Maximality in the ⍺-C.A. Degrees

Abstract

In [4], Downey and Greenberg define the notion of totally ⍺-c.a. for appropriately small ordinals ⍺, and discuss the hierarchy this notion begets on the Turing degrees. The hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a definable antichain in the c.e. degrees. Following on from the work of [4], we solve problems which are left open in the aforementioned relating to this hierarchy. Our proofs are all constructive, using strategy trees to build c.e. sets, usually with some form of permitting. We identify levels of the hierarchy where there is absolutely no collapse above any totally ⍺-c.a. c.e. degree, and construct, for every ⍺ ≼ ε0, both a totally ⍺-c.a. c.e. minimal cover and a chain of totally ⍺-c.a. c.e. degrees cofinal in the totally ⍺-c.a. c.e. degrees in the cone above the chain's least member