{ "artifactId": "artifact.library.mathlib.cantor-theorem.v001", "candidateOnly": false, "category": "Logic and set theory", "claimBoundary": "This target indexes Mathlib's diagonal theorem that no function from a type to its power set is surjective. It does not separately state the cardinal inequality or claim a new proof.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Function.cantor_surjective {α : Type*} (f : α → Set α) : ¬Function.Surjective f", "family": "Diagonal arguments", "id": "library.mathlib.cantor-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/cantor-theorem/", "entryData": "/data/library-theorems/cantor-theorem.json", "evidence": "/proofs/artifact.library.mathlib.cantor-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.cantor-theorem.v001.evidence.json", "source": "/sources/upstream/cantor-theorem/" }, "landmark": { "completeFormalType": "theorem Function.cantor_surjective {α : Type*} (f : α → Set α) : ¬Function.Surjective f", "concepts": [ "power set", "surjectivity", "diagonal argument", "fixed point", "complementation" ], "plainLanguage": "Given any attempted list of all subsets of a type, form the subset that reverses membership at each element's own listed position. That diagonal subset cannot appear anywhere in the list.", "poster": { "alt": "An ivory logic poster shows a green domain, cobalt subset folio, gold diagonal witness, the exact non-surjectivity statement, fixed-point bridge, and complement obstruction.", "byteSize": 2967998, "caption": "No mapping from a type reaches every subset of that same type.", "description": "The poster makes the hypothetical enumeration, the diagonal subset, and the complement fixed-point contradiction explicit while keeping the formal endpoint at non-surjectivity onto Set α.", "derivatives": [ { "byteSize": 131566, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-poster-v2-640.webp", "sha256": "sha256:30982f6b47ebe74c346ed35237f65525efd5d4b0bf4fd14468337f9fc3f18baa", "width": 640 }, { "byteSize": 320932, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-poster-v2-1024.webp", "sha256": "sha256:45defb51019d5cb8d5f7593652a20450722ce59e0444862df85d9550da1f3a74", "width": 1024 } ], "generatedAt": "2026-07-22T04:35:06Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-poster-v2.png", "sha256": "sha256:768bf762b93fcf6c8e7122a885bb296318540b3b768a68e4de4e09c7680a64ab", "title": "Cantor's Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "LOGIC · DIAGONAL ARGUMENT\nCANTOR'S THEOREM\nNo function reaches every subset of its own domain\nf : α → Set α is never onto\nTHE DIAGONAL SUBSET ESCAPES EVERY ENUMERATION\nHOW THE CHECKED ROUTE MOVES\n1 · If f were onto, every endofunction would have a fixed point\n2 · Complementation has no fixed point\nEXACT SCOPE\nNo function α → Set α is onto. No separate cardinal inequality is stated." }, "visual": { "alt": "A green source constellation maps into layered cobalt subsets while one recurring gold diagonal subset escapes through a single restrained red contradiction fracture.", "byteSize": 2807896, "caption": "The diagonal subset reverses membership at each proposed preimage, so no enumeration reaches every subset.", "description": "A single mapping tapestry connects source elements to candidate subsets. The gold diagonal construction follows those assignments and reverses each self-membership decision, making its own subset unavailable as an image of the function.", "derivatives": [ { "byteSize": 62758, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:15998c5fcac5393faadfd8a4afa8097bf1c93d5e75e281c5d9c134210b8cda0d", "width": 640 }, { "byteSize": 177856, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:29f9266d8f1ca3f5adc83c3c604bb118e5ab76d3c5d95e6b90dde4f4fc01e68f", "width": 1200 } ], "generatedAt": "2026-07-21T23:50:51Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cantor-theorem/theorem-schematic-v1.png", "sha256": "sha256:de3fb2eee42f5cfd8385c9041c5f1b81db3b199ad313eb43245cc84b30a6a2bc", "title": "Cantor's Theorem schematic", "width": 1536 }, "walkthrough": { "introduction": "Mathlib proves that no function from a type to its power set is surjective by first showing that a hypothetical surjective family would force