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\begin{document}
\huge

\begin{center}
\textsf{\bfseries\larger A Method for Companionability,\\
  Applied to Group Actions\\
	and Valuations}
	
	\emph{with Ay\c se Berkman and with\\
	\"Ozlem Beyarslan, Daniel Max Hoffmann, and G\"onen\c c Onay}
	
\textsc{David Pierce}

{\smaller Mimar Sinan G\"uzel Sanatlar \"Universitesi, \.Istanbul}

\vfill
{Delphi, July, 2017}
\vfill
\smaller
\gr{>`Hn strate'uhtai >ep`i P'ersas,
meg'alhn >arq`hn min katal'usein}\hfill\mbox{}

\mbox{}\hfill---Oracle to Croesus, as reported by Herodotus (\textsc i.53)
\end{center}

\pagebreak

Three theories of education:
\begin{enumerate}
\item
Learn the word of God.
\item
Learn skills.
\item
Learn freedom.
\end{enumerate}
\vfill
%Conflict between the first two has led to trouble today.
%\vfill
Mathematics teaches freedom by being
\begin{enumerate}[(a)]
\item
\emph{personal}
(none can command you to accept a theorem),
\item
\emph{universal}
(disagreement must be settled peacefully).
\end{enumerate}
\vfill
\begin{quote}\smaller{}
[Formal proof] was one of the great discoveries of the early 20th century, 
largely due to Frege, Russell, and Whitehead\lips 
This discovery has had a profound impact on mathematics, 
because it means that 
\emph{any dispute about the validity 
of a mathematical proof can always be resolved.}\\
\mbox{}\hfill---Timothy Gowers, 
\emph{Mathematics: A Very Short Introduction}
\end{quote}

\pagebreak

\begin{center}
  \includegraphics[height=105mm]{nmk-2016-07-smaller.eps} %A5 height 148mm

  Nesin Mathematics Village, \c Sirince\\
  Sel\c cuk, \.Izmir (Ephesus, Ionia), July 9, 2016
\end{center}

\pagebreak

\textsc{A problem of model theory:}
identify complete theories and their properties,
such as being axiomatizable or not.

\textbf{Presburger 1930.}
$\Th{\N,+}$ is axiomatizable.

\textbf{G\"odel 1931.}
$\Th{\N,+,\times}$ is not.
\vfill
%\begin{center}
\textsc{Useful definitions:}
%\end{center}
\begin{align*}
\diag{\str A}&=\Th{\text{structures in which $\str A$ embeds}},\\	
\Trand_{\forall}&=\Th{\text{structures embedding in models of $\Trand$}}.
\end{align*}
\textbf{A. Robinson 1956.}
A theory $\Trand$ is \textbf{model-complete} if,
whenever $\str A\models\Trand$,
then $\Trand\cup\diag{\str A}$ is complete.

\textbf{``Eli Bers''
%Barwise--Eklof--Robinson--Sabbagh 
1969.}  A model-complete theory $\Trand^*$ is the \textbf{model-companion}
of any theory $\Trand$ for which $\Trand^*{}_{\forall}=\Trand_{\forall}$.

\pagebreak

Fields may have
\parbox[t]{0.69\textwidth}
{\begin{enumerate}[1)]
\item
a valuation ring $\valr$ (with max. ideal $\maxi$),
\item
a derivation $\delta$,
\item
an automorphism $\sigma$.
\end{enumerate}}
\vfill
\begin{tabular}{rcr}
the theory of fields&\multicolumn2{l}{has model companion\qquad\qquad source}\\\hline
---&$\acf$&Tarski 1948\\
with $\valr$&$\acvf$&Robinson 1956\\
of char.\ $0$ with $\delta$&$\dcf_0$&Robinson 1959\\
of char.\ $p$ with $\delta$&$\dcf_p$&Wood 1973\\
with $\sigma$&$\acfa$&$\left\{\text{\parbox{87mm}{\flushright Macintyre 1997\\
Chatzidakis--Hrushovski 1999}}\right.$\\
of char.\ $0$ with $\delta$, $\sigma$&yes&\\
of char.\ $p$ with $\delta$, $\sigma$&no&\parbox{105mm}{\flushright\raisebox{0.3\baselineskip}[0pt][0pt]{\parbox{104mm}{$\left.\vphantom{\text{\parbox{0pt}{M9\\C9}}}\right\}$\hfill P. 2004}}}\\
with $\valr$, $\sigma$&yes&Beyarslan--Hoffmann--Onay--P. 201?\\\hline
\end{tabular}

\pagebreak

\textsc{Theorem} (generalizing \textbf{Robinson 1957}).
If it exists,
the model-companion $\Trand^*$ of a theory $\Trand$
is axiomatized by $\Trand_{\forall}$ and sentences
\begin{equation*}
\Forall{\bm x}\Forall{\bm y}\Exists{\bm z}\bigl(\theta(\bm x,\bm y)\lto\phi(\bm x,\bm z)\bigr),
\end{equation*}
where
\vfill
\begin{itemize}
\item
$\phi$ is a system of atomic and negated atomic formulas,
\vfill
\item
$\theta$ is from a set $\Theta_{\phi}$ of formulas,
\vfill
\item
for all models $\str M$ of $\Trand_{\forall}$
with parameters $\bm a$, 
\vfill
\begin{center}
$\theta(\bm a,\bm y)$ is soluble in $\str M$ for some $\theta$ in $\Theta_{\phi}$
$\iff$\\
\vfill
$\phi(\bm a,\bm z)$ is soluble in a model of $\Trand_{\forall}\cup\diag{\str M}$.
\end{center}

\end{itemize}
\vfill
\emph{Not every system $\phi$ need be considered, but ``enough'' of them.}

\newpage

To axiomatize $\dcf_0$, Robinson considered all systems.

\textbf{Blum 1968:} one-variable systems are enough.

For the model-companion of any theory $\Trand$,
it is enough to consider \emph{unnested} systems.

\textsc{Example.}  Over a field $K$ with $\sigma$,
$\valr$, and $\maxi$,
one need only understand systems
\begin{equation*}
\bigwedge_{f\in I_0}f=0
\land\bigwedge_{i<m}X_i{}^{\sigma}=X_{\tau(i)}
\land\bigwedge_{\ell\in\lambda}X_{\ell}\in\valr
\land\bigwedge_{k\in\kappa}X_k\in\maxi,
\end{equation*}
where, for some $n$ in $\upomega$,
\begin{itemize}
\item
$I_0$ is a finite subset of $K[X_j\colon j<n]$,
\item
$m\leq n$ and $\tau\colon m\rightarrowtail n$,
\item
$\kappa\included\lambda\included n$.
\end{itemize}

\pagebreak

\textsc{Example.}
A \textbf{group action} is $(P,A)$, where
\begin{enumerate}[1)]
\item
$P=\{\text{functions}\}$,
\item
$A=\{\text{points}\}$,
\item
there is
$(\xi,y)\mapsto\myop{\xi}y$
from $P\times A$ to $A$ whereby
\begin{enumerate}
\item
functions have inverses:\hfill
$\Forall{\xi}\Exists{\eta}\Forall z(\myop{\xi}\myop{\eta} z=z\land\myop{\eta}\myop{\xi} z=z)$;
\item
two functions have a composite:\hfill
$\Forall{\xi}\Forall{\eta}\Exists{\zeta}\Forall v
\myop{\xi}\myop{\eta} v=\myop{\zeta} v$;
\item
there is an identity:\hfill
$\Exists{\xi}\Forall y\myop{\xi} y=y$.
\end{enumerate}
\end{enumerate}
\vfill
Let $\GA=\Th{\text{group actions}}$.

Then
$\GA_{\forall}$ is that
functions are injective:
\begin{equation*}
\Forall{\xi}\Forall y\Forall z(y\neq z\lto\myop{\xi} y\neq\myop{\xi} z).
\end{equation*}
\vfill
\textsc{Key observation:}
Compositions not preserved in extensions.

