There are topics in mathematics that are well known and at the same time recognized by tradition as too difficult (or unimportant) to include in compulsory education: custom refers them to elective, additional, special, etc. p. In the list of such topics, there are several that remain there now solely due to inertia. One of them is Gdel's theorem .. Despite the fact that many mathematicians (and non-mathematicians) have heard about it, few of them can explain what the statement of Gdel's theorem is, and even more so how it is proved. At the same time, the result is so important, and the reasons for the inexorable incompleteness (t.) e. The impossibility of ensuring that every true statement is provable is so simple that Gdel's theorem could be expounded in the youngest courses. Furthermore, , To understand the proof, one only needs to become familiar with the simplest terminology of set theory (the words "set" ,). "function", "area of definition" and the like) and some habit of perceiving mathematical reasoning , So it is quite accessible to a trained schoolboy . The method of proving Gdel's theorem set out in this pamphlet is different from the method proposed by Gdel himself, and relies on elementary concepts of algorithm theory. All the necessary information from this theory is communicated in the course of the case, so that the reader simultaneously gets acquainted with the basic facts of the theory of algorithms. Brochure written on the basis of the author's article in the journal "Successes of Mathematical Sciences", 1974, volume 29, issue 1 (175). Naturally, the change in the circle of prospective readers made it necessary to revise it. In particular, some more specific questions, as well as bibliographic references to original publications, are excluded, and an inquisitive reader may find them in the author's cited article. At the same time, the section devoted to the relationship between the semantic and syntactic formulations of the incompleteness theorem was expanded, and annexes were added devoted to Tarski's theorem on the inexpressibility of the concept of truth and the justification of the axiom of arithmetic.