every endofunction to have a fixed point, then applying that bridge to complementation, which has no fixed point.", "pages": [ { "alt": "An engraved green source constellation maps into a cobalt folio while one recurring gold diagonal thread selects a candidate fixed point.", "byteSize": 3461434, "caption": "Evaluating a surjective family on its own index constructs an element fixed by any chosen endofunction.", "description": "The checked helper exists_fixed_point_of_surjective assumes that a family f : α → α → β is surjective. For an endofunction g : β → β, surjectivity selects an index for the diagonal function a ↦ g (f a a); evaluating at that index produces a fixed point of g.", "derivatives": [ { "byteSize": 62682, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-01-v1-640.webp", "sha256": "sha256:4076c1efb40b3e1592ca75939b63c705ccdaff91c62b894032a2763492ad317b", "width": 640 }, { "byteSize": 240462, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-01-v1-1200.webp", "sha256": "sha256:9fd1ca4c6342290424b5a2f5057edb2bdf281aacca7522019dfdbd5637d58f2f", "width": 1200 } ], "generatedAt": "2026-07-21T23:28:42Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-01-v1.png", "sha256": "sha256:ecfbee430dde4d16105e70358ebdf2b431ba7ece792367781c4a933c02c92cc8", "title": "A surjection would force every endofunction to have a fixed point", "width": 1536, "equation": "Surjective f → ∀ g : β → β, ∃ x, g x = x", "kind": "proof_walkthrough_page", "sequenceIndex": 1 }, { "alt": "The same green constellation, cobalt folio, and gold diagonal thread culminate in an ivory-to-charcoal complement reversal marked by one restrained red fracture.", "byteSize": 2985393, "caption": "Applying the fixed-point bridge to set complementation would make a subset equal to its own complement, an impossibility.", "description": "Function.cantor_surjective feeds the hypothetical surjection f : α → Set α and complementation into the helper. The resulting equality would identify a set with its complement, so not_iff_self closes the contradiction and rejects surjectivity.", "derivatives": [ { "byteSize": 49708, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-02-v1-640.webp", "sha256": "sha256:5b900fc13591498eafc963a1e1e5c5de48eebb111873ca8d8dad91446896678f", "width": 640 }, { "byteSize": 171412, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-02-v1-1200.webp", "sha256": "sha256:f52c1ec57d642060c54006d01da7ffd7f00612b4089179e84fcd00fe22705c31", "width": 1200 } ], "generatedAt": "2026-07-22T01:02:31Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/cantor-theorem/proof-walkthrough-02-v1.png", "sha256": "sha256:1f0f7b167245f28e289bde50a27d1169a266efb9448a2b658f1f4cf5a1b26519", "title": "Complementation reverses membership at the diagonal", "width": 1536, "equation": "∀ (f : α → Set α), ¬ Function.Surjective f", "kind": "proof_walkthrough_page", "sequenceIndex": 2 } ], "title": "Cantor's diagonal contradiction" }, "whyLandmark": "Cantor's diagonal argument transformed the study of infinity and became a reusable template across logic, computability, and mathematics. The selected Mathlib theorem captures the exact non-surjectivity statement." }, "nonClaims": [ "Proof Atlas did not originate Cantor's Theorem or Mathlib's declaration.", "The selected declaration is non-surjectivity onto Set α, not a separately stated cardinal inequality or an assertion that no functions exist.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.cantor-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "2 of 3 presentation reviews recorded", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "No function from a type to its power set is surjective.", "targetId": "target.library.mathlib.cantor-theorem.v001", "title": "Cantor's Theorem", "upstreamOrigin": { "declarationName": "Function.cantor_surjective", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Logic/Function/Basic.lean#L258", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:5dffbb69147f9bf2cfc8de6083b7eab88d0b297762052e5e81d19a77f346dc97", "sourceByteLength": 49259, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Logic/Function/Basic.lean", "sourceLine": 258, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }