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\title{A global version of a theorem of Ben-David and Magidor
\thanks{1991 Mathematics Subject Classifications:
primary 03E35; secondary 03E05, 03E55}
\thanks{Keywords: Square, weak square,
stationary reflection, Radin forcing,
large cardinals}}
\date{July 27, 1999}
\author{Arthur W.~Apter
\thanks{Supported by the Volkswagen-Stiftung
(RiP-program at Oberwolfach). In addition, this research was partially supported
by PSC-CUNY Grants 665337 and 667379.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010
USA\\
awabb@cunyvm.cuny.edu\\
http://math.baruch.cuny.edu/$\sim$apter\\
\\
James Cummings
\thanks{Supported by the Volkswagen-Stiftung
(RiP-program at Oberwolfach). In addition, this research was partially supported
by NSF Grant DMS-9703945.}\\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
Pittsburgh PA 15213
USA\\
jcumming@andrew.cmu.edu\\
http://www.math.cmu.edu/users/jcumming}
\begin{document}
\maketitle
\begin{abstract}
We prove a consistency result about square
principles and stationary reflection which
generalises the result of Ben-David and
Magidor \cite{BDM}.
\end{abstract}
\baselineskip=24pt
\section{Introduction}
In this paper we prove a consistency result about square principles
and stationary reflection, which is a generalisation of a theorem
from Ben-David and Magidor's paper \cite{BDM}. We begin by giving some
pertinent facts and definitions.
\begin{definition} Let $\gk$ be an uncountable regular cardinal,
let $S \subseteq \gk$ be stationary, and let $\ga < \gk$. Then
$S$ {\em reflects at $\ga$\/} iff $\cf(\ga) > \go$ and $S \cap \ga$
is stationary in $\ga$. $S$ {\em reflects\/} iff there exists $\ga <\gk$
such that $S$ reflects at $\ga$. $S$ is {\em non-reflecting} iff $S$
does not reflect. A sequence $\vec S = \seq{S_i: i < \gb}$ of stationary
subsets of $\gk$ {\em reflects simultaneously\/} iff there exists
$\ga < \gk$ such that $S_i$ reflects at $\ga$ for every $i < \gb$;
$\vec S$ {\em reflects simultaneously to cofinality $\gm$\/} iff there exists
$\ga$ such that $\cf(\ga) = \gm$ and $S_i \cap \ga$ is stationary for all
$i < \gb$.
\end{definition}
Large cardinals can be used to get instances of simultaneous reflection. In
particular we will use the following fact (due to Solovay). Since we could not
find a reference, we sketch the proof.
\begin{fact}
\label{solovayfact}
If $\gk$ is $\gk^+$ supercompact and $\vec S$ is a sequence of
stationary subsets of $\setof{\ga < \gk^+}{\cf(\ga) < \gk}$ such that
$\lh(\vec S) < \gk$, then
there are unboundedly many $\gm < \gk$ such that
\begin{enumerate}
\item $\gm$ is the successor of an inaccessible.
\item $\vec S$ reflects simultaneously to cofinality $\gm$.
\end{enumerate}
\end{fact}
\begin{proof} Let $\vec S = \seq{S_\ga: \ga < \gn}$ where $\gn < \gk$ and
each $S_\ga$ is a stationary subset of $\setof{\ga < \gk^+}{\cf(\ga) < \gk}$.
Let $\gr < \gk$. Let $j: V \lra M$ be an elementary embedding such that
$\crit(j) = \gk$, $j(\gk) > \gk^+$ and ${}^{\gk^+} M \subseteq M$.
Let $\gl = \sup(j`` \gk^+)$, where it follows from the closure of $M$ that
$\cf_M(\gl) = \gk^+ = \gk^+_M$ and $\gl < j(\gk^+)$.
