# Automatic Ordinals

1 ELM - Équipe de Logique Mathématique 2 IMJ - Institut de Mathématiques de Jussieu 3 Department of Mathematics - University of Toronto

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.

keyword : ordinals $\omega$-tree-automatic structures $\omega^n$-automatic structures ordinals.

Author: Olivier Finkel - Stevo Todorcevic -

Source: https://hal.archives-ouvertes.fr/