KnowraAxiom of FoundationLinked fromLinked fromThe 9 pages that link to Axiom of Foundation, each with the reason it gives.All 9Related 8Compared with 1Zermelo–Fraenkel set theoryRelated: It makes membership well-founded and excludes sets that contain themselves through cycles.Well-founded relationRelated: Applied to membership, it ensures a minimal element in every nonempty collection of sets.Cumulative hierarchyRelated: It rules out membership cycles and supports the hierarchy's well-founded picture.Axiom of unionCompared with: Foundation restricts membership structure, whereas union guarantees a set formed from existing members.Axiom of infinityRelated: It shapes the set-theoretic setting in which the empty set and its successors form ordinals.Axiom schema of specificationRelated: Foundation constrains membership structure, whereas specification constrains subset formation.Von Neumann universeRelated: Foundation ensures membership descends through the hierarchy without infinite regress.Von Neumann–Bernays–Gödel set theoryRelated: Foundation constrains membership among NBG’s sets, not the full class universe.Mostowski collapse lemmaRelated: Foundation makes the membership relation well-founded in standard set theory.