{
  "artifactId": "artifact.library.mathlib.downward-lowenheim-skolem-theorem.v001",
  "candidateOnly": false,
  "category": "Logic and foundations",
  "claimBoundary": "This target indexes Mathlib's Downward Löwenheim–Skolem theorem for a nonempty first-order L-structure M, a prescribed seed set s, and an infinite cardinal κ. After the explicit universe lifts, the seed and language cardinalities must be at most κ and κ must be at most the ambient cardinality. The endpoint gives an elementary L-substructure containing s with lifted cardinality exactly κ. It is not an unrestricted countable-submodel claim or the upward theorem.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem FirstOrder.Language.exists_elementarySubstructure_card_eq (L : FirstOrder.Language.{u, v}) {M : Type w} [Nonempty M] [L.Structure M] (s : Set M) (κ : Cardinal.{w'}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} κ) (h3 : Cardinal.lift.{w'} L.card ≤ Cardinal.lift.{max u v} κ) (h4 : Cardinal.lift.{w} κ ≤ Cardinal.lift.{w'} #M) : ∃ S : L.ElementarySubstructure M, s ⊆ S ∧ Cardinal.lift.{w'} #S = Cardinal.lift.{w} κ",
  "family": "Elementary submodels and cardinality",
  "id": "library.mathlib.downward-lowenheim-skolem-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/downward-lowenheim-skolem-theorem/",
    "entryData": "/data/library-theorems/downward-lowenheim-skolem-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.downward-lowenheim-skolem-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.downward-lowenheim-skolem-theorem.v001.evidence.json",
    "source": "/sources/upstream/downward-lowenheim-skolem-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem FirstOrder.Language.exists_elementarySubstructure_card_eq (L : FirstOrder.Language.{u, v}) {M : Type w} [Nonempty M] [L.Structure M] (s : Set M) (κ : Cardinal.{w'}) (h1 : ℵ₀ ≤ κ) (h2 : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} κ) (h3 : Cardinal.lift.{w'} L.card ≤ Cardinal.lift.{max u v} κ) (h4 : Cardinal.lift.{w} κ ≤ Cardinal.lift.{w'} #M) : ∃ S : L.ElementarySubstructure M, s ⊆ S ∧ Cardinal.lift.{w'} #S = Cardinal.lift.{w} κ",
    "concepts": [
      "first-order structure",
      "elementary substructure",
      "Skolem functions",
      "cardinality",
      "universe lift",
      "model theory"
    ],
    "plainLanguage": "Start with a nonempty first-order structure M, a set s of elements that must be retained, and an infinite size κ. If both the seed and the language fit within κ, and κ itself fits within M, then M has an elementary substructure S that contains every element of s and has exactly size κ, with Mathlib's universe lifts made explicit.",
    "poster": {
      "alt": "An ivory model-theory poster lists the infinite-cardinal, seed, language, and ambient bounds; nests an elementary substructure S containing s inside M; and states that the lifted cardinality of S equals the lifted κ.",
      "byteSize": 2487147,
      "caption": "When the seed and language fit inside an infinite κ and κ fits inside M, an elementary substructure containing the seed exists with exactly that lifted cardinality.",
      "description": "The poster keeps all four hypotheses visible above a nested M and S diagram, then follows the checked source route through Skolem functions, an ambient κ-sized set, closure in the expanded language, the elementary reduct, and the final cardinal squeeze.",
      "derivatives": [
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          "height": 960,
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          "width": 640
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/downward-lowenheim-skolem-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:0059b2fa4b6a799e8d1d143e3b03547bdb60402594670db2473723a3fd5a6271",
          "width": 1024
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      ],
      "generatedAt": "2026-07-25T16:26:56-04:00",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/downward-lowenheim-skolem-theorem/theorem-poster-v1.png",
      "sha256": "sha256:64e0724318ca015a5c82852122808cb45081d9aef4c5df0292ec1edc81c2c137",
      "title": "Downward Löwenheim–Skolem theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "MODEL THEORY · ELEMENTARY SUBSTRUCTURES\nDOWNWARD LÖWENHEIM–SKOLEM THEOREM\nL-structure M · seed s ⊆ M · cardinal κ\nℵ₀ ≤ κ\nAFTER THE STATED UNIVERSE LIFTS\n#s ≤ κ · L.card ≤ κ · κ ≤ #M\nTHEN THERE EXISTS S\nS is an elementary L-substructure of M\ns ⊆ S · lifted #S = lifted κ\nHOW THE CHECKED ROUTE MOVES\n1 · Add Skolem functions to L\n2 · Choose κ many elements of M and adjoin s\n3 · Close under the expanded functions\n4 · Take the elementary L-reduct and squeeze its size to κ\nEXACT SCOPE\nNonempty M and all four bounds required. No canonical or algorithmic S."
    },
    "visual": {
      "alt": "A dark-green ambient structure surrounds an ivory elementary substructure with the entire green seed cluster at its center, while matched relation arcs and three finite point clusters compare ambient, intermediate, and compact scales.",
      "byteSize": 2965695,
      "caption": "The seed is retained inside an elementary substructure whose chosen infinite size accommodates both the seed and the language and remains within the ambient structure.",
      "description": "A wordless nested model-theory engraving distinguishes a broad ambient structure, an inner elementary substructure, and a prescribed seed cluster wholly inside the inner region. Matched witness arcs signal elementarity, while three finite-looking point clusters compare broad, intermediate, and compact scales without substituting an infinity symbol for the theorem's cardinal conditions.",
      "derivatives": [
        {
          "byteSize": 61334,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/downward-lowenheim-skolem-theorem/theorem-schematic-v2-640.webp",
          "sha256": "sha256:a2ca86dc7444e04ba73f82dd2f6cade2b99de50d55dbd730229613f24ffcc5a8",
          "width": 640
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        {
          "byteSize": 188340,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/downward-lowenheim-skolem-theorem/theorem-schematic-v2-1200.webp",
          "sha256": "sha256:6b11c95b38ce7d828bd3b865d7d208b75faf4cba0aa73e41f390b1af620e921e",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T16:26:56-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/downward-lowenheim-skolem-theorem/theorem-schematic-v2.png",
      "sha256": "sha256:c8cbb5753ec01963f0c180b0e2a1b0c7a8ccb9f77cb123de720b3948320db178",
      "title": "Downward Löwenheim–Skolem theorem schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The theorem is a central size-control principle of model theory: elementary truth can be retained inside a substructure whose cardinality is chosen exactly within explicit seed, language, and ambient bounds."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Downward Löwenheim–Skolem theorem or Mathlib's declaration.",
    "The theorem does not give a countable elementary substructure unless the displayed hypotheses permit choosing the relevant cardinal to be countable.",
    "The selected declaration is not the upward Löwenheim–Skolem theorem and does not construct a larger elementary extension.",
    "The theorem does not make the elementary substructure unique, canonical, proper, or algorithmically computable.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.downward-lowenheim-skolem-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 sufficiently large nonempty structure has an elementary substructure of any prescribed infinite cardinality between the seed-and-language bounds and the ambient size.",
  "targetId": "target.library.mathlib.downward-lowenheim-skolem-theorem.v001",
  "title": "Downward Löwenheim–Skolem Theorem",
  "upstreamOrigin": {
    "declarationName": "FirstOrder.Language.exists_elementarySubstructure_card_eq",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/ModelTheory/Skolem.lean#L122",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:85c5fa3fd4f76381b02d46a8bfd515cceb14254543e0afae95004c1fc07137b0",
    "sourceByteLength": 6227,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/ModelTheory/Skolem.lean",
    "sourceLine": 122,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}