Axiom of Extensionality in set theory
E459317
The Axiom of Extensionality in set theory states that a set is completely determined by its members, meaning two sets are equal if and only if they have exactly the same elements.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Axiom of Extensionality | 0 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
axiom of set theory
ⓘ
logical axiom schema ⓘ |
| alternativeFormulation | ∀A∀B(A = B ↔ ∀x(x ∈ A ↔ x ∈ B)) ⓘ |
| appearsIn | standard presentations of ZF axioms ⓘ |
| assumes | equality is a primitive logical notion ⓘ |
| category | foundational principle of mathematics ⓘ |
| compatibleWith |
axiom of choice
ⓘ
axiom of foundation ⓘ axiom of infinity ⓘ axiom schema of replacement ⓘ axiom schema of separation ⓘ |
| constrains | equality of sets ⓘ |
| contrastsWith | intensional characterizations of objects ⓘ |
| ensures | uniqueness of sets determined by membership ⓘ |
| expresses | extensionality principle for sets ⓘ |
| field | set theory ⓘ |
| formalStatement | ∀A∀B(∀x(x ∈ A ↔ x ∈ B) → A = B) ⓘ |
| hasConsequence |
there is at most one empty set
ⓘ
there is at most one set with a given membership profile ⓘ |
| hasVariant | extensionality for classes in class theories ⓘ |
| historicalOrigin | introduced by Ernst Zermelo in axiomatizations of set theory ⓘ |
| holdsIn | the cumulative hierarchy V of sets ⓘ |
| implies |
any property of a set is determined by properties of its elements in pure set-theoretic representation
ⓘ
if two sets have different elements then they are not equal ⓘ no two distinct sets have exactly the same elements ⓘ |
| isIndependentOf | other ZF axioms given suitable formalization ⓘ |
| language | first-order language of set theory with ∈ and = ⓘ |
| logicalForm | first-order sentence with equality and membership ⓘ |
| modelTheoreticRole | restricts possible interpretations of the membership relation ⓘ |
| necessaryFor |
identifying natural numbers with specific sets in set-theoretic constructions
ⓘ
identifying ordered pairs with specific sets in set-theoretic constructions ⓘ |
| philosophicalInterpretation | identifies sets with their extension rather than their intension ⓘ |
| relatesConcept |
membership relation
ⓘ
set equality ⓘ |
| role |
identifies sets with their membership structure
ⓘ
rules out urelements in pure set theories unless explicitly allowed ⓘ |
| states |
sets are determined solely by their members
ⓘ
two sets are equal if and only if they have the same elements ⓘ |
| symbolUses |
equality symbol =
ⓘ
membership symbol ∈ ⓘ |
| usedIn |
Morse–Kelley set theory
NERFINISHED
ⓘ
Zermelo set theory NERFINISHED ⓘ Zermelo–Fraenkel set theory NERFINISHED ⓘ most standard axiomatizations of set theory ⓘ von Neumann–Bernays–Gödel set theory NERFINISHED ⓘ |
| usedToProve |
uniqueness of set-theoretic constructions defined by comprehension-like conditions
ⓘ
uniqueness of the empty set ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.