Maximal totally α-c.a. degrees
Rodney G. Downey, Noam Greenberg · Princeton University Press eBooks · 2020
This chapter shows that at every level of the main hierarchy, there are maximal degrees (Theorem 4.1). Thus, for example, there are maximal degrees with respect to not bounding a critical triple, namely, maximal totally ω-c.a. degrees. Since the totally ω-c.a. degrees are naturally definable, one obtains a naturally definable antichain in the c.e. degrees. The only previously known such antichain consisted of the maximal contiguous degrees. On the other hand, the chapter demonstrates (Theorem 4.12) that maximality cannot go too far, that is, to the next level. It also investigates bounding by maximal degrees. For example, there are totally ω-c.a. degrees bounded by no such maximal degrees.