The Field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic

Recently, Schlicht and Stephan lifted the notion of automatic-structures to the notion of (finite-word) ordinal-automatic structures. These are structures whose domain and relations can be represented by automata reading finite words whose shape is some fixed ordinal α. We lift Delhommé’s relative-growth-technique from the automatic and tree-automatic setting to the ordinal-automatic setting. This result implies that the random graph is not ordinal-automatic and infinite integral domains are not ordinal-automatic with respect to ordinals below ω1 + ω ω where ω1 is the first uncountable ordinal.