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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.


\begin{displaymath}\exists x \forall y \hspace{.1in} y \not\,\in x \end{displaymath}

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.


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