Sets, logic, and axiomatic theories

THIS BOOK is an introduction to the nature of modern abstract mathematics. It is intended to bridge the gap between the false image of mathematics as solely a computational theory and the true image of mathematics as the science of abstract form and structure. It explains the basic role of set theory for mathematics generally, the modern attitude regarding the axiomatic method in mathematics, and the role of symbolic logic in developing axiomatic theories. Intuitive set theory is treated in detail with numerous examples and exercises. The elementary part of symbolic logic, the statement calculus, is fully developed, and the first-order predicate calculus is sketched to the point where its role in the formulation and the investigation of formal axiomatic theories can be examined. As an illustration of the axiomatic method in practice, the elementary part (including the representation theorem) of the theory of Boolean algebras is discussed in detail. This book is intended for use in a one-semester course devoted to the foundations of mathematics, as a text for courses designed to introduce high school teachers to modern mathematics, and as a reference book. It contains selected portions from a forthcoming textbook which treats the foundations of modern abstract mathematics in a more comprehensive manner.