The book describes axiomatic set theory and methods for proving the compatibility of set theory statements: Gödel method, Cohen forcing method, Boolean-valued model method, Frenkel-Mostovsky method. Using these methods, models are built for many well-known hypotheses of set theory: the generalized continuum hypothesis, the negation of the continuum hypothesis, the negation of the axiom of choice, the Suslin hypothesis and its negation, etc.