Diverges

formal source

  56def Stabilizes (c : ℝ) : Prop := defect c = 0
  57
  58/-- A configuration diverges if its defect goes to infinity. -/
  59def Diverges (c : ℝ) : Prop := ∀ C : ℝ, defect c > C
  60
  61/-! ## Self-Referential Stabilization Queries -/
  62
  63/-- A self-referential stabilization query is a configuration c that
  64    "encodes" the assertion "I do not stabilize."
  65
  66    The key property: c is "true" iff ¬Stabilizes(c).
  67
  68    We model this as: c has an associated "truth value" that is
  69    the negation of its stabilization status.
  70
  71    Formally: c is a SelfRefQuery if there exists a "decoder" function
  72    that maps c to the proposition ¬Stabilizes(c). -/
  73structure SelfRefQuery where
  74  /-- The configuration -/
  75  config : ℝ
  76  /-- The self-referential property: c encodes "I don't stabilize" -/
  77  self_ref : (defect config = 0) ↔ ¬(defect config = 0)
  78
  79/-- **THEOREM 1**: Self-referential stabilization queries are contradictory.
  80
  81    If c encodes "c ⟺ ¬Stab(c)", then assuming c has any definite
  82    stabilization status leads to contradiction.
  83
  84    This is the heart of the Gödel dissolution: such c cannot exist
  85    as consistent configurations. -/
  86theorem self_ref_query_impossible : ¬∃ q : SelfRefQuery, True := by
  87  intro ⟨q, _⟩
  88  -- q.self_ref says: (defect config = 0) ↔ ¬(defect config = 0)
  89  -- This is impossible: P ↔ ¬P has no model