Zermelo–Fraenkel set theory

E13857

axiomatic set theory formal system foundational system for mathematics

Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.

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Observed surface forms (7)

Surface form	Occurrences
Zermelo–Fraenkel set theory with Choice	3
Zermelo–Fraenkel axioms	2
Zermelo–Fraenkel set theory with choice	2
Zermelo–Fraenkel set theory (ZF) under suitable assumptions	1
Zermelo–Fraenkel set theory (indirectly)	1
Zermelo–Fraenkel set theory with Choice (ZFC) under suitable assumptions	1
Zermelo–Fraenkel set theory without choice	1

Statements (48)

Predicate	Object
instanceOf	axiomatic set theory ⓘ formal system ⓘ foundational system for mathematics ⓘ
abbreviation	ZF ⓘ
associatedWith	von Neumann universe ⓘ surface form: von Neumann cumulative hierarchy
assumes	all objects are sets ⓘ
canBeAugmentedWith	axiom of choice ⓘ
consistencyStatus	not known to be provably consistent within itself ⓘ
designedToAvoid	Russell’s paradox ⓘ surface form: Russell paradox set-theoretic paradoxes ⓘ
excludesByDefault	axiom of choice ⓘ
extension	ZF ⓘ surface form: ZFC Zermelo–Fraenkel set theory self-linksurface differs ⓘ surface form: Zermelo–Fraenkel set theory with choice
field	mathematical logic ⓘ set theory ⓘ
goal	avoid set-theoretic paradoxes ⓘ provide rigorous axioms for set theory ⓘ
hasAxiom	axiom of empty set ⓘ axiom of extensionality ⓘ axiom of infinity ⓘ axiom of pairing ⓘ axiom of power set ⓘ axiom of regularity ⓘ axiom of union ⓘ axiom schema of replacement ⓘ axiom schema of separation ⓘ
hasIndependenceResults	continuum hypothesis independence from ZFC ⓘ
hasModelType	transitive model ⓘ
hasPrimitiveRelation	membership relation ⓘ
hasVariant	ZF ⓘ surface form: ZFC
historicalPrecursor	naive set theory ⓘ
implies	existence of integers ⓘ existence of many transfinite cardinals ⓘ existence of natural numbers ⓘ existence of rational numbers ⓘ existence of real numbers ⓘ
isBaseTheoryFor	development of classical mathematics ⓘ most of modern set theory ⓘ
language	single-sorted language of sets ⓘ
logicalFramework	first-order logic ⓘ
namedAfter	Abraham Fraenkel ⓘ Ernst Zermelo ⓘ
refines	Zermelo set theory ⓘ
relatedConcept	cumulative hierarchy of sets ⓘ
strengthens	axiom schema of replacement over separation alone ⓘ
symbolForPrimitiveRelation	∈ ⓘ
timePeriodOfDevelopment	early 20th century ⓘ
usedAs	standard foundation for much of modern mathematics ⓘ

Referenced by (27)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gödel's incompleteness theorems → appliesTo → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory with Choice

Cantor’s paradox → avoidedIn → Zermelo–Fraenkel set theory ⓘ

Abraham Fraenkel → coDeveloperOf → Zermelo–Fraenkel set theory ⓘ

axiom schema of separation → componentOf → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel axioms

von Neumann–Bernays–Gödel set theory → extends → Zermelo–Fraenkel set theory ⓘ

Zermelo–Fraenkel set theory → extension → Zermelo–Fraenkel set theory self-linksurface differs ⓘ

this entity surface form: Zermelo–Fraenkel set theory with choice

ZF → fullName → Zermelo–Fraenkel set theory ⓘ

set theory → hasAxiomSystem → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory with Choice

Cantor’s theorem → holdsIn → Zermelo–Fraenkel set theory ⓘ

axiom of choice → independentOf → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory without choice

Cantor’s paradox → influenced → Zermelo–Fraenkel set theory ⓘ

Frege’s system in "Grundgesetze der Arithmetik" → influenced → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory (indirectly)

Russell’s paradox → influenced → Zermelo–Fraenkel set theory ⓘ

Zermelo set theory → influenced → Zermelo–Fraenkel set theory ⓘ

von Neumann–Bernays–Gödel set theory → isConservativeOver → Zermelo–Fraenkel set theory ⓘ

von Neumann–Bernays–Gödel set theory → isEquiconsistentWith → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory with choice

Morse–Kelley set theory by class–set distinction → isRelatedTo → Zermelo–Fraenkel set theory ⓘ

Morse–Kelley set theory by class–set distinction → isStrongerThan → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory with Choice

Abraham Fraenkel → knownFor → Zermelo–Fraenkel set theory ⓘ

Ernst Zermelo → knownFor → Zermelo–Fraenkel set theory ⓘ

Burali-Forti paradox → motivatedDevelopmentOf → Zermelo–Fraenkel set theory ⓘ

Ernst Zermelo → notableIdea → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel axioms

von Neumann universe → satisfies → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory (ZF) under suitable assumptions

von Neumann universe → satisfies → Zermelo–Fraenkel set theory ⓘ

this entity surface form: Zermelo–Fraenkel set theory with Choice (ZFC) under suitable assumptions

axiom schema of separation → usedIn → Zermelo–Fraenkel set theory ⓘ

Zermelo set theory → weakerThan → Zermelo–Fraenkel set theory ⓘ