I want to tell the reader about the theory of sets in the same way, in which I learned it, by following the "corridor” course of study. Thus, our attention will be focused mainly on giving clear presentations of problems, discussing un­ expected or surprising examples, quite often giving contra­ dictory "naive” discussions. We shall find that the theory of functions of a real variable is richly endowed with all these. And if, after he has read this book, a high-school or college student wants to study the theory of sets or the theory of functions of a real variable more deeply, the author will feel that his book has been a success.Professor Vilenkin has produced a small masterpiece that can be read with profit and delight by students of mathematics and laymen with an interest in mathematics. Slightly more than half the book explores the notion of cardinality of sets and the re­ mainder traces the evolution of some of the most important concepts of m athem atics such as function, curve, surface and dimension.The exposition combines informality with integrity of presenta­ tion and there is a wealth of unusual examples illustrating the paradoxical properties of curves and surfaces. Professor Vilenkin’s essay provides a royal road to the important concepts with which it is concerned.