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Full Idea
In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα.
Clarification
A Vα is a stage of the hierarchy
Gist of Idea
The Axiom of Foundation says every set exists at a level in the set hierarchy
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.3)
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.60
13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
13039 | Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |