Epsilon-omega order on the set of natural numbers

Abstract

The limit of the sequence $\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots$, denoted $\varepsilon_0$, is the smallest epsilon number, i.e. ordinal number $\alpha$ such that $\omega^\alpha=\alpha$. While all the epsilon numbers with countable indices are known to be order types of countable sets, it is a highly counter-intuitive property. The goal of this paper is to provide better insight into the structure of epsilon ordinals by constructing $\varepsilon_0, \varepsilon_1, \ldots, \varepsilon_\omega$ orders on the set of natural numbers. The presented construction can be easily extended to even larger epsilon numbers.