Publisher Summary This chapter introduces the problems of the theory of numberings, a developing branch of the algorithm theory. In the theory of numberings the necessary system of notions is being developed, natural questions are being arisen, and corresponding answers are being found. Those are the questions of dependency of various properties of a set on the choice of a numbering, an existence (uniqueness) of a numbering of a set possessing certain properties, and others. And as usual, in development of the theory, the inherent problems arise leading to unexpected beautiful and difficult theorems. Results obtained in the theory of numberings proved to be important for an understanding of some of the difficulties encountered in modern computer science. For instance, an important problem of contemporary programming is the problem of effective construction of a program to compute a function on a particular computer, given a program that computes the same function on the other computer. Practical realization of these translations for a pair of universal computers proves to be too difficult to be realized.
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