JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.Our main result is: if b and c are recursively enumerable degrees such that b < c, then there exists a recursively enumerable degree d such that b < d < c. By degree we mean degree of recursive unsolvability as defined by Kleene and Post [2]. A degree is called recursively enumerable if it is the degree of a recursively enumerable set. The upper semi-lattice of recursively enumerable degrees has a least member, 0, the degree of all recursive sets, and a greatest member, 0', the degree of all complete sets. Post [4] asked: does there exist a recursively enumerable degree d such that 0 < d < 0'? Friedberg [1] and Muchnik [3] answered Post's question in the affirmative. Sacks [5] showed that every countable, partially ordered set is imbeddable in the upper semi-lattice of recursively enumerable degrees. Muchnik [3] announced that if c is a non-zero, recursively enumerable degree, then there exists a recursively enumerable degree d such that 0 < d < c. The arguments of Friedberg [1], Muchnik [3] and Sacks [5]have a great deal in common. All three authors make use of a method which may be roughly described as follows (a precise abstract description is given in Sacks [5]): One or more sets are to be recursively enumerated. Certain requirements are to be met. A typical requirement is: the set B is not recursive in the set A with G6del number e. Unfortunately, the requirements tend to conflict. Thus we may wish to add n to B to insure that B t {e}A, but the addition of n to B may ruin what we did earlier in the enumeration to insure A t {f }B. We resolve all conflicts by appealing to a system of priorities assigned to the requirements before the enumeration begins. Then it can be shown that each requirement is "ruined" only finitely often, and that, consequently, each requirement is eventually met.In the argument below we assign priorities to requirements, but we are forced to permit each requirement to be "ruined" infinitely often. Nonetheless, we are still able to meet each requirement. Thus there is an important combinatorial difference between the method of Friedberg [1] and Muchnik [3J and the method we use below. The difference is a consequence of the fact that Friedberg and Muchnik deal only with finite, initial segments of functions while we are forced to deal indirectly with entire functions. Almost all
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.
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