Categoricity may fail late
{Saharon Shelah}\thanks{\ The first author's research was partially supported by `BSF' (USA-Israel); Publication no 648 in the first author's publication list. The second author was sponsored by Colciencias and Semilleros de Investigaci\'on, Universidad Nacional de Colombia.}
Department of Mathematics
{Andr\'es Villaveces}
Universidad Nacional de Colombia
We build an example that generalizes \cite{HaSh:323} to uncountable cases. In particular, our example yields a sentence $\psi\in {\cal L}_{(2^\lambda)^+,\omega}$ that is categorical in $\lambda,\lambda^+,\dots,\lambda^{+k}$ but not in $\beth_{k+1}(\lambda)^+$. This is connected with the \L o\'s Conjecture and with Shelah's own conjecture and construction of excellent classes for the $\psi\in {\cal L}_{\omega_1,\omega}$ case. %% We generalize \cite{HaSh:323} to larger contexts. [We hope to look for %% examples of $\l$-good frames which are not $\l$-good$^+$, or other %% variations on the subject matter of \cite{Sh:576}, \cite{Sh:600} and %% \cite{Sh:705}.]
math0404258.tex
% To: "Saharon Shelah's Office" <shlhetal@math.huji.ac.il>
% Subject: 648 update
% Date: Sun, 20 Oct 2002 22:01:27 -0400 (EDT)
% From: Andres Villaveces <villavec@andrew.cmu.edu>
% In-Reply-To: <20021019143348.B9364@dsc235.dsc.unomaha.edu>
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\markright{{{\bf Categoricity may fail late}}\hfill \today\ \ \ \ \ }

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%\setcounter{section}{-1}

\title{Categoricity may fail late}
\author{{\bf Saharon Shelah}\thanks{\
The first author's research was partially supported
by `BSF' (USA-Israel); Publication no 648 in the first author's
publication list. The second author was sponsored by Colciencias and
Semilleros de Investigaci\'on, Universidad Nacional de Colombia.}\\ 
Institute of Mathematics, The Hebrew University\\
Department of Mathematics, Rutgers University
\and
{\bf Andr\'es Villaveces}\\
Universidad Nacional de Colombia,\\
Carnegie Mellon University\\
\scriptsize{villavec@matematicas.unal.edu.co}
}
\date{%revisions: July 26, August 20, Nov. 18, 1993\\
%last revision (after Referee rep.): Fall, 1995\\
%last revision (after Referee rep.): April, 1996\\
%last corrections introduced: May- Sept 1996\\
%last corrections introduced: Dec 1996\\
printed: \today}

\begin{document}
%\baselineskip=20pt
\maketitle

\begin{abstract}
We build an example that generalizes \cite{HaSh:323} to uncountable
cases. In particular, our example yields a sentence $\psi\in {\cal
L}_{(2^\lambda)^+,\omega}$ that is categorical in
$\lambda,\lambda^+,\dots,\lambda^{+k}$ but not in
$\beth_{k+1}(\lambda)^+$. This is connected with the \L o\'s
Conjecture and with Shelah's own conjecture and construction of
excellent classes for the $\psi\in {\cal L}_{\omega_1,\omega}$ case.

%% We generalize \cite{HaSh:323} to larger contexts. [We hope to look for
%% examples of $\l$-good frames which are not $\l$-good$^+$, or other
%% variations on the subject matter of \cite{Sh:576}, \cite{Sh:600} and
%% \cite{Sh:705}.]
\end{abstract}

\section{The \L o\'s Conjecture, without excellence}

Early results on the Categoricity Spectrum launched the development of
Stability Theory and Classification Theory for first order logic. In
the natural quest for generalizing the powerful results of those
theories to Nonelementary Classes, some questions on Categoricity --
specifically, the status of the \L o\'s Conjecture (=Morley Theorem in
First Order) -- became a crucial test question.  Among the specific
issues studied, the following is central.

\begin{question}
{\bf Old folklore question:} what is the status of the \L o\'s Conjecture (=
Morley Theorem in the first order case) for $\psi \in {\cal L}_{\w_1,\w}$?
\end{question}

Keisler in \cite{Ke70} first provided a condition under which the
Morley analysis works (and thus provides a positive answer to the
question) ). However, Marcus in \cite{Mar} showed that this condition
is unfortunately not necessary.

In the landmark papers \cite{Sh:87a} and \cite{Sh:87b}, Shelah
provided a positive answer to this question under weak set
theoretical assumptions such as weak versions of the GCH
(specifically, weak diamonds). He identified the class of
`excellent sentences' and showed that if an ${\cal
L}_{\w_1\w}$-sentence $\psi$ is excellent then the
\L o\'s Conjecture holds for $\Mod(\psi)$ (and the Morley analysis can
be carried to this situation). In those two Shelah papers, 
% (really, one long paper split into two
% parts by the publishers),
the need for these assumptions seemed to be implied.  Harrington
explicitly voiced a natural question: he asked whether those
assumptions were really needed (or in what sense could 
one weaken them).  An answer to Harrington's question appeared in
\cite{HaSh:323}: they provide the first examples of ${\cal L}_{\w_1\w}$
sentences which were categorical up to some level but not excellent.

Excellent classes turned out to be of greater mathematical relevance
than one would perhaps think. Not only did their definition
provide the early conceptual tools for generalizing the work on
categoricity to various nonelementary contexts, but also a version of
the Main Gap was proved for them by Grossberg and Hart (see
\cite{GrHa89}), they are actually at the heart of Zil'ber's recent
study of fields with pseudoexponentiation and the Schanuel
Conjecture (see \cite{Zilber}). An excellent expository paper that
connects excellent classes to other domains in the classification
theory for nonelementary classes is the `Bilgi' paper by Grossberg
(\cite{Gr02}).

The `old folklore question' of course admits many generalizations of
the kind

\begin{question}
\begin{description}
\item What is (the status of) the \L o\'s Conjecture for $\psi \in
L_{\k,\w}$ (suitable $\k$)?
\item What is (the status of) the \L o\'s Conjecture for ${\frak K}$
(say) a PC-class? 
\item What is (the status of) the \L o\'s Conjecture for ${\frak K}$ a
non-elementary class?
\end{description}
\end{question}

Of course, those questions (especially the last one) are way too
general to be settled in short. In particular, the long series of
papers \cite{Sh:88},
\cite{MaSh:285}, \cite{KlSh:362}, \cite{Sh:472}, \cite{Sh:394},
\cite{Sh:576}, \cite{Sh:600}, \cite{ShVi:635}, \cite{Sh:705} may
be regarded as a (long and until now far from complete) organized
attempt to solve the third of these generalizations. Or at least, an
attempt to provide enough set theoretical or model theoretical
hypotheses to obtain {\it positive} solutions to the \L o\'s
Conjecture. These include various large cardinal hypotheses, weak
diamond style hypotheses, on the set theoretical side, and
amalgamation properties, existence of `large' models, axioms for
building `nonforking' frames, etc. on the model theoretical side.

Clearly, the long-run intention is to obtain not just very general
settings in which the \L o\'s Conjecture (or some decent enough
version of it) holds, but also to obtain more `dividing lines' to
pursue the program of classification theory for nonelementary
classes. In this respect, the attempts done in the papers mentioned in
the last paragraph all address what could be called `the first test
question'.

On what one could call `the {\it negative} complementary
perspective\footnote{Model theoretic phenomena dealing with cases
where the \L o\'s Conjecture fails},' there are so far very few
published results. The paper by Hart and Shelah is one the most
important of these.  Section 6 of \cite{Sh:88} also addresses some of
these `negative' aspects. The purpose of looking at some of these
negative cases must be seen in constrast with the positive results:
\cite{HaSh:323} must be read with a \cite{Sh:87a} and
\cite{Sh:87b} background. 

Ideally, this paper should be read with a \cite{Sh:88}
and \cite{Sh:394} background.
As more positive results 
appear, it is natural to attempt to `fill the gap' between positive and
negative cases, and to provide new tools for understanding the blurry
boundary between model (or set) theoretic situations where the \L o\'s
Conjecture holds or fails.

In this paper, we focus our energy in generalizations of Hart and
Shelah's counterexample to the \L o\'s Conjecture in
\cite{HaSh:323}. A second aim of this paper is to try to {\bf explain}
what can be expected (and why) on this `boundary region' between
positive and negative answers to the \L o\'s Conjecture.
The following table summarizes the previous remarks and the status
of research on the subject. \L C stands for the \L o\'s Conjecture.

\vspace{1cm}

\centerline{\begin{tabular}{|c||c|c|c|c|}
\hline
 & FOL	& ${\cal L}_{\w_1\w}$ & ${\cal L}_{\l^+\w}$	&	AEC	\\
\hline
\hline
Full \L C & Morley	& Shelah \cite{Sh:87a}	& Shelah \cite{Sh:705}	& many partial  \\
holds &	& and \cite{Sh:87b}	& and consequences	& results \\
\hline
Some \L C &---	& \cite{HaSh:323}	& {\bf here}		& ?	\\
fails 	& 	&	 &	 & \\
\hline
\end{tabular}
}

\bigskip
Finally, we thank Rami Grossberg, Alexei Kolesnikov and John Baldwin for
several remarks concerning (directly or less directly) this work.

%\newpage

\section{Toward the counterexample}\label{basisi}

We construct here a generalization of the proof by Hart and Shelah in 
\cite{HaSh:323}, shifting the focus of their example from
$\aleph_0,\aleph_1,\dots,\aleph_k$ to $\l,\l^+, \dots,\l^{+k}$ and on
the way switching the logic for which the class of models is
elementary from ${\cal L}_{\w_1\w}$ to ${\cal L}_{(2^\l)^+\w}$.  The
construction first follows a path parallel to their older proof, and
then takes care of new complications arising at the time of setting up
the $\pmin(n)$-diagrams (as in \cite{Sh:87a}) amalgamation family. The
main differences with the construction in \cite{HaSh:323} are the need
here for the use of a regular filter $\frak D$ and the replacement of
$\w$ there by much larger groups here.

As in \cite{HaSh:323}, we first describe {\bf canonical models} $M_I$,
$M_{I,f}$ 
and in a second stage we extract from them the sentence $\psi$ in
${\cal L}_{(2^\l)^+,\w}$. 

\subsection{The construction}

\begin{context}
For this section we fix $\l$ an infinite cardinal and $k<\w$ ($k\geq
2$).
\end{context}

%The following are the ``building pieces'' of our models: certain groups
%defined on the set $S$ of finite subsets of $\l$.
The canonical models will be built using groups defined on the set $S$
of finite subsets of $\lambda$.

