Zermelo-Fraenkel set theory (ZF)

URI: https://gptkb.org/entity/Zermelo-Fraenkel_set_theory_(ZF)

GPTKB entity

Statements (50)

Predicate	Object
gptkbp:instanceOf	gptkb:set_theory
gptkbp:abbreviation	gptkb:ZF
gptkbp:alternativeTo	gptkb:Morse–Kelley_set_theory gptkb:Tarski–Grothendieck_set_theory gptkb:Kripke–Platek_set_theory gptkb:New_Foundations
gptkbp:basisFor	most of modern mathematics
gptkbp:excludes	gptkb:Axiom_of_Choice
gptkbp:extendsTo	gptkb:Zermelo-Fraenkel_set_theory_with_Choice_(ZFC)
gptkbp:field	gptkb:mathematics gptkb:set_theory
gptkbp:hasAxiom	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:hasModel	gptkb:von_Neumann_universe
https://www.w3.org/2000/01/rdf-schema#label	Zermelo-Fraenkel set theory (ZF)
gptkbp:introducedIn	early 20th century
gptkbp:isAxiomSystemFor	sets
gptkbp:isConsistentIf	no contradiction can be derived from its axioms
gptkbp:isCumulativeHierarchy	true
gptkbp:isFirstOrderTheory	true
gptkbp:isIncomplete	due to Gödel's incompleteness theorems
gptkbp:isStandardFormulationOf	gptkb:set_theory
gptkbp:namedAfter	gptkb:Ernst_Zermelo gptkb:Abraham_Fraenkel
gptkbp:prevention	gptkb:Russell's_paradox
gptkbp:publishedIn	gptkb:Mathematische_Annalen
gptkbp:relatedTo	gptkb:continuum_hypothesis gptkb:Peano_axioms gptkb:category_theory gptkb:Zermelo_set_theory gptkb:large_cardinal_axioms gptkb:Gödel's_constructible_universe axiom of choice model theory proof theory axiom schema
gptkbp:replacedBy	gptkb:naive_set_theory
gptkbp:usedBy	gptkb:mathematician logicians philosophers of mathematics
gptkbp:usedFor	foundation of mathematics
gptkbp:bfsParent	gptkb:Zermelo-Fraenkel_set_theory_with_the_axiom_of_choice_(ZFC)
gptkbp:bfsLayer	6