The reader is invited to a book by the famous American logician E. Nagel and the experienced popularizer of science J.R. Newman, dedicated to Gödel's incompleteness theorem. This theorem was presented in a short article by K. Gödel, which subsequently played a decisive role in the history of logic and mathematics. The authors of this book, without trying to give a general outline of the ideas and methods of mathematical logic, build their presentation around the central, from their point of view, problems of this science - the problems of consistency and completeness. Proof of the fact that for sufficiently rich mathematical theories these requirements are incompatible is the astonishing discovery of Gödel, to which the book is dedicated. Without requiring any prior knowledge from the reader, the authors successfully explain to him the essence of one of the most remarkable and profound theorems of mathematics and logic. For specialists in mathematical logic, undergraduate and graduate students, as well as all interested readers.