Summary

After a discussion of what sets are useful for, a list is given of set operations and constructions which are in normal use by mathematicians. Then there is a complete list of the Zermelo–Fraenkel axioms, followed by discussion of the meaning, application and significance of each axiom individually, including reference to historical development. Normal mathematics can be developed within formal set theory, and the basis of this process is described. A system of abstract natural numbers is defined within ZF set theory and demonstrated to satisfy Peano's axioms. As an alternative to ZF, the von Neumann–Bernays system VNB of set/class theory is described and its usefulness and its relationships with ZF are discussed. Finally, some of the logical and philosophical aspects of formal set theory are described, including consistency and independence results.

The reader is presumed to be familiar with the algebra of sets and with standard set constructions and notation. Some experience with abstract algebraic ideas is useful. Section 1.1 is referred to, but this chapter is essentially independent of Chapters 2 and 3. No knowledge of mathematical logic is assumed.

What is a set?

On the face of it, the notion of set is one of the simplest ideas there can be. It is this simplicity and freedom from restrictive particular properties which make the notion so suitable for use in abstract mathematics. Indeed, ‘set’ itself is an abstraction which means little in isolation.