Recursively Enumerable Degrees and the Degrees Less Than 0(1)

Abstract

This chapter presents results to strengthen Shoenfield's result by proving that there is a degree a between 0 and 0( 1 ), which is incomparable with every recursively enumerable (r.e.) degree between 0 and 0 ( 1 ). Shoenfield proved that there is a degree between 0 and 0 ( 1 ), which is not recursively enumerable. The chapter focuses on a string of a finite sequence of ones and twos. The number of elements of a string σ will be denoted by length ( σ ), the n -th element by σ ( n ), and the initial segment of σ , which has length n by σ [ n ]. Because a set T is identified with its representing function and this in turn can be considered as an infinite sequence of ones and twos, σ ⊂ T to mean that σ represents T on the segment of the integers, which has length equal to length ( σ ). There exist a number of obvious recursive enumerations of the set of all strings and recursion theory can be proved with strings just as easily as with integers. This is usually done indirectly by explicitly representing strings by integers.