Limit Computable Sets and Degrees

This paper will study sets and degrees containing sets that are determined as limits of computable approximations. By the Shoenfield Limit Lemma, the limit computable sets are precisely the degrees below ∅′. In particular, the paper will study limit computable sets by classifying them according to bounds to the number of changes to elements in various approximations of the sets. This leads to the n-c.e., ω-c.e., and ∆2 classifications. The paper will show these characterizations of sets and degrees are proper at various levels. Properties of n-c.e. sets particularly concerning the non-existence of n-c.e. minimal degrees will be developed in this paper. These classifications will also provide some insights into the structures of the Truth-Table Turing Degrees.