ZFC (Zermelo–Fraenkel set theory with Choice)

URI: https://gptkb.org/entity/ZFC_(Zermelo–Fraenkel_set_theory_with_Choice)

GPTKB entity

Statements (53)

Predicate	Object
gptkbp:instanceOf	gptkb:set_theory
gptkbp:abbreviation	gptkb:ZFC
gptkbp:alternativeTo	gptkb:Zermelo–Fraenkel_set_theory_(ZF) Morse–Kelley set theory (MK) Von Neumann–Bernays–Gödel set theory (NBG)
gptkbp:axiomOfChoiceStatus	included
gptkbp:consistencyUndecidableBy	gptkb:Gödel's_incompleteness_theorems
gptkbp:fieldOfStudy	gptkb:logic gptkb:set_theory
gptkbp:formedBy	early 20th century
gptkbp:fullName	gptkb:Zermelo–Fraenkel_set_theory_with_the_Axiom_of_Choice
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_Union gptkb:Axiom_Schema_of_Replacement gptkb:Axiom_Schema_of_Separation
gptkbp:hasIndependenceResults	gptkb:Continuum_Hypothesis gptkb:Martin's_Axiom gptkb:Axiom_of_Determinacy gptkb:Axiom_of_Constructibility_(V=L) gptkb:Large_Cardinal_Axioms gptkb:Suslin's_Hypothesis gptkb:Whitehead's_Problem Diamond Principle
gptkbp:hasModel	gptkb:constructible_universe_(V=L) inner models
gptkbp:hasModelIf	ZFC has a model if ZF has a model
https://www.w3.org/2000/01/rdf-schema#label	ZFC (Zermelo–Fraenkel set theory with Choice)
gptkbp:influencedBy	gptkb:Russell's_paradox gptkb:Cantor's_set_theory
gptkbp:isConsistentIf	ZFC is consistent if ZF is consistent
gptkbp:isCountable	true (as a theory, not as a model)
gptkbp:isCumulativeHierarchy	true
gptkbp:isFirstOrderTheory	true
gptkbp:isSecondOrderTheory	false
gptkbp:isStandardFoundation	true
gptkbp:isWellFounded	true
gptkbp:namedAfter	gptkb:Ernst_Zermelo gptkb:Abraham_Fraenkel
gptkbp:standardFoundationFor	most of modern mathematics
gptkbp:usedBy	gptkb:mathematician logicians philosophers of mathematics
gptkbp:usedFor	formalizing mathematics
gptkbp:usedIn	foundations of mathematics
gptkbp:bfsParent	gptkb:ZF_(Zermelo–Fraenkel_set_theory_without_choice) gptkb:cardinal_characteristics_of_the_continuum
gptkbp:bfsLayer	7