Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a condition for infinite $\Delta^0_\alpha$-dimension

An old classical result in computable structure theory is Ash's theorem stating that for every computable ordinal $\alpha\ge2$, under some additional conditions, a computable structure is $\Delta^0_\alpha$-categorical iff it has a computable $\Sigma_\alpha$ Scott family. We construct a counterexample revealing that the proof of this theorem has a serious error. Moreover, we show how the error can be corrected by revising the proof. In addition, we formulate a sufficient condition under which the $\Delta^0_\alpha$-dimension of a computable structure is infinite.