{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-set",
  "semantic_family_id": "sf-set",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "集合",
    "en": "Set"
  },
  "summary": {
    "zh-Hant": "集合把指定脈絡中的若干對象視為一個整體，並以元素隸屬來描述其內容。"
  },
  "statement": {
    "zh-Hant": "在指定的論域或型別中，集合 A 由哪些對象是其元素來刻畫；具有完全相同元素的集合視為相同。"
  },
  "formula": {
    "tex": "x\\in A",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "x",
        "glyph": "x"
      },
      "set": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "fac41f5df7e2995de6d6b48a0cce188696ab53b3a4ebdbcc87c451009145c802"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "討論中的對象型別或論域已經指定。"
    },
    {
      "id": "a2",
      "zh-Hant": "A 是該脈絡中的集合，x 是候選元素。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "element_candidate",
      "role_zh": "候選元素",
      "scope": "concept"
    },
    {
      "token": "A",
      "role": "set",
      "role_zh": "集合",
      "scope": "concept"
    },
    {
      "token": "∈",
      "role": "membership_relation",
      "role_zh": "隸屬關係",
      "scope": "concept"
    }
  ],
  "dependencies": [],
  "explanation": {
    "paragraphs_zh": [
      "集合讓我們把多個對象作為一個整體討論。對指定的 x 與 A，x∈A 表示 x 是 A 的元素；這個式子是集合語言的使用介面，不是完整的集合論公理。",
      "集合的外延觀點要求：若兩個集合恰好具有相同元素，就把它們視為同一集合。元素出現的順序或重複書寫不會產生不同集合。",
      "並非每個可寫出的條件都能毫無限制地形成集合。完整理論必須說明允許的集合建構原則；本入門條目不選定 ZF、ZFC 或其他公理系統。"
    ],
    "common_misconception_zh": "常見誤解是把元素 x 與只含 x 的集合混為一談，或認為任意條件都能不受限制地定義一個集合。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本條目是概念候選稿；研究來源不是 OCME Evidence，尚未建立公理化或 Lean 形式證明。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "把入門語義映射到 Mathlib 的 Set α 與隸屬記法，並決定外延性應由哪個後續公理或定理 MKO 承擔。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-set.md 的來源與語義決定整理；內容為 OCME 原創改寫，待人工審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-08-07"
  }
}