Automatic Ordinals

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

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