A robuster Scott rank
Antonio Montalbán · Proceedings of the American Mathematical Society · 2015
We give a new definition of Scott rank motivated by our main theorem: For every countable structure $\mathcal {A}$ and ordinal $\alpha <\omega _1$, we have that: every automorphism orbit is infinitary $\Sigma _\alpha$-definable without parameters if and only if $\mathcal {A}$ has an infinitary $\Pi _{\alpha +1}$ Scott sentence, if and only if $\mathcal {A}$ is uniformly boldface $\bf {\Delta }^0_\alpha$-categorical. As a corollary, we show that a structure is computably categorical on a cone if and only if it is the model of a countably categorical infinitary $\Sigma _3$ sentence.