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Full Idea
In the second half of the twentieth century there emerged the opinion that foundation is the heart of the way to do set theory.
Gist of Idea
In the modern view, foundation is the heart of the way to do set theory
Source
William D. Hart (The Evolution of Logic [2010], 3)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.79
A Reaction
It is foundation which is the central axiom of the iterative conception of sets, where each level of sets is built on previous levels, and they are all 'well-founded'.
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] |