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 \newtheorem*{lt}{\L o\'{s} Test}
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\newcommand{\bN}{\mathbb{N}}
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\begin{document}
 
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%Binary operators: http://www.access2science.com/latex/Binary.html
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\title{Math 141b Problem Set 2}
\author{Due Feb. 12, 2019}

\maketitle
 
For simplicity, we will assume all our language $\tau$ are at most countable. 
 
\begin{enumerate}
	\item Characterize the models of $T_L$ as we did with $T_S$.  The following outline might be helpful:
	\begin{enumerate}
		\item $M \vDash T_L$ consists of a copy of $\bN$ and an \emph{ordered} set of $\bZ$-chains
		\item The ordering of these $\bZ$-chains characterizes the models up to isomorphism
		\item For each linear order $I$, find a model of $T_L$ whose $\bZ$-chains are ordered like $I$
	\end{enumerate}
	\item Show that addition is not definable in $\bN_L$
	\item Show that an eventually definable set $A \subset \bN$ is definable in Presburger arithmetic $\bN_P$.
	\item Work in the language consisting of a single binary relation $<$ (and equality).  Let $T$ be the following theory (often called `dense linear orders without endpoints'):
	\begin{enumerate}
	\item Antireflexivity: $\forall x \neg (x < x)$
	\item Antisymmetry: $\forall x, y (x < y \to y \not < x)$
	\item Transitivity: $\forall x,y,z (x < y \wedge y < z \to x < z)$
	\item Trichotomy: $\forall x,y (x < y \vee x = y \vee x> y)$
	\item Dense: $\forall x,y (x < y \to \exists z (x<z<y))$
	\item No Endpoints: $\forall x \exists y, z (y < x < z)$
	\end{enumerate}
	$(\bQ, <)$ and $(\bR, <)$ are the most familiar models of this theory.  Show $T$ has quantifier elimination.
\end{enumerate}

\end{document}