$j(\vec S) = \seq{j(S_\ga) : \ga < \gn}$ and $j(\gr) = \gr < \gk$,
and also $\gk$ is inaccessible in $M$. It will therefore suffice to show
that
\[
M \models \hbox{``$j(S_\ga) \cap \gl$ is stationary for every $\ga < \gn$''}
\]
and then appeal to the elementarity of $j$ to see that
\[
V \models \hbox{``there is an inaccessible $\gz > \gr$ such that $\vec S$ reflects to cofinality $\gz^+$''}
\]
Clearly $j`` S_\ga \subseteq j(S_\ga) \cap \gl$. If $D \subseteq \gl$ is club then
it is easy to see that $j^{-1}`` D$ is $<\gk$-club in $\gk^+$, so that
there is some $\gb \in S_\ga$ with $j(\gb) \in D$. Thus
$j(S_\ga) \cap \gl$ is stationary (even in $V$!) and we are done.
\end{proof}
\begin{definition} Let $\gk$ be an infinite cardinal, and let $\gl \le\gk$
be a cardinal. Then a {\em $\square_{\gk, \gl}$-sequence\/} is a
sequence $\seq{ {\cal C}_\ga : \lim(\ga), \ga < \gk^+}$ such that
\begin{enumerate}
\item \label{size} $1 \le \card{ {\cal C}_\ga } \le \gl$.
\item For every $C \in {\cal C}_\ga$
\begin{enumerate}
\item $C$ is club in $\ga$.
\item $\ot{C} \le \gk$.
\item For all $\gb \in \lim(C)$, $C \cap \gb \in {\cal C}_\gb$.
\end{enumerate}
\end{enumerate}
We say that ``$\square_{\gk, \gl}$ holds'' iff there exists a
$\square_{\gk, \gl}$-sequence. A {\em
$\square_{\gk,<\gl}$-sequence} is defined as above with clause
\ref{size} replaced by ``$1 \le \card{ {\cal C}_\ga } < \gl$''.
\end{definition}
The principle $\square_{\gk, \gl}$ was defined by Schimmerling
\cite{Sch} in his work on the core model for one Woodin cardinal.
It is a common generalisation of two principles
studied by Jensen \cite{JeFS}, the principles $\square_\gk =\square_{\gk, 1}$
and $\square^*_\gk = \square_{\gk, \gk}$.
With these definitions in hand we are ready to state the main
theorem of this paper.
\begin{theorem} \label{mainthm}
Let GCH hold and let $\kappa$ be a
$\kappa^{+5}$-supercompact cardinal. Then there exists a forcing poset
$\FP$ such that in $V^\FP$
\begin{enumerate}
\item $\kappa$ is $\kappa^{+5}$-supercompact.
\item For every singular cardinal $\lambda < \kappa$
\begin{enumerate}
\item There exists $S \subseteq \gl^+$ stationary such that
if $\vec S$ is a sequence of stationary subsets of $S$ and
$\lh(\vec S) < \cf(\gl)$ then $\vec S$ reflects to cofinality
$\gm$ for unboundedly many $\gm < \gl$.
\item The combinatorial principle $\square_{\gl, \cf(\gl)}$
holds.
\end{enumerate}
\end{enumerate}
\end{theorem}
Notice that truncating the generic extension by $\FP$ at $\gk$
will give a set model of the theory
``ZFC + there exists a proper class of cardinals $\delta$ which are
$\delta^{+4}$-supercompact + every singular cardinal $\gl$
has the above properties''.
This theorem is (as we explain below) a generalisation of the
following result of Ben-David and Magidor.
\begin{fact}[Ben-David and Magidor \cite{BDM}]
\label{BDMthm}
If $\gk$ is $\gk^+$-supercompact there is a generic extension
in which
\begin{enumerate}
\item $\gk = \ha_\go$, $\gk^+ = \ha_{\go+1}$.
%\item There is an ultrafilter $U$ on $\ha_{\go+1}$
% which is $\gl$-indecomposable for each $\gl$ with
% $\go < \gl < \ha_0$.
