The Recursively Enumerable Degrees

Degrees of Unsolvability

“…The one-quantifier theory is decidable by Sacks' result that any finite partial ordering can be embedded into R (and, as already mentioned in Section 9 above, by a very sophisticated argument, Lempp and Lerman [Lempp and Lerman, 1996] show that decidability is preserved if n-jump relations are added for all n). For a more thorough discussion of the possible design of a decision procedure for ∀∃ − T h(R) and possible obstacles, see Lerman [1996] and Shore [1999]. In fact, this question has to be solved for embeddings preserving both least and greatest elements.…”

The global structure of computably enumerable sets

“…Even given Cooper's claim that there is an automorphism of the computably enumerable degrees, the biinterpretability conjecture with parameters remains a strong possibility to globally understand the computably enumerable degrees (R) [see Cooper, 1999;Shore, 1999;Slaman, 1999].…”

Mass problems and density

“…Inspired by these results of Sacks, the study of structural properties of E T has played a leading role in recursion theory from the 1960s to the present. See for instance the recent paper [1] and the survey papers [12,23]. Also in the mid-20th century but largely ignored in the West, Medvedev [13] and Muchnik [14] introduced more general notions of degrees of unsolvability.…”