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.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Zermelo–Fraenkel set theory with choice | 1 |
| Zermelo’s axioms for set theory | 1 |
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 ⓘ |
Referenced by (7)
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