\pagebreak

To find $\GA^*$, 
one need only consider systems
\begin{equation*}
\bigwedge\myop{\alpha}x=y
\land\bigwedge\myop{\xi}t=u,
%\land\bigwedge\myop{\xi}a=b
%\land\bigwedge\myop{\xi}a=y
%\land\bigwedge\myop{\xi}x=b
%\land\bigwedge\myop{\xi}x=y.
\end{equation*}
where $t$ and $u$ are point variables or constants.

$\GA^*$ is complete and says,
\begin{enumerate}[1)]	
\item
any $n!$ distinct functions
act like $\Sym n$ on some $n$ points;
\item
on any $n$ distinct points,
some $n!$ functions act like $\Sym n$;
\item
there are at least two points.
\end{enumerate}
\vfill
$\GA$ includes $\Th{\text{\textbf{parametrized permutations}}}$,
axiomatized by
\begin{align*}
\Forall{\xi}\Forall y\Forall z(y\neq z&\lto\myop{\xi} y\neq\myop{\xi} z),&	
\Forall{\xi}\Forall y\Exists z\myop{\xi}z&=y.
\end{align*}
\vfill
\textbf{Shelah 1993:} $\Tfeq$, namely
\begin{center}
$\Th{\text{\textbf{parametrized equivalence relations}}}$.
\end{center}

\pagebreak

$\Tfeq{}^*=\Th{\text{Fra\"iss\'e limit
of the class of finite models of $\Tfeq$}}$.
\vfill
It was shown that $\Tfeq{}^*$ has $\TP2$ 
and, ultimately, $\NSOP1$.
\vfill
Thus $\Tfeq{}^*$ occupies an undivided region of the 
\emph{Map of the Universe}
(\textbf{Conant 2013--,} \url{forkinganddividing.com}).
\vfill
As for $\GA^*$,
so for for $\Tfeq{}^*$,
one can obtain axioms:
\begin{enumerate}[1)]
\item
a partition of $n$ points is effected by some relation,
\item
the intersection of classes of $n$ distinct relations is nonempty,
\item
there are $n$ relations and $n$ classes of each.
\end{enumerate}
\vfill
Like $\Tfeq{}^*$, $\GA^*$ has $\TP2$ and $\NSOP1$.

\pagebreak

\textbf{Chernikov--Ramsey 2016:}
In a finite relational signature,
\begin{description}
\item[if ] the theory of the Fra\"iss\'e limit 
of a Fra\"iss\'e class with Strong Amalgamation is simple,
\item[then ] the theory of \emph{parametrized} models has $\NSOP1$,
\item[because ] it has a certain independence relation $\ind$
with independent amalgamation of types.
\end{description}
\vfill
\textsc{Theorem.}
$\GA^*$ also has $\NSOP1$,
because of $\ind$ given by
\begin{multline*}
(\Rho,A)\ind_{(\Tau,C)}(\Sigma,B)\iff\\
\Rho\cap\Sigma\included\Tau\And{}
\gen{A\cup C}_{\Rho\cup\Tau}\cap\gen{B\cup C}_{\Sigma\cup\Tau}\included\gen C_{\Tau},
\end{multline*}
\vfill
where
$\gen X_{\Xi}=\{\myop{\xi^n}x\colon\xi\in\Xi\And n\in\Z\And x\in X\}$.

\newpage

%\mbox{}\vfill
%\mbox{}\hfill
\includegraphics[height=78mm]{cicada-closeup-marmara-2012-07.eps}
\hfill
\includegraphics[height=78mm]{cicada-marmara-2012-07.eps}
%\hfill\mbox{}

\centerline{\gr{T'ettix} on tree,
  Marmara Island (Proconnesus), July 26, 2012}
\vfill
\smaller
\gr{t~h| d`e presbut'ath| Kalli'oph| ka`i t~h| met'' a>ut`hn O>uran'ia| 
to`us >en filosof'ia| di'agont'as te ka`i tim~wntas 
t`hn >eke'inwn mousik`hn >agg'ellousin, 
a<`i d`h m'alista t~wn Mous~wn per'i te o>uran`on ka`i l'ogous o>~usai je'ious te ka`i >anjrwp'inous <i~asi kall'isthn fwn'hn}
\hfill---Plato, \emph{Phaedrus} 259\textsc d

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