Post's program and incomplete recursively enumerable sets.
Abstract
A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a firstorder property, Q(X), definable in E, the lattice of r.e. sets under inclusion, such that (i) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete and (ii) there exists an r.e. set A satisfying Q(A). This resolves a long open question stemming from Post's program of 1944, and it sheds light on the fundamental problem of the relationship between the algebraic structure of an r.e. set A and the (Turing) degree of information that A encodes.
 Publication:

Proceedings of the National Academy of Science
 Pub Date:
 November 1991
 DOI:
 10.1073/pnas.88.22.10242
 Bibcode:
 1991PNAS...8810242H