Editing Axiom of extensionality (section)

== In set theory with ur-elements ==
{{Unreferenced section|date=November 2024}}
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, <math>B \in A</math> makes no sense if <math>A</math> is an ur-element, so the axiom of extensionality simply applies only to sets.

Alternatively, in untyped logic, we can require <math>B \in A</math>  to be false whenever <math>A</math> 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:

:<math>\forall A \, \forall B \, ( \exists X \, (X \in A) \implies [ \forall Y \, (Y \in A \iff Y \in B) \implies A = B ] \, ).</math>

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 <math>A</math> itself to be the only element of <math>A</math>
whenever <math>A</math> is an ur-element. While this approach can serve to preserve the axiom of extensionality, the [[axiom of regularity]] will need an adjustment instead.