mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the axiom of power set is one of the Zermelo–Fraenkel axioms of

axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

. In the

formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symb ...

of the Zermelo–Fraenkel axioms, the axiom reads: :

\forall x \, \exists y \, \forall z \, \in y \iff \forall w \, (w \in z \Rightarrow w \in x) /math>

where ''y'' is the

Power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

of ''x'',

\mathcal(x)

. In English, this says: :

Given any In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''x'',

there is English grammar is the set of structural rules of the English language. This includes the structure of words, phrases, clauses, sentences, and whole texts. This article describes a generalized, present-day Standard English – a form of speech an ...

a set

\mathcal(x)

such that In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Definin ...

, given any set ''z'', this set ''z'' is a member of

\mathcal(x)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

every element of ''z'' is also an element of ''x''. More succinctly: ''for every set

x

, there is a set

\mathcal(x)

consisting precisely of the subsets of

x

.'' Note the

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

relation

\subseteq

is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership,

\in

. By the

axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...

, the set

\mathcal(x)

is unique. The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although

constructive set theory Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a con ...

prefers a weaker version to resolve concerns about

predicativity In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more com ...

Consequences

The power set axiom allows a simple definition of the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...

of two sets

X

and

Y

: :

X \times Y = \.

Notice that :

x, y \in X \cup Y

\, \ \in \mathcal(X \cup Y)

and, for example, considering a model using the Kuratowski ordered pair, :

(x, y) = \ \in \mathcal(\mathcal(X \cup Y))

and thus the Cartesian product is a set since :

X \times Y \subseteq \mathcal(\mathcal(X \cup Y)).

One may define the Cartesian product of any

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

collection Collection or Collections may refer to: * Cash collection, the function of an accounts receivable department * Collection (church), money donated by the congregation during a church service * Collection agency, agency to collect cash * Collectio ...

of sets recursively: :

X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_) \times X_n.

Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the

Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its fo ...

References

Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...

, ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . *Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. . {{Set theory Axioms of set theory de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC