📂Set Theory

Axiom of Pairs

Axioms

$$ \forall A \forall B \exists U ( A \in U \land B \in U ) $$ For any two sets $A$ and $A$, there exists a set $U$ that has $A$ and $B$ as elements.

Explanation

When first encountering the axiom of pairs (and indeed, this applies to most axioms), one might wonder why such an axiom is necessary at all. However, the pairing axiom can actually be seen as playing a crucial role in elevating the concept of a set to the realm of mathematics.

In the definition of a set, a collection of distinctly identifiable objects is referred to as a set. Yet, it has not been explicitly stated whether the set itself consists of distinctly identifiable objects. Assuming the axiom of pairs, any given set $X$ and $X$ would be elements of a singleton set $\left\{ X , X \right\} = \left\{ X \right\}$, thereby declaring that all sets themselves are ‘distinctly identifiable objects’.

Now, $U = \left\{ A, B \right\}$ can be interpreted to mean that $A$ and $B$ can be dealt with unambiguously in mathematical terms. Handling $A$ and $B$ mathematically means that discussions can incorporate $B$ along with $A$, covering aspects that $A$ alone could not address. As a result, the existence of entities like unions, power sets, and ordered pairs, which are ‘bigger than the existing $A$ and naturally expected to exist,’ cannot be doubted. Although it’s possible to conceive of concepts involving larger sets without $U$, the foundation would be substantially unstable without the existence of $U$.