Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC)
GPTKB entity
Statements (103)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:set_theory
|
| gptkbp:abbreviation |
gptkb:ZFC
|
| gptkbp:basisFor |
most of mathematics
|
| gptkbp:consistencyRelativeTo |
gptkb:Zermelo-Fraenkel_set_theory_(ZF)
|
| gptkbp:field |
gptkb:logic
gptkb:set_theory |
| gptkbp:formedBy |
early 20th century
|
| gptkbp:hasAxiom |
gptkb:set_theory
gptkb:Axiom_of_Choice gptkb:Axiom_of_Extensionality gptkb:Axiom_of_Infinity gptkb:Axiom_of_Pairing gptkb:Axiom_of_Power_Set gptkb:Axiom_of_Regularity gptkb:Axiom_of_Replacement gptkb:Axiom_of_Separation gptkb:Axiom_of_Union |
| https://www.w3.org/2000/01/rdf-schema#label |
Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC)
|
| gptkbp:isConsistentIf |
no contradiction can be derived from its axioms
|
| gptkbp:isCountablyAxiomatizable |
true
|
| gptkbp:isFirstOrderTheory |
true
|
| gptkbp:isFoundationFor |
gptkb:algebraic_geometry
gptkb:logic gptkb:probability_theory gptkb:Banach_spaces gptkb:Hilbert_spaces gptkb:set-theoretic_topology gptkb:descriptive_set_theory complex analysis computability theory functional analysis measure theory model theory number theory proof theory recursion theory fields trees combinatorics groups mathematical objects graph theory homotopy theory algebraic varieties categories graphs homological algebra ideals modules real analysis real numbers sequences vector spaces filters differentiable manifolds rings topological spaces natural numbers functions lattices partially ordered sets relations ultrafilters cohomology theories metric spaces schemes cardinals higher category theory smooth manifolds functors model categories natural transformations manifolds measure spaces probability spaces sigma-algebras Boolean algebras compact spaces locally compact spaces topoi sheaves connected spaces discrete spaces measurable spaces ordinals presheaves well-ordered sets |
| gptkbp:isIncomplete |
true
|
| gptkbp:isUndecidable |
true
|
| gptkbp:isWeakerThan |
gptkb:Kelley–Morse_set_theory
gptkb:Zermelo-Fraenkel_set_theory_(ZF) Von Neumann–Bernays–Gödel set theory (NBG) |
| gptkbp:namedAfter |
gptkb:Ernst_Zermelo
gptkb:Abraham_Fraenkel |
| gptkbp:standardFoundationFor |
modern mathematics
|
| gptkbp:usedIn |
gptkb:algebra
gptkb:topology gptkb:category_theory analysis model theory number theory |
| gptkbp:bfsParent |
gptkb:Continuum_Hypothesis
|
| gptkbp:bfsLayer |
6
|