\chapter*{Preface}

These notes started out as transcriptions of lectures given for Math
406, `Introduction to
Mathematical Logic and Model-theory', at METU in 2004.  I have
expanded on some points and rearranging some topics.

I assume that the reader already knows something of certain topics,
as covered in Math 111, `Fundamentals of 
Mathematics':
\begin{mylist}
  \item
formal logic (predicate and first-order);
\item
sets, relations, and functions;
\item
induction and recursion on the set of natural numbers.
\end{mylist}

%Background reading includes \cite{111-2004}.  

In writing these notes, I attempt to distinguish notationally between
\defn{constants} and \defn{variables.}  However, what is a constant in
one context is a variable in another.

In a tradition at least as old as Descartes's \emph{Geometry}
\cite{Descartes-Geometry}, originally published in French in 1637),
letters from the beginning of the Latin alphabet stand for known
quantities; letters from the end, unknown.  Hence, if we are asked to
solve the equation
\begin{equation}\label{eqn:eqn}
  ax^2+bx+c=0,
\end{equation}
we know that we are expected to come up with the equation
\begin{equation*}
  x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},
\end{equation*}
rather than, say,
\begin{equation*}
  a=-\frac{bx+c}{x^2}.
\end{equation*}
In Equation~\eqref{eqn:eqn}, the letters $a$, $b$, and $c$, are
understood to stand for particular numbers; the letter $x$ does not
stand for a particular number, but for the `possibility' of a number.
Grammatically, Equation~\eqref{eqn:eqn} is `incomplete' or
`elliptical'.  The equation might stand for the instructions, 
\begin{quote}
  Find
every number $d$
such that Equation~\eqref{eqn:eqn} becomes a true statement when $d$
replaces $x$.
\end{quote}
Alternatively, the equation stands for the set
\begin{equation}\label{eqn:set}
  \{x:ax^2+bx+c=0\},
\end{equation}
whose members would be obtained by following the instructions.

The expression on Line~\eqref{eqn:set} is grammatically a noun.  It
represents a sort of `completion' of Equation~\eqref{eqn:eqn} by means
of an additional use of the letter $x$.  A different letter, such as
$z$, could be used in place of $x$ in Line~\eqref{eqn:set} without
changing the set indicated.  Such an observation identifies the letter
$x$ in Equation~\eqref{eqn:eqn} as a \defn{variable.}