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The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the
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The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although
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The existence of the
Cartesian product can be proved without using the power set axiom, as in the case of the
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and is not used in the formal language of the
Zermelo–Fraenkel axioms. Rather, the subset relation
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1092:{\displaystyle X_{1}\times \cdots \times X_{n}=(X_{1}\times \cdots \times X_{n-1})\times X_{n}.}
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1199:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
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but in other models of ZF set theory could contain sets that are not constructible.
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900:{\displaystyle (x,y)=\{\{x\},\{x,y\}\}\in {\mathcal {P}}({\mathcal {P}}(X\cup Y))}
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973:{\displaystyle X\times Y\subseteq {\mathcal {P}}({\mathcal {P}}(X\cup Y)).}
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This article incorporates material from Axiom of power set on
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Set Theory: The Third
Millennium Edition, Revised and Expanded
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of the
Zermelo–Fraenkel axioms, the axiom of power set reads:
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784:{\displaystyle \{x\},\{x,y\}\in {\mathcal {P}}(X\cup Y)}
670:{\displaystyle X\times Y=\{(x,y):x\in X\land y\in Y\}.}
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The power set axiom allows a simple definition of the
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1227:Set Theory: An Introduction to Independence Proofs
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460:{\displaystyle \forall x\,\exists y\,\forall z\,}
1248:Creative Commons Attribution/Share-Alike License
794:and, for example, considering a model using the
1137:"Axiom of power set | set theory | Britannica"
910:and thus the Cartesian product is a set since
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1166:. Berlin: Springer-Verlag. pp. 56–57.
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96:The elements of the power set of the set
80:Learn how and when to remove this message
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43:This article includes a list of general
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49:it lacks sufficient corresponding
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270:{\displaystyle {\mathcal {P}}(x)}
185:{\displaystyle {\mathcal {P}}(x)}
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27:Concept in axiomatic set theory
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714:{\displaystyle x,y\in X\cup Y}
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139:. It guarantees for every set
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1207:(Springer-Verlag edition).
337:{\displaystyle \subseteq }
309:{\displaystyle \subseteq }
1568:One-to-one correspondence
1305:
1104:Kripke–Platek set theory
796:Kuratowski ordered pair
497:In English, this says:
344:is defined in terms of
282:constructive set theory
242:axiom of extensionality
159:the existence of a set
133:Zermelo–Fraenkel axioms
64:more precise citations.
1526:Constructible universe
1353:Constructibility (V=L)
1225:Kunen, Kenneth, 1980.
1160:Devlin, Keith (1984).
1117:constructible universe
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368:. Given this, in the
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1917:Axioms of set theory
1823:Burali-Forti paradox
1578:Set-builder notation
1531:Continuum hypothesis
1471:Symmetric difference
1210:Jech, Thomas, 2003.
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361:{\displaystyle \in }
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1536:Diagonal argument
1435:Cartesian product
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590:{\displaystyle Y}
570:{\displaystyle X}
555:Cartesian product
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322:formal set theory
233:{\displaystyle x}
209:{\displaystyle x}
152:{\displaystyle x}
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16:(Redirected from
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1886:Bertrand Russell
1876:John von Neumann
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1856:Richard Dedekind
1818:Suslin's problem
1729:Cantor's theorem
1446:De Morgan's laws
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1680:
1679:Subset
1626:hereditarily
1588:Venn diagram
1546:ordered pair
1461:Intersection
1405:Axiom schema
1382:
1241:
1240:
1226:
1211:
1196:
1177:. Retrieved
1162:
1155:
1144:. Retrieved
1140:
1131:
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1101:
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680:Notice that
679:
557:of two sets
552:
549:Consequences
541:
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471:
469:
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1871:Thomas Jech
1714:Alternative
1693:Uncountable
1647:Ultrafilter
1506:Cardinality
1410:replacement
1358:Determinacy
1193:Paul Halmos
1110:Limitations
525:, this set
277:is unique.
125:mathematics
62:introducing
1866:Kurt Gödel
1851:Paul Cohen
1688:Transitive
1456:Identities
1440:Complement
1427:Operations
1388:Regularity
1326:Adjunction
1285:Set theory
1244:PlanetMath
1146:2023-08-06
1123:References
988:collection
244:, the set
45:references
1799:Paradoxes
1719:Axiomatic
1698:Universal
1674:Singleton
1669:Recursive
1612:Countable
1607:Amorphous
1466:Power set
1383:Power set
1341:dependent
1336:countable
1179:8 January
1074:×
1063:−
1052:×
1049:⋯
1046:×
1017:×
1014:⋯
1011:×
956:∪
930:⊆
924:×
886:∪
860:∈
773:∪
757:∈
706:∪
700:∈
656:∈
650:∧
644:∈
611:×
519:such that
502:Given any
446:∈
440:⇒
434:∈
421:∀
417:⟺
410:∈
397:∀
390:∃
383:∀
356:∈
332:⊆
316:is not a
304:⊆
240:. By the
194:power set
117:inclusion
1911:Category
1803:Problems
1707:Theories
1683:Superset
1659:Infinite
1488:Concepts
1368:Infinity
1292:Overview
512:there is
70:May 2020
1741:General
1736:Zermelo
1642:subbase
1624: (
1563:Forcing
1541:Element
1513: (
1491:Methods
1378:Pairing
218:subsets
113:ordered
58:improve
1632:Filter
1622:Finite
1558:Family
1501:Almost
1346:global
1331:Choice
1318:Axioms
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985:finite
514:a set
470:where
192:, the
127:, the
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1724:Naive
1654:Fuzzy
1617:Empty
1600:types
1551:tuple
1521:Class
1515:large
1476:Union
1393:Union
1637:base
1231:ISBN
1216:ISBN
1201:ISBN
1181:2023
1168:ISBN
577:and
1598:Set
505:set
320:in
220:of
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