Zermelo set theory
E87093
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Zermelo set theory canonical | 8 |
| Zermelo–Fraenkel set theory with choice | 1 |
| Zermelo’s axioms for set theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T694071 — 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: Zermelo set theory Context triple: [Zermelo–Fraenkel set theory, refines, Zermelo set theory]
-
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.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
-
C.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
D.
von Neumann 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.
-
E.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
- 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: Zermelo set theory Target entity description: 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.
-
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.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
-
C.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
D.
von Neumann 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.
-
E.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic set theory
ⓘ
formal system ⓘ |
| abbreviation | Z ⓘ |
| addressesProblem |
Burali-Forti paradox
ⓘ
Russell’s paradox ⓘ
surface form:
Russell paradox
|
| aim |
avoid known set-theoretic paradoxes
ⓘ
provide rigorous axiomatization of set theory ⓘ |
| allowsConstructionOf |
natural numbers
ⓘ
ordinal numbers below certain ranks ⓘ real numbers ⓘ |
| assumes |
all objects in the domain are sets
ⓘ
extensionality of sets ⓘ |
| basedOn | naive set theory ⓘ |
| category | foundations of mathematics ⓘ |
| clarifies | use of the axiom of choice ⓘ |
| consistentRelativeTo |
Peano arithmetic
ⓘ
surface form:
Peano arithmetic (under standard assumptions)
|
| doesNotInclude |
axiom of foundation in its original form
ⓘ
axiom of replacement ⓘ |
| excludes | proper classes as objects ⓘ |
| field |
mathematical logic
ⓘ
set theory ⓘ |
| formalizes | Cantorian set-theoretic ideas ⓘ |
| hasAxiom |
axiom of choice
ⓘ
axiom of empty set ⓘ axiom of extensionality ⓘ axiom of infinity ⓘ axiom of pairing ⓘ axiom of power set ⓘ axiom of union ⓘ axiom schema of separation ⓘ |
| hasModel | von Neumann universe up to certain ranks ⓘ |
| historicalRole | first widely accepted axiomatization of set theory ⓘ |
| includes | axiom of choice ⓘ |
| influenced |
Zermelo–Fraenkel set theory
ⓘ
modern axiomatic set theories ⓘ |
| introducedBy | Ernst Zermelo ⓘ |
| language | first-order logic ⓘ |
| namedAfter | Ernst Zermelo ⓘ |
| permits |
development of algebra and topology at an elementary level
ⓘ
development of basic analysis ⓘ |
| publicationYear | 1908 ⓘ |
| restricts | set formation by separation from existing sets ⓘ |
| strengthComparedTo | weaker than Zermelo–Fraenkel set theory with replacement ⓘ |
| subsetOf |
Zermelo set theory
self-linksurface differs
ⓘ
surface form:
Zermelo–Fraenkel set theory with choice
|
| symbolForMembership | ∈ ⓘ |
| uses | axiom schema to restrict comprehension ⓘ |
| usesConcept | membership relation ⓘ |
| weakerThan | Zermelo–Fraenkel set theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Zermelo set theory Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Zermelo’s axioms for set theory
this entity surface form:
Zermelo–Fraenkel set theory with choice