Axiom of empty set

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In set theory, the axiom of empty set is one of the axioms of Zermelo-Fraenkel set theory.

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

or in words:

There is a set such that no set is a member of it.

We can use the axiom of extensionality to show that there is only one such set. Since it is unique we can name it. It is called the empty set, and also denoted by {}. Thus the essence of the axiom is:

An empty set exists.

The axiom of empty set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity. On the other hand, there are other formulations of that axiom that don't presuppose the existence of an empty set. Also, the ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.

Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. That said, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. However, if separation is derived as a theorem schema from the axiom schema of replacement (as is sometimes done), then that derivation requires the axiom of empty set. So it could not be used to eliminate the axiom of empty set.