\documentclass[a4paper,10pt,draft]{article}
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\title{First-order Compactness exercises}
\author{Math 406}
\date{2004.11.22}

\newtheorem{problem}{Problem}
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\begin{document}
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\begin{problem}
Show that every Archimedean ordered field is elementarily equivalent
to some \emph{countable, non-Archimedean} 
ordered field.  
  \end{problem}

\begin{problem}
  Show that every non-Archimedean ordered field contains
  \defn{infinitesimal} elements, that is, positive elements $a$ that
  are less than every positive rational number.
\end{problem}

\begin{problem}
  Find an example of a non-Archimedean ordered field.
\end{problem}

\begin{problem}
The \defn{order} of an element $g$ of a group is the size of the
subgroup $\{g^n:n\in \Z\}$ that $g$ generates.  In a \defn{periodic}
group, all elements have finite order.  Suppose $G$ is a periodic
group in which there is no finite upper bound on the orders of
elements.  Show that $G\equiv H$ for some non-periodic group $H$.
\end{problem}

\begin{problem}
  Suppose $(X,<)$ is an infinite total order in which $X$ is
  \emph{well-ordered}
  by $<$.  Show that there is a total order $(X^*,<^*)$ such that
  \begin{equation*}
    (X,<)\equiv(X^*,<^*),
  \end{equation*}
but $X^*$ is \emph{not} well-ordered by $<^*$.
\end{problem}

\end{document}

