Axiom of empty set

In axiomatic set theory, the axiom of empty set,^[1]^[2] also called the axiom of null set^[3] and the axiom of existence,^[4]^[5] is a statement that asserts the existence of a set with no elements.^[3] 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.^[6]

Formal statement

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

\exists A\,\forall x\,(x\notin A)

.^[1]^[2]^[5]

Or, alternatively, $\exists x\,\lnot \exists y\,(y\in x)$ .^[7]

In words:

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

^ ^a ^b 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.
^ ^a ^b "Set Theory | Internet Encyclopedia of Philosophy". Retrieved 2024-06-10.
^ ^a ^b 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
^ 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.
^ ^a ^b "AxiomaticSetTheory". www.cs.yale.edu. Retrieved 2024-06-10.
^ 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.
^ "Set Theory > Zermelo-Fraenkel Set Theory (ZF) (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2024-06-10.

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.