Mountain Math Software
home consulting videos book QM FAQ contact

Completed second draft of this book

PDF version of this book
next up previous contents
Next: Axiom of unordered pairs Up: Axioms of Set Theory Previous: Axiom of extensionality   Contents

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.


\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.

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




Mountain Math Software
home consulting videos book QM FAQ contact
Email comments to: webmaster@mtnmath.com