%\item The transfer principle $(\ha_1, \ha_0) \longrightarrow (\ha_{\go+1}, \ha_\go)$
% holds.
\item $\square_{\ha_\go}$ fails.
\item $\square^*_{\ha_\go}$ holds.
\end{enumerate}
\end{fact}
The model of \cite{BDM} is built by using a modification of
Magidor's
``supercompact Prikry forcing with interleaved forcing''
from \cite{SCH1}. Apter and Henle \cite{AH} showed that a somewhat similar proof
can be made to work using $\gk$ which is only $\gk^+$-strongly
compact. Using the ideas of \cite{AH} and methods of Gitik for iterating Prikry-type
forcing, Apter was able to show
\begin{fact}[Apter \cite{SCBDM}]
Con(ZFC + GCH + $\gk$ is an inaccessible limit of cardinals
$\gd$ which are $\gd^+$ strongly compact)
$\implies$
Con(ZFC + $\gk$ is inaccessible + For every cardinal
$\gl < \gk$, there exists a stationary
$S \subseteq \gl^{+ \go + 1}$ such that if
$S' \subseteq S$ is stationary, then $S'$ reflects at
$\gd$ for unboundedly many $\gd < \gl^{+ \go + 1}$
(so $\neg \Box_{\gl^{+ \go}}$) +
For every cardinal $\gl < \gk$, $\Box_{\gl^{+ \go}, \go})$.
\end{fact}
In the model of \cite{BDM}
there is a uniform ultrafilter on $\ha_{\go+1}$ which is
$\gl$-indecomposable for $\ha_0 < \gl < \ha_\go$.
This implies
\cite[Lemma 2.2]{BDM}
that every stationary subset
of $S =\setof{\ga < \ha_{\go+1}}{\cf(\ga) > \go}$ reflects, and
from this stationary reflection principle it follows \cite[Lemma 2.1]{BDM} that
$\square_{\ha_\go}$ fails.
In the model of \cite{BDM} the transfer principle $(\ha_1, \ha_0) \lra (\ha_{\go+1}, \ha_\go)$
holds.
It follows from this transfer principle that there is a special
$\ha_{\go+1}$-Aronszajn tree, and hence by work of Jensen
\cite{JeFS} that $\square^*_{\ha_\go}$ holds.
Our model for Theorem \ref{mainthm} is built using Foreman and
Woodin's ``supercompact Radin forcing with interleaved forcing''
from their consistency proof \cite{FW} for ``ZFC + GCH fails
everywhere''. Their work builds on ideas from \cite{SCH1} and
Radin's paper \cite{Radin}.
It is worth remarking that the similarity between our model for
Theorem \ref{mainthm} and the model of \cite{BDM}
is even more pronounced than has been shown so far.
The methods of Section \ref{reflectionsection} of this paper can be
used
to show that in the model of
\cite{BDM} any finite sequence of stationary subsets
of $S$ reflects simultaneously, and it follows from
Fact \ref{ernesto} that $\square_{\ha_\go, \go}$ holds in the model of
\cite{BDM}.
The machinery that we use to get $\square_{\gl, \cf(\gl)}$ to hold in the
model of Theorem \ref{mainthm} is based on the following distinctive property of
the forcing $\FP$: every singular cardinal of $V^\FP$ below $\gk$ is inaccessible in $V$.
This idea originates in the proof of Fact \ref{ernesto}. Before
stating Fact \ref{ernesto} we make a technical definition.
\begin{definition} Let $\gk$ be regular and uncountable. $X \subseteq
\gk$ is a {\em $>\go$-club subset of $\gk$} if and only if $X$ is
unbounded in $\gk$ and $X$ is closed under suprema of uncountable
cofinality.
\end{definition}
\begin{remark} It is easy to see that the $> \go$-club subsets of
$\gk$ generate a normal filter on $\gk$. We shall refer to this
filter as the ``$>\go$-club filter''.
