Material set theory

Axiom of Extensionality

The Axiom of Extensionality is a possible axiom in Material set theory that seems to follow directly from Cantor’s definition of the set: zf

(∀𝔐⁡𝐴)(∀𝔐⁡𝐵)[𝐴=𝐵⟺(∀𝑥)[𝑥∈𝐴⟺𝑥∈𝐵]]

which is to say, two sets are the same iff they have the same elements.

Relation to other axioms

• The Axiom of Extensionality seems to give a complete definition of set equality, but if we are dealing with ill-founded sets it may reduce to tautologous 𝐴 =𝐵 ⟺ 𝐴 =𝐵. Thus Extensionality is compatible with Aczel’s Antifoundation Axiom.

Axiom of Extensionality for classes

In a material set theory with classes have an identical axiom with sethood replaced with classhood: nbg

(∀ℭ𝔩𝔰⁡𝐴)(∀ℭ𝔩𝔰⁡𝐵)[𝐴=𝐵⟺(∀𝑥)[𝑥∈𝐴⟺𝑥∈𝐵]]

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