Axiom of empty set

In axiomatic set theory, the axiom of empty set, also called the axiom of null set and the axiom of existence, is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo set theory and Zermelo–Fraenkel set theory, with or without the axiom of choice.

Formal statement

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

${\displaystyle \exists A\,\forall x\,(x\notin A)}$.

Or, alternatively, ${\displaystyle \exists x\,\lnot \exists y\,(y\in x)}$.

In words:

There is a set such that no element 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 ∅). The axiom, stated in natural language, is in essence:

An empty set exists.

This formula is a theorem and considered true in every version of set theory. The only controversy is over how it should be justified: by making it an axiom; by deriving it from a set-existence axiom (or logic) and the axiom of separation; by deriving it from the axiom of infinity; or some other method.

In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity. However, there are other formulations of that axiom that do not presuppose the existence of an empty set. 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. However, 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, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem.

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

1. Cunningham, Daniel W. (2016). Set theory: a first course. Cambridge mathematical textbooks. New York, NY: Cambridge University Press. p. 24. ISBN 978-1-107-12032-7.
2. "Set Theory | Internet Encyclopedia of Philosophy". Retrieved 2024-06-10.
3. Bagaria, Joan (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Set Theory", The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-06-10
4. Hrbacek, Karel; Jech, Thomas J. (1999). Introduction to set theory. Pure and applied mathematics (3. ed., rev. and expanded, [Repr.] ed.). Boca Raton, Fla.: CRC Press. p. 7. ISBN 978-0-8247-7915-3.
5. "AxiomaticSetTheory". www.cs.yale.edu. Retrieved 2024-06-10.
6. Jech, Thomas J. (2003). Set theory (The 3rd millennium ed., rev. and expanded ed.). Berlin: Springer. p. 3. ISBN 3-540-44085-2. OCLC 50422939.
7. "Set Theory > Zermelo-Fraenkel Set Theory (ZF) (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2024-06-10.

Further reading

• Burgess, John, 2005. Fixing Frege. Princeton Univ. Press.
• 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.