Zermelo–Fraenkel set theory with choice (ZFC)

URI: https://gptkb.org/entity/Zermelo–Fraenkel_set_theory_with_choice_(ZFC)

GPTKB entity

Statements (49)

Predicate	Object
gptkbp:instanceOf	gptkb:logic gptkb:set_theory
gptkbp:abbreviation	gptkb:ZFC
gptkbp:alternativeTo	gptkb:Morse–Kelley_set_theory gptkb:Tarski–Grothendieck_set_theory gptkb:Kripke–Platek_set_theory gptkb:New_Foundations gptkb:Zermelo–Fraenkel_set_theory_(ZF)
gptkbp:axiomOfChoiceStatus	included
gptkbp:basisFor	gptkb:category_theory model theory most of classical mathematics
gptkbp:consistencyUndecidable	gptkb:Gödel's_incompleteness_theorems
gptkbp:formedBy	early 20th century
gptkbp:generalizes	gptkb:Zermelo_set_theory
gptkbp:hasAxiom	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
gptkbp:hasAxiomCount	9 or 10 (depending on formulation)
gptkbp:hasModel	gptkb:von_Neumann_universe
gptkbp:hasNoRussellSet	true
gptkbp:hasNoUniversalSet	true
gptkbp:hasWellFoundedSets	true
https://www.w3.org/2000/01/rdf-schema#label	Zermelo–Fraenkel set theory with choice (ZFC)
gptkbp:independentStatement	gptkb:Continuum_Hypothesis gptkb:Axiom_of_Constructibility_(V=L)
gptkbp:isConsistentIf	ZFC is consistent if ZF is consistent
gptkbp:isCountable	false
gptkbp:isCumulativeHierarchy	true
gptkbp:isFinitelyAxiomatizable	false
gptkbp:isFoundationFor	gptkb:mathematics modern mathematics
gptkbp:isUndecidable	true
gptkbp:language	gptkb:first-order_logic
gptkbp:namedAfter	gptkb:Ernst_Zermelo gptkb:Abraham_Fraenkel gptkb:Axiom_of_Choice
gptkbp:numberOfIssues	false
gptkbp:usedIn	gptkb:logic gptkb:set_theory foundations of mathematics
gptkbp:bfsParent	gptkb:Axiom_of_choice
gptkbp:bfsLayer	5