Automatic ordinals

Abstract

We prove that the injectively ω-tree-automatic ordinals are the ordinals smaller than ωωω . Then we show that the injectively ωn -automatic ordinals, where n ≥ 1 is an integer, are the ordinals smaller than ωωn . This strengthens a recent result of Schlicht and Stephan who considered in [21] the subclasses of finite word ωn -automatic ordinals. As a by-product we obtain that the hierarchy of injectively ωn -automatic structures, n ≥ 1, which was considered in [9], is strict.