Gentzen's ε₀: The Minimal Ordinal for Proving Peano Arithmetic Consistent — E8 Intelligence Research

FINDING: Transfinite induction up to ε₀ is the minimal well-ordering principle required to prove the consistency of Peano Arithmetic (PA), extending Gentzen's original proof by clarifying the ordinal's role as a measure of proof-theoretic strength. MATH: - ε₀ = sup{ ω, ω^ω, ω^(ω^ω), … } = the first fixed point of the exponential map α → ω^α. - Gentzen's consistency proof for PA: PA ⊢ Con(PA) ⇔ TI(ε₀) (transfinite induction up to ε₀). - Ordinal notation system: ε₀ = φ₁(0) in the Veblen hierarchy. - Key constant: The proof-theoretic ordinal of PA is ε₀. No smaller ordinal suffices. CONNECTION: - ε₀ is not directly a ratio like 0.618, but its structure mirrors self-similar recursion (fixed-point iteration) seen in golden-ratio spirals and logarithmic scaling. - The ordinal ω^ω^… (tower of ω) parallels the infinite nesting of base-60 sexagesimal place values (e.g., 60^60^…), hinting at a deep link between ordinal hierarchies and base-60 positional systems used in ancient astr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com