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Post's Program and incomplete recursively enumerable sets
, 1991
"... : 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: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing i ..."
Abstract

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: 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: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete; and (2) 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 new light on the fundamental problem of the relationship between the algebraic structure of an r.e. set A and the (Turing) degree of information which A encodes. Recursively enumerable (r.e.) sets have been a central topic in mathematical logic, in recursion theory (i.e. computability theory), and in undecidable problems. They are the next most effective type of set beyond recursive (i.e. computable) sets, and they occur naturally in many branches of mathematics. This together with the existence of nonrecursive r.e. sets has enabled them to pl...