On Skolem’s exponential functions below $2^{2^{X}}$

Abstract:

A result of Ehrenfeucht implies that the smallest class of number-theoretic functions $f:{\mathbf {N}} \to {\mathbf {N}}$ containing the constants $0,1,2, \ldots$, the identity function $X$, and closed under addition, multiplication and $f \to {f^X}$, is well-ordered by the relation of eventual dominance. We show that its order type is ${\omega ^{{\omega ^\omega }}}$, and that for any two nonzero functions $f,g$ in the class the quotient $f(n)/g(n)$ tends to a limit in ${E^ + } \cup \{ 0,\infty \}$ as $n \to \infty$, where ${E^ + }$ is the smallest set of positive real numbers containing $1$ and closed under addition, multiplication and under the operations $x \to {x^{ - 1}},x \to {e^x}$.