Axiom of empty set

In set theory, the axiom of empty set is one of the axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory.

Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads::$exist\; x,\; forall\; y,\; lnot\; (y\; in\; x)$or in words::There is a set such that no set is a member of it.

Interpretation

We can use the axiom of extensionality to show that there is only one empty set. Since it is unique we can name it. It is called the "empty set" (denoted by { } or ∅). 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 do not 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. This is true, since the empty set is a subset of any set consisting of those elements that satisfy a contradictory formula.

In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms, then the existence of the empty set is a theorem and one does not need a separate axiom for it.

If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema. The formulation used in the axiom schema of replacement article only allows to construct the image "F" ["a"] when "a" is contained in the domain of the class function "F"; then the derivation of separation requires the axiom of empty set. On the other hand, the constraint of totality of "F" is often dropped from the replacement schema, in which case it implies the separation schema without using the axiom of empty set (or any other axiom for that matter).

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.