# Axiom of power set
The **axiom of power set** is one of the axioms of [Zermelo–Fraenkel set theory](https://www.edgepedia.org/zermelo-fraenkel-set-theory) (ZF, and with Choice, ZFC). It asserts that for every set \( A \) there exists a set, denoted \( \mathcal{P}(A) \) and called the power set of \( A \), whose elements are all the subsets of \( A \).[1](https://plato.stanford.edu/entries/set-theory/) It is one of the set-existence principles that make the universe of ZFC rich enough to host classical mathematics, and it is the axiom behind [Cantor's theorem](https://www.edgepedia.org/cantors-theorem), the result that no set is in bijective correspondence with its power set.[1](https://plato.stanford.edu/entries/set-theory/) • [2](https://encyclopediaofmath.org/index.php?title=ZFC)
| Key fact | Detail |
|---|---|
| Statement | For every set \( A \) there exists a set \( \mathcal{P}(A) \) whose elements are all the subsets of \( A \)[1](https://plato.stanford.edu/entries/set-theory/) |
| Formal form | \( \forall x \, \exists y \, \forall z \,( z \in y \leftrightarrow z \subseteq x ) \), where \( z \subseteq x \) abbreviates \( \forall w \,( w \in z \rightarrow w \in x ) \)[3](https://plato.stanford.edu/archives/spr2017/entries/set-theory/ZF.html) |
| Origin | Included among the seven axioms of Ernst Zermelo's 1908 axiomatization of set theory[4](https://mathshistory.st-andrews.ac.uk//Biographies/Zermelo/) |
| Main consequence | Cantor's theorem: \( |A| < |\mathcal{P}(A)| \) for every set \( A \), proved by the diagonal argument[5](https://notes.amittai.studio/real-analysis/foundations/sets-logic-functions) |
| Role in the hierarchy | The Power Set axiom is used to obtain \( V_{\alpha+1} = \mathcal{P}(V_\alpha) \) from \( V_\alpha \) in the cumulative hierarchy of sets[1](https://plato.stanford.edu/entries/set-theory/) |
| Without it | ZFC minus power set (with replacement) has models in which \( \omega_1 \) is singular and the Łoś ultrapower theorem fails; replacing replacement by collection repairs these deficits[6](https://arxiv.org/abs/1110.2430) |
| Constructive status | The definition is impredicative, so constructive set theories replace power set with exponentiation (Myhill) or subset collection (CZF)[7](https://plato.stanford.edu/entries/mathematics-constructive/) |
## Statement
In the first-order language of set theory, whose only non-logical symbol is the membership relation \( \in \), the axiom says that for any set \( x \) there is a set \( y \) containing exactly those sets whose members are all elements of \( x \), that is, all the subsets of \( x \):[3](https://plato.stanford.edu/archives/spr2017/entries/set-theory/ZF.html)
\[ \forall x \, \exists y \, \forall z \,[\, z \in y \leftrightarrow \forall w \,( w \in z \rightarrow w \in x ) \,] \]
Writing \( z \subseteq x \) for \( \forall w \,( w \in z \rightarrow w \in x ) \), this simplifies to \( \forall x \, \exists y \, \forall z \,( z \in y \leftrightarrow z \subseteq x ) \).[3](https://plato.stanford.edu/archives/spr2017/entries/set-theory/ZF.html) Extensionality guarantees that such a \( y \) is unique, so the notation \( \mathcal{P}(x) \) is justified for every set \( x \).[3](https://plato.stanford.edu/archives/spr2017/entries/set-theory/ZF.html)
The axiom is a single axiom, not a schema: unlike Separation or Replacement, it does not quantify over formulas. Its content is a genuine existence claim, since the logic alone does not provide a set collecting all subsets of an infinite set.[2](https://encyclopediaofmath.org/index.php?title=ZFC)
## History
The first axiomatization of set theory was given by [Ernst Zermelo](https://www.edgepedia.org/ernst-zermelo) in his 1908 paper "Untersuchungen über die Grundlagen der Mengenlehre, I", undertaken after the appearance of [Russell's paradox](https://www.edgepedia.org/russells-paradox) around 1903 and motivated also by the need to spell out the principles underlying his proof of the [well-ordering theorem](https://www.edgepedia.org/well-ordering-theorem).[8](https://plato.stanford.edu/entries/zermelo-set-theory/) • [1](https://plato.stanford.edu/entries/set-theory/) Zermelo gave seven axioms: extensionality, elementary sets, separation, power set, union, choice, and infinity.[4](https://mathshistory.st-andrews.ac.uk//Biographies/Zermelo/) In his 1904 well-ordering proof he had already listed two postulates he explicitly depended on, a version of separation and the power set axiom.[8](https://plato.stanford.edu/entries/zermelo-set-theory/)
