ZF
E84442
ZF is the standard axiomatic framework for set theory that underpins much of modern mathematics.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| ZFC | 6 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic set theory
ⓘ
first-order theory ⓘ formal system ⓘ |
| allowsConstructionOf |
complex numbers
ⓘ
integers ⓘ natural numbers ⓘ rational numbers ⓘ real numbers ⓘ |
| assumes | all mathematical objects are sets ⓘ |
| consistencyQuestion | relative to large cardinal axioms ⓘ |
| developedInCentury | 20th century ⓘ |
| differsFrom |
ZF
self-linksurface differs
ⓘ
surface form:
ZFC
|
| extends | Zermelo set theory ⓘ |
| field |
mathematical logic
ⓘ
set theory ⓘ |
| fullName | Zermelo–Fraenkel set theory ⓘ |
| hasAxiom |
axiom of empty set
ⓘ
axiom of extensionality ⓘ axiom of foundation ⓘ axiom of infinity ⓘ axiom of pairing ⓘ axiom of power set ⓘ axiom of union ⓘ axiom schema of replacement ⓘ axiom schema of separation ⓘ |
| hasAxiomSchema |
replacement
ⓘ
separation ⓘ |
| hasCumulativeHierarchy | von Neumann universe ⓘ |
| hasModelType |
countable model
ⓘ
transitive model ⓘ |
| hasNonLogicalSymbol | binary relation symbol ∈ ⓘ |
| hasUndecidableStatement |
axiom of choice
ⓘ
continuum hypothesis ⓘ generalized continuum hypothesis ⓘ |
| isExtendedBy |
ZF
self-linksurface differs
ⓘ
surface form:
ZFC
|
| isFormalizedIn |
Hilbert-style deductive systems
ⓘ
natural deduction systems ⓘ |
| isIncompletenessSubjectTo |
Gödel's incompleteness theorems
ⓘ
surface form:
Gödel incompleteness theorems
|
| isStandardFrameworkFor | axiomatic set theory in mainstream mathematics ⓘ |
| language | first-order logic with equality ⓘ |
| namedAfter |
Abraham Fraenkel
ⓘ
Ernst Zermelo ⓘ |
| omitsAxiom | axiom of choice ⓘ |
| standardIn | foundations of mathematics ⓘ |
| underpins | much of modern mathematics ⓘ |
| usedFor |
foundations of algebra
ⓘ
foundations of analysis ⓘ foundations of topology ⓘ |
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
ZFC
this entity surface form:
ZFC
this entity surface form:
ZFC
this entity surface form:
ZFC
this entity surface form:
ZFC
this entity surface form:
ZFC