The Ordinal of Skolem + Tetration Is τ ₀

Abstract

In [1], we proved that a certain family of number theoretic functions S _* is well-ordered by the majorisation relation ‘≼’. Furthermore, we proved that a lower bound on the ordinal O(S _*, ≼ ) of this well-order is the least critical epsilon number τ ₀. In this paper we prove that τ ₀ is also an upper bound for its ordinal, whence our sought-after result,

$$O(S_*,\,\preceq)=\tau_0,$$

is an immediate consequence.