%\begin{notation}
%\end{notation}
\begin{definition}
\end{definition}
\begin{enumerate}
\item Let $S = S_\l = [\l]^{<\al_0}$,
\item $\gd = \gd_\l = \{ A\subset S | \exists u_A\in S \forall v\in
S(u_A\subset v \to
v\in A)\}$, the regular filter on $S$ generated by sets of the form
$\langle u\rangle = \{ v\in S | u\subset v\}$.
\item $G^+ = G^+_\l = {}^S(\bbz_2)$, as a group with the natural operation
$(f+g)(v) = f(v)+_{\bbz_2}g(v)$,
\item $G = G_\l = \{ f\in {}^S(\bbz_2) | \ker(f) = \{ u\in S | f(u) =
0 \} \in \gd
\}$, as a subgroup of $G^+$: $G\lhd G^+$, as if $f,g \in G$ then
$\ker(f),\ker(g)\in \gd$, so $\ker(f+g) \supset \ker(f)\cap \ker(g)\in
\gd$, so $\ker(f+g)\in \gd$ and $f+g\in G$. Note that $|G|=2^\lambda$.
\end{enumerate}

It is worth keeping in mind that the vocabulary for the construction
of $M_I$ and the idea of the definition of $\psi$ depends on these
basic notions.

\begin{definition}\label{defM_I}{\bf The construction of the model $M_I$.}
\end{definition}
For a fixed set $I$ we first define the following objects.
\begin{description}
\item[(a)] $H = H_I$ is the group with set of elements $[\ialak]^{<\al_0}$,
with `addition' given by $S+T:= S\Delta T$ (symmetric difference).
(We will also regard sometimes $h\in H$ as a function $\ialak \to \bbz_2$,
with the usual translation by means of the characteristic function of $h$;
clearly, this is in our case a function with value 0 except at finitely many
places.)
\item[(b)] the model $M = M_I$:
\begin{description}
\item[Universe:] 
\[ |M_I| = I \cup \ \ialak \cup \ \ialakmasuno \cup (\ialak \times
S\times H )\cup (\ialak \times S\times \bbz_2) \cup H \cup (\ialakmasuno
\times G).\] 

One way of thinking about the model is as
\[ |M_I|\]
\begin{center}
consisting of
\end{center}
\[ \overbrace{I \cup \ \ialak \cup \
\ialakmasuno}^{\mbox{\footnotesize `control part'}}
 \cup \overbrace{(\ialak \times
S\times H )\cup (\ialak \times S\times \bbz_2) \cup H \cup (\ialakmasuno
\times G)}^{\mbox{\footnotesize `zeroless copies of $H_I$, $\bbz_2$
and $G$'}}.\] 

Notice that we actually get an empty intersection between all those
pieces of the model.

Before putting structure on $|M_I|$, we try to provide the reader with
a general description of how our model will be. The relations and
functions below will make specific the following vague description: in
addition to the indices $I$ and $k$ and $k+1$ tuples of indices from
the `control region' and a copy of $H$, we have `zeroless versions' of
the groups $H_I$, $\bbz_2$ and a copy of $G$ for each $k+1$-tuple, one
for each $k$-tuple from $I$ and finite set $s\in S$.

\begin{quotation}
{\footnotesize 
\noindent
In a way analog to \cite{HaSh:323}, we will get the
interesting behavior in the models by predicates $Q_s$ connecting (for
each $s\in S$) one copy of $H$ with $k$ copies of $\bbz_2$ and one
copy of $G$.}
\end{quotation}

%% So, inside $M_I$, the only information missing in order to compute how
%% the model is constructed from $I=P_0^M$ are the zeros in each copy of
%% $H$, $\bbz_2$, $G$. We will show that this ``freedom level'' does not
%% affect categoricity {\bf only} up to level $\l{+k}$.

\noindent
{\bf Remark:} these functions capture our idea of building the model
with `shifted' copies of the groups $G$, $H_I$ and $\bbz_2$ -- through
maps between these groups and $\ialak$. 

\item[Basic Relations:]
$P_0^M,P_{1,1}^M,P_{1,2}^M,P_2^M,(P_{2,s}^M)_{s\in S}, 
P_3^M,(P_{3,s}^M)_{s\in S}, P_4^M,P_5^M$.\\ These are defined by
\begin{description}
\item $P_0^M = I$,
\item $P_{1,1}^M = \ialak$,
\item $P_{1,2}^M = \ialakmasuno$,
\item $P_2^M = \ialak \times S\times H$ (a copy of $H$ for each
$b\in \ialak$, $s\in S$ - recall that $H=[{}^kI]^{\aleph_0}$),
\item for $s\in S$, $P_{2,s}^M = \{ (u,s,h) \in P_2^M | u\in \ialak,
h\in H\} = \ialak \times \{ s\} \times H$,
\item $P_3^M = \ialak \times S\times \bbz_2$ (a copy of $\bbz_2$ for each
$b\in \ialak$, $s\in S$),
\item for $s\in S$, $P_{3,s}^M = \{ (u,s,i) \in P_3^M | u\in \ialak,
i\in \bbz_2\} = \ialak \times \{ s\} \times \bbz_2$,
\item $P_4^M = H$,
\item $P_5^M = \ialakmasuno\times G$
\end{description}
\begin{quotation}
{\footnotesize 
\noindent
The intended meanings are:
\begin{description}
\item $P_2^M \approx (H_{v,s})_{v\in \ialak,s\in S}$, 
\item $P_3^M\approx ((\bbz_2)_{v,s})_{v\in \ialak,s\in S}$,
\item $P_5^M\approx (G_u)_{u\in \ialakmasuno}$.
\end{description}}
\end{quotation}
\item[Partial Functions:]
$(\pi_\ell)_{\ell<k},F_2^M,F_3^M,F_4^M,F_5^M,(F^M_{3,g^*})_{g^*\in 
G}$, given by
\begin{description}
\item For $\ell <k$, the projections $\pi_\ell:\ialak \to I$ given by 
\[\pi_\ell(\bar{a}) = a_\ell\]
and $\pi_k:\ialakmasuno \to I$ given by
\[\pi_k(\bar{a}) = a_k.\]
\item A unary function $F_2^M$ with domain $P_2^M$, given by
\[F_2^M(u,s,h) = u,\]
\item A unary function $F_3^M$ with domain $P_3^M$, given by
\[F_3^M(u,s,i) = u,\]
\item A unary function $F^M_5$ with domain $P^M_5$, given by 
\[F_5^M(u,g) = u,\]
\item A binary function $F^M_4$ with domain $P^M_2\times
P_4^M$, given (on our intended `zeroless copies of $H$') by 
\[F_4^M\Big( (v,s,h),h_1\Big) = (v,s,h+_Hh_1)\]
\item for $g^*\in G$, a unary function $F^M_{3,g^*}$ with domain
$P^M_5$, given by \[F^M_{3,g^*}(u,g) = (u,g^*+g),\]
\end{description}
\begin{quotation}
{\footnotesize
\noindent
The intended meanings are a mixture of projections and
functions that tell us `in which' copy we are. Also notice that
\begin{description}
\item $F_4^M(F_4^M ((u,s,h),h_1),h_2) = F_4^M((u,s,h),h_3)$ if and
only if $H\models h_1+h_2=h_3$. Hence, $+^H$ is definable.
\item Notice also that if we replace $\bbz_2$ by a larger group, we
may need an analog to $F_4$ for copies of $\bbz_2$.
%% \item $F_4^M ((u,s,h),h_1) = F_4^M((u,s,h'),h_1)$ iff $h+_Hh_1 =
%% h'+_Hh_1$ iff $h=h'$, so $F_4$ codes in a weak way the group structure
%% of $H$.
\end{description}}
% So, in a sense, we do not have real groups, but maps of sorts between $G$, $H$
% and $[I]^k$, or `abelian groups without zero.'
\end{quotation}

\item[A predicate] $Q_s$, for each $s\in S$, that we shall relate later
to $\pmin (n)$-diagrams and their amalgamation, speaking about the
`$s$th copy of one copy of $H$, $k$ copies of $\bbz_2$ and one copy of
$G$'. It is interpreted in $M_I$ as
the set of tuples 
\[ \langle a_0,\dots,a_k,u_0,\dots,u_k,x_0,\dots,x_{k-1},y_k,z \rangle \]
satisfying (for fixed $s\in S$!!) $h_k \in H$, $i_\ell \in
\bbz_2 (\ell \leq k)$, $g\in G$:
\begin{description}
\item[$(\a)$] $a_\ell \in I$ with no repetitions ($\ell \leq k$),
\item[$(\b)$] $u_\ell=\langle a_m |m\not= \ell\rangle\in P_{1,1}^M$
($\ell \leq k$),
\item[$(\g)$] $y_k = (u_k,s,h_k)\in P_2^M$,
\item[$(\delta)$] $x_\ell$ has the form $(u_\ell,s,i_\ell)\in P_3^M$
($\ell < k$) so $i_\ell\in\bbz_2$, 
\item[$(\epsilon)$] $z$ is of the form $(u,g)\in P_5^M$,
where $u=(a_0,\dots,a_k)\in \ialakmasuno$ and
\item[$(\z)$] {\bf (main point)}
\[\bbz _2 \models \sum_{\ell<k}i_\ell = h_k(u_0) + g(s).\]
\end{description}
\end{description}
\end{description}

Part $(\z)$ of the definition provides the connection between $k$
copies of $\bbz_2$, one copy of $H$, one copy of $G$ and the $k+1$
$k$-element subsets of a set of size $k+1$ in $I$.
%% The point we want to stress with part $(\z)$ of the definition
%% is that the $k+1$-tuple from $\bbz_2$ sums up to the `characteristic
%% function' of $u_0$ in $h_k$ plus the value of $g$ at $s\in S$.

The fact that in $(\z)$ we choose $0$ as subindex for $u$ is not
important; it could have been any $\ell \leq k$; note that $u_k$
appears only in $y_k$.

We want to show that there are few $M$ quite equivalent to $M_I$ in the
relevant sizes, and we
shall give a full characterization of these models.

