Axiom of extensionality

The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a set is. Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.

Formal statement

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

$\forall x\forall y[\forall z\;(\left.z\in x\right.\leftrightarrow \left.z\in y\right.)\rightarrow x=y]$

or in words:

If the sets $x$ and $y$ have the same members, then they are the same set.

In pure set theory, all members of sets are themselves sets, but not in set theory with urelements. The axiom's usefulness can be seen from the fact that, if one accepts that $\exists A\,\forall x\,(x\in A\iff \Phi (x))$, where $A$ is a set and $\Phi (x)$ is a formula that $x$ occurs free in but $A$ doesn't, then the axiom assures that there is a unique set $A$ whose members are precisely whatever objects (urelements or sets, as the case may be) satisfy the formula $\Phi (x)$.

The converse of the axiom, $\forall x\forall y[\forall z\;x=y\rightarrow (\left.z\in x\right.\leftrightarrow \left.z\in y\right.)]$, follows from the substitution property of equality.

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 the substitution property,

$\forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\implies \forall Y\,(A\in Y\iff B\in Y)\,),$

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.

Alternatively, in untyped logic, we can require $B\in A$ to be false whenever $A$ is an ur-element. In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set. To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:

$\forall A\,\forall B\,(\exists X\,(X\in A)\implies [\forall Y\,(Y\in A\iff Y\in B)\implies A=B]\,).$

That is:

Given any set A and any set B, if A is a nonempty set (that is, if there exists a member X of A), then if A and B have precisely the same members, then they are equal.

Yet another alternative in untyped logic is to define $A$ itself to be the only element of $A$ whenever $A$ is an ur-element. While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

See also

• Extensionality for a general overview.

References

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.