\documentclass[%
version=last,%
a5paper,
10pt,%
headings=small,%
bibliography=totoc,%
twoside,%
reqno,%
cleardoublepage=empty,%
parskip=half,%
draft=true,%
DIV=classic,%
%DIV=12,%
headinclude=false,%
%titlepage=true,%
pagesize]
{scrartcl}
\usepackage{hfoldsty,url}
\usepackage[neverdecrease]{paralist}

\usepackage{amsmath,amssymb}

\newcommand{\F}{\mathbb F}
\newcommand{\ecmodel}{\models_{\mathrm{ec}}}
\newcommand{\thy}[1]{\mathrm{#1}}
\newcommand{\ACFA}{\thy{ACFA}}	
\newcommand{\mDF}{\text{$m$-$\thy{DF}$}} % fields with m derivations
\newcommand{\mpDF}{\text{$(m+1)$-$\thy{DF}$}}  % fields with m+1 derivations
\newcommand{\oDF}{\text{$\upomega$-$\thy{DF}$}} % fields with omega derivations
%\newcommand{\mDPF}{\text{$m$-$\thy{DPF}$}} % diff. perf. fields with m derivations
\newcommand{\DCF}{\thy{DCF}}           % differentially closed fields
\newcommand{\mDCF}{\text{$m$-$\thy{DCF}$}}  % differentially closed fields
				% with m derivations
\newcommand{\oDCF}{\text{$\upomega$-$\thy{DCF}$}}  % differentially closed fields
				% with omega-many derivations
\newcommand{\mpDCF}{\text{$(m+1)$-$\thy{DCF}$}}  % differentially closed fields
				% with m+1 derivations

\newcommand{\Mod}[1]{\operatorname{Mod}(#1)}

\newcommand{\size}[1]{\lvert#1\rvert}
\newcommand{\ts}[1]{\operatorname S(#1)}
\newcommand{\Forall}[1]{\forall{#1}\;}
\newcommand{\Exists}[1]{\exists{#1}\;}

\newcommand{\ec}{e.c.}

\usepackage{upgreek}
\usepackage{mathrsfs}
\newcommand{\sig}{\mathscr S}

\newcommand{\str}[1]{\mathfrak{#1}}
\newcommand{\diag}[1]{\operatorname{diag}(#1)}

\newcommand{\included}{\subseteq}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\theequation}{\fnsymbol{equation}}
\renewcommand{\phi}{\varphi}
\renewcommand{\setminus}{\smallsetminus}
\renewcommand{\emptyset}{\varnothing}

\usepackage{amsthm}
\newtheorem*{theorem}{Theorem}

\begin{document}
\title{Chains of theories}
\author{David Pierce}
\publishers{Mathematics Department\\
Mimar Sinan Fine Arts University\\
Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}
\maketitle

This is work with \"Ozcan Kasal.  There is some parallel work by Alice Medvedev (presented in Ol\'eron, June, 2011) concerning $\ACFA$.

Suppose
\begin{equation*}
T_0\included T_1\included T_2\included\dotsb,
\end{equation*}
all theories, closed under entailment, so their signatures also form a chain:
\begin{equation*}
\sig_0\included\sig_1\included\sig_2\included\dotsb
\end{equation*}
In one example of interest, $T_m$ is $\mDF$, the theory of fields with
$m$ commuting derivations $\partial_0$, \dots, $\partial_{m-1}$; their
union is $\oDF$.

In general, what properties are preserved in
$\bigcup_{k\in\upomega}T_k$?  Compare:

\begin{theorem}[Chang, \L o\'s--Suszko]
For a fixed theory $T$, the following are equivalent:
\begin{compactenum}
  \item
$T$ is $\forall\exists$-axiomatizable.
\item
$\Mod T$ is closed under taking unions of chains
\begin{equation*}
  \str A_0\included\str A_1\included\str A_2\included\dotsb.
\end{equation*}
\end{compactenum}
\end{theorem}

Again, if $T_0\included T_1\included T_2\included\dotsb$, then among
possible properties of the theories $T_k$,
\begin{compactenum}[1)]
\item
Preserved by $\bigcup_{k\in\upomega}T_k$ are: 
\begin{inparaenum}[(a)]
\item
consistency,
\item
completeness, 
\item
quantifier elimination, 
\item
model-completeness, 
\item
stability,
\item \dots;
\end{inparaenum}
\item
not preserved (but this is not obvious) are: companionability,
$\upomega$-stability, 
superstability, \dots 
\end{compactenum}

\begin{asparaenum}[(a)]
  \item
Consistency is preserved, by compactness.
\item
Completeness is preserved, because every
sentence of the union $\bigcup_{k\in\upomega}\sig_k$ is a sentence of
some $\sig_k$. 
\item
Likewise for quantifier-elimination.
\item
\textbf{Model-completeness} of a theory $T$ may be usually remembered
as 
\begin{equation*}
\str A\included\str B\implies\str A\preccurlyeq\str B
\end{equation*}
within $\Mod T$.  Equivalently (the theory axiomatized by)
\begin{equation*}
T\cup\diag{\str A}
\end{equation*}
is always complete when $\str A\models T$, where
\begin{equation*}
\diag{\str A}=\{\sigma\in\operatorname{Th}(\str A_A)\colon\sigma\text{
    is quanfifier-free}\}
\end{equation*}
the theory of
the structures in which $\str A$ embeds.  A sufficient (and obviously
necessary) condition is (Abraham) \textbf{Robinson's Condition,} 
\begin{equation*}
\str A\included\str B\implies\str A\preccurlyeq_1\str B
\end{equation*}
where the conclusion means every quantifier-free formula over $\str A$ soluble in $\str B$ is soluble in $\str A$.  Robinson's Condition is, equivalently,
\begin{equation*}
\str A\ecmodel T
\end{equation*}
---every model of $T$ is an \textbf{existentially closed} model.  For
this it is sufficient (and in fact necessary) that $T$ admit
quantifier-elimination down to $\exists$ formulas.  Therefore
model-completeness is preserved in unions of chains. 
\item
A \emph{complete} theory $T$ is
\begin{compactitem}
\item
\textbf{$\kappa$-stable,} if $\kappa\geq\size T$ and
\begin{equation*}
\size A\leq\kappa\implies\size{\ts A}\leq\kappa
\end{equation*}
for all parameter-sets $A$ of models of $T$;
\item
\textbf{superstable,} if $\kappa$-stable for $\kappa$ large enough;
\item
\textbf{stable,} if $\kappa$-stable for some $\kappa$.
\end{compactitem}
When $\size T=\upomega$, then
\begin{compactitem}
\item
superstability implies $\kappa$-stability when $\kappa\geq2^{\upomega}$;
\item
stability implies $\kappa$-stability when $\kappa=\kappa^{\upomega}$.
\end{compactitem}
(Note that if $\operatorname{cof}(\kappa)=\upomega$, as when
$\kappa=\aleph_{\upomega}$, then $\kappa<\kappa^{\upomega}$.)

In fact \emph{in}stability of $T$ is equivalent to the presence of a
formula $\phi(\vec x,\vec y)$ defining an infinite linear order in
some model of $T$, so that, for all $n$ in $\upomega$, 
\begin{equation*}
T\vdash\Exists{(\vec x_0,\dots,\vec x_n)}\Biggl(\bigwedge_{0\leq i\leq
  j\leq n}\phi(\vec x_i,\vec x_j)\land\bigwedge_{0\leq j<i\leq
  n}\lnot\phi(\vec x_i,\vec x_j)\Biggr). 
\end{equation*}
If $T=\bigcup_{k\in\upomega}T_k$, then these sentences are all in some
$\sig_k$, and then (assuming $T_k$ is complete) $T_k$ will be
instable. 
\end{asparaenum}

An arbitrary theory $T$ is \textbf{companionable} if, for some theory
$T^*$ of its signature, 
\begin{compactitem}
\item
$T_{\forall}=T^*{}_{\forall}$,
\item
$T$ is model-complete.
\end{compactitem}
In this case, $T^*$ is the \textbf{model-companion} of $T$.  If
\begin{equation*}
T_0\included T_1\included T_2\included\dotsb,
\end{equation*}
and each $T_k$ has the model-companion $T_k{}^*$, \emph{and}
\begin{equation}\label{*}
T_0{}^*\included T_1{}^*\included T_2{}^*\included\dotsb,
\end{equation}
then $\bigcup_{k\in\upomega}T_k{}^*$ is the model-companion of $\bigcup_{k\in\upomega}T_k$.  However \eqref{*} may fail.

\begin{theorem}[McGrail]
$\mDF_0$ (in characteristic $0$) has a model-companion, $\mDCF_0$, which admits quantifier-elimination and is $\upomega$-stable.
\end{theorem}

\begin{theorem}[P.]
$\mDF$ has a model-companion, $\mDCF$.
Nevertheless, $\bigcup_{m\in\upomega}\mDF$ is not companionable.
\end{theorem}

For the last part (non-companionability), if $j\in\upomega$, let $K_j$ be an \ec\ (\emph{existentially closed}) model of $\oDF$ (that is, $\bigcup_{m\in\upomega}\mDF$), with
\begin{align*}
\F_p(\alpha)&\included K_j,&
\alpha&\notin\F_p{}^{\mathrm{alg}},&
\partial_i\alpha&=\updelta_{ij}=
\begin{cases}
1,&\text{ if }i=j,\\
0,&\text{ if }i\neq j.	
\end{cases}
\end{align*}
Then $\alpha$ has no $p$-th root in $K_j$, since
\begin{align*}
\partial_j\alpha&=1,&
\partial_j(x^p)&=p\cdot x^{p-1}\cdot\partial_jx=0.
\end{align*}
Therefore $\alpha$ has no $p$-th root in a nonprincipal ultraproduct
\begin{equation*}
\prod_{j\in\upomega}K_j/\mathfrak p,
\end{equation*}
even though, in this, $\partial_i\alpha=0$ for all $i$ in $\upomega$, so $\alpha$ has a $p$-th root in some extension.  Thus the ultraproduct is not \ec.  Therefore the class of \ec\ models of $\oDF$ is not elementary.

\begin{theorem}[P.]
\begin{equation*}
\mDCF_0\included\mpDCF_0,
\end{equation*}
and therefore $\oDF_0$ has a model-companion, $\oDCF_0$, which is stable, but not superstable.
\end{theorem}

This is established by means of:

\begin{theorem}[folklore, P.]
Assuming $T_0\included T_1$, each $T_k$ having signature
$\sig_k$, consider:
\begin{compactenum}[$A$.]
\item\label{ext}
If
\begin{align*}
\str A&\models T_1,&
\str B&\models T_0,&
\str A\restriction\sig_0&\included\str B,  
\end{align*}
then there is $\str C$ such that
\begin{align*}
\str C&\models T_1,&
\str A&\included\str C,&
\str B&\included\str C\restriction\sig_0.
\end{align*}
\item\label{red}
For all $\str A$,
\begin{equation*}
\str A\ecmodel T_1\implies\str A\restriction\sig_0\ecmodel T_0.
\end{equation*}
\item\label{AP}
$T_0$ has the \emph{Amalgamation Property:} if one model embeds in two others, then those two in turn embed in a fourth model, compatibly with the original embeddings.
\item\label{ec}
$T_1$ is $\forall\exists$ (so that every model embeds in an \ec\ model).
\end{compactenum}
We have the two implications
\begin{align*}
\ref{ext}&\implies\ref{red},&
\ref{red}\And\ref{AP}\And\ref{ec}&\implies\ref{ext},
\end{align*}
but there is no implication among the four conditions that does not follow from these.  This is true, even if $T_1$ is required to be a \emph{conservative} extension of $T_0$, so that $T_1\restriction\sig_0=T_0$.
\end{theorem}

\begin{proof}  (Can be left as exercise.)
Suppose $\ref{ext}$ holds.  Let
\begin{align*}
\str A&\ecmodel T_1,&
\str B&\models T_0,&
\str A&\restriction\sig_0\included\str B.
\end{align*}
We show
\begin{equation*}
\str A\restriction\sig_0\preccurlyeq_1\str B
\end{equation*}
(\emph{i.e.} every existential formula over $\str A\restriction\sig_0$ soluble in $\str B$ is soluble in $\str A\restriction\sig_0$).
By hypothesis, there is a model $\str C$ of $T_1$ such that
\begin{align*}
\str A&\included\str C,&
\str B&\included\str C\restriction\sig_0.
\end{align*}
Then
\begin{gather*}
\str A\preccurlyeq_1\str C,\\
\str A\restriction\sig_0\preccurlyeq_1\str C\restriction\sig_0,\\
\str A\restriction\sig_0\preccurlyeq_1\str B.
\end{gather*}
Therefore $\str A\restriction\sig_0$ must be an \ec\ model of $T_0$.  Thus $\ref{red}$ holds.

