We define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

We show that every nonzero enumeration degree bounds a nonsplitting nonzero enumeration degree.

This paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .