\documentclass[12pt]{amsart}
\usepackage{amsmath, amsthm, amssymb, amsfonts}
\usepackage{enumerate}
\usepackage{verbatim}
%\numberwithin{equation}{section}

%%%Document Configuration
\pagestyle{headings}
%\addtolength{\topmargin}{-.5in}
\textwidth = 6in 
\oddsidemargin = .25in
\evensidemargin = .25in
\voffset = -1in
\textheight = 9.5 in
%\textwidth = 6in
%\parindent = 0in
%\headsep = 0in
\relpenalty = 9999
\binoppenalty = 9999
%linespacing
\renewcommand{\baselinestretch}{1.25}
\reversemarginpar     
%%%

%%%Theorem Configuration
\theoremstyle{plain}
\newtheorem*{main}{Main~Theorem}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{exer}[thm]{Exercise}
\newtheorem{fact}[thm]{Fact}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{remark}[thm]{Remark}
\newtheorem{hw}[section]{}
%\newtheorem{hw}[section]{HW}
\newtheorem{dfn}{Definition}[section]
%%%

%%%Bibliography Configuration
\bibliographystyle{plain}
%%%

%%%semantic definitions
\DeclareMathOperator*{\Aut}{Aut}
\DeclareMathOperator*{\Dom}{Dom}
\DeclareMathOperator*{\Mod}{Mod}
\DeclareMathOperator*{\Th}{Th}
\DeclareMathOperator*{\tp}{tp}
\DeclareMathOperator*{\acl}{acl}
\renewcommand{\div}{\mid}
\renewcommand{\emptyset}{\varnothing}
\renewcommand{\implies}{\Rightarrow}
\renewcommand{\natural}{\mathbb{N}}
\newcommand{\Acl}[1]{\obar{#1}}
\newcommand{\assign}{\leftarrow}
\newcommand{\caseif}{\Rightarrow}
\newcommand{\caseonlyif}{\Leftarrow}
\newcommand{\cantprove}{\nvdash}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\field}{\mathbb{F}}
\newcommand{\minus}{\setminus}
\newcommand{\power}[1]{{\mathcal{P}}({#1})}
\newcommand{\nonnegative}{\mathbb{N}}
\newcommand{\integer}{\mathbb{Z}}
\newcommand{\intersect}{\cap}
\newcommand{\isomorphic}{\simeq}
\newcommand{\obar}[1]{\overline{#1}}
\newcommand{\proves}{\vdash}
\newcommand{\rational}{\mathbb{Q}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\union}{\cup}
\newcommand{\Union}{\bigcup}
\newcommand{\units}[1]{{\left(#1\right)}^{\times}}
%%%

%%%local definitions
\newcommand{\EH}{{Ehrenfeucht}}
\newcommand{\eleq}{\equiv}
\newcommand{\elleq}{\preccurlyeq}
\newcommand{\lang}[1]{{\mathcal{#1}}}
\newcommand{\Los}{{\L o\'s}}
\newcommand{\LS}{{L\"{o}wenheim-Skolem }}
\newcommand{\mathscr}[1]{{\mathcal{#1}}}
\newcommand{\st}[1]{{\mathcal{#1}}}
\newcommand{\subst}{\preceq}
\newcommand{\substp}{\prec}
\newcommand{\structure}[1]{{\mathcal{#1}}}
\newcommand{\struct}[1]{\left\langle#1\right\rangle}
\newcommand{\symAnd}{\mathop{\bigwedge}}
\newcommand{\symand}{\wedge}
\newcommand{\symimply}{\rightarrow}
\newcommand{\symiff}{\leftrightarrow}
\newcommand{\symnot}{\neg}
\newcommand{\symor}{\vee}
\newcommand{\WFF}{well-formed formula}
\newcommand{\0}{\textbf{0}}
\newcommand{\suc}{\textbf{S}}
\newcommand{\A}{\mathfrak{A}}
\newcommand{\B}{\mathfrak{B}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\NNN}{\mathfrak{N}}
\newcommand{\E}{\textbf{E}}
\newcommand{\Cn}{\textrm{Cn}}
\newcommand{\res}{\upharpoonright}
%%%

\begin{document}

\title{Model Theory II Homework 5}
\author{Dakota Blair}
\maketitle

\setcounter{section}{10}

\begin{hw}
A countable theory $T$
has a countable atomic model $\st{N}$ that is not minimal
if and only if
$T$ has an atomic model $\st{M}$ of power $\aleph_1$.
\end{hw}

\begin{proof}
If $\st{N}$ is not minimal,
then there is a structure $\st{M}$
such that $\st{M} \isomorphic \st{N}$
and $\st{M} \substp \st{N}$.
Let $f$ be an isomorphism from $\st{N}$ to $\st{M}$.
Therefore $f$ is an elementary map,
so by isomorphic correction
we can find $\st{N}_1$ such that $N \substp N_1$
by interpreting $\st{M}$ as a copy of $\st{N} = \st{N}_0$.
Further since $\st{N}_0$ is countable and atomic,
it is homogeneous, therefore $\st{N}_1$ is also atomic.
This process forms a proper elementary chain
\[
\st{N}_0 \substp \st{N}_1 \substp
\cdots \substp \st{N}_\alpha \substp \cdots
\]
from which we can form
\[
\st{N}_{\aleph_1} = \Union_{\alpha < \aleph_1} \st{N}_\alpha.
\]
This is an $\aleph_1$ union of countable sets, hence
$|\st{M}_{\aleph_1}| = \aleph_1$.
By homogeneity of $\st{N}_\alpha$ for $\alpha < \aleph_1$,
the model $\st{M}_{\aleph_1}$ is constructible and therefore atomic.

Conversely assume  $\st{N}$ is atomic with $|\st{N}| = \aleph_1$.
Then by \EH's Omitting Types Theorem,
there exists an $\st{M}$ which is countable and atomic.
Therefore $\st{M}$ is prime, and hence elementarily embeddable in $\st{N}$.
Then $(\st{N},\st{M})$ form an $(\aleph_0, \aleph_1)$ elementary pair.
By \LS and \EH's theorem there exists an elementary pair
$(\st{N}',\st{M}') \eleq (\st{N}',\st{M}')$ which is countable and atomic.
Then $\st{M'}$ and $\st{N'}$ are countable and elementarily equivalent,
even to $\st{M}$, consequently they are isomorphic.
Thus $\st{N'}$ is a countable atomic model which is not minimal.
\end{proof}

\begin{hw}
Prove $(\integer/4\integer)^{\omega}$ is almost strongly minimal.
\end{hw}

\begin{proof}
Identify the elements of $\integer/4\integer$ with the elements of $4$.
The formula $\varphi$ that says $2x \neq 0$ is strongly minimial.
It consists of elements which are
sequences with finitely many $1$s and $3$s.
The only complete types this model realizes are
$x = 0$,  $x \neq 0$, $2x = 0$, $2x \neq 0$ and $4x = 0$.
Thus $\varphi$ cannot be split into an infinite coinfinite definable set.
Further $\acl \varphi$ is everything,
therefore the model is almost strongly minimal.
\end{proof}

\end{document}