Now, we direct our attention to the language, in order to extract
our sentence $\psi$.

\begin{definition}\label{la6.2}

Let $\tau^-$ be the vocabulary implicit in all the construction above,
without including $\{ Q_s|s\in S\}$ and $\tau$ be the full vocabulary
implicit above. Specifically,

\[ \tau^- = \langle P_0,P_{1,1},P_{1,2},P_2,(P_{2,s})_{s\in
S},P_3,(P_{3,s})_{s\in S},P_4,P_5, \qquad \qquad \] 
\[ \qquad \qquad \qquad\pi_0,\dots,\pi_{k-1},F_2,F_3,F_4,F_5,
(F_{3,g^*})_{g^*\in G} \rangle \]
\[ \tau = \tau^- \cup \{ Q_s|s\in S\}.\]
\end{definition}

Notice that $|\tau| = |G|+|H|+|S|+\aleph_0 = 2^\l + |I|$,
since $|G_\l| =2^\l$.

\begin{definition}\label{defM_If}
\end{definition}
For a set $I$ and a function
\[ f: \ialakmasuno\times S \to \bbz_2 \]
we define a model $M_{I,f}$ as follows:
\begin{description}
\item[Vocabulary:] also $\tau$. The model will then be like $M_I$,
only the interpretation of $Q_s$ changes. So, we let
\[M_{I,f}\restriction \tau^- = M_I\restriction \tau^-,\]
and in $M_{I,f}$ the interpretation of $Q^{M_{I,f}}_s$ is the set  of
tuples  
\[ \langle a_0,\dots,a_k,u_0,\dots,u_k,x_0,\dots,x_{k-1},y_k,z \rangle \]
as in \ref{defM_I}, only equation $(\z)$ becomes here
\[ (\z)_f^* \qquad \qquad \bbz_2 \models \sum_{\ell<k} i_\ell =
h_k(u_0) + g(s) + f(u,s). \]
$f(u,s)$ then gives us the `correction' for any other copy of the groups.
\item $M^-_{I,f}$ is defined like $M_I$ but omitting the predicates
$Q_s$. Its vocabulary is $\tau^-$ from \ref{la6.2}.
\item $M$ is {\bf strongly standard} if $M\restriction \tau^- =
M_I\restriction \tau^-$ for $I = P_0^M$,
\end{description}

We leave the following claim with no proof.

\begin{claim}
 For any $M\models \psi$, $M_I \approx M_{I,{\bf 0}}$.
\end{claim}

\begin{claim}\label{allisostrstrd}
Every $M\models \psi$ is isomorphic to a strongly standard $M$.
\end{claim}

\noindent
Next, a straightforward observation. 

\begin{claim}
$M_{I,f}$ is strongly standard.
\end{claim}

\begin{definition} {\bf (The first class and the sentence)}
\end{definition}
\begin{enumerate}
\item $K_1 := \{ M | M\approx M_{I,f}$ for some infinite set $I$, for
some $f$ as in \ref{defM_If}$\}$ so $K_1$ is a
class of $\tau$-models, the vocabulary $\tau$ from
\ref{defM_If}. 
\item Our sentence $\psi \in {\cal L}_{(2^\l)^+, \w}(\tau)$ is defined
by using the following sentences:
\begin{enumerate}
\item $T_0$ consists of {\bf all} the first order sentences which
every $M_I$ satisfies (as $I$ is infinite, all the $M_I$ are
elementarily equivalent in first order),
\item $\psi_G \equiv \forall z_1z_2 ([P_5(z_1)\wedge P_5(z_2)\wedge
F_5(z_1)=F_5(z_2)] \to \bigvee_{g^*\in G}F_{3,g^*}(z_1)=z_2$,
\item $\psi_{\bbz_2}\equiv \forall y(P_2(y)\leftrightarrow
\bigvee_{s\in S}P_{2,s}(y)$,
\item $\psi_H \equiv \forall y(P_3(y)\leftrightarrow \bigvee_{s\in
S}P_{3,s}(y)$,
\end{enumerate}
Then we define our sentence $\psi \in {\cal L}_{(2^\l)^+,\w}(\tau)$ by
\[ \psi \equiv \bigwedge T_0 \wedge \psi_G \wedge \psi_{\bbz_2}\wedge
\psi_H \]
%\[ M\models \psi \Leftrightarrow \bigvee_{I,f}M\approx M_{I,f} .\]

\begin{quotation} 
{\footnotesize
\noindent
$\psi_G$ says that the copies of $G$ are really copies of $G$, so that 
The construction of the models if used to prove that our sentence
$\psi$ has the desired properties.
Note that, although there are $2^{2^\l}$ sentences in
the logic, we are only using $2^\l$ of them, as witnessed by
$|G|=2^\l$.}
\end{quotation}

%% Another description is
%% \[ \psi := \bigwedge _{I,f} Th_{{\cal L}_{\w_1\w}}(M_{I,f}).\]
%% We can bound the sizes of $I$ in the previous to $\l$.

%\medskip
%\hrule

%% \medskip
%% {\tiny Can start with $\psi := \bigwedge _{I,f} Th_{{\cal
%% L}_{(2^\l)^+,\w}}(M_{I,f})$. Really, $M_I$ or $M_{I,f}$? Should be
%% $M_{I,f}$. Of course, this depends on the proof later. Be aware.
%% Why are there few such sentences? Again, the only way out is looking
%% at Hart's construction and adapt it. Maybe... all f.o. + the few
%% necessary... Notice that for $I$ infinite, all the $M_I$ are
%% elementarily equivalent. Check from later (\ref{labase}) *which*
%% equivalent ones.
%% }
%% \medskip
%% \hrule

\item $K_2 := \Mod(\psi)$. 
\item $M$ from $K_2$ is {\bf standard} if $P_{1,1}^M = [P_0^M]^k$ and
$P_{1,2}^M = [P_0^M]^{k+1}$ and the $\pi_\ell$'s are natural.
%$P_{1,2}$ with natural $\in^M$ between those two sets.

\end{enumerate}

%% \begin{fact}
%% $K_1\subset K_2$.
%% \end{fact}

%% \Proof 
%% \qed

%\newpage

\section{Recovering lost zeros}

We now start the second stage of our proof, preparing the desired
categoricity cases for $\psi$. For this, it is enough to show that
every model in the suitable cardinals is isomorphic to a standard one.
We will describe choices and correction functions, that will be used
in counting the models in sizes $\l,\l^+,\dots,\l^{+m}$, $m<k$.

This comes down to trying to recover the `lost' zero of the copies of
the groups. To this end, we define `choices' (depending on the model
$M$ and on various subsets of $P_{1,1}^M$) of relevant elements for
the crucial equation, and `correction functions' for these equations.

Being isomorphic to standard means that we can `make choices' with
zero corrections but we have freedom in choosing the choice
function. Expanding choices from partial to global ones is the crux of
the proof.

\begin{definition}
\end{definition}
\begin{enumerate}
\item For $M\models \psi$, we say $\chxyz$ is a 
{\bf partial $M$-$(J_0,J_1,J_2)$-choice} if
\begin{description}
\item[(a)] $J_0, J_1\subset P^M_{1,1}$, $J_2\subset P^M_{1,2}$,\\
{\footnotesize 
(the intended meaning here is for standard models: $J_0,J_1\subset
\ialak$, $J_2\subset  \ialakmasuno$)
}
\item[(b)] $\bar{x} = \langle x_{u,s} | s\in S, u\in J_0\rangle$, where
\[ x_{u,s}\in (\bbz_2)^M_{u,s} := \{ z\in P_{3,s}^M | F^M_3(u,s,z) =
u\} \subset P_{3,s}^M.\]
\item[(c)] $\bar{y} = \langle y_{u,s} | s\in S, u\in J_1\rangle$,
\[ y_{u,s}\in H^M_{u,s} := \{ y\in P_{2,s}^M | F^M_2(u,s,y) = u\}
\subset P_{2,s}^M.\]
\item[(d)] $\bar{z} = \langle z_u | u\in J_2\rangle$,
\[ z_u\in G_u^M := \{ z|F_5(z) = u\} \subset P_5^M.\]
\end{description}
(So, informally, $\bar{x}$ chooses an element $i$ in each copy of
$\bbz_2$, $\bar{y}$ chooses a $h$ in each copy of $H$, $\bar{z}$
chooses a $g$ in each copy of $G$, for each relevant $(u,s)$, so
$x_{u,s}$ is {\it some} element in the `fiber' of $u$ via $F_3^M$, and
analogously for $\bar{y}$ and $\bar{z}$)

\item Call $\chxyz$ a
{\bf partial $M$-$J$-choice} if it is an $M$-$(J,J,J^M_*)$-choice, where

\[J_*^M :=
\Big\{ a \in P_{1,2}^M \Big| \bigwedge_{m\leq k} \exists b\in J [\bigwedge
_{\ell<m} 
(\pi_\ell(a) = \pi_\ell(b) \wedge \bigwedge _{\ell\in [m,k[} \pi_\ell(b) =
\pi_{\ell+1} (a)]\Big\} .\] 

\begin{quotation}
{\footnotesize
\noindent
The idea is that elements of $J_*^M$ must have all
$k$-subsets in them for amalgamation of the $\pmin(n)$-diagram later and
this happens through this representation of the sets.}
\end{quotation}

If $M$ is standard, we have that
\[J^M_* = \Big\{ \langle a_\ell | \ell \leq k \rangle \Big| \bigwedge
_{m\leq k} \langle a_\ell |\ell \not= m \rangle \in J \Big\} .\]
Finally, we say that $\chxyz$ is a {\bf global $M$-choice} if it is a
partial $M$-$P_{1,1}^M$-choice. We will sometimes just say $M$-choice
when meaning a global $M$-choice.

% \item For $M$ strongly standard, let its {\bf correction function} be
% $f = f^M$ with $dom(f) = P^M_{1,2}$ with 
% \[ f(\bar{a}) = \left \{
% \begin{tabular}{ll}
% 0	&	if $M_I \models Q_s[\bar{a}]$\\
% 1	&	otherwise
% \end{tabular}
% \right . \]
\item Fix a standard $M$ and a
$M$-$(J_0,J_1,J_2)$-choice $\chxyz$. Then we let the {\bf correction
function} $f$ for $M$ and $\chxyz$ be the function such that
\begin{enumerate} 
\item $\Dom (f)$ is the set of pairs
$(u,s)$ such that
\begin{description}
\item[$(\a)$] $u = \langle a_\ell | \ell\leq k\rangle \in J_2\subset
P^M_{1,2}$,
\item[$(\b)$] if $u_m := \langle a_\ell | \ell\leq k, \ell\not= m\rangle$,
$u_\ell \in J_0$ for $\ell<k$, $u_k\in
J_1\subset P^M_{1,1}$,
\end{description}
\item $\ran(f) \subset \bbz_2$, and 
\item (recall $x_{u_\ell,s},y_{u_k,s},z_{u_k}$ are from the choice)
\[ f(u,s) = 0 \Leftrightarrow \langle
a_0,\dots,a_k,u_0,\dots,u_k,x_{u_0,s},
\dots,x_{u_{k-1},s},y_{u_k,s},z_{u_k}\rangle
\in Q_s^M.\]
\end{enumerate}
The definition makes sense as the $\lambda$'s are from the choice.
\item If $f$ is an correction function for some $M_I$ and a
$M_I$-$(J_0,J_1,J_2)$-choice $\chxyz$, then $f$ we call $f$ a
$(I,J_0,J_1,J_2)$-correction function.
\item $C(I,J_0,J_1,J_2)$ denotes the set of all
$(I,J_0,J_1,J_2)$-correction functions.
\end{enumerate}