\end{remark}
\begin{fact}[Cummings and Schimmerling \cite{CuSch}]
\label{ernesto}
Let $V \models ``\gk$ is inaccessible''.
Let $W \supseteq V$ be a model such that
\begin{enumerate}
\item $\gk$ and $\gk^+$ are cardinals in $W$.
\item There is $E \in W$ such that
\begin{enumerate}
\item $o.t.(E) = \omega$.
\item For all $D \in V$, if
$V \models ``D$ is a $> \omega$-club subset of
$\gk$'' then $E \cap D \neq \emptyset$.
\end{enumerate}
\end{enumerate}
Then $W \models \square_{\gk, \omega}$.
% Let the cardinal $\gk$ be measurable. Let $U$ be a normal measure on
%$\gk$ and let $\FP$ be the Prikry forcing derived from the normal
%measure $U$. Then
% $\forces_\FP \square_{\gk, \go}$.
\end{fact}
\begin{remark} In particular if $\gk$ is measurable in $V$, and $W$ is a
generic extension by Prikry forcing at $\gk$, then $\square_{\gk, \go}$
holds in $W$.
\end{remark}
In the context of Theorem \ref{mainthm} we are changing cofinalities to values other
than $\go$, and will have to prepare the ground model in order to get an analogue
to Fact \ref{ernesto}. This preparation uses ideas of Baumgartner.
We now quote some facts whose proofs can be found in
\cite{CFM}.
Taken together
they indicate that the result of Theorem \ref{mainthm}
is close to being optimal.
\begin{fact}[Cummings, Foreman and Magidor \cite{CFM}]
Let $\gl$ be singular. If $\square_{\gl, \gm}$ holds for $\gm < \gl$
and $S \subseteq \gl^+$ is stationary then there exists $\vec T = \seq{T_i : i < \cf(\gl)}$
such that each $T_i$ is a stationary subset of $S$ and $\vec T$ does not reflect simultaneously
to cofinality $\gm$ for any $\gm$ with $\cf(\gl) < \gm < \gl$.
\end{fact}
\begin{fact}[Schimmerling]
Let $\gl$ be singular.
If $\square_{\gl, \gm}$ holds for $\gm < \cf(\gl)$ and $S \subseteq \gl^+$ is stationary,
then there exists $T \subseteq \gl$ stationary and $\gd < \gl$
such that $T$ does not reflect at any point of cofinality greater than $\gd$.
\end{fact}
Our forcing notation is fairly standard. We write $p \le q$ when $p$ is a stronger condition
than $q$. ${\rm Add}(\gk, \gl)$ is the poset to add $\gl$ Cohen subsets of $\gk$, ${\rm Coll}(\gk, <\gl)$
is the Levy collapse to make every ordinal in $[\gk, \gl)$ have cardinality $\gk$. A poset is
$\gk$-closed when every descending sequence of length less than $\gk$ has a lower bound, and
is $\gk$-directed closed when every directed subset of size less than $\gk$ has a lower bound.
${\rm RO}(\FP)$ is the complete Boolean algebra corresponding to the poset $\FP$.
The second author would like to thank Matt Foreman for telling him Fact \ref{jimforcing}, and showing him a
proof that if $\gk$ is $\gk^+$-supercompact then any finite sequence of stationary
subsets of $\gk^+$ reflects after doing Prikry forcing at $\gk$.
\section{Building squares}
In this section we discuss the machinery we use to get
$\square_{\gl, \cf(\gl)}$-sequences in the model of Theorem \ref{mainthm}.
We will define
a combinatorial principle $\square^B_\gk$ (due to Baumgartner) and
prove the following lemma, which is a generalisation of Fact \ref{ernesto}.
