Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication

Extract

In this paper we shall prove three theorems about recursively enumerable sets. The first two answer questions posed by Myhill [1].

The three proofs are independent and can be presented in any order. Certain notations will be common to all three. We shall denote by “R_e” the set enumerated by the procedure of which e is the Gödel number. In describing the construction for each proof, we shall suppose that a clerk is carrying out the simultaneous enumeration of R₀, R₁, R₂, …, in such a way that at each step only a finite number of sets have begun to be enumerated and only a finite number of the members of any set have been listed. (One plan the clerk can follow is to turn his attention at Step a to the enumeration of R_e where e+1 is the number of prime factors of a. Then each R_e receives his attention infinitely often.) We shall denote by “R_e^a” the set of numbers which, at or before Step a, the clerk has listed as members of R_e. Obviously, all the R_e^a are finite sets, recursive uniformly in e and a. For any a we can determine effectively the highest e for which R_e^a is not empty, and for any a, e we can effectively find the highest member of R_e^a, just by scanning what the clerk has done by Step a. Additional notations will be introduced in the proofs to which they pertain.