Asymptotic analysis of Skolem's exponential functions

TL;DR

This paper analyzes the asymptotic structure of Skolem's exponential functions, proving they have order type omega within each archimedean class, and improves bounds on their complexity using surreal numbers.

Contribution

It establishes that the set of asymptotic classes within any archimedean class of Skolem functions has order type omega, and refines the bounds on the order type of these functions.

Findings

Asymptotic classes within any archimedean class have order type omega.

Provides upper bounds for the fragment below 2^{n^x} for each positive integer n.

Improves the epsilon-zero bound for the fragment below 2^{x^x} using surreal numbers.

Abstract

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^x}$. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type $\omega$. As a consequence we obtain, for each positive integer $n$, an upper bound for the fragment below $2^{n^x}$. We deduce an epsilon-zero upper bound for the fragment below $2^{x^x}$, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.