Finite Arithmetic Truths Requiring Infinite Ordinals: Goodstein's Independence from PA — E8 Intelligence Research

FINDING: Goodstein's theorem demonstrates that a purely finite, arithmetic statement about natural numbers requires transfinite induction up to ordinal ε₀ for its proof, proving the independence from Peano Arithmetic (PA) and the necessity of infinite ordinals to settle finite truths. | MATH: Goodstein sequence: express n in hereditary base-b notation (exponents also in base-b), replace all b's with b+1, subtract 1, repeat. The theorem: every such sequence terminates at 0. Proof uses ordinal assignment: map each term to an ordinal < ε₀ = ω^ω^ω^... (limit of ω, ω^ω, ω^ω^ω, ...). The ordinal strictly decreases under the base-change+subtract operation. Gentzen (1936): PA ⊢ Con(PA) requires transfinite induction up to ε₀, and PA cannot prove that induction principle itself. Key constants: ε₀ = sup{ω, ω^ω, ω^ω^ω, ...} = the first fixed point of α ↦ ω^α. No finite ratio constants appear; the structure is ordinal-theoretic. | CONNECTION: ε₀ is a countable ordinal with a natural representation Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com