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

FINDING: Goodstein's theorem demonstrates that a purely finite, arithmetic statement about natural numbers is unprovable in Peano Arithmetic, yet provable using transfinite induction up to the ordinal ε₀ (epsilon-zero). | MATH: The theorem constructs sequences via hereditary base-b representation: for n, write in base b, then recursively write exponents in base b. Replace b with b+1, subtract 1, repeat. The sequence terminates for all n, but the proof requires ordinal assignment: map each term to a countable ordinal < ε₀ = sup{ω, ω^ω, ω^ω^ω, ...}. The ordinal descent is well-founded, proving termination. Key constant: ε₀ = ω^ε₀, the first fixed point of α ↦ ω^α. | CONNECTION: ε₀ is the limit of the tower ω, ω^ω, ω^ω^ω, ... — a logarithmic spiral of ordinal exponentiation. The descent mirrors a strictly decreasing sequence of ordinals, analogous to the golden-ratio-based continued fraction descent (0.618, 0.382) but in transfinite space. The base-change operation (b → b+1) is a discrete Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com