Goodstein's Theorem: Unprovable in Peano Arithmetic, Provable via ε₀ Induction — E8 Intelligence Research

FINDING: Goodstein's theorem demonstrates that every Goodstein sequence terminates at 0, yet this statement is unprovable within Peano Arithmetic (PA), requiring transfinite induction up to the ordinal ε₀ for proof. | MATH: Goodstein sequence: start with m in base 2, write in hereditary base notation (exponents also in base 2), replace base 2 with 3, subtract 1, repeat with base increasing by 1 each step. Termination proven via ordinal assignment: map each hereditary base-b representation to a Cantor normal form ordinal < ε₀ = sup{ω, ω^ω, ω^(ω^ω), ...}. The ordinal decreases strictly under the base-change-minus-one operation. Key constants: ε₀ = ω^(ω^(ω^...)) — the first fixed point of α ↦ ω^α. PA proves transfinite induction for all α < ε₀ but not for ε₀ itself (Gentzen 1943). | CONNECTION: The ordinal ε₀ is intimately tied to the structure of rooted trees and the hydra game (Kirby–Paris). The hereditary base representation is a tree of exponents — its "height" corresponds to ordinal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com