``High'' is definable in the partial order of the Turing degrees of the
recursively enumerable sets
Theodore A. Slaman
A recursively enumerable set A is high if and only if A' has the same
Turing degree as 0''. Martin proved that A is high if and only if
there is a maximal recursively enumerable set in the Turing degree of
A. Thus, A's being high has a lattice theoretic characterization. As
indicated in the following theorem, there is also a characterization
expressed purely in terms of the Turing partial ordering.
Theorem (Nies, Shore, and Slaman) ``High'' is first-order definable in
the partial order of the Turing degrees of the recursively enumerable
sets.