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\title{Basic Model-Theory (Math 736)}
\date{Fall 2001}
\author{David Pierce}
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 \noindent\emph{Text:}  There is no official text, besides the lectures
and some notes I will provide on some topics; but most of the material
can be found in the first half of \cite{Poizat}.  Other references
include the more recent \cite{Hodges}, and the older
\cite{Chang--Keisler}.

\noindent\emph{About the subject:}  A \tech{structure} is a set, possibly
equipped with some distinguished functions and relations.  Examples
include groups, rings, and linear orders.  \tech{Model-theory} is the
study of structures, as such or as \tech{models} of particular
\tech{theories}. (In this context, a group is just a model of
group-theory, in a technical sense of the word \men{theory}---a sense
which is \emph{not} intended in the term \men{model-theory} itself.)

Model-theory has been called \men{algebraic geometry without fields.}  If
algebraic geometry is about polynomial equations over fields, then
model-theory is about analogous formulas over arbitrary structures.

Model-theory has also been called \men{the geography of tame
mathematics.}  The notions of \men{tame} and its opposite, \men{wild},
are not precisely defined; but the structure $(\mathbf{N},+,\cdot)$ of
the natural numbers is wild (by G\"odel's Incompleteness Theorem), while
the structure $(\mathbf{C},+,\cdot)$ of the complex numbers is tame for
various reasons, which model-theory identifies and looks for in other
structures as well.

\noindent\emph{About the course:}  The first theorem will be Compactness
(the model-theoretic version of G\"odel's Completeness Theorem).  This is
a model-existence result, saying for example that the theory of finite
fields has infinite models.  We shall define and examine---with
motivating examples---\emph{theories} that are:\ complete,
model-complete, quantifier-eliminable, and categorical; and
\emph{structures} that are:\ prime, minimal, universal, saturated, and
stable.

\noindent\emph{Prerequisites:}  No specific background is required, just
some familiarity with some part of mathematics or logic.

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\bibitem{Chang--Keisler}
C.~C. Chang and H.~J. Keisler.
\newblock {\em Model theory}.
\newblock North-Holland Publishing Co., Amsterdam, third edition, 1990.

\bibitem{Hodges}
Wilfrid Hodges.
\newblock {\em Model Theory}.
\newblock Cambridge University Press, 1993.

\bibitem{Poizat}
Bruno Poizat.
\newblock {\em A course in model theory}.
\newblock Springer-Verlag, New York, 2000.
%\newblock An introduction to contemporary mathematical logic, Translated from
%  the French by Moses Klein and revised by the author.

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