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Axiom of the empty set

Before defining any structure we need an axiom that asserts the existence of the empty set. This axiom 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 $x$. We also introduce the notation $x \not\,\in y$ to indicate 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.

Before we can define the integers we need to give two axioms for constructing finite sets.