Recursive Enumerability

One of the fundamental contributions of mathematical logic has been the precise definition and study of algorithms and the closely associated study of recursively enumerable sets. A subset A £ co is recursive (decidable) if there is an algorithm for computing its characteristic function cA and recursively enumerable (r.e.) if there is an algorithm for generating its members. Nonrecursive r.e. sets have played a crucial role in undecidability results beginning with Gödel's incompleteness theorem [2] and more recently in number theory and group theory. Matiyasevic showed undecidability of Hubert's tenth problem by proving that every r.e. set A is Diophantine (namely there is a polynomial p(x9 y) with integral coefficients such that x£A iff (3y)[p(x9 y) = 0])9 and Boone, Clapham and Fridman each independently proved that every r.e. degree is the degree of the word problem for a finitely presented group (thus generalizing the Boone-Novikov result that the word problem is unsolvable). For sets A9B^œ (the set of nonnegative integers), A is recursive in (Turing reducible to) B9 written A^TB9 if there is an algorithm for computing cA given cB9 and A=TB if A^TB and B^TA. The degree of A, dg (A)9 is the equivalence class {B: B=TA}9 dg (A)^ôg(B) if A^TB9 and a degree is r.e. if it contains an r.e. set. The classification of r.e. sets was initiated by Post [10] who posed the problem: does there exist more than one nonrecursive r.e. degree? The existence of infinitely many such degrees implies for example that there are infinitely many genuinely different unsolvable word problems for finitely presented groups.