In this paper we show that there is a pair of superhigh r.e. degree that forms a minimal pair. An analysis of the proof shows that a critical ingredient is the growth rates of certain order functions. This leads us to investigate certain high r.e. degrees, which resemble ∅’ very closely in terms of ∅’-jump traceability. In particular, we will construct an ultrahigh degree which is cappable.
"On very high degrees." J. Symbolic Logic 73 (1) 309 - 342, March 2008. https://doi.org/10.2178/jsl/1208358755