Lattices of α-Recursively Enumerable Sets

Abstract

Lattices of a-recursively enumerable sets provide an excellent setting for the study of recursion theory on admissible ordinals. Many theorems of ordinary recursion theory fail in this setting, providing examples which distinguish those properties of w which are significant for generalized recursion theory from those properties which are particular to ordinary recursion theory. These differences have stimulated the discovery of genuinely new constructions which are not liftings of proofs or constructions from ordinary recursion theory. Yet there is enough similarity with ordinary recursion theory so that the difficult methods of a-recursion theory which were introduced to lift theorems come into play in the study of the lattices. Infact, some of these methods were discovered by considering Lattice-theoretic questions. We will survey lattices of u-resets, including the case a = w , in this paper. few of the easier proofs will be presented in order to provide the reader with the flavor of the methods used in the subject. We will, however, avoid the more difficult proofs. Reduction to a-recursion theory and lattices of ure sets. I n Section 2, we will introduce a quotient of the lattice of a-resets in which many questions of interest are more easily worked.