Arbitrary Unions

Suppose we have an infinite collection of sets

A = \{b_0,b_1,b_2,\dots\}.

For a finite set, we could simply union b_1 \cup b_2 \cup \dots \cup b_n, but for an infinite set, this would require an infinite amount of steps. For this reason we upgrade our union axiom to be

Union Axiom

For any set A, there exists a set B whose elements are exactly the members of the members of A.

\forall x[ x \in B \iff (\exists b \in A)x \in b].

In our more common symbolic notion, we would write

x \in \bigcup A \iff (\exists b \in A) x \in b.

We needed to upgrade our union axiom for infinite sets, but maybe suprisngly we don’t need a new axiom for infinite intersections. This is because intersections of sets have less elements, and we can use the separation axiom.

Proposition

For any nonempty set A, there exists a set unique set B such that for any x,

x \in b \iff (\forall a \in A) x \in a

Proof

Let A be nonempty. Simply apply the subset axioms to create a set B where

x \in B \iff (\forall c \in A) x \in c