Suppose conversely $\ref{red}\And\ref{AP}\And\ref{ec}$ holds.  Let
\begin{align*}
\str A&\models T_1,&
\str B&\models T_0,&
\str A\restriction\sig_0&\included\str B. 
\end{align*}
We establish the consistency of
\begin{equation*}
T_1\cup\diag{\str A}\cup\diag{\str B}.
\end{equation*}
It is enough to show the consistency of
\begin{equation*}
T_1\cup\diag{\str A}\cup\{\Exists{\vec x}\phi(\vec x)\},
\end{equation*}
where $\phi$ is an arbitrary quantifier-free formula of $\sig_0(A)$ such that
\begin{equation*}
\str B\models\Exists{\vec x}\phi(\vec x).
\end{equation*}
By $\ref{ec}$, there is $\str C$ such that
\begin{align*}
\str C&\ecmodel T_1,&
\str A&\included\str C.
\end{align*}
By $\ref{red}$ then,
\begin{align*}
\str C\restriction\sig_0&\ecmodel T_0,&
\str A\restriction\sig_0&\included\str C\restriction\sig_0.
\end{align*}
By $\ref{AP}$, both $\str B$ and $\str C\restriction\sig_0$ embed over $\str A\restriction\sig_0$ in a model $\str D$ of $T_0$.  In particular,
\begin{equation*}
\str D\models\Exists{\vec x}\phi(\vec x).
\end{equation*}
Therefore $\phi$ is already soluble in $\str C\restriction\sig_0$ itself.  Thus
\begin{equation*}
\str C\models T_1\cup\diag{\str A}\cup\{\Exists{\vec x}\phi(\vec x)\}.
\end{equation*}
Therefore $\ref{ext}$ holds.

For the rest, 11 (counter-)examples are found\dots
\end{proof}

Now suppose
\begin{gather*}
(L,\partial_0,\dots,\partial_{m-1})\models\mDF_0,\\
K\included L,\\
(K,\partial_0\restriction K,\dots,\partial_{m-1}\restriction
K,\partial_m)\models\mpDF_0,\\
a\in L\setminus K.
\end{gather*}
We shall define a differential field
\begin{equation*}
(K\langle a\rangle,\tilde{\partial}_0,\dots,\tilde{\partial}_m),
\end{equation*}
where $a\in K\langle a\rangle$, and for each $i$ in $m$,
\begin{equation}\label{eqn:rest}
\tilde{\partial}_i\restriction K\langle a\rangle\cap L=\partial_i\restriction K\langle a\rangle\cap L,
\end{equation}
and $\tilde{\partial}_m\restriction K=\partial_m$.

Considering $\upomega^{m+1}$ as the set of $(m+1)$-tuples of natural
numbers, we shall have 
\begin{equation*}
K\langle a\rangle=K(a^{\sigma}\colon\sigma\in\upomega^{m+1}),
\end{equation*}
where
\begin{equation}\label{eqn:sigma}
a^{\sigma}=\tilde{\partial}_0{}^{\sigma(0)}\dotsm\tilde{\partial}_m{}^{\sigma(m)}a.
\end{equation}
In particular then, by \eqref{eqn:rest}, we must have
\begin{equation}\label{eqn:m0}
\sigma(m)=0\implies
a^{\sigma}=\partial_0{}^{\sigma(0)}\dotsm\partial_{m-1}{}^{\sigma(m-1)}a. 
\end{equation}
Using this rule, we make the definition
\begin{equation*}
K_1=K(a^{\sigma}\colon\sigma(m)=0).
\end{equation*}
Recursively, we can define
\begin{equation*}
K_j=K(a^{\sigma}\colon\sigma(m)<j)  
\end{equation*}
as desired.  If $L\setminus K\langle a\rangle\neq\emptyset$, we can
repeat, as necessary.

It may not be possible to make $L$ itself closed under $\tilde{\partial}_m$.
\end{document}