Vol. 40, No. 3, 1972

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Degrees of members of Π10 classes

Carl Groos Jockusch, Jr. and Robert Irving Soare

Vol. 40 (1972), No. 3, 605–616

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 0but 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.

Mathematical Subject Classification
Primary: 02F40
Received: 27 October 1970
Published: 1 March 1972
Carl Groos Jockusch, Jr.
Robert Irving Soare