\documentstyle{article}

\textwidth=6.25in
\textheight=8.0in
\oddsidemargin=0.25in
\evensidemargin=0.25in
\topmargin=.5in
\renewcommand{\baselinestretch}{1.025}

\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\I{\mathop{\cal I}\nolimits}
\def\A{\mathop{\cal A}\nolimits}
\def\B{\mathop{\cal B}\nolimits}
\def\N{\mathop{\cal N}\nolimits}
\def\F{\mathop{\cal F}\nolimits}
\def\G{\mathop{\cal G}\nolimits}
\def\La{\mathop{\cal L}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\gra{\mathop{\alpha}\nolimits}     
\def\grS{\mathop{\Sigma}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\implies{\mathop{\to}\nolimits}     
\def\deduces{\mathop{\vdash}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


\begin{document}
\begin{center} {\bf The Zermelo-Fraenkel Axioms for Set Theory}\\
Math 609  \end{center}
\bigskip\bigskip

The language for set theory is the language $\La = ( \in )$
with a single binary function symbol.
By ZF (Zermelo-Fraenkel) Set Theory, we mean the set of
axioms (or axiom schemata) 1 - 9 below. By Z (Zermelo) Set
Theory is meant the set of axioms 1 - 7 and the axiom schema 8. 
By ZFC (ZF with Choice) Set Theory, we mean the following list of axioms.
 

\begin{description}
\item[1. Extensionality.] 
$\all x, y \; (x \doteq y \ssi \all z \, (z \in x \ssi z \in y)).$ 
\item[2. Null set.]  $\exists x\, \all y ( \neg y \in x).$
\item[3. Pairs.]  $\all x, y\, \exists z\, \all u (u \in z \ssi u \doteq x
\or u \doteq y).$ 
\item[4. Unions.] $\all x\, \exists y\, \all z (z \in y \ssi 
\exists w \, (z \in w \and w \in y)).$ 
\item[5. Power set.] $\all x\, \exists y\, \all z (z \in y \ssi 
\all w \, (w \in z \implies w \in x)).$
\item[6. Infinity.] 
$\exists x\, [\exists y\, ((y \in x) \and \exists z\, (y \in z \and z \in x))].$
\item[7. Regularity.] $\all x\, [ \exists y\, (y \in x \and 
\neg \exists z\, (z \in y \and z \in x))].$
\item[8$_{\grf}$. Definable subsets.] $\all x \, \exists y \, \all z 
(z \in y \ssi z \in x \and \grf ( z, x_1, \ldots, x_n )),$ where $\grf$ is a 
formula in which $y$ does not occur.
\item[9$_{\grf}$. Collection.]  $\all x\, [x \in u  \implies 
\exists z \grf ( x, z, u, x_1, \ldots, x_n )] \implies 
\exists y\, \all x\, [x \in u \implies 
\exists z\, (z \in y \and \grf ( x, z, u, x_1, \ldots, x_n ))],$
where $\grf$ is a formula in which $y$ does not occur.
\item[10. Choice.] $\all x\, \exists y\, [(y \mbox{ is a function with domain }
x) \and \all z\, ( z \in x \and \exists u\, u \in z \implies y(z) \in z)].$
\end{description}


\end{document}