The power set axiom, together with the axiom of infinity, is one of the four central axioms of Zermelo's system, since together they show the existence of uncountable sets.[8](https://plato.stanford.edu/entries/zermelo-set-theory/) Later work by Fraenkel and Skolem in the early 1920s added the [Replacement axiom](https://www.edgepedia.org/axiom-schema-of-replacement), and von Neumann added Foundation, producing the standard system ZFC.[1](https://plato.stanford.edu/entries/set-theory/)
## Consequences
**Cantor's theorem.** The most immediate consequence is Cantor's theorem: for every set \( A \), \( |A| < |\mathcal{P}(A)| \); in particular there is no surjection \( A \to \mathcal{P}(A) \).[5](https://notes.amittai.studio/real-analysis/foundations/sets-logic-functions) The proof uses only the Power Set axiom, to form \( \mathcal{P}(A) \), and Separation, to form the diagonal set \( B = \{ x \in A : x \notin g(x) \} \) for a given function \( g : A \to \mathcal{P}(A) \); the assumption \( B = g(x_0) \) yields \( x_0 \in B \) if and only if \( x_0 \notin B \), a contradiction.[5](https://notes.amittai.studio/real-analysis/foundations/sets-logic-functions) • [9](https://alphabetamath.cc/item/thm-cantor-powerset) For a finite set of size \( n \) the power set has size \( 2^{n} \), and Cantor's theorem extends the strict inequality to infinite sets.[5](https://notes.amittai.studio/real-analysis/foundations/sets-logic-functions) Since the power set of any set is bigger than the set itself, for any infinite set there is a bigger one, giving an infinity of sizes of infinity.[10](https://www.andrew.cmu.edu/user/scheines/80-110/lectures/apr28/apr28.htm) Cantor famously showed that a set can never be put into bijective correspondence with its power set, a result yielding an endless hierarchy of transfinite cardinalities.[2](https://encyclopediaofmath.org/index.php?title=ZFC)
**The number systems.** With the set of natural numbers available through the axiom of infinity, the power set axiom provides enough sets to represent the rationals, the reals, and functions on the reals; the reals arise as sets of subsets of the rationals (Dedekind cuts).[8](https://plato.stanford.edu/entries/zermelo-set-theory/) • [1](https://plato.stanford.edu/entries/set-theory/)
**Cardinality beyond Zermelo's theory.** The axiom of infinity and the power set axiom together allow the creation of sets of cardinality \( \geq \aleph_{n} \) for each natural number \( n \), but, in the absence of a result showing \( 2^{\aleph_{0}} > \aleph_{n} \) for every \( n \), they do not guarantee a set of power \( \aleph_{\omega} \). This gap, pointed out by Fraenkel and Skolem in the early 1920s, motivated the Axiom of Replacement.[8](https://plato.stanford.edu/entries/zermelo-set-theory/)
## Role in the cumulative hierarchy
The ZF axioms are justified a posteriori by their use in building the cumulative hierarchy \( V \), defined by transfinite recursion: \( V_{0} = \varnothing \), \( V_{\alpha+1} = \mathcal{P}(V_{\alpha}) \), and \( V_{\alpha} = \bigcup \{ V_{\beta} : \beta < \alpha \} \) at limit ordinals. The Power Set axiom is used to obtain \( V_{\alpha+1} \) from \( V_{\alpha} \), while Replacement and Union handle limit stages.[1](https://plato.stanford.edu/entries/set-theory/)
## Set theory without power set
[Set theory](https://www.edgepedia.org/set-theory) without the power set axiom is used in arguments and constructions throughout the subject, and it has an abundance of natural models, including the structures \( H_{\kappa} \) of sets whose hereditary size is less than an uncountable regular cardinal \( \kappa \).[11](https://victoriagitman.github.io/publications/2011/12/09/what-is-the-theory-zfc-without-powers-set.html)
