Generalizations of Enumeration Reducibility Using Recursive Infinitary Propositional Sentences

Abstract Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 (1992) 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.