Axiom of extensionality

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In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.

Contents

Formal statement

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

or in words:

(It is not really essential that C here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)

The converse, \forall A \, \forall B \, (A = B \Rightarrow \forall C \, (C \in A \iff C \in B) ), of this axiom follows from the substitution property of equality.

Interpretation

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 if and only if they have precisely the same members. The essence of this is:

The axiom of extensionality can be used with any statement of the form \exist A \, \forall B \, (B \in A \iff P(B) \, ), where P is any unary predicate that does not mention A or B, to define a unique set A whose members are precisely the sets satisfying the predicate P. We can then introduce a new symbol for A; it's in this way that definitions in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.

The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, 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 is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is

and it becomes this axiom that is referred to as the axiom of extensionality in this context.

In set theory with ur-elements

An ur-element is a member of a set that is not itself a set. In the Zermelo-Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory. Ur-elements can be treated as a different logical type from sets; in this case, B \in A makes no sense if A is an ur-element, so the axiom of extensionality simply applies only to sets.

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