Computability in arbitrary domains and bases

In this issue, in the section “Turning over the old pages", the editorial board offers readers the article by A, P, Ershov “Computability in arbitrary holds and bases", Reading this article many years ago, I was struck at that time, and now I am still amazed at how A, P, Ershov was able to specifically, not abstractly, foresee that due to the development of programming, the mathematical logic will be an applied discipline. For programmers-eneoders, this article may seem to be uninteresting and unnecessary. Those, who want to understand, even if not completely, may wish to better understand the basis of the modern programming, to understand and realize the connections of set theory, theory of algorithms and mathematical logic and their role in the development of programming in the coming decades. As an epigraph to this publication, an extract from the same article begs: The spread of computers and programming make mathematical logic an applied science. The caste of mathematical logicians, the priest-keepers of the sacred fire of the foundations of mathematics, to their amazement finds themselves face to face with the invasion of hordes of programmers, whose mathematical culture is close to barbarism, but who, nevertheless, inspired by their prophets of structural programming, seek to light a torch from the life-giving fire and drag it to them in order to illuminate the corners of their units clogged with lamps, OS / 360 instructions, Fortran forms and other symbols of idolatry, Nevertheless, in order to establish a union of such different cultures, each side - logicians and programmers - needs to overcome their inferiority complexes, to develop a capacity for mutual understanding and, what is more important, to become convinced in the interest to the other side. The text is technically somewhat out of date, for example, the symbols of idolatry among programmers are now different, but the thoughts expressed are true today. Indeed, the era of programming, based largely on the mathematical logic is beginning, Logie is slowly but surely becoming an applied science, and programming systems in the mass-scale programming are gradually beginning to be replaced by systems for the automatic construction of algorithms and programs on the bases of the axiomatic description of a subject domain. There appear systems, in which an algorithm and a program are taken from the axiomatic description of the subject area, see, for example, Charm and LuXA systems (Victor Malyshkin, Active Knowledge, LuXA and Literacy for Oncoming Centuries // Springer, LXCS, Vol, 9465, P, 292 -303), Nowadays the students, who are trying to minimize their knowledge of mathematics, may have problems in finding a job in the future: they simply will not be able to understand the formulation of problems and how to solve them.