This paper deals with the
degrees of members of Π10 classes of sets or function which lack recursive members.
For instance, it is shown that if a is a degree and 0 < a ≦ 0′, then there exists a Π10
class of sets which has a member of degree a but none of degree 0. By way of contrast
there is no Π10 class of functions which has members of all nonzero r.e. degrees but
no recursive members. Furthermore, each nonempty Π10 class of sets has a member
of r.e. degree but not necessarily of r.e. degree less than 0′. As corollaries results
are derived about degrees of theories and degrees of models such as: (1)
There is an axiomatizable, essentially undecidable theory with a complete
extension of minimal degree; (2) (ScottTennenbaum): There is a complete
extension of Peano arithmetic of degree 0′ but none of r.e. degree < 0′;(3)
There is no nonstandard model of Peano arithmetic of r.e. degree < 0′.
The recursion theorem is applied to yield new information about standard
constructions such as Yates’ simple nonhypersimple set of given nonzero
r.e. degree.
