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By Bradley Jackson, Dmitri Thoro

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Lachlan, A. H. (1966). Lower bounds for pairs of r. e. degrees. Proc. London Math. Soc. , 16, no. 3, pp. 537-569. Lachlan, A. H. (1968). Complete recursively enumerable sets. Proc. Amer. Math. , 19, pp. 99-102, Lachlan, A. H. (1972). Embedding nondistributive lattices in the recursively enumerable degrees. Conf. Math. Logic, London London 1970. , no. 255, SpringerVerlag, Berlin and New York, 1972, pp. 149-177. Lachlan, A. H. (1973). The priority method for the construction of recursively enumerable sets.

Shoenfield, J. R. (1971). Degrees of Unsolvability.. NorthHolland, Amsterdam. R. I. (1978). The generalized diamond theorem. (Abstract) Recursive Function Theory Newsletter, 19, no. 219. I. (1976). The infinite injury priority method. J. Symbolic Logic, 41, pp. 513-530. I. (1977). Computational complexity, speedable and levelable sets. J. 545-563. I. (1978). Recursively enumerable sets and degrees. Bull. S. , 84, no. 6, pp. 1149-1181. I. (1980). Constructions in the recursively enumerable degrees.

U = {x;xETi & (3 s > sx)(3z < x)[z e W s+1 - Wi Now if U were finite then W. A since for almost every z, z 5T z e W. iff z e W. swhere x = (µy > z)[y e T1]. Hence, U is in1 x finite; step 1 is performed on each x e U at some stage tx > sx; s]}. , 46 , and x is eligible at every stage s > tX. Now since It x :xeU} is Ar. e. J, s+1 - Wj, s ]}. But V is infinite else WJ < T A. Now for at most finitely many x e V is rX restrained from A with priority higher than Rn. Hence, Rn eventually receives attention via some x e V - B, becomes satisfied; and remains satisfied thereafter (unless injured by a higher priority requirement R m , m < n).

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