Some Effectively Infinite Classes of Enumerations

Goncharov, S., A. Yakhnis and V. Yakhnis, Some effectively infinite classes of enumerations, Annals of Pure and Applied Logic 60 (1993) 207-235. This research partially answers the question raised by Goncharov about the size of the class of positive elements of a Roger’s semilattice. We introduce a notion of effective infinity of classes of computable enumerations. Then, using finite injury priority method, we prove five theorems which give sufficient conditions to be effectively infinite for classes of all enumerations without repetitions, positive undecidable enumerations, negative undecidable enumerations and all computable enumerations of a family of r.e. sets. These theorems permit to strengthen the results of Pour-El, Pour-El and Howard, Ershov and Khutoretskii about existence of enumerations without repetitions and positive undecidable enumerations.