The point here is that when the cofinality of $\gk$ is changed to be uncountable
we need some amount of square in the ground model to see that the desired
form of square holds in the extension.
\begin{lemma}
\label{squarelemma}
Let $V \models \hbox{``$\gk$ is inaccessible and
$\square^B_\gk$''}$. Let $W \supseteq V$ be a model such that
\begin{enumerate}
\item $\gk$ and $\gk^+$ are cardinals in $W$ and $\gl$
is regular in $W$.
\item \label{technical-condition}
There is $E \in W$ such that
\begin{enumerate}
\item $E$ is closed and unbounded in $\gk$.
\item $\ot{E} = \gl$.
\item For all $D \in V$,
if $V \models \hbox{``$D$ is a $>\go$-club subset of $\gk$''}$
then $E \cap D \neq \emptyset$.
\end{enumerate}
\end{enumerate}
Then $W \models \square_{\gk, \gl}$.
\end{lemma}
\begin{remark}
If $\gk$ is measurable and $A \subseteq \gk$ is a $>\go$-club
subset of $\gk$ then $A \in U$ for every normal measure $U$
on $\gk$. It follows that if $C$ is a generic club of order
type $\gl$ added by any of the standard forcings for changing
cofinality (Prikry forcing \cite{Prikry}
[where the generic set $C$, having
order type $\omega$, is trivially considered as being
club in $\gk$],
Magidor forcing
\cite{Magidor} or Radin forcing \cite{Radin}) then the technical
condition \ref{technical-condition} in the statement of Lemma
\ref{squarelemma} holds; in fact $C$ will be eventually contained
in every $>\go$-club subset of $\gk$ from the ground model.
This will also hold for the modified Radin forcing $\FP_a$ which
we define in the proof of Theorem \ref{mainthm}.
It is also easy to see that condition \ref{technical-condition}
holds if $W \models \cf(\gk) = \gl \ge \ha_2$. We do not know whether
it holds always when $W \models \cf(\gk) = \gl = \ha_1$ or
$W \models \cf(\gk) = \gl = \omega$.
\end{remark}
Before giving the proof of this lemma we discuss some ideas that
are used in the proof, define $\square^B_\gk$ and show that this
principle is consistent with $\gk$ being a large cardinal.
\subsection{Good matrices}
We will use the idea of a {\em good matrix}, which
comes from the proof of Fact \ref{ernesto}.
In the interest of making this paper self-contained we have
reproduced some results from \cite{CuSch}.
\begin{definition} Let $\gk$ be inaccessible and let
$S = \setof{\ga < \gk^+}{\cf(\ga) < \gk}$.
A {\em good matrix} is an array of sets
\[
\seq{C_{\ga i}: \ga \in S, i \in X_\ga}
\]
such that
\begin{enumerate}
\item $C_{\ga i}$ is club in $\ga$.
\item $X_\ga$ contains a $>\go$-club subset of $\gk$.
\item $\ot{C_{\ga i}} < \gk$.
\item If $i \in X_\ga$ and $\gb \in \lim C_{\ga i}$ then
$i \in X_\gb$ and $C_{\ga i} \cap \gb = C_{\gb i}$.
\item If $i, j \in X_\ga$ and $i < j$ then $C_{\ga i} \subseteq
C_{\ga j}$.
\item If $\ga, \gb \in S$ with $\gb < \ga$ then $\gb \in \lim C_{\ga
i}$ for some $i \in X_\ga$ (and thus for all large $i \in X_\ga$ by
the preceding clause).
\end{enumerate}
\end{definition}
\begin{fact}[Cummings and Schimmerling \cite{CuSch}] \label{good-matrix}
Let the cardinal $\gk$ be inaccessible.
Then there exists a good matrix for $\gk$.
\end{fact}
\begin{remark}
When building a good matrix the hardest
stages are those of the form $\gb + \go$ where $\cf(\gb) = \gk$,
which are treated in Case 5 of the inductive definition below. To
prepare for these stages, we need to make sure in Case 4 that if $\go < \cf(\ga) < \gk$ then $X_\ga$ is as large as possible.
