more from this thinker     |     more from this text

Single Idea 13039

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

Full Idea

Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains.

Gist of Idea

Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y)))

Source

Kenneth Kunen (Set Theory [1980], §3.4)

Book Ref

Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.100

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]