The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's lemma, the continuum hypothesis, Martin's axiom, the diamond principle ◊, and elements of forcing. The choice of the topics and the way in which they are presented is subordinate to one purpose – to get the tools that are most useful in applications, especially in abstract geometry, analysis, topology, and algebra. In particular, most of the methods presented in this course are accompanied by many applications in abstract geometry, real analysis, and, in a few cases, topology and algebra. Thus the text is dedicated to all readers that would like to apply set-theoretic methods outside set theory.

The course is presented as a textbook that is appropriate for either a lower-level graduate course or an advanced undergraduate course. However, the potential readership should also include mathematicians whose expertise lies outside set theory but who would like to learn more about modern set-theoretic techniques that might be applicable in their field.

The reader of this text is assumed to have a good understanding of abstract proving techniques, and of the basic geometric and topological structure of the n-dimensional Euclidean space ℝn. In particular, a comfort in dealing with the continuous functions from ℝn into ℝ is assumed. A basic set-theoretic knowledge is also required.