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 first-order 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.