Classes of Recursively Enumerable Sets and Degrees of Unsolvability

Computability theory, algorithmic randomness and Turing's anticipation

“…For example, it was shown that maximal sets (that is coatoms in the quotient structure of the computably enumerable sets modulo finite sets-a notion from Post's paper) always have high degrees (meaning A ≡ T ∅ ) and include all high degrees; and form an orbit in the automorphism group of the lattice of computably enumerable sets. (Martin [100] and Soare [135], respectively.) Harrington and Soare [68] used the infinite injury method and great ingenuity to show that there is a definable property of the lattice of computably enumerable sets which solves Post's problem.…”

Automorphisms of the lattice of $\Pi _1^0$ classes; perfect thin classes and anc degrees

“…In the lattice of computably enumerable sets, Martin [21] established that the high c.e. degrees were invariant since the were precisely the degrees of the maximal (and hyperhypersimple) sets, which are definable in E. We would like to demonstrate that the anc degrees are an invariant class for L(Q).…”

“…As is well known, Post sought a thinness property of the lattice of computably enumerable sets which guaranteed Turing incompleteness. In the deep paper [28], Soare demonstrated that this was impossible since all maximal sets were automorphic, and Martin [21] had earlier proved that the degrees containing maximal sets (indeed hyperhypersimple sets) were precisely the collection of all high degrees.…”

Highness and bounding minimal pairs

“…For the high sets (A ′ ≡ T ∅ ′′ ) and degrees, the trend of results has been that anything possible happens. Here the classic examples are Martin's [14] theorem that every high degree contains a maximal set and Cooper's [2] result that there is a minimal pair (a, b = 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and [20] have shown that other important phenomena (the special triples of Slaman [21] and the nonsplitting pairs of Lachlan [12], respectively) occur below every high degree.…”

“…(They also supply a proof of Harrington's extension of the Robinson splitting theorem to the situation where c is assumed low 2 and d can be an arbitrary degree below c.) In E * , these results have supplied various characterizations of the high and low 2 degrees. The high ones, for example, are precisely the ones containing maximal sets (Martin [14]). The low 2 degrees are precisely those containing sets with no maximal superset (Lachlan [11] and Shoenfield [17]).…”