Geodesic
Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a~condition for infinite $\Delta^0_\alpha$-dimension
Algebra i logika, Tome 54 (2015) no. 5, pp. 551-574.

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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.
Keywords: computable structure, Ash's theorem, $\Delta^0_\alpha$-categorical structure, $\Sigma_\alpha$ Scott family, $\Delta^0_\alpha$-dimension of a computable structure. 
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     author = {P. E. Alaev},
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P. E. Alaev. Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a~condition for infinite $\Delta^0_\alpha$-dimension. Algebra i logika, Tome 54 (2015) no. 5, pp. 551-574. http://geodesic.mathdoc.fr/item/AL_2015_54_5_a0/

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