Automatic Ordinals

TL;DR

This paper characterizes the class of ordinals that are injectively omega-tree-automatic and omega^n-automatic, establishing their bounds and demonstrating the strict hierarchy of these automatic structures.

Contribution

It proves the bounds of injectively omega-tree-automatic and omega^n-automatic ordinals, extending previous results and showing the hierarchy of such structures is strict.

Findings

Injectively omega-tree-automatic ordinals are smaller than ω^{ω^ω}.

Injectively ω^n-automatic ordinals are smaller than ω^{ω^n}.

The hierarchy of injectively ω^n-automatic structures is strict.

Abstract

We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $\omega^{\omega^\omega}$. Then we show that the injectively $\omega^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $\omega^{\omega^n}$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $\omega^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $\omega^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.