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:category	formal semantics
gptkbp:consistency	cannot be proven within ZF (by Gödel's incompleteness theorems)
gptkbp:excludes	gptkb:Axiom_of_Choice
gptkbp:extendsTo	gptkb:Zermelo–Fraenkel_set_theory_with_Choice_(ZFC)
gptkbp:field	gptkb:mathematics gptkb:set_theory
gptkbp:formedBy	early 20th century
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:constructible_universe_(L)
https://www.w3.org/2000/01/rdf-schema#label	Zermelo–Fraenkel set theory (ZF)
gptkbp:influencedBy	gptkb:Ernst_Zermelo gptkb:Thoralf_Skolem gptkb:Abraham_Fraenkel
gptkbp:isFoundationFor	gptkb:logic gptkb:category_theory gptkb:set-theoretic_topology abstract algebra functional analysis measure theory model theory modern mathematics number theory theoretical computer science combinatorics real analysis
gptkbp:language	gptkb:first-order_logic
gptkbp:namedAfter	gptkb:Ernst_Zermelo gptkb:Abraham_Fraenkel
gptkbp:prevention	gptkb:Russell's_paradox
gptkbp:relatedTo	gptkb:Morse–Kelley_set_theory gptkb:Peano_axioms gptkb:Von_Neumann–Bernays–Gödel_set_theory
gptkbp:replacedBy	gptkb:naive_set_theory
gptkbp:standardFormulation	8 or 9 axioms
gptkbp:usedFor	foundation of mathematics
gptkbp:bfsParent	gptkb:Zermelo–Fraenkel_set_theory_with_the_axiom_of_choice_(ZFC) gptkb:ZFC_(with_choice) gptkb:Zermelo–Fraenkel_set_theory_with_choice_(ZFC) gptkb:Axiom_of_Collection
gptkbp:bfsLayer	6