Collapse in a Transfinite Hierarchy of Turing Degrees

In [2], Downey and Greenberg use the ordinals below ε0 to bound the number of mind-changes of computable approximations. This gives rise to a new transfinite hierarchy in the c.e. degrees; the totally α-c.a. degrees. This hierarchy is significant because it unifies the combinatorics of many constructions as well as giving natural definability results in the c.e. Turing degrees. We study the structure of this hierarchy; in particular we investigate collapse in upper cones. We give a proof in which we build a c.e. set using a strategy tree to show there is no uniform way to find a maximal totally ω^2-c.a. degree above a given totally ω-c.a. degree. Then we discuss extensions of this result.