On infinite direct sums of minimal numberings of functional families

The paper discusses two approaches to defining the computability of numberings of families of total functions. We consider both the classical definition of computable numbering of a family of computable functions, according to which the number of a function in this numbering effectively provides its G"odel number, and, expanding the previous one, a definition based on the uniform application of the concept of the left-c.e. element of Baire space. The main question studied in the paper is the possibility of generating all computable numberings of a family by the closure with respect to the reducibility of infinite direct sums of uniform sequences of its single-valued, positive, and minimal numberings.