The Zermelo-Fraenkel Axioms
This chapter provides the first three axioms of set theory as proposed by Ernst Zermelo and modified by others, including Abraham Fraenkel. The theory is described as Zermelo-Praenkel set theory – ZF for short. The chapter shows how to construct some of the basic building blocks of the theory on the basis of these axioms, for instance how to represent ordered pairs and functions as sets. Richard Dedekin's construction of the real numbers, along with the associated axioms for the reals, completes the process of putting the calculus on a rigorous footing. It is important to realize that there are schools of mathematics that would reject 'standard' real analysis and, along with it, Dedekind's work. Gottlob Frege's hope was to be able to define the natural numbers in terms of pure logic and sets, and the sets themselves from pure logic.