Material set theory

Well-founded set

A material set 𝐴 is said to be ill-founded iff it is the start of a descending ∈-chain¹, set i.e. there exists some function 𝑓 :ℕ0 →𝐸 with 𝑓(0) =𝐴 and 𝑓(𝑖 +1) ∈𝑓(𝑖) for every 𝑖 ∈ℕ0. A set is well-founded iff it is not ill-founded.

Ill-founded sets are forbidden by the Axiom of Foundation, and hence in ZF. A strong negation of the axiom of foundation is Aczel’s Antifoundation Axiom.

Properties

1. A set 𝐴 is well-founded iff its powerset P(𝐴) is well-founded.
2. A set 𝐴 is well-founded iff all of its elements are well-founded.

Proof

proof

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Footnotes

1. 2006. Notes on set theory, ¶11.26, p. 166 ↩