From the journal "Successes of Mathematical Sciences", 1974 . t. XXIX, vol. 1(175), p. 3-47. Gdel's incompleteness theorem denies the possibility of such a system of axioms and rules of inference. In other words, the statement , from the axioms , coincides with the set of true statements of arithmetic (moreover, for any non-contradictory system the true can be effectively constructed, but not a derivable statement). The article is devoted to the proof of this theorem, based on the concepts and methods of algorithm theory; the necessary information from algorithm theory is reported as needed. The article does not require any special knowledge (in particular, from the field of mathematical logic), but involves only acquaintance with elementary general terminology and symbolism.