib.first-order-compactness-theorem.v001", "title": "Compactness Theorem for First-Order Logic", "upstreamOrigin": { "declarationName": "FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/ModelTheory/Satisfiability.lean#L100", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:0abb92d531851a57909945b740981d79a4cbb29238f2a3d21cb5fa57aa143edb", "sourceByteLength": 25036, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/ModelTheory/Satisfiability.lean", "sourceLine": 100, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }