📂Set Theory

Empty Set Axiom

Axioms ¹

$$ \exists X \forall x \left( \lnot \left( x \in X \right) \right) $$ A set $X$ that does not contain any elements exists, and this set $X$ is defined as the empty set.

Explanation

The empty set is typically denoted as $\emptyset$. Meanwhile, the empty set can also be viewed as a set with $0$ elements, and sets that can be defined in terms of the number of elements include the following:

1. Singleton Set: A set with only one element is called a singleton set.
2. Finite Set: If the number of elements in a set belongs to $\mathbb{N}$, it is called a finite set.
3. Infinite Set: If a set is neither an empty set nor a finite set, it is called an infinite set.

Here, the definitions of finite and infinite sets are somewhat messy, but they are rigorously redefined later.

Note that a singleton set $\left\{ x \right\}$ is, after all, a set, and $x$ is distinctly different as an element of $\left\{ x \right\}$. Furthermore, in modern mathematics, even definitions like $x := \left\{ x \right\}$ are not permitted.

The reason to differentiate between the axiom of the empty set and the definition of the empty set is precisely because the two are different. The empty set itself can certainly be defined regardless of the empty set axiom. However, whether it truly exists is another matter. We intuitively understand the existence of the empty set, but a mere definition cannot guarantee that. This is similar to the axiom of completeness in analysis.

The non-obviousness of the existence of the empty set might become clear when considering the definition of a set. We said that a set is a collection of distinguishable objects as the subjects of our intuition or thought, and the objects belonging to a set are called elements. According to this definition, the empty set should not have any ‘distinguishable entities’ at all, yet having a collection without anything to collect is clearly odd. Nevertheless, as humans are too familiar with the ‘presence of absence’, we add such axioms in order to deal with the empty set.

Regardless of understanding or empathizing with this non-obviousness, the existence of the empty set can be derived from other axioms. Since fewer axioms are generally better, textbooks usually omit the empty set axiom.