Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 173–184. We consider the relation between sets A and B that for every set S if A is Σ0α in S then B is Σ0β in S. We show that this is equivalent to the condition that B is definable from A in a particular way involving recursive infinitary propositional sentences. When α = β = 1, this condition is that B is (...) enumeration reducible to A. We establish further generalizations involving infinitely many sets and ordinals. (shrink)

. We prove a jump inversion theorem for the enumeration jump and a minimal pair type theorem for the enumeration reducibilty. As an application some results of Selman, Case and Ash are obtained.