Axiom of union

Axiom of union

Axiom of union

Motivation

While we have described various sets, we have not said that they exist. That is, if $$A$$ and $$B$$ both exist, then currently we cannot ensure $$A\land B$$ exists, just that it can be described.

The axiom of union enables us to ensure all sets from unions and intersections exist.

Axiom of union

$$\forall a\exists b \forall c [c\in b\leftrightarrow \exists d(c\in d\land d\in a)]$$

That is, for every set $$a$$, there exists a set $$b$$ where all the elements in $$b$$ are the elements of the elements in $$a$$.

Here, $$b$$ is the union of the sets in $$a$$.