The theory \( \mathsf{ZFC}^{-} \), the usual axioms of ZFC with the power set axiom removed and choice taken as the well-ordering principle, is weaker than commonly supposed. Gitman, Hamkins, and Johnstone showed that it is inadequate to establish several basic facts: there are models of \( \mathsf{ZFC}^{-} \) in which \( \omega_{1} \) is singular, in which every set of reals is countable yet \( \omega_{1} \) exists, and in which there are sets of reals of every size \( \aleph_{n} \) but none of size \( \aleph_{\omega} \); the Łoś ultrapower theorem and the Gaifman theorem can fail for such models.[6](https://arxiv.org/abs/1110.2430) These deficits are repaired by strengthening the theory to \( \mathsf{ZFC}^{-} \) with the collection scheme in place of replacement, since without power sets the replacement and collection schemes cease to be equivalent.[6](https://arxiv.org/abs/1110.2430) • [12](https://victoriagitman.github.io/publications/2022/08/03/zfc-without-powerset-ii-reflection-strikes-back.html) The theory \( \mathsf{ZFC}^{-} \) is equiconsistent with second-order [Peano arithmetic](https://www.edgepedia.org/peano-arithmetic).[13](https://doi.org/10.3390/axioms14120865)
## Constructive alternatives
The definition of the power set \( \{ S : S \subset X \} \) is impredicative, and most constructive mathematicians hesitate to regard it as a set.[7](https://plato.stanford.edu/entries/mathematics-constructive/) Myhill observed that power set is not needed for Bishop-style constructive mathematics: it can be replaced by the exponentiation axiom, which asserts that for sets \( A \) and \( B \) the set of all mappings from \( A \) into \( B \) exists. Classically the two are equivalent, since subsets of a set are represented by characteristic functions, but without the law of excluded middle they are not.[7](https://plato.stanford.edu/entries/mathematics-constructive/) In constructive Zermelo–Fraenkel set theory (CZF), the subset collection schema takes the place of the power set axiom; adding full power set to CZF exceeds the strength of classical Zermelo set theory, bringing the theory to a fully impredicative setting.[14](https://plato.stanford.edu/archives/sum2012/entries/set-theory-constructive/axioms-CZF-IZF.html) • [7](https://plato.stanford.edu/entries/mathematics-constructive/)
## References
1. [Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/set-theory/)
2. [ZFC, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=ZFC)
3. [Zermelo-Fraenkel Set Theory (supplement), Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/archives/spr2017/entries/set-theory/ZF.html)
4. [Ernst Zermelo (1871–1953), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Zermelo/)
5. [Sets, Logic, and Functions: Cantor's theorem](https://notes.amittai.studio/real-analysis/foundations/sets-logic-functions)
6. [Gitman, V., Hamkins, J. D., Johnstone, T. A. (2011). What is the theory ZFC without power set? arXiv](https://arxiv.org/abs/1110.2430)
7. [Constructive Mathematics, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/mathematics-constructive/)
8. [Zermelo's Axiomatization of Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/zermelo-set-theory/)
9. [Cantor's theorem: A ≺ P(A), Alphabeta Math](https://alphabetamath.cc/item/thm-cantor-powerset)
10. [80-110: The Nature of Mathematical Reasoning, Carnegie Mellon](https://www.andrew.cmu.edu/user/scheines/80-110/lectures/apr28/apr28.htm)
11. [What is the theory ZFC without power set?, Victoria Gitman](https://victoriagitman.github.io/publications/2011/12/09/what-is-the-theory-zfc-without-powers-set.html)
12. [ZFC without Power Set II: Reflection Strikes Back, Victoria Gitman](https://victoriagitman.github.io/publications/2022/08/03/zfc-without-powerset-ii-reflection-strikes-back.html)
13. [Notes on the Equiconsistency of ZFC Without the Power Set Axiom and Second-Order Arithmetic, Axioms](https://doi.org/10.3390/axioms14120865)
14. [Axioms of CZF and IZF, Stanford Encyclopedia of Philosophy archive](https://plato.stanford.edu/archives/sum2012/entries/set-theory-constructive/axioms-CZF-IZF.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Axiomatic set theories › Zermelo–Fraenkel axioms › Axiom of power set*
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