{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-real-number","semantic_family_id":"sf-real-number","type":"concept","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"實數","en":"Real number"},
  "summary":{"zh-Hant":"實數包含有理數，並以完備化補入有理數系統中缺少的極限點。"},
  "statement":{"zh-Hant":"實數系 R 可由有理數的 Cauchy 完備化建構；每個有理數都可嵌入 R，而不同建構仍須證明彼此等價。"},
  "formula":{"tex":"x\\in R","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></math>","semantic_ast":{"type":"membership","element":{"type":"symbol","name":"x","glyph":"x"},"set":{"type":"symbol","name":"R","glyph":"R"}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"aa61977ab46f2f852f17857fe89047ba4f95313cdb5d8388dfb088543542b4c6"}},
  "assumptions":[{"id":"a1","zh-Hant":"採用包含有理數嵌入的實數 Cauchy 完備化語境。"}],
  "symbols":[{"token":"x","role":"real_candidate","role_zh":"待判定或操作的數","scope":"representative_formula"},{"token":"R","role":"real_number_system","role_zh":"實數系代表記號","scope":"representative_formula"}],
  "dependencies":[{"id":"mko-rational-number","title_zh":"有理數","reason_zh":"本條目以有理數的嵌入與完備化建立實數入口。"}],
  "explanation":{"paragraphs_zh":["每個整數與有理數都可視為實數；有限小數也可轉成精確分數，因此不是另一種獨立數系。","實數可由有理數 Cauchy 序列的等價類建構；『補入極限點』是建構直覺，不取代完備性的精確定義與證明。","本 MKO 先建立有理點在實數線上的精確表示、比較與區間語言；無理數分類、序完備性與連續性留待後續。"],"common_misconception_zh":"常見誤解是把有限小數視為無理數、把近似小數當成精確值，或只說『沒有空隙』便宣稱完備性已證明。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"有限個有理點的精確重算只驗證指定表示、順序與區間，不構成實數完備性或全部無理數性質的證明。","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":"綁定 Mathlib Real 的 Cauchy completion、有理數嵌入、線性序與完備性聲明及版本證據。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-real-number.md 整理；待人工與形式化審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-23"}
}