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Single Idea 13495

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

Full Idea

The usual statement of Foundation is that any nonempty set has a member disjoint from it. This phrasing is ordinal-free and closer to the primitives of ZFC.

Gist of Idea

Foundation Axiom: an nonempty set has a member disjoint from it

Source

William D. Hart (The Evolution of Logic [2010], 3)

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.80

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]