Some theorems on classes of recursively enumerable sets

(2) to start a classification of recursively enumerable classes which parallels the classification of recursively enumerable sets initiated by Post in [15]. A mapping of numbers (or ordered n-tuples of numbers) onto numbers is called a function. Numbers and functions are denoted by small Latin letters, sets by small Greek letters and classes by capital Latin letters. 'a-,B' stands for the set of all numbers which belong to ax, but not to ,B, while 'A -B' stands for the class of all sets which belong to A, but not to B. We denote the range of the function f(x) by 'pf(x)' or 'pf'. If a is the range of the everywhere defined function f(x), we say that a can be generated by f(x). Let g(n, x) be defined for every ordered pair of numbers and let A be the class of all sets which occur at least once in the sequence pg(O, x), pg(l, x), We say that A can be generated by g(n, x). A set is recursively enumerable (r.e.) if it is empty or it can be generated by a recursive function of one variable. Similarly, a class of r.e. sets is recursively enumerable, if it is empty or consists only of the empty set, or the class of its nonempty members can be generated by a recursive function of two variables. Let 'F' stand for the class of all r.e. sets and 'Q' for the class of all finite sets. Then Q is a proper subclass of F and it is easily seen that both F and Q are r.e. classes [2, T2.2 ]. A set is called decidable or recursive if there exists a recursive procedure which enables us to test membership in the set, i.e., if its characteristic function is recursive. A decidable set is said to have a solvable decision problem, and an undecidable set an unsolvable decision problem. The class of all recursive sets is denoted by 'E'; this class properly includes Q and is properly