- fundamentals of set theory
- material set theory
- presentations of set theory
- structuralism in set theory
- class-set theory
- constructive set theory
- algebraic set theory

In material set theory as a foundation of mathematics, the axiom of union is an important axiom needed to get the foundations off the ground (to mix metaphors). It states that unions exist.

The **axiom of union** states the following:

If $\mathcal{X}$ is a (material) set, then there exists a set $U$ such that $a \in U$ whenever $a \in B \in \mathcal{X}$.

Using the axiom of separation (bounded separation is enough), one can prove the existence of a particular set $U$ such that the members of the members of $\mathcal{X}$ are the *only* members of $U$. Using the axiom of extensionality, we can then prove that this set $U$ is unique; it is usually denoted $\bigcup\mathcal{X}$ and called the **union** of (the elements of) $\mathcal{X}$.

A slightly different notation may be used when $\mathcal{X}$ is (Kuratowski)-finite; for example, $\bigcup\{A,B,C\}$ may be denoted $A \cup B \cup C$. If $(B_k \;|\; k \in I)$ is a family of sets, then we may write $\bigcup_{k \in I} B_k$ (and the usual variations for a sequence of sets) for $\bigcup \{B_k \;|\; k \in I\}$; however, we require the axiom of replacement to prove that the latter set (the range of the family) exists in general.

If $\mathcal{X}$ is given as a collection of subsets of some ambient set $S$, then the axiom of union is not necessary; $S$ itself already satisfies the conclusion of the hypothesis (and then bounded separation gives us the union that we want). This is the only case when unions are taken in structural set theory. However, structural set theory makes use of *disjoint unions*, and predicative mathematics requires an axiom giving their existence. (In impredicative mathematics, we can construct disjoint unions from power sets and cartesian products.)

One could also include union structure in the set theory.

category: foundational axiom

Last revised on December 12, 2022 at 13:05:53. See the history of this page for a list of all contributions to it.