Material set theory

Axiom of Union

The Axiom of Union is a possible axiom of Material set theory: zf

(∀𝔐⁡E)(∃𝔐⁡𝐵)[𝑥∈𝐵⟺(∃𝑋∈E)[𝑥∈𝑋]]

which is to say, for any set E there exists a union 𝐵 consisting of the elements of the elements of E. It follows from the Axiom of Extensionality that such a 𝐵 is unique, and we denote it by ⋃E.

In a material set theory with classes like NBG, the existence of a union class is already guaranteed by other axioms, but one requires the above axiom to guarantee that the union of sets is itself a set.

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