  Goodstein sequences are numerical sequences in which a natural number m,
expressed as the complete normal form to a given base a, is modified by
increasing the value of the base a by one unit and subtracting one unit from
the resulting expression. As initially defined, the first term of the Goodstein
sequence is the complete normal form of m to base 2. Goodstein's Theorem states
that, for all natural numbers, the Goodstein sequence eventually terminates at
zero. Goodstein's Theorem was originally proved using the well-ordered
properties of transfinite ordinals. The theorem was also shown to be
unprovable-in-PA (Peano Arithmetic) using transfinite induction and Godel's
Second Incompleteness Theorem. This article describes a proof of Goodstein's
Theorem in first-order arithmetic that contradicts the theorem's
unprovability-in-PA. The proof uses mathematical induction and is applied (via
the super-exponential function) to a generalized version of the Goodstein
sequences. Such a proof demonstrates the inconsistency of classical set theory,
more precisely the combination of the Zermelo-Fraenkel axioms and the axiom of
choice (ZFC).
