A degree is saturated bounding if it can compute a saturated model for any complete, decidable theory whose types are all computable. All high and PA degrees are saturated bounding (following from results of Jockusch and MacIntyre-Marker.) We show that for every n, no lown c.e. degree is saturated bounding, extending the previous known result that 0 is not saturated bounding, by Millar.

Important examples of Π 1 classes of functions f ∈ ω are the classes of sets (elements of 2) which separate a given pair of disjoint r.e. sets: S2(A0, A1) := { f ∈ 2 : (∀i < 2)(∀x ∈ Ai)f(x) 6= i }. A wider class consists of the classes of functions f ∈ k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: Sk(A0, . . . , Ak−1) := { f ∈...

A nontrivial automorphism of the Turing degrees is constructed.

We show that every generalized high Turing degree is the join of two minimal degrees, thereby settling a conjecture of Posner’s from the 70s.

We show that any finite structure consistent with the axioms of jump uppersemilattices with least element can be embedded into the jump uppersemilattice of Turing degrees with least element. This result implies that the existential theory of this structure is decidable.

Unequal access to higher medical degrees, which are important for career advancement, is a problem that is likely to plague UK medicine as integration within Europe proceeds. This paper analyses the characteristics of M.D.s at the Royal Free Hospital School of Medicine and proposes a solution of creating a common European doctorate.

Let G ≤ Sn be a permutation group of degree n with a quotient isomorphic to Ak. We show that if n < (1/2 − o(1))k then the extension splits. Motivated by this result, Guralnick and Liebeck have shown that the conclusion fails to hold if we allow n to be n = 2k(k − 1), establishing a tight quadratic growth rate for the smallest degree of a faithful permutation representation of a non-split exten...