In the formal language of the ZermeloFraenkel axioms, the axiom reads:
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal iff they have precisely the same members. The essence of this is:
The axiom of extension can be used with any statement of the form
The axiom of extension is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory. However, it may require modifications for some purposes, as below.
In predicate logic without equality
The axiom given above assumes that equality is a primitive symbol in predicate logic. Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality. Then it's necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol, and it now these axioms that are referred to as the axioms of extension.
In set theory with urelements
An urelement[?] is a member of a set that is not itself a set. In the ZermeloFraenkel axioms, there are no urelements, but some alternative axiomatisations of set theory have them. Urelements can be treated as a different logical type[?] from sets; in this case, C ∈ A makes no sense if A is an urelement, so the axiom of extension simply applies only to sets.
Alternatively, in untyped logic, we can require C ∈ A to be false whenever A is an urelement. In this case, the usual axiom of extension would imply that every urelement is equal to the empty set. To avoid this, we can modify the axiom of extension to apply only to nonempty sets, so that it reads:
Search Encyclopedia

Featured Article
