Affiliations: [a] Volga Region Scientific-Educational Centre of Mathematics, Kazan (Volga Region) Federal University, 35, Kremlevskaya Str., Kazan, Russia | [b] Faculty of computer science and engineering, Innopolis University, 1, Universitetskaya Str., Innopolis, Russia
Abstract: We prove various sufficient conditions for the effective infinity of classes of computable numberings. Then we apply them to show that for every computable family of left-c.e. reals without the greatest element the class of its Friedberg computable numberings is effectively infinite. In particular, this result covers the families of all left-c.e. and all Martin-Löf random left-c.e. reals whose Friedberg computable numberings have been constructed by Broadhead and Kjos-Hanssen in their paper (In Mathematical Theory and Computational Practice, CiE 2009 (2009) 49–58 Springer). In addition, for every infinite computable family of left-c.e. reals we prove that the classes of all its computable, positive and minimal numberings are effectively infinite.
