law_of_existence

formal source

 229def Exists (x : ℝ) : Prop := 0 < x ∧ J x = 0
 230
 231/-- The Law of Existence: x exists ⟺ x = 1. -/
 232theorem law_of_existence {x : ℝ} (hx : 0 < x) : Exists x ↔ x = 1 := by
 233  simp only [Exists, J_eq_zero_iff hx, and_iff_right hx]
 234
 235/-- Unity is the unique existent. -/
 236theorem unity_is_unique_existent : ∀ x : ℝ, Exists x ↔ x = 1 := by
 237  intro x
 238  by_cases hx : 0 < x
 239  · exact law_of_existence hx
 240  · simp only [Exists]
 241    constructor
 242    · intro ⟨hpos, _⟩; exact absurd hpos hx
 243    · intro heq; subst heq; exact ⟨one_pos, by simp [J]⟩
 244
 245/-! ## Deriving MP from Cost -/
 246
 247/-- **Meta-Principle (Derived)**: "Nothing cannot recognize itself."
 248
 249In the cost framework, "Nothing" corresponds to the limit x → 0.
 250Recognition requires a finite cost, but J(0) → ∞, so recognition
 251of "Nothing" by "Nothing" would require infinite cost—impossible.
 252
 253This makes MP a **derived theorem**, not a primitive axiom. -/
 254theorem mp_from_cost :
 255    ∀ M : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → J x > M := by
 256  exact J_arbitrarily_large_near_zero
 257
 258/-- Alternative formulation: No finite-cost state can approach Nothing. -/
 259theorem nothing_costs_infinity :
 260    ¬∃ C : ℝ, ∀ x, 0 < x → J x ≤ C := by
 261  push_neg
 262  intro C