The next three claims are general observations on correction functions
and choices.

\begin{claim}\label{zerosforchoices}
\begin{enumerate}
\item If $\chxyz$ is a global $M$-choice, $M\models \psi$, and $f$ is
the $M$-correction function for $\chxyz$, and $f$ is identically zero,
then $M\approx M_I$ for some $I$.
\item If $f$ above is zero on $P_{1,1}^M$, $P_{1,2}^M$ and
$f=f'\restriction J_2\times S$, then $M\approx M_{P_1,f'}$.
\end{enumerate}
\end{claim}

\begin{corollary}
The correction function for $M_{I,f}$ and the canonical $M$-choice
$\chxyz$ is $f$.
%\ojoA{read paper with Vaisanen - like EXT the use of groups...}
\end{corollary}

\Proof Similar to the above: add zeroes to $f$ as
in~\ref{zerosforchoices}. \qed


%Say $N\approx M_{I,f}$ i.e. up to isomorphism $Mod (\psi) = \{ M_{I,f}
%| I,f$ as above$\}


\begin{claim}
For every $M\in \Mod(\psi)$, there is an $M$-choice
$\chxyz$.
\end{claim}

\Proof Immediate: just construct the tuples. There 
demands are on each choice separately. There are no demands connecting
different choices. \qed

The next lemma is a crucial step. It shows how to build if possible
isomorphisms from arbitrary $N$ in the class $K_2$ to canonical models
$M_{I,f}$.

\begin{lemma}\label{labase}
For every $N\in \Mod(\psi)$ and global $N$-choice
$\chxyz$ with correction function $f$ there are $I$ and  $\bf h$ such that
$\bf h$ is an isomorphism from $N$ onto $M_{I,f}$, ($I=P_0^M=P_0^N$
except if unwanted equations hold - e.g. those failing in some $M_I$) and
\[ {\bf h}(x_{u,s}) = ({\bf h}(u),s,0_{\bbz_2}), \quad {\bf h}(y_{u,s}) =
({\bf h}(u),s,0_{H_I}), \quad {\bf h}(z_u) = ({\bf h}(u),0_G) .\]
\end{lemma}



\Proof First, extract the predicates: this provides $I=P^M_0$. Clearly
$P^M_0 = P^N_0$. Then, by the definition of $M_{I,f}$, we get that

\medskip
$\left . \begin{tabular}{l}
$x\in P_{1,1}\cup P_{1,2}$\\
$\ell < k$, $\pi_\ell(x) = x_\ell$
\end{tabular}
\right \} \Rightarrow {\bf h}(x) = ({\bf h}(x_0),\dots ,{\bf
h}(x_{k-1}))$

\medskip
\noindent
and thus the construction of $\bf h$ on $P_2^N$, $P_3^N$, $P_4^N$ and
$P_5^N$ should respect the predicates $P_{2,s}^N$ and $P_{3,s}^N$.
 So we have

\medskip
$\left . \begin{tabular}{l}
$x\in P^N_2$\\
$F_2^N(x) = u\in P_{1,1}^N$\\
$F_4^N(x,h) = x'$
\end{tabular}
\right \} \Rightarrow F_2^N({\bf h}(x)) = {\bf h}(F_2^N(x)) = {\bf
h}(u)$

\noindent
and $F_4^N({\bf h}(x),{\bf h}(h))={\bf h}(x')$.

\medskip
\noindent
So, 
\[ {\bf h}(x) = ({\bf h}(u),s,-).\]
As the definition of ${\bf
h}\restriction P_2^N$
is free in the choice of the `third coordinate' element of $H$
this part will only be tied by the correction function.
Lastly, we need to check that this definition works together fine
with the predicates $Q_s$. Fix $s\in S$. Finally, checking
\[ Q^N_s(a_0\dots a_ku_0\dots u_kx_0\dots x_{k-1}y_kz)\]
\[ \Updownarrow\]
\[ Q^{M_{I,f}}_s({\bf h}(a_0)\dots {\bf h}(a_k){\bf h}(u_0)\dots {\bf
h}(u_k){\bf h}(x_0)\dots {\bf h}(x_{k-1}){\bf h}(y_k){\bf h}(z)).\]

amounts to answering the question

\[ \bbz_2 \models \sum_{\ell<k}i_\ell = h_k(u_0) + g(s) + f(u,s) \]
\[ \Updownarrow ?\]
\[ \bbz_2 \models \sum_{\ell<k}{\bf h}(i_\ell) = h_k({\bf h}(u_0)) +
g({\bf h}(s)) + f({\bf h}(u),{\bf h}(s)) \]

%$a_\ell \in P_0^N$, $u_\ell \in P_{1,s}^N$ does not make sense.
%By decree, let
%\[ Q^N_s(a_0\dots) \Leftrightarrow Q^{M_{I,f}}_s(h(a_0)\dots) \]
%... which {\it does not make sense: $N$ is given a priori so somehow
%we should let...}

%% Of course, it depends on $h$. But since the previous equivalence
%% really means that
%% \[ Q^N_s(\bar{a},\bar{u},\bar{x},y_k,z)\]
%% \[ \Updownarrow \]
%% \[ \bbz_2\models \sum_{\ell<k} h(i_\ell) = h(h_k)(h(u_0)) + h(g)(h(s))
%% + f(h(u),h(s)), \]

\noindent
Choosing 

\medskip

$\qquad \left \{ \begin{tabular}{l}
${\bf h}(x_{u,s}) = ({\bf h}(u),s,0_{\bbz_2})$,\\
${\bf h}(y_{u,s}) =
({\bf h}(u),s,0_H),$\\
${\bf h}(z_u) = ({\bf h}(u),0_G)$
\end{tabular} \right .$

\medskip
\noindent
works for these equations: we are `choosing' $0$ on the third
coordinates -- at the $x_{u,s}$, $y_{u,s}$, $z(u)$
that had already been selected by the choice function.

% Get {\it third coordinates} of $h(x_\ell)$, $h(y_k)$, {\it second
% coordinate} of $h(z)$... $h(s)$ is just $s$, as this is independent of
% the specific model we are taking,
% \[ h(u) = h(F^N_5(z))\]
% the definition suggested by Saharon for
% \[ h(x_{u,s}) = (h(u),s,0_H)\]
% forces us to `choose' $0$ on third coordinates -- at the chosen
% $x_{u,s}$, $y_{u,s}$, $z_u$...

\noindent
So, this turns the equation at the choices into
\[ \bbz_2 \models 0=\sum_{\ell<k}0 = 0(\star) + 0(\star) +f(\star).\]
But, since $f$ was a correction function for our choice,
\[ f(u,s) = 0 \Leftrightarrow \langle
a_0,\dots,a_k,u_0,\dots,u_k,x_{u_0,s},
\dots,x_{u_{k-1},s},y_{u_k,s},z_{u_k}\rangle
\in Q_s^N,\]
the definition of $\bf h$ works.
\qed

%\newpage

\section{Canonical choices}

\begin{definition}
Fix $M=M_{I,f}$, and let $\chxyz$ be the $M$-choice
given by
\[ x_{u,s} = (u,s,0_{\bbz_2}),\]
\[ y_{u,s} = (u,s,0_{H_I}),\]
\[ z_u = (u,0_G).\]
This is by definition the {\bf canonical} $M$-choice\footnote{So the
choices 
act like `variations' on the group structure of $G$, $H$ and
$\bbz_2$.}.
\end{definition}



\medskip
\noindent
Here is the crucial lemma.

\begin{lemma}\label{modset}
If $M_1$ and $M_2$ are strongly standard, and
$\chxyz_\ell$ is an $M_\ell$-choice for $M_\ell$ ($\ell =1,2$),
$P_0^{M_1} = P_0^{M_2}$ with
correction function $f_\ell$ for $\ell = 1,2$ then the following are
equivalent:
\begin{description}
\item[(a)] there is an isomorphism from $M_1$ onto $M_2$ over the
identity on $P_0^{M_1}\cup
P_1^{M_1}$ 
\item[(b)$_1$] there is an $M_2$-choice $\chxyz$
whose correction function is $f_1$,
\item[(b)$_2$] there is an $M_1$-choice $\chxyz$
whose correction function is $f_2$,
%\ojoA{?`si tiene sentido esto? also
%see page $\gamma$ of 4B of Jun 30 2000}
\item[(c)] there are functions $g_0$, $g_1$, $g_2$ (this is to
correct the choice of zeros), with
\begin{enumerate}
%\item $\Dom(g_1) = \{ (u,s) | s\in S, u \in {}^{k+1}I \}$, (like the
%$x$s)
%\item $\Dom(g_2) = \{ (u,s) | s\in S, u \in {}^kI \}$, (like the
%$y$s)
%\item $\Dom(g_3) = {}^{k+1}I$,
%\item $g_1(u,s) \in H_I$,
%\item $g_2(u,s) \in \bbz_2$,
%\item $g_3(u) \in G$, such that
\item $g_1:\ialak \times S \to \bbz_2$ (like the $x_{u,s}$'s above),
\item $g_2:\ialak \times S \to I_H$ (like the $y_{u,s}$'s above),
\item $g_3:\ialakmasuno  \to G$ (like the $z_u$'s above),
\item if $\langle a_0\dots a_ku_0\dots u_kx_0\dots
x_{k-1}y_k,z\rangle$ are like in Definition \ref{defM_I} for $M_1$,
or $M_2$ {\sf then} 
\[ \bbz_2 \models \sum_{\ell<k} i_\ell - h_k(u_0) - g(s) = \sum
_{\ell<k} g_1(u_\ell,s) - g_2(u_k,s)(u_0) - g_3(u)(s) \]
\end{enumerate}
%% \item[(d)] {\tiny {\bf relate the $g$s to the $f$s!:} no need for
%% models, just looking at the $\ialak$ and $\ialakmasuno$ is enough. So,
%% we have abelian groups of the $f$s, subgroups of these generated by
%% $\bar{g}$. The quotient represents the isomorphism type.
%% }
\end{description}
\end{lemma}

