Geodesic
Computable families of sets in Ershov hierarchy without principal numberings
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 1, pp. 54-62

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There was constructed example of computable family of sets without $\Sigma^{-1}_{a}$-principal numberings, for any $a\in\mathcal{O}$.
Keywords: computable numbering, Ershov hierarchy. 
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     author = {S. S. Ospichev},
     title = {Computable families of sets in {Ershov} hierarchy without principal numberings},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {54--62},
     publisher = {mathdoc},
     volume = {15},
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     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2015_15_1_a4/}
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S. S. Ospichev. Computable families of sets in Ershov hierarchy without principal numberings. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 1, pp. 54-62. http://geodesic.mathdoc.fr/item/VNGU_2015_15_1_a4/