Schock Rolf. Some remarks on Russell's treatment of definite descriptions. Logique et analyse , n.s. vol. 5 (1962), pp. 77–80.

This is an English translation of the second edition (XXVIII 111(1)) of Trakhtenbrot's booklet and is one of a series of translations from the Russian series, "Popular lectures in mathematics," which are being prepared under a project at the University of Chicago. The popular character of the exposition means in the present case that no knowledge of mathematics beyond intermediate algebra is presupposed. But, as the translators explain in their preface, there are several passages in which the reader is expected to follow a rather complex train of logical inference. The main topic is the abstract theory of computability, based a t first on an informal notion of algorithm, and then on the Turing machine. There are, however, two chapters (pp. 38-51) which give a descriptive account of actual computing machines and of programs for them. The booklet includes a detailed account of the Turing machine and of the universal Turing machine, and proofs are given of the unsolvability of the self-computability problem for Turing machines and of the word problem for semigroups. (The author says associative calculus rather than semigroup, because the Russian terminology reserves the name semigroup for semigroups with cancellation.) The definition of computability by means of the Turing machine — to the exclusion of other notions of computability, which are only mentioned — is advantageous in a popular account, because it provides a unified treatment, and because it brings the account into closer relation with that of computing machines as actually constructed. In Chapter 10 (pp. 77-79) the author introduces what he calls "the basic hypothesis of the theory of algorithms," that "all algorithms can be given in the form of functional matrices and executed by the corresponding Turing machines." No source is given for the hypothesis. As regards general recursiveness as definition of computability, it should have been credited to the reviewer (Bulletin of the American Mathematical Society, vol. 41 (1935), p. 333), and as regards the Turing machine as definition of computability, to Turing in his paper of 1936 (see review I I 42(4)). In correction, or in clarification, of page 86, it should also be said that the first examples of unsolvable decision problems were given by the reviewer (I 73(2), I 74(1)), followed closely by the independently obtained similar results of Turing. And in connection with the definition of computability and the "basic hypothesis," the paper of Post, II 43(1), also requires mention. ALONZO CHURCH