\Proof \begin{description}
\item[(a) $\to$ (b)$_1$] Recall that $M_1\restriction \tau^- =
M_2\restriction \tau^-$, so $M_1$ and $M_2$ have the same universes.
Fix $F:M_1 \stackrel{\approx}{\longrightarrow}_{P_0^{M_1}
\cup P_1^{M_1}} M_2$. 
%% We have freedom on how to copy $H^{M_1}_{u,s}$ onto itself, as well as
%% $(\bbz_2)^{M_1}_{u,s}$ (really, of course, the interpretation of $P_2$
%% and $P_3$).
We have, since $f_1$ is a correction function for $M$ for
the choice $\chxyz_1$, that
\[ f_1(u,s) = 0 \Leftrightarrow
\langle a_0\dots a_k u_0\dots u_k x^1_{u_0s} \dots x^1_{u_{k-1}s} y^1_
{u_ks} z^1_{u_k} \rangle \in Q_s^{M_1}.\]
But the right hand side holds iff
\[ \langle a_0\dots a_k u_0\dots u_k F(x^1_{u_0s}) \dots F(x^1_{u_{k-1}s})
F(y^1_{u_ks}) F(z^1_{u_k}) \rangle \in Q_s^{M_2},\]
since $F$ is an isomorphism fixing $P_0^{M_1}\cup P_1^{M_1}$, and
$a_0,\dots ,a_k\in P_0^{M_1}$. This gives 
us the $M_2$-choice for which $f_1$ is a correction function: given
$u_\ell \subset u$, $u_\ell\in \ialak$, $u\in \ialakmasuno$, let
$x_{u_\ell,s}' = F(x_{u_\ell,s}^1)$, $y_{u_,s}' = F(y_{u_k,s}^1)$,
$z_{u_k}' = F(z_{u_k}^1)$.
\item[(a) $\to$ (b)$_2$] Same.
\item[(b)$_\ell$ $\to$ (c)] ($\ell=1,2$) The point of (c) is that we
may find concrete 
representations $g_1,g_2,g_3$, that
act {\it independently from $M$ or $N$}  as `corrected choice functions'
for the zeros for $f_1$ and $f_2$. So, suppose we have 
a $M_2$-choice $\chxyz$ with correction function $f_1$. Then for any
$u \in P_0^{M_2}$ and any $s\in S$, if $\langle a_0\dots a_ku_0\dots
u_kx_0\dots 
x_{k-1}y_k,z\rangle$ are like in Definition \ref{defM_I} 
\[ \langle a_0\dots a_k u_0\dots u_k x_{u_0s} \dots x_{u_{k-1}s} y_
{u_ks} z_{u_k} \rangle \in Q_s^{M_2}\]
\[ \Updownarrow \]
\[ f_1(u,s) = 0.\]
But since $f_1$ is also a correction function for the $M_1$-choice
$\chxyz_1$,
\[ f_1(u,s) = 0\]
\[ \Updownarrow \]
\[ \langle a_0\dots a_k u_0\dots u_k x^1_{u_0s} \dots x^1_{u_{k-1}s}
y^1_{u_ks} z^1_{u_k} \rangle \in Q_s^{M_1}.\]
So, we have both $\bbz \models \sum_{\ell <k}i_\ell = h_k(u_0)+g(s)$
and $\bbz \models \sum_{\ell <k}i^1_\ell = h^1_k(u_0)+g^1(s)$, so
setting

\[ g_1(u_\ell,s) = i^1_\ell,\quad g_2(u_k,s) = h^1_k,\quad g_3(u) =
g^1\]
yields
\[ \bbz_2 \models \sum_{\ell<k} i_\ell - h_k(u_0) - g(s) = \sum
_{\ell<k} g_1(u_\ell,s) - g_2(u_k,s)(u_0) - g_3(u)(s). \]
Since $f_1$ does this for all possible $k+1$-tuples, we have
all the compability we need.
%% The role of the functions $g_0$, $g_1$ and $g_2$ we
%% are looking for here is to correct the equation in Definition
%% \ref{defM_I}. 
%% It all boils down to coding the predicates $Q^2_s$.

\item[(c) $\to$ (a)] If the predicates are the same modulo $g_1$, $g_2$
and $g_3$ then obtaining {\bf (a)} becomes a matter of building $F:M_1
\stackrel{\approx}{\longrightarrow}_{P_0^{M_1}\cup P_1^{M_1}} M_2$. Clearly
we can start by
$F\restriction P_0^{M_1} = id$, and then extend its definition to all the other
portions of the model. The only strong restriction to the extension of this
to the whole model is given by the relations $Q_s^{M_1}$ and $Q_s^{M_2}$.
\end{description} 
\qed

\begin{remark}
\begin{enumerate}
\item We shall use `simple' versions of $\langle g_1,g_2,g_3\rangle$,
usually to prove isomorphism (two of them zero).
\item The number of isomorphism types count has some similarity to
$Ext(G,\bbz)$, in particular to the work of Shelah and V\"ais\"anen in
\cite{ShVs:646}. Here $I(\lambda,\psi)$ is counted by the group of
correction functions, derived from {\it some} $g_1$, $g_2$, $g_3$:
\[ I(\lambda,\psi) = \Big \{ f\in C(I,J_0,J_1,J_2) \Big| f(u,s) =
\sum_{\ell<k}g_1(u_\ell,s) 
-g_2(u_0,s) -g_3(u)\Big \} .\]
\item In the isomorphism proof, we will use the regularity of the
filter: we will put together $\lambda$ demands.
\end{enumerate}
\end{remark}

The next lemma is the first step in the categoricity proof. It
provides conditions for extending partial $M$-choices to global
$M$-choices for combination $(\l,m)$, $m < k$.

\begin{lemma}\label{existxyz}{\bf (Extension property for $W$ of size
$m<k$, $|P_0^M|\leq \l$.)}
%\begin{table}
%\end{table}
\\
Assume $m<k$, $M\models \psi$, $M$ is strongly standard, $|P_0^M|\leq
\l$, 
$W\subset P_0^M$, $W=\{ b_\ell |\ell < m\}$ with no
repetition, $J=\{ u\in P^M_{1,1} |  W\not\subset u\}$
(note that $u\in [P^M_0]^k$, as
$M$ is standard), $\chxyz$ is an $M$-$J$-choice
with correction function $f_0$, identically zero.
{\sf Then,} we can extend $\chxyz$ to an $M$-choice
with correction function identically zero.
\end{lemma}

\Proof
\begin{description}
\item[Part A:]
Without loss of generality, by~\ref{allisostrstrd}, since $M$ is
strongly standard, $I=P^M_0$.
Let $\langle \bar{a}^\a | \a<\b^*\rangle$ list $P_{1,1}^M$ with $\langle
\bar{a}^\a | \a < \a^*\rangle$ listing $J$ (we have also used $u$ for
naming these $\bar{a}^\a$'s).
Let $\langle \bar{b}^\g | \g<\g^*\rangle$ list $\{ \bar{a}\in {}^{k+1}I |
\bar{a}$ with no repetition and $W\subset \ran (\bar{a})\}$ and $\g^*
<\l^+$. Let, for $\a <\a^*$,
\begin{description}
 \item $x_{\bar{a}^\a,s} = (\bar{a}^\a,s,i_{\a,s})\in
 (\bbz_2)_{\bar{a}^\a,s}$, $i_{\a,s}\in \bbz_2$,
\item $y_{\bar{a}^\a,s} = (\bar{a}^\a,s,h_{\a,s})\in
 (H)_{\bar{a}^\a,s}$, $h_{\a,s}\in H_I$,
\item $z_{\bar{b}^\g} = (\bar{b}^\g,g)$, $g \in G$.
\end{description}
Our hypothesis is then that we have choice functions for all $u\in
P_{1,1}^M$ such that
$u\not\supset W$. We use a zero correction function - as we
don't know yet how to take care of $u\supset W$.

We will now choose $x_{\bar{a}^\a,s} = (\bar{a}^\a,s,i_{\a,s})$,
$y_{\bar{a}^\a,s} = (\bar{a}^\a,s,h_{\a,s})$, $z_{\bar{b}^\g} =
(\bar{b}^\g,g)$ for $\a^* \leq \a <\b^*$ and appropriate $\g$.
% $\a\in [\a^*,\b^*]\setminus \{ \b^*\}$.

Without loss of generality, $\b^*\leq \a^*+\l$, $\g^*\leq
\l$. (Remember $S=[\l]^{<\aleph_0}$.)

\item[Part B:]
\begin{description}
\item[First,] we choose $i_{\a,s}=0_{\bbz_2}$ for $\a^* \leq \a < \b^*$,
$s\in S$.
\item[Second,] we are now at the level of consistently `choosing
$h$'. We try to choose $h_{\a,s}$ for $\a^* \leq \a < \b^*$
and $s\in S$ such that
\begin{description}
\item[(*)] if $\g \in s\subset \l$, $\bar{b}^\g = \langle b^\g_\ell |
\ell\leq k \rangle$, 
$u_n^\g = \langle b^\g_\ell | \ell \leq k, \ell\not= n\rangle$, let
$\varepsilon(\gamma,n) < \b^*$ be such that $u^\gamma_n =
\bar{a}^{\epsilon(\g,n)}$ then
\end{description}
\[ h_{\epsilon(\g,k),s}(\bar{a}^{\epsilon(\g,0)}) = 0\]
\[\Updownarrow \]
\[ \langle b_0^\g,\dots ,b_k^\g,u_0^\g,\dots,u_k^\g,x_{\epsilon(\g,0)},
\dots,x_{\epsilon(\g,k-1)},(\bar{a}^{\epsilon(\g,k)},s,0_H),
(u^\g,0_G)\rangle \in Q^M_s .\]

\end{description}
{\bf Note} that all the elements in the bottom part are defined.

\begin{quotation}
{\footnotesize
\noindent
Let $t(\gamma,s)$ be $0$ if the bottom statement is true, $1$
otherwise (so we are using $\bbz_2$ to code). This gives $|s|$
demands, one for each $\gamma \in s$. The sequence $\langle
\bar{a}^{\varepsilon(\gamma,0)} | \gamma \in s\rangle$ is without
repetition (see part {\bf B}). So we have to show that the set of
equations in the variable $h$ varying on $H = [{}^kI]^{<\aleph_0}$,
considered as a set of characteristic functions is
\[ \{ h(\bar{a}^{\varepsilon(\gamma,0)}) = t(\gamma,s) | \gamma \in
s\}.\]
By the definition, it is solvable by the characteristic function of
the subset $\{ \bar{a}^{\varepsilon(\gamma,0)}|\gamma \in s\}$ of
$\ialak$. 
%% So the regularity was used as indexing device to put on all
%% ... finitely many demands.
}
\end{quotation}

The decisions are done for each $s$ separately, also fixing $s$
we can deal with one $\a\in [\a^*,\b^*]\setminus \{ \b^*\}$
e.g. choosing $h_{\a,s}$
we have to consider only $\g<\g^*$ such that $\{ \epsilon(\g,\ell)|
\ell<k\} \subset s$;
%\ojoA{Sah thinks not necessary!}
there are here only finitely many $\g$'s, and if
$\g_1\not= \g_2\in s$ (and $\epsilon(\g_1,k) = \a = \epsilon(\g_2,k)$
necessarily $\epsilon(\g_1,0) \not= \epsilon(\g_2,0)$ (as $\bar{a}^\g$
is reconstructible from $\a$ and $\epsilon(\g_1,0)$), i.e. if equality
holds then $\bar{b}^{\g_1} = \bar{b}^{\g_2}$) and by the choice of
$H$ we can find $h_{\epsilon(\g,k),s}$.

\item[Part C:] We now `glue' the choices, for fixed $\gamma$.
 For each $\bar{b}\in {}^{k+1}I$,
$\bar{b}=\bar{b}^\g$ for some $\g<\g^*$, so
\[ S^*_\g = \Big\{ s\in S \Big| M\models Q_s(b_0^\g,\dots,b_k^\g,
\bar{a}^{\epsilon(\g,0)},\dots,\bar{a}^{\epsilon(\g,k)},
x_{u^\g _0,s},\dots,x_{u^\g _{k-1},s},
y_{u^\g _k,s},(u^\g,0_S))\Big\} \]
belongs to $\frak D$ (by the regularity of $\frak D$).

Next choose $z_{\bar{b}_1} = (\bar{b},g)$ by

\medskip
\qquad $g(s) = \left \{ \begin{tabular}{ll}
$0$	&	if $s\in S^*_\g$\\
$1$	&	if $s\not\in S^*_\g$
\end{tabular} \right . $

\medskip
Now then, with these $x$, $y$ and $z$, the equation holds.
\end{description}
\qed$_{\ref{existxyz}}$



We now prove the general extension property.

\begin{lemma}\label{fullext2002}{\bf (Full extension)}\\
Let $M\models \psi$ be strongly canonical,
$J_1\subset J_2\subset P_0^M$, with $|J_2| <
\l^{+k-1}$ and $\chxyz$ an $M$-$J_1$-choice with correction function
identically zero. Then $\chxyz$ can be extended to an $M$-$J_2$-choice
with correction function identically zero.
\end{lemma}

\Proof Without loss of generality, $J_2 = J_1 \cup \{ b\}$. If $J_1$
has size $\leq \l$,
%\ojoA{check cardinals!}
this
is lemma \ref{existxyz}. Now suppose $|J_1| = \l^{+m_1}$ (for
$m_1<k$),
so enumerate $J_1$ 
as $\langle a_\b | \b<\l^{+m_1}\rangle$. Let $J_1^\a = \{
a_\b | \b<\a\}$, and let $\chxyz_\a$ be the restriction
of $\chxyz$ to an $M$-$J_1^\a$-choice. We define by induction
$M$-$J^\a_1$ choices with correction function identically zero
$\chxyz_\a' \supset \chxyz_\a$.
Use lemma \ref{gencase} for $m_2 = 2$
to extend $\chxyz_\a' \cup \chxyz_{\a+1}$ to an $M$-$J_1^{\a+1}\cup\{
b\}$-choice with correction function identically zero. At limits take
unions; finally,
\[ \Big(
\bigcup_{\a<\l^{+m_1}}\bar{x}_\a',\bigcup_{\a<\l^{+m_1}}\bar{y}_\a',
\bigcup_{\a<\l^{+m_1}}\bar{z}_\a'\Big) \]
turns out to be an $M$-$J_2$-solution extending $\chxyz$.
\qed


\begin{quotation}
{\footnotesize

\begin{claim}
In \ref{existxyz}, we can allow $|P^M_0|\leq \l^{+m_1}$ if $W = \{ b_\ell |
\ell<m_2\}$ and $m_1+m_2<k$.
\end{claim} 


\Proof We prove this by induction on $m_1$. The proof is quite parallel
to some of the proofs in \cite{Sh:87a} and to \cite{HaSh:323}. This part
has few changes. We include versions of those proofs adapted to our
context.

For $m_1=0$, this was done in \ref{existxyz}.
Suppose it holds for $m_1$ ($<k$), and
$m_2$ is such that $m_1+m_2 < k$. Consider $W\subset P_0^M$,
\[ W = \{ b_\ell | \ell < m_2\},\]
$(m_1+1)+m_2 <k$, $J = \{ u\in P_1^M | W\not\subset u\}$, and let
$\chxyz$ be an $M$-$J$-choice with correction function identically 0.

Suppose, for $M\models \psi$ strongly standard, $A_\emptyset \subset
P_0^M$, $a_0, \dots,a_{m_2-1}$ different elements of $P_0^M\setminus
A_\emptyset$,

\begin{definition}
$\langle A_s,\chxyz_s | s\in {\cal P}^-(m_2) \rangle$ is a compatible
$\l^{+m_1}$-${\cal P}^-(m_2)$-system of choices iff
\begin{enumerate}
\item $\bigcup_{s\in {\cal P}^-(m_2)}A_s = A_\emptyset \cup \{
a_0,\dots,a_{m_2-1} \}$, $|A_\emptyset| \leq \l^{+m_1}$, $A_s
= A_\emptyset \cup \{ a_t | t\in s\}$.
\item $\chxyz_s$ is a $M$-$A_s$-choice, $\forall s\in {\cal P}^-(m_2)$.
\item For every $s,t\in {\cal P}^-(m_2)$, $s\subset t \Rightarrow \chxyz_s
\subset \chxyz_t$\footnote{here, of course, we are abusing notation - by
$\chxyz_s \subset \chxyz_t$ we mean $\bar{x}_s\subset \bar{x}_t$,
$\bar{y}_s\subset \bar{y}_t$ and $\bar{z}_s\subset \bar{z}_t$.}.
\end{enumerate}
\end{definition}

\begin{lemma}\label{0case}
If $\langle A_s,\chxyz_s | s\in {\cal P}^-(m_2)\rangle$ is a
compatible $\l$-${\cal P}^-(m_2)$-system with $m_2<k$ then there is an
$M$-$\bigcup_{s\in {\cal P}^-(m_2)} A_s$-choice $\chxyz$ extending all
the $\chxyz_s$, for $s\in \pmin(m_2)$.
\end{lemma}

\Proof If $\langle A_s,\chxyz_s | s\in {\cal P}^-(m_2)\rangle$ is a
compatible $\l$-${\cal P}^-(m_2)$-system with $m_2<k$, if
\[ u\in [\bigcup_{s\in {\cal P}^-(m_2)} A_s]^k \setminus
\bigcup_{s\in {\cal P}^-(m_2)}[A_s]^k,\]
then $\{ a_0\dots a_{m_2-1}\} \subset u$. As $m_2<k$, there must be
some $b\in u\setminus \{ a_0\dots a_{m_2-1}\}\subset u$. Now, if $c\in
\bigcup_{s\in {\cal P}^-(m_2)}A_s\setminus u$ then

\[ (u\setminus \{ b\}) \cup \{ c\} \notin \bigcup_{s\in {\cal
P}^-(m_2)} [A_s]^k\]
hence if $u\subset v$ where $v \in [\bigcup_{s\in {\cal P}^-(m_2)}
A_s]^{k+1}$ then
there must exist $u'\subset v$, $|u'|=k$, $u\not= u'$ such that
$u'\notin
\bigcup_{s\in {\cal P}^-(m_2)}[A_s]^k$.
\qed$_{\ref{0case}}$

\begin{lemma}\label{gencase}
If $\langle A_s,\chxyz_s | s\in \pmin(m_2)\rangle$ is a compatible
$\l^{+m_1}$-$\pmin(m_2)$-system of choices with $m_1+m_2<k$ then there
is a $\bigcup_{s\in \pmin(m_2)}A_s$-choice $\chxyz$ such that
$\chxyz_s \subset \chxyz$ for every $s\in \pmin(m_2)$.
\end{lemma}

\Proof Induct on $m_1$. For $m_1=0$, this is lemma \ref{0case}. For
$m_1 >0$, suppose $A_s = A_\emptyset \cup \{ b_t|t\in s\}$. Enumerate
$A_\emptyset$, $\langle a_\b | \b <\l^{+m_1}\rangle$ and let
$A_\emptyset^\a = \{ a_\b |\b<\a\}$. Now let $A_s^\a = A_\emptyset^\a
\cup \{ b_t|t\in s\}$ for every $s\in \pmin(m_2)$ and $\chxyz_s^\a$
the restriction of $\chxyz_s$ to an $M$-$A^\a_s$-choice. The point is
to get $\chxyz_\a$ (increasing with $\a$ for $\a < \l^{+m_1}$ such
that $\chxyz_\a$ is an $M$-$\bigcup_{s\in \pmin(m_2)}A_s^\a$-choice,
with $\chxyz_\a\supset \chxyz^\a_s$ (the obvious meaning, again) for
every $s\in \pmin(m_2)$. With this, taking $\chxyz = \bigcup_{\a<
\l^{+m_1}}\chxyz_\a$ we are done. [Why is the construction possible?
At limits, just take unions. At successors, use the induction
hypothesis.]
\qed$_{\ref{gencase}}$


}
\end{quotation}

\begin{theorem}
If $M\models \psi$ is strongly canonical
and $|M|<\l^{+k}$ then there is an $M$-choice with
correction function identically zero.
\end{theorem}

\Proof Use the previous lemmas and induction .





\begin{conclusion}\label{psicateginlow}{\bf (Categoricity and
amalgamation up to $\lambda^{+(k-1)}$)}\\
\begin{enumerate}
\item For $m<k$, $\Mod(\psi)$ has a unique model $M$, $|P_0^M| =
\l^{+m}$.
\item For $m<k-2$, if $2^\l \leq \l^{+m}$, then $\Mod(\psi)$ has
amalgamation in $\l^{+m}$.
\item If $m<k$, $\l^{+m} > 2^\l$, then $\psi$ is categorical in
$\l^{+m}$.
\end{enumerate}
\end{conclusion}

\Proof This is just a summary of the previous arguments. Notice,
however, that we require $2^\l < \l^{+m}$ or $2^\l \leq \l^{+m}$ for
our conclusions. This is due to the fact that our models are large:
they contain copies of $G$, so they have size at least $2^\l$.
% we have here in some way a dimension/cardinal dichotomy
\qed

%% \begin{theorem}\label{theend}
%% Also the failure of categoricity and amalgamation has NO NEW
%% POINT. So, we have fulfilled the positive half of our
%% aims. Generalizing \cite{HaSh:323} at the price of having $\psi \in
%% L_{(2^\l)^+,\w}$ (or $K$ is abstract elementary class, with
%% $LS(K)=2^\l$. See later a proof.
%% \end{theorem}



\section{$\psi$ is not categorical above $\beth_{k+1}(\lambda)^+$}


We proved in \ref{psicateginlow} that $\psi$ is categorical in
$\l^{+m}$  if $m<k$ and $2^\l <\l^{+m}$. We now prove it is not
totally categorical, and we will actually get the 
maximal number of models.
%\ojo{need to complete the many models part}.