Accordingly when $\go < \cf(\ga) < \gk$ we choose $X_\ga$ as the ``maximally fat'' set
of indices $i$ on which it is possible to define $C_{\ga
i}$. $X_\ga$ will be in the $> \go$-club filter on $\gk$
but is not necessarily an actual $> \go$-club subset of $\gk$.
\end{remark}
\begin{proof}[Fact \ref{good-matrix}]
We construct a good matrix by induction on $\ga \in S$.
\bigskip
\noindent {\bf Case 1:} $\ga = \go$. We set $X_\go = \gk$ and
$C_{\go i} = \go$ for all $i$.
\bigskip
\noindent {\bf Case 2:} $\ga = \gb + \go$ for limit $\gb$ with
$\cf(\gb) < \gk$ (that is to say $\gb \in S$).
We set $X_\ga = X_\gb$ and $C_{\ga i} = C_{\gb i} \cup [\gb, \ga)$
for all $i \in X_\ga$.
Clearly $C_{\ga i}$ is club in $\ga$. $X_\ga = X_\gb$ and so
$X_\ga$ is in the $> \go$-club filter, and $\ot{C_{\ga i}} =
\ot{C_{\gb i}} +
\go < \gk$.
If $i \in X_\ga$ and $\gg \in \lim C_{\ga i}$ then either $\gg \in
\lim C_{\gb i}$ or $\gg = \gb$. In the former case we have by
induction that $i \in X_\gg$ and $C_{\gg i} = C_{\gb i} \cap \gg$,
in the latter that $i \in X_\gb = X_\gg$ and $C_{\gg i} = C_{\gb i}$:
in either case $C_{\ga i} \cap \gg = C_{\gg i}$.
If $i,j \in X_\ga$ with $i < j$ then by induction
$C_{\gb i} \subseteq C_{\gb j}$, so that $C_{\ga i} \subseteq
C_{\ga
j}$. Finally if $\gg \in S \cap \ga$ then either $\gg \in S \cap \gb$
or $\gg = \gb$: if $\gg \in S \cap \gb$ then by induction
$\gg \in \lim C_{\gb i}$ for some $i$ and then $\gg \in \lim C_{\ga
i}$ for
the
same $i$, while if $\gg = \gb$ then $\gg \in \lim C_{\ga i}$ for every $i
\in X_\ga$.
\bigskip
\noindent {\bf Case 3:} $\cf(\ga) = \go$ and $\ga$ is a limit of limit
ordinals. We choose $\seq{\ga_m: m < \go}$ an increasing sequence of
ordinals in $S$ which is cofinal in $\ga$. We set
\[
X_\ga =
\setof{i < \gk}{\hbox{$\forall m < \go \;\; i \in X_{\ga_m}$
and $\forall m < n < \go \;\; \ga_m \in \lim C_{\ga_n i}$}}.
\]
$X_\ga$ is in the $>\go$-club filter because it is a final segment of
$\bigcap_j X_{\ga_j}$.
We observe that if $i \in X_\ga$ then $C_{\ga_m i} = C_{\ga_n i} \cap
\ga_j$ for all $m < n < \go$. We now set
$C_{\ga i} = \bigcup_m C_{\ga_m i}$ for all $i \in X_\ga$.
$C_{\ga i}$ is club in $\ga$ because every initial segment is an
initial segment of $C_{\ga_m i}$ for some $m$. A similar argument
shows that $\ot{C_{\ga i}} < \gk$. If $\gb \in \lim C_{\ga i}$ then
$\gb \in \lim C_{\ga_m i}$ for some $m$, and by induction
$i \in X_\gb$ and $C_{\gb i} = C_{\ga_m i} \cap \gb = C_{\ga i} \cap
\gb$.
If $i, j \in X_\ga$ with $i < j$ then by induction $C_{\ga_m i}
\subseteq C_{\ga_m j}$ for all $m < \go$, so that $C_{\ga i}
\subseteq C_{\ga j}$. Finally if $\gb \in S \cap \ga$ then
$\gb \in S \cap \ga_m$ for some $m$, and so by induction
$\gb \in \lim C_{\ga_m i}$ for all large $i \in X_{\ga_m}$;
it follows that $\gb \in \lim C_{\ga i}$ for any large enough
$i \in X_\ga$.
\bigskip
\noindent{\bf Case 4:} $\go < \cf(\ga) < \gk$. Let $\cf(\ga) = \gr$ say.
As in Case 3 we fix $\seq{\ga_m: m < \gr}$ an increasing and continuous
sequence of
members of $S$ which is cofinal in $\ga$. We define
\[
Y_\ga =
\setof{i < \gk}{\hbox{$\forall m < \gr \;\; i \in X_{\ga_m}$
and $\forall m < n < \gr \;\; \ga_m \in \lim C_{\ga_n i}$}}.
\]
Note that the precise nature of $Y_\ga$ depends on the sequence
$\seq{\ga_m : m < \rho}$
used in its definition.
Exactly as in Case 3 $Y_\ga$ is in the $>\go$-club filter, and
if $i \in Y_\ga$ then $C_{\ga_m i} = C_{\ga_n i} \cap \ga_m$
for all $m < n < \gr$.
We now let
\[
X_\ga = \setof{i < \gk}{\hbox{$\exists E$ club in $\ga \; \forall
\gg \in \lim(E) \; (i \in X_\gg$ and $E \cap \gg = C_{\gg i})$}}.
\]
If $i \in Y_\ga$ and we let $E = \bigcup_m C_{\ga_m i}$ then
it is easy to check that $E$ witnesses $i \in X_\ga$, so that
$Y_\ga \subseteq X_\ga$.
We observe that
\begin{itemize}
\item $X_\ga$ is independent of the choice of the sequence
$\seq{\ga_m : m < \gr}$.
\item By its definition and clause 4 in the definition of a good
matrix, $X_\ga$ is the ``maximally fat'' set of
indices $i$ for which we can hope to define $C_{\ga i}$.
\end{itemize}
Suppose that $i \in X_\ga$ and $E$, $E'$ are both clubs in $\ga$
witnessing this. Then $E \cap E'$ is club in $\ga$ and
\[
E = \bigcup\limits_{\gg \in \lim(E \cap E')} C_{\gg i} = E'.
\]
For each $i \in X_\ga$, we now define $C_{\ga i}$ to be the unique $E$ which
is club in
$\ga$ and is such that
$\forall \gg \in \lim(E) \; E \cap \gg = C_{\gg i}$.
Notice that if $i \in Y_\ga$ then automatically
$C_{\ga i} = \bigcup_m C_{\ga_m i}$.
Since every initial segment of $C_{\ga i}$ is an initial segment
of $C_{\gg i}$ for some $\gg < \ga$, $\ot{ C_{\ga i} } < \gk$.
If $\gb \in \lim C_{\ga i}$ then $\gb \in \lim C_{\gg i}$
for some $\gg \in \lim C_{\ga i}$, and we have by induction that $i \in X_\gb$
and $C_{\gb i} = C_{\gg i} \cap \gb = C_{\ga i} \cap \gb$.
Let $i, j \in X_\ga$ with $i < j$. Let $C_{\ga i} = E$ and $C_{\ga
j} = F$. Then
\[
E = \bigcup\limits_{\gg \in \lim(E \cap F)} C_{\gg i} \subseteq
\bigcup\limits_{\gg \in \lim(E \cap F)} C_{\gg j} = F,
\]
that is to say that $C_{\ga i} \subseteq C_{\ga j}$.
