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Single Idea 13493

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

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'.

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]