As we will see, the control of isomorphism here is relatively easier
than lifting isomorphism in the previous section.

The following fact is a consequence of \ref{modset}. We will rely on a
variant of it.

%\begin{fact}
%\begin{enumerate}
%\item Every $M\models \psi$ is isomorphic to some $M_{I,f}$\ojo{not really}.
%\item 
%If $M_{I_\ell,c_\ell}\models \psi$ for $\ell=1,2$, $h:I_1\to
%I_2$ is one to one onto, {\sf then} there is an isomorphism
%$h^+:M_{I_1,c_1}\to M_{I_2,c_2}$, onto, extending $h$ iff ... {\bf ?}
%{\bf ill phrased}
%\end{enumerate}
%\end{fact}

\begin{fact}\label{modset2}
If $M_{1,f_1}$ and $M_{2,f_2}$ are models of $\psi$, and $h:I_1\to
I_2$ is one-to-one and onto, {\sf then} there is an isomorphism
$h^+:M_{1,f_1}\to M_{2,f_2}$ extending $h$ iff 
there are functions $g_0$, $g_1$, $g_2$ (this is to
correct the choice of zeros), with
\begin{enumerate}
\item $g_1:\ialak \times S \to \bbz_2$ (like the $x_{u,s}$'s above),
\item $g_2:\ialak \times S \to I_H$ (like the $y_{u,s}$'s above),
\item $g_3:\ialakmasuno  \to G$ (like the $z_u$'s above),
\item if $\langle a_0\dots a_ku_0\dots u_kx_0\dots
x_{k-1}y_k,z\rangle$ are like in Definition \ref{defM_I} for $M_1$,
or $M_2$ {\sf then} 
\[ \bbz_2 \models \sum_{\ell<k} i_\ell - h_k(u_0) - g(s) = \sum
_{\ell<k} g_1(u_\ell,s) - g_2(u_k,s)(u_0) - g_3(u)(s) \]
\end{enumerate}
\end{fact}


%% \medskip
%% {\tiny
%% Need of course in \ref{modset} explain clearly the translation from
%% model theory to combinatorics.
%% }

\begin{claim}\label{7.2}
Let $f:[I]^{k+1}\times S\to \bbz_2$ be such
that 
\[ \bigotimes \qquad \qquad 
 \{ s\in S | f_u \not= 0\} \in {\frak D}, \forall u\in [I]^{k+1},\]
for $f_u: S\to \bbz_2$ with $f_u(s) = f(u,s)$.\\
%for $u\in \ialakmasuno$ are not zero, for $\frak D$-many elements $s$
%of $S$.\\
{\sf Then,} the following is a sufficient condition for
\[M_{I,f}\not\approx M_I:\]

% To take advantage of the combinatorial description of the
% isomorphism...

\begin{description}
\item[(*)] if $F_1:\ialak \to [I]^{\leq \l}$, $F_2: \ialak \to
{}^S(\bbz_2)$ $\pi$ a permutation of $I$, {\sf then} we can find
$t_0,\dots,t_k\in I$ with no
repetitions such that
\begin{description}
\item[($\alpha$)] $t_\ell \not\in F_1[\{ t_0\dots t_k\} \setminus \{
t_\ell\}]$ if $\ell = k$,
\item[($\beta$)] $f_{\pi\{ t_0,\dots,t_k\}} - \sum_{\ell<k}
F_2(\{ t_0,\dots,t_k\}\setminus \{ t_\ell\}) \not\in G$\\
%%(Recall $u\in \ialakmasuno \to f_u\in {}^S(\bbz_2)$, $G\subset
%%{}^S(\bbz_2)$.) 
\begin{quotation}
{\footnotesize
\noindent
So, by the definition of $G$, 
%for ${\frak D}$-few elements of $S$,
\[ \{ s\in S | f_{\pi(t_0,\dots,t_k)}(s) = \sum_{\ell<k}F_2(\{ t_0,\dots,t_k\}
\setminus \{ t_\ell\} )(s)\} \notin {\frak D}.\]
}
\end{quotation}
\end{description}
\end{description}
\end{claim}

Before proving~\ref{7.2}, we note some facts.

\begin{definition}
$f:[I]^{k+1}\times S\to \bbz_2$ is an $I$-function iff it satisfies
$\bigotimes$ above.
\end{definition}

\begin{fact}
If $f_1$, $f_2$ are $I$-functions, and $f= f_1-f_2$ (coordinatewise)
satisfies (*), then $M_{I,f_1}\not\approx M_{I,f_2}$.
\end{fact}

\Proof By \ref{modset}, we can apply the criterion (*) to the model.
\qed


\begin{fact}
In \ref{7.2}, the assumption on $f$ does not entail a loss of
generality, as for every $f$ there is a $f'$ as above such that
$M_{I,f}\approx M_{I,f'}$.
\end{fact}

%The isomorphism is not just by choosing $\chxyz$ (can improve
%notation) but {\bf mainly} also a permutation $\pi$ of $I$.

Notice the role of the permutation $\pi$ of $I$ in the combinatorics
that follows.

\Proof of \ref{7.2}. Assume that $\chxyz$ witnesses $M_{I,f} \approx
M_I$, with correction function identically zero. We show that $(*)$
of~\ref{7.2} cannot hold, for the following choice of $F_1$ and $F_2$.

Define $F_1: [I]^k \to [I]^{\leq \l}$ by
\[ F_1(u) = \bigcup \{ v\in \ialak | \mbox{ for some } s_1\in S,
y_{u,s_1}(v)\not= 0\} .\]
This is well defined, as $F_1(u)$ is a union of $|S|$ finite sets.
Also, set
\[ F_2(u) = \langle x_{u,s} | s\in S\rangle.\]
Let now $t_0,\dots,t_k\in I$ (with no repetitions) satisfy
$(\a)+(\b)$. Let as usual $u=\{ t_0,\dots,t_k\}$, $u_\ell = u\setminus
\{ t_\ell\}$. By $(\a)$, 
\[ y_{u_k,s} (u_0) = 0.\]
{\footnotesize [Just notice that $(\a)$ asks that $t_k\notin F_1(u_k)
= \bigcup \{ v\in \ialak | \mbox{ for some } s_1\in S,
y_{u,s_1}(v)\not= 0\}$, so for all $v\in [I]^k$, if $t_k\in v$, then
for all $s_1\in S$ we have $y_{u_k,s_1}(v)=0$. In particular, as
$t_k\in u_0$, $y_{u_k,s_1}(u_0)=0$.]}

So, by the choice $\chxyz$ and since we chose our correction function
to be identically zero, for each $z$, we have that
\[ \bbz_2\models x_{u_0,s} +  x_{u_1,s} + \dots  x_{u_{k-1},s} -
y_{u_k,s} -  z_u =  f_{\pi(u_0),s} + \dots + f_{\pi(u_{k-1}),s} -
f_{u_k,s}(u_0) - f_u(s) .\]

But we also have that $f_u(s)$ is not zero (initial assumption) and
$z_u(s) = 0$ for the $\gd$-majority of $s\in S$ (by the definition of
$G$). Also, $y_{u_k,s}(u_0) = 0$, by the choice
of the $t$'s (clause $(\a)$). So,
\begin{description}
\item[(*)] For the $\gd$-majority of $s\in S$
\[ \sum_{\ell<k} x_{u_\ell,s} = f_{\pi(u)}(s) .\]
\end{description}

%% So\ojoA{(or define)}, recall for $v\in \ialak$, $f^*_v = \langle
%% x_{u_\ell,s}|s\in S\rangle$, $f^*_u\in {}^S(\bbz_2)$.

%% So, we have $\sum_{\ell<k}f^*_{u_\ell} = f_u \mod G\subset
%% {}^S(\bbz_2)$. Let $F_2:\ialak \to {}^S(\bbz_2)$ be given by $F_2(v) =
%% f^*_v = \langle x_{v_\ell} | s\in S\}$. 
But this contradicts $(\b)$. \qed

\begin{remark} 
We can then regard $F_2$ as
\[F_2:\ialak \to {}^S(\bbz_2)/G .\]
\end{remark}

\begin{conclusion}\label{psinotcatup}
For $\mu = \beth_{k+1}(\l)^+$, $\psi$ is not categorical.
\end{conclusion}

This is not optimal ($\mu$ is large) but is enough for our main
aim. In a possible continuation, we will address this issue.

\Proof We take advantage of the combinatorial reduction
from~\ref{7.2}.

\noindent
{\sc Stage A:} First, let $k$ be even. There is $f$ an $I$-function
(as in \ref{7.2}). Now, assume that $F_1$, $F_2$ are as in \ref{7.2},
and derive a contradiction.
first find $E\subset \mu$ club such that
\[\a_0<\dots <\a_k\in E \Longrightarrow 
\left \{ \begin{tabular}{l}
$F_1(\a_0,\dots ,\a_{k-1})\subset \a_k$,\\
$\pi(\a_0),\dots,\pi(\a_{k-1}) < \a_k$.
\end{tabular}
\right . \]
% can be done the usual way, by hand
Apply then Erd\"os-Rado to $F_2$ in order to get $\a_0<\dots <\a_k$ in
$E$ with $u=\{ \a_0,\dots,\a_k\}$, $u_\ell = u\setminus \{ \a_\ell\}$,
$\langle F_2(u_\ell) | \ell<k\rangle$ constant.

But then
\[ \sum_{\ell<k}F_2(u_\ell) = 0 ,\]
as our group is of order 2 (and $k$ was chosen to be an even number).

\noindent
{\sc Stage B:} More generally, choose $\mu$ such that
\begin{description}
\item[$\bigotimes_1$] $\mu \to (\w)^k_{2^\l}$,
\item[$\bigotimes_2$] $\mu \not\to (\w)^{k+1}_{2^\l}$,
\item[$\bigotimes_3$] $\mu$ regular.
\end{description}
{\sf Then,} use $f$ as in \ref{7.2} exemplifying
$\bigotimes_2$. Looking at $f$ as $(u\mapsto f_u/G \in
({}^S(\bbz_2)/G)$, toward contradiction, if
$F_1$, $F_2$ are as in \ref{7.2}, we let $E\subset \mu$ club as above,
$\a_0,\dots,\a_n,\dots$ exemplifying $\mu \to
(\w)^k_{2^\l}$ for the coloring $F_2$. So $F_2\restriction
[\{ \a_0,\dots,\a_n,\dots \}]^k$ is constant, say for increasing
$k$-tuples from $E$, hence also the coloring by $c$ by the argument as
above, contradicting its choice.

