Effectively infinite classes of numberings and fixed point theorems
Effectively infinite classes of numberings
Keywords:
computable numbering, complete numbering, precomplete numbering, recursion theorem, effective infinityAbstract
In this paper, we prove a sufficient condition for the effective infinity of classes of complete and precomplete numberings, as well as numberings satisfying the recursion theorem, of computable families. A sufficient condition for the effective infinity of classes of non-precomplete numberings of computable families satisfying the recursion theorem is also obtained. These conditions are satisfied by the family of all c.e. sets and the family of graphs of all partially computable functions. For finite families of c.e. sets, we prove a criterion for the effective infinity of classes of their numberings that satisfy the recursion theorem. Finally, it is established that the classes of complete and precomplete numberings of finite families of c.e. sets are not effectively infinite.
