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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII

[axiom saying all sets have a preceding basis]

6 ideas
Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ∃z(z∈x ∧ z∈y))) [Kunen]
In the modern view, foundation is the heart of the way to do set theory [Hart,WD]
Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD]
The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]