Γ₀ as the Least Impredicative Ordinal: Calibrating Predicative Proof-Theoretic Strength — E8 Intelligence Research

FINDING: Ordinal analysis calibrates the proof-theoretic strength of formal systems by assigning each a proof-theoretic ordinal; Γ₀ (Feferman–Schütte) is the least impredicative ordinal, marking the limit of predicative mathematics. | MATH: Γ₀ = φ_{εΩ+1}(0) = sup{φ_α(0) : α < Γ₀}; Veblen hierarchy φ_α(β) with φ_0(β)=ω^β; ε-numbers: ε_α = φ_1(α), fixed points of β↦ω^β; Γ-numbers: Γ_α = φ1+α(0), fixed points of α↦φ_α(0). ω₁^CK = Church–Kleene ordinal = sup of all recursive ordinals = least non-recursive ordinal. Consistency strengths: PA ≡ ε₀; ATR₀ ≡ Γ₀; ID₁ ≡ φ_{εΩ+1}(0) = Γ₀; Π¹₁-CA₀ ≡ ψ₀(Ω_ω); Π¹₂-CA₀ ≡ ψ₀(Ω_{ΩΩ_...}). | CONNECTION: The Veblen hierarchy φ_α is a transfinite generalization of the golden-ratio-like fixed-point iteration x↦ω^x; ε₀ is the first fixed point of the exponential map, analogous to how φ=(1+√5)/2 is the first fixed point of x↦1+1/x. The hierarchy of fixed points (ε-numbers, Γ-numbers) mirrors the self-similar nesting seen in continued fractions and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com