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Single Idea 18193

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII ]

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

The 6 ideas with the same theme [axiom saying all sets have a preceding basis]:

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]