We are going to sketch how the half-way intuitive and half-way axiomatic development given in the first eleven sections can be transformed into a rigorously axiomatic development. To follow this sketch, an elementary acquaintance with the basics of mathematical logic is needed. As this book considers only set theory, it is beyond its scope to provide the necessary background in mathematical logic. To facilitate matters, we will, however, explain the notation and clarify what is meant by an axiomatic development of set theory.

In what follows, we will denote by L0 a first-order language that, in addition to the variable symbols contains two two-place predicates: = (equality) and ε (being and element of). L0 is called the language of the Zermelo–Fraenkel axiom system (see Section Al). In a strictly formal presentation, variable symbols would be specified as, say, υ0, υ1, υ2, … For easier readability, we will use a variety of letters to denote variables. In particular, unless otherwise indicated, the letters x, y, z, u, v, w, A, B, C (possibly with subscripts) will always denote variables.

Later we will consider languages L that are extensions of the language L0. Such languages may contain other predicate and function symbols. We assume that the reader knows how to define terms and well-formed formulas in a language. The set of terms and well-formed formulas in a language L are denoted by Term(L) and Wff(L), respectively.