- Axiom of empty set
In

set theory , the**axiom of empty set**is one of theaxiom s ofZermelo–Fraenkel set theory and one of the axioms ofKripke–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 theaxiom 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.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Empty set**— ∅ redirects here. For similar looking symbols, see Ø (disambiguation). The empty set is the set containing no elements. In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality… … Wikipedia**Axiom of pairing**— In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Frankel axioms, the … Wikipedia**Axiom schema of specification**— For the separation axioms in topology, see separation axiom. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or… … Wikipedia**Axiom of choice**— This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of … Wikipedia**set theory**— the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… … Universalium**Axiom of regularity**— In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo Fraenkel set theory and was introduced by harvtxt|von Neumann|1925. In first order logic the axiom reads::forall A (exists B (B in A)… … Wikipedia**Set theory**— This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects … Wikipedia**Axiom of infinity**— In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Fraenkel axioms,… … Wikipedia**Axiom schema of replacement**— In set theory, the axiom schema of replacement is a schema of axioms in Zermelo Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite… … Wikipedia**Axiom of countable choice**— The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory, similar to the axiom of choice. It states that any countable collection of non empty sets must have a choice function. Spelled out, this means… … Wikipedia