Subjects: set theory axiomatic set theory logic antinomy In the development of set theory, which is the basis for the construction of most mathematical disciplines, extremely complex problems of consistency have arisen. The book is the most comprehensive of the existing reviews of the studies brought to life by this problem, it describes and compares all the most important systems of axiomatic set theory. Much attention is paid to the application of ideas and methods of mathematical logic in various areas of research on the basis of mathematics (logicism, intuitionism, formalism). The book, equipped with an extensive list of literature, is valuable for mathematicians dealing with the foundations of mathematics and related issues of mathematical logic, as well as for philosophers and representatives of other specialties related to methodological problems of mathematics.