nLab
axiom of union

The axiom of union

Idea

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.

Statement

The axiom of union states the following:

Axiom (union)

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

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

A slightly different notation may be used when 𝒳\mathcal{X} is (Kuratowski)-finite; for example, {A,B,C}\bigcup\{A,B,C\} may be denoted ABCA \cup B \cup C. If (B k|kI)(B_k \;|\; k \in I) is a family of sets, then we may write kIB k\bigcup_{k \in I} B_k (and the usual variations for a sequence of sets) for {B k|kI}\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 𝒳\mathcal{X} is given as a collection of subsets of some ambient set SS, then the axiom of union is not necessary; SS 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.)

Revised on September 5, 2011 16:12:11 by Toby Bartels (75.88.82.16)