\noindent
{\sc Stage C:} For larger cardinals this obviously works, as the
criterion is monotonic.
\qed

\begin{remark}
Here are some of the main differences between the structure of this
proof and that of \cite{HaSh:323}:
\begin{enumerate}
\item The use of the filter $\gd$ - it is not needed there.
\item The way the group itself is used is slightly different at the
end of the proof.
\end{enumerate}
\end{remark}

\begin{remark}
We get even many models (maximal number) but later...
\end{remark}

%% \begin{claim}
%% $\psi \vdash $ a complete $L_{(\l^{\al_0})^+,\w}$-theory\ojoA{timtsa
%% makom nakhon le ze}.
%% \end{claim}

\Proof By categoricity, or directly back and forth.

%\bibliographystyle{amsalpha}
%\bibliography{listb,listx}
\def\germ{\frak} \def\scr{\cal} \ifx\documentclass\undefinedcs
  \def\bf{\fam\bffam\tenbf}\def\rm{\fam0\tenrm}\fi % f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{mathfrak}{\frak}
  \defaultdefine{mathbb}{\bf} \defaultdefine{mathcal}{\cal}
  \defaultdefine{beth}{BETH}\defaultdefine{cal}{\bf} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt>
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{She01b}

\bibitem[GH89]{GrHa89}
Rami Grossberg and Bradd Hart, \emph{{The classification of excellent
  classes}}, Journal of Symbolic Logic \textbf{54} (1989), 1359--1381.

\bibitem[Gro02]{Gr02}
Rami Grossberg, \emph{{Classification theory for abstract elementary classes}},
  {Proceedings of Bilgi Conference, Istanbul, Turkey, 2001}, Perspectives in
  Mathematical Logic, Springer-Verlag, 2002.

\bibitem[HS90]{HaSh:323}
Bradd Hart and Saharon Shelah, \emph{{Categoricity over $P$ for first order $T$
  or categoricity for $\phi\in{\rm L}_ {\omega_ 1\omega}$ can stop at $\aleph_
  k$ while holding for $\aleph_ 0,\cdots,\aleph_ {k-1}$}}, {Israel Journal of
  Mathematics} \textbf{70} (1990), 219--235, math.LO/9201240.

\bibitem[Kei70]{Ke70}
Jerome~H. Keisler, \emph{{Logic with the quantifier "there exist uncountably
  many"}}, Annals of Mathematical Logic \textbf{1} (1970), 1--93.

\bibitem[KS96]{KlSh:362}
Oren Kolman and Saharon Shelah, \emph{{Categoricity of Theories in
  $L_{\kappa,\omega}$, when $\kappa$ is a measurable cardinal. Part 1}},
  {Fundamenta Mathematicae} \textbf{151} (1996), 209--240, math.LO/9602216.

\bibitem[Mar72]{Mar}
Leo Marcus, \emph{{A minimal prime model with an infinite set of
  indiscernibles}}, Israel Journal of Mathematics \textbf{11} (1972), 180--183.

\bibitem[MS90]{MaSh:285}
Michael Makkai and Saharon Shelah, \emph{{Categoricity of theories in $L_
  {\kappa\omega},$ with $\kappa$ a compact cardinal}}, {Annals of Pure and
  Applied Logic} \textbf{47} (1990), 41--97.

\bibitem[Shea]{Sh:600}
Saharon Shelah, \emph{{Categoricity in abstract elementary classes: going up
  inductive step}}, {Preprint}, math.LO/0011215.

\bibitem[Sheb]{Sh:705}
\bysame, \emph{{Toward classification theory of good $\lambda$ frames and
  abstract elementary classes}}.

\bibitem[She83a]{Sh:87a}
\bysame, \emph{{Classification theory for nonelementary classes, I. The number
  of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part A}},
  {Israel Journal of Mathematics} \textbf{46} (1983), 212--240.

\bibitem[She83b]{Sh:87b}
\bysame, \emph{{Classification theory for nonelementary classes, I. The number
  of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part B}},
  {Israel Journal of Mathematics} \textbf{46} (1983), 241--273.

\bibitem[She87]{Sh:88}
\bysame, \emph{{Classification of nonelementary classes. II. Abstract
  elementary classes}}, {Classification theory (Chicago, IL, 1985)}, {Lecture
  Notes in Mathematics}, vol. 1292, {Springer, Berlin}, 1987, {Proceedings of
  the USA--Israel Conference on Classification Theory, Chicago, December 1985;
  ed. Baldwin, J.T.}, pp.~419--497.

\bibitem[She99]{Sh:394}
\bysame, \emph{{Categoricity for abstract classes with amalgamation}}, {Annals
  of Pure and Applied Logic} \textbf{98} (1999), 261--294, math.LO/9809197.

\bibitem[She01a]{Sh:576}
\bysame, \emph{{Categoricity of an abstract elementary class in two successive
  cardinals}}, {Israel Journal of Mathematics} \textbf{126} (2001), 29--128,
  math.LO/9805146.

\bibitem[She01b]{Sh:472}
\bysame, \emph{{Categoricity of Theories in $L_{\kappa^* \omega}$, when
  $\kappa^*$ is a measurable cardinal. Part II}}, {Fundamenta Mathematicae}
  \textbf{170} (2001), 165--196, math.LO/9604241.

\bibitem[SV99]{ShVi:635}
Saharon Shelah and Andres Villaveces, \emph{{Toward Categoricity for Classes
  with no Maximal Models}}, {Annals of Pure and Applied Logic} \textbf{97}
  (1999), 1--25, math.LO/9707227.

\bibitem[SV00]{ShVs:646}
Saharon Shelah and Pauli Vaisanen, \emph{{On the number of
  $L_{\infty,\omega_1}$-equivalent non-isomorphic models}}, {Transactions of
  the American Mathematical Society} \textbf{353} (2000), 1781--1817,
  math.LO/9908160.

\bibitem[Zil01]{Zilber}
B.I. Zilber, \emph{Analytic and pseudo-analytic structures}, preprint, 2001.

\end{thebibliography}



\shlhetal
\end{document}

math0404258.extracted_bib
\begin{thebibliography}{She01b}

\bibitem[GH89]{GrHa89}
Rami Grossberg and Bradd Hart, \emph{{The classification of excellent
  classes}}, Journal of Symbolic Logic \textbf{54} (1989), 1359--1381.

\bibitem[Gro02]{Gr02}
Rami Grossberg, \emph{{Classification theory for abstract elementary classes}},
  {Proceedings of Bilgi Conference, Istanbul, Turkey, 2001}, Perspectives in
  Mathematical Logic, Springer-Verlag, 2002.

\bibitem[HS90]{HaSh:323}
Bradd Hart and Saharon Shelah, \emph{{Categoricity over $P$ for first order $T$
  or categoricity for $\phi\in{\rm L}_ {\omega_ 1\omega}$ can stop at $\aleph_
  k$ while holding for $\aleph_ 0,\cdots,\aleph_ {k-1}$}}, {Israel Journal of
  Mathematics} \textbf{70} (1990), 219--235, math.LO/9201240.

\bibitem[Kei70]{Ke70}
Jerome~H. Keisler, \emph{{Logic with the quantifier "there exist uncountably
  many"}}, Annals of Mathematical Logic \textbf{1} (1970), 1--93.

\bibitem[KS96]{KlSh:362}
Oren Kolman and Saharon Shelah, \emph{{Categoricity of Theories in
  $L_{\kappa,\omega}$, when $\kappa$ is a measurable cardinal. Part 1}},
  {Fundamenta Mathematicae} \textbf{151} (1996), 209--240, math.LO/9602216.

\bibitem[Mar72]{Mar}
Leo Marcus, \emph{{A minimal prime model with an infinite set of
  indiscernibles}}, Israel Journal of Mathematics \textbf{11} (1972), 180--183.

\bibitem[MS90]{MaSh:285}
Michael Makkai and Saharon Shelah, \emph{{Categoricity of theories in $L_
  {\kappa\omega},$ with $\kappa$ a compact cardinal}}, {Annals of Pure and
  Applied Logic} \textbf{47} (1990), 41--97.

\bibitem[Shea]{Sh:600}
Saharon Shelah, \emph{{Categoricity in abstract elementary classes: going up
  inductive step}}, {Preprint}, math.LO/0011215.

\bibitem[Sheb]{Sh:705}
\bysame, \emph{{Toward classification theory of good $\lambda$ frames and
  abstract elementary classes}}.

\bibitem[She83a]{Sh:87a}
\bysame, \emph{{Classification theory for nonelementary classes, I. The number
  of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part A}},
  {Israel Journal of Mathematics} \textbf{46} (1983), 212--240.

\bibitem[She83b]{Sh:87b}
\bysame, \emph{{Classification theory for nonelementary classes, I. The number
  of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part B}},
  {Israel Journal of Mathematics} \textbf{46} (1983), 241--273.

\bibitem[She87]{Sh:88}
\bysame, \emph{{Classification of nonelementary classes. II. Abstract
  elementary classes}}, {Classification theory (Chicago, IL, 1985)}, {Lecture
  Notes in Mathematics}, vol. 1292, {Springer, Berlin}, 1987, {Proceedings of
  the USA--Israel Conference on Classification Theory, Chicago, December 1985;
  ed. Baldwin, J.T.}, pp.~419--497.

\bibitem[She99]{Sh:394}
\bysame, \emph{{Categoricity for abstract classes with amalgamation}}, {Annals
  of Pure and Applied Logic} \textbf{98} (1999), 261--294, math.LO/9809197.

\bibitem[She01a]{Sh:576}
\bysame, \emph{{Categoricity of an abstract elementary class in two successive
  cardinals}}, {Israel Journal of Mathematics} \textbf{126} (2001), 29--128,
  math.LO/9805146.

\bibitem[She01b]{Sh:472}
\bysame, \emph{{Categoricity of Theories in $L_{\kappa^* \omega}$, when
  $\kappa^*$ is a measurable cardinal. Part II}}, {Fundamenta Mathematicae}
  \textbf{170} (2001), 165--196, math.LO/9604241.

\bibitem[SV99]{ShVi:635}
Saharon Shelah and Andres Villaveces, \emph{{Toward Categoricity for Classes
  with no Maximal Models}}, {Annals of Pure and Applied Logic} \textbf{97}
  (1999), 1--25, math.LO/9707227.

\bibitem[SV00]{ShVs:646}
Saharon Shelah and Pauli Vaisanen, \emph{{On the number of
  $L_{\infty,\omega_1}$-equivalent non-isomorphic models}}, {Transactions of
  the American Mathematical Society} \textbf{353} (2000), 1781--1817,
  math.LO/9908160.

\bibitem[Zil01]{Zilber}
B.I. Zilber, \emph{Analytic and pseudo-analytic structures}, preprint, 2001.

\end{thebibliography}