von Neumann universe

E14977

class cumulative hierarchy proper class set-theoretic universe

The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.

Aliases (2)

Statements (49)

Predicate	Object
instanceOf	class → cumulative hierarchy → proper class → set-theoretic universe →
alsoKnownAs	cumulative hierarchy of sets →
builtByTransfiniteRecursionOn	ordinals →
choiceAxiomMayHoldIn	von Neumann universe →
contains	all sets (in ZF/ZFC) as elements of some level V_α →
containsAsSubstructure	cumulative hierarchy of hereditarily finite sets →
cumulativeProperty	for all α, V_α = ⋃_{β<α} V_β for limit α and P(V_{α−1}) for successors →
definedIn	axiomatic set theory →
extensionalityAxiomHoldsIn	von Neumann universe →
firstInfiniteLevel	V_ω →
foundationAxiomHoldsIn	von Neumann universe →
hasProperty	cumulative → rank-initial segment structure → transitive → well-founded →
historicallyIntroducedBy	John von Neumann in the 1920s →
infinityAxiomHoldsIn	von Neumann universe →
isTransitiveClass	von Neumann universe →
isUnionOf	V_α for all ordinals α →
levelNotation	V_0 = ∅ → V_{α+1} = P(V_α) → V_λ = ⋃_{β<λ} V_β for limit ordinal λ →
membershipRelationRestrictedTo	V forms a well-founded relation →
namedAfter	John von Neumann →
pairingAxiomHoldsIn	von Neumann universe →
powerSetAxiomHoldsIn	von Neumann universe →
rankFunctionCharacterization	x ∈ V_α iff rank(x) < α →
rankFunctionCodomain	ordinals →
rankFunctionDomain	all sets →
relatedConcept	Grothendieck universe → constructible universe L → rank hierarchy →
replacementAxiomHoldsIn	von Neumann universe →
satisfies	Zermelo–Fraenkel set theory (ZF) under suitable assumptions → Zermelo–Fraenkel set theory with Choice (ZFC) under suitable assumptions →
separationSchemaHoldsIn	von Neumann universe →
subsetRelation	for each α, V_α ⊂ V →
symbol	V →
unionAxiomHoldsIn	von Neumann universe →
usedAs	standard model of the set-theoretic universe →
usedIn	forcing arguments (as ambient universe) → inner model theory → relative consistency proofs →
V_0Equals	empty set →
V_1Contains	all subsets of the empty set →
V_ωContains	all hereditarily finite sets →

Referenced by (14)

Subject (surface form when different)	Predicate
von Neumann universe ("cumulative hierarchy of sets") →	alsoKnownAs
Zermelo–Fraenkel set theory ("von Neumann cumulative hierarchy") →	associatedWith
von Neumann universe →	choiceAxiomMayHoldIn
von Neumann universe →	extensionalityAxiomHoldsIn
von Neumann universe →	foundationAxiomHoldsIn
ZF →	hasCumulativeHierarchy
von Neumann universe →	infinityAxiomHoldsIn
von Neumann universe →	isTransitiveClass
John von Neumann →	notableConcept
von Neumann universe →	pairingAxiomHoldsIn
von Neumann universe →	powerSetAxiomHoldsIn
von Neumann universe →	replacementAxiomHoldsIn
von Neumann universe →	separationSchemaHoldsIn
von Neumann universe →	unionAxiomHoldsIn