# Axiom:Axiom of Unions

## Axiom

### Set Theory

For every set of sets $A$, there exists a set $x$ (the **union set**) that contains all and only those elements that belong to at least one of the sets in the $A$:

- $\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

### Class Theory

Let $x$ be a set (of sets).

Then its union $\bigcup x$ is also a set.

## Also known as

The **axiom of unions** is in fact most frequently found with the name **axiom of union**.

However, in some treatments of axiomatic set theory and class theory, for example Morse-Kelley set theory this name is used to mean something different.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ specifically uses the plural form **axiom of unions** for this, and reserves the singular form **axiom of union** for that.

Other terms that can be found to refer to the **axiom of unions**:

- the
**axiom of the sum set** - the
**axiom of amalgamation** - the
**union axiom**.

## Set Theoretical and Class Theoretical Formulations

Equivalence of Formulations of Axiom of Unions notwithstanding, the two formulations have a subtle difference.

The purely set theoretical (formulation 1) version starts with a given set (of sets), and from it allows the **creation** of its union by providing a rule by which this may be done.

The class theoretical (formulation 2) version accepts that such a construct is already constructible in the context of the union of a class, and is itself a class.

What formulation 2 then goes on to state is that if $x$ is actually a set (of sets), then $\bigcup x$ is itself a set.

This is consistent with how:

- the philosophy of
**axiomatic set theory**defines the constructibility of sets from nothing

differs from

- the
**class theoretical**approach, in which classes may be considered to be already in existence, and it remains a matter of determining which of these classes are actually sets.

## Also see

- Results about
**the axiom of unions**can be found**here**.

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Union