III.99 The Zermelo–Fraenkel Axioms

The Zermelo–Fraenkel, or ZF, axioms are a collection of axioms that provide a foundation for set theory. They may be viewed in two ways. The first is as a list of the “allowed operations” on sets. For example, there is an axiom that states that, given sets x and y, there exists a “pair set,” whose members are precisely x and y.

One of the reasons the ZF axioms are important is that it is possible to reduce all of mathematics to set theory, so the ZF axioms can be regarded as a foundation for mathematics as a whole. Of course, for this to be the case it is vital that the operations allowed by the ZF axioms do indeed allow one to perform all of the usual mathematical constructions. Some of the axioms are rather subtle as a result.

The other way to view the ZF axioms is as giving us just what we need to “build up” the world of all sets, starting with just the empty set. One can look at the various ZF axioms and see that each one plays an essential role as we create the set-theoretic universe. Equivalently, they are “closure rules” that any universe of sets, or more precisely any model of set theory, should obey. So, for example, there is an axiom that says that every set has a power set (the set of all its subsets), and this axiom allows us to build up a huge collection of sets starting with just the empty set: one obtains the power set of the empty set, the power set of the power set of the empty set, and so on. Indeed, the universe of all sets could (in a sense) be described as the closure of the empty set under all the allowable operations of ZF.

The ZF axioms are written in the language of FIRST- ORDER LOGIC [IV.23 §1]. So each axiom can mention variables (which are interpreted as ranging over all sets), as well as the usual logical operations, and also one “primitive relation,” namely membership. For example, the pair-set axiom above would be formally written as

By convention, the ZF axioms do not include the AXIOM OF CHOICE [III.1]; when one does include the axiom of choice, the axioms are usually called the “ZFC axioms.”

For a more detailed discussion of the ZF axioms see SET THEORY [IV.22 §3.1].