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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Axiom of Extensionality in set theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4645484 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Axiom of Extensionality in set theory Context triple: [Basic Law V, relatedTo, Axiom of Extensionality in set theory]
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A.
Zermelo–Fraenkel set theory
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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B.
Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
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C.
axiom schema of separation
The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
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D.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
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E.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Axiom of Extensionality in set theory Target entity description: 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.
-
A.
Zermelo–Fraenkel set theory
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.
-
B.
Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
-
C.
axiom schema of separation
The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
-
D.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
-
E.
Cantor–Bernstein–Schröder theorem
The Cantor–Bernstein–Schröder theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
- F. None of above. chosen
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
ⓘ
Zermelo set theory ⓘ Zermelo–Fraenkel set theory ⓘ most standard axiomatizations of set theory ⓘ von Neumann–Bernays–Gödel set theory ⓘ |
| usedToProve |
uniqueness of set-theoretic constructions defined by comprehension-like conditions
ⓘ
uniqueness of the empty set ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Axiom of Extensionality in set theory Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.