# 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.