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 the empty Up: Axioms of Set Theory Previous: Axioms of Set Theory   Contents

Axiom of extensionality

Before giving the axioms that will allow us to construct the integers we give the axiom of set theory that defines what we mean by `$=$'. Without this axiom there would be little point in defining the integers or anything else. The axiom of extensionality tells us that sets are uniquely defined by their members.


\begin{displaymath}\forall x \forall y \hspace{.1in} (\forall z \hspace{.1in} z \in x \equiv z \in y) \equiv (x=\index{\protect=}y) \end{displaymath}

$a \equiv b$ means $a$ and $b$ have the same truth value or are equivalent. They are either both true or both false. It is the same as $(a\rightarrow b)\wedge (b \rightarrow a)$. This axiom says a pair of sets $x$ and $y$ are equal if and only if they have exactly the same members.




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