Zermelo–Fraenkel set theory

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:set_theory
gptkbp:abbreviation	gptkb:ZF
gptkbp:alternativeName	gptkb:ZF gptkb:ZFC_(with_Axiom_of_Choice)
gptkbp:basisFor	gptkb:logic foundations of mathematics modern set theory
gptkbp:developedBy	gptkb:Ernst_Zermelo gptkb:Abraham_Fraenkel
gptkbp:field	gptkb:mathematics gptkb:set_theory
gptkbp:formedBy	early 20th century
gptkbp:hasAxiom	gptkb:Axiom_of_Choice gptkb:Axiom_of_Empty_Set 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
gptkbp:hasSubgroup	ZFC (if Axiom of Choice is included)
https://www.w3.org/2000/01/rdf-schema#label	Zermelo–Fraenkel set theory
gptkbp:isConsistentIf	no contradiction can be derived from its axioms
gptkbp:isCumulativeHierarchy	yes
gptkbp:isFirstOrderTheory	yes
gptkbp:isFoundationFor	most of modern mathematics
gptkbp:isIncomplete	due to Gödel's incompleteness theorems
gptkbp:isStandardFormulationOf	gptkb:set_theory
gptkbp:notation	gptkb:ZF gptkb:ZFC_(with_choice)
gptkbp:purpose	foundation for mathematics
gptkbp:relatedTo	gptkb:Gödel's_incompleteness_theorems gptkb:Russell's_paradox gptkb:Kuratowski–Zorn_lemma gptkb:Well-ordering_theorem
gptkbp:solvedBy	paradoxes in naive set theory
gptkbp:usedIn	gptkb:logic gptkb:category_theory foundations of mathematics model theory proof theory
gptkbp:bfsParent	gptkb:Ernst_Zermelo gptkb:Russell's_paradox gptkb:logic
gptkbp:bfsLayer	4