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\begin{document}
\begin{center} {\bf Peano's Axioms for Arithmetic}\\
Math 609  \end{center}
\bigskip\bigskip

The language for arithmetic is $\La = (+, \cdot, S, 0)$
where $+$ and $\cdot$ are binary function symbols, $S$
is a unary function symbol and $0$ is a constant symbol.
By PA (Peano Arithmetic), we mean the following set of
axioms.


\begin{description}
\item[1.] $\neg (S(x) \doteq 0).$
\item[2.] $S(x) \doteq S(y) \implies x \doteq y.$
\item[3.] $x + 0 \doteq x.$
\item[4.] $x + S(y) \doteq S (x + y).$
\item[5.] $x \cdot 0 \doteq 0.$
\item[6.] $x \cdot S(y) \doteq (x \cdot y) + x.$
\item[7$_{\grf}$.] For each formula $\grf ( x_0, x_1, \ldots, x_n )$
where $x_0$ does not occur bound in $\grf,$ the axiom
$$[\grf ( 0, x_1, \ldots, x_n ) \and 
\all x_0 \,(\grf ( x_0, x_1, \ldots, x_n ) \implies 
\grf ( S(x_0), x_1, \ldots, x_n ))] \implies \;
\all x_0\, \grf ( x_0, x_1, \ldots, x_n ).$$
\end{description}


\end{document}