PDF version of this book

Axiom of the empty set

The empty set must be defined before any other set can be defined. The axiom of the empty set uses the existential quantifier ($\exists$). $\exists$$x\, g(x)$ means there exists some set $x$ for which $g(x)$ is true. Here $g(x)$ is any expression that includes $x$.

The notation $x \not\,\in y$ indicates that $x$ is not a member of set $y$.

The axiom of the empty set is as follows.

$$\exists x \forall y \hspace{.1in} y \not\,\in x$$

This says there exists an object $x$ that no other set belongs to. $x$ contains nothing. The empty set is denoted by the symbol $\emptyset$.

The definition of the integers requires two axioms for constructing finite sets.