Finally we may argue as in Case 3 that $S \cap \ga \subseteq
\bigcup_{i \in Y_\ga} \lim C_{\ga i}$, which suffices since
$Y_\ga \subseteq X_\ga$.
\bigskip
\noindent{\bf Case 5:} $\ga = \gb + \go$ where $\cf(\gb) = \gk$.
We fix $\seq{\gb_i: i < \gk}$ an increasing and continuous sequence
of members of $S$ which is cofinal in $\gb$. Let
\[
Z = \setof{i < \gk}{\hbox{$\forall j < i \;\; i \in X_{\gb j}$ and
$\forall j < k < i \;\; \gb_j \in \lim C_{\gb_k i}$}}.
\]
We claim that $Z$ is in the $>\go$-club filter. To see this first observe
that if $D = \setof{i < \gk}{\forall j < i \;\; i \in X_{\gb_j}}$ then $D$ is
a diagonal intersection
of sets in the $>\go$-club filter, and therefore is in the
$>\go$-club filter. Define
$F:[\gk]^2 \lra \gk$ by setting $F(j, k)$ equal to the least $i \in X_{\gb_k}$
with
$\gb_j \in \lim C_{\gb_k i}$, and let $C$ be the club set of $i < \gk$
such that $F`` [i]^2 \subseteq i$. If $i \in D \cap C$ then
\begin{enumerate}
\item Since $i \in D$, $\forall j < i \;\; i \in X_{\gb_j}$.
\item If $j,k < i$ then since $i \in C$ we have $F(j, k) < i$,
and by definition $F(j, k) \in X_{\gb_k}$ and $\gb_j \in \lim
C_{\gb_k F(j,
k)}$.
Since $i \in D$ we also have $i \in X_{\gb_k}$, and so by the properties
of a good matrix $C_{\gb_k F(j, k)} \subseteq C_{\gb_k i}$ and
so $\gb_j \in \lim C_{\gb_k i}$.
\end{enumerate}
It follows that $D \cap C \subseteq Z$, and $D \cap C$ is easily
seen to be in the $>\go$-club filter.
We let
$X_\ga = \setof{\gz \in D \cap C}{\cf(\gz) > \go}$. Let $i \in X_\ga$ and consider the
construction at level $\gb_i$; since $\cf(i) > \go$ and the sequence
$\seq{\gb_n: n < \gk}$ is continuous, $\cf(\gb_i) = \cf(i) > \go$ and
the relevant clause of the definition is Case 4.
If we let $E = \bigcup_{j < i} C_{\gb_j i}$ then the fact that $i \in Z$
and the coherence properties of the good matrix imply that $\forall \gg \in
\lim(E) \; E \cap \gg = C_{\gg i}$,
so that by the definition of $X_{\gb_i}$ and $C_{\gb_i i}$ from Case 4 $i
\in X_{\gb_i}$ and
$C_{\gb_i i} = \bigcup_{j < i} C_{\gb_j i}$.
Note that if $i \in Z$,
the sequence
$B = \seq{\gb_j : j < i}$ is very likely
different from the sequence
$A = \seq{\ga_m : m < \cf(i)}$
used in the definition of $Y_{\gb_i}$.
Even though $A$ and $B$ agree on a club
of order type $\cf(i)$,
this isn't necessarily enough to allow
us to infer that $i \in Y_\ga$ for every
$\ga \in A$, something that would be critical
in allowing us to infer that $i \in Y_{\gb_i}$.
The ``maximal fatness'' of $X_{\gb_i}$,
however, ensures this isn't a problem
and that $i \in X_{\gb_i}$.
We define
\[
C_{\ga i} = C_{\gb_i i} \cup \{ \gb_i \} \cup [\gb, \ga).
\]
Clearly $C_{\ga i}$ is club in $\ga$, and $\ot{C_{\ga i}} =
\ot{C_{\gb_i i}} + \go < \gk$. If $\gg \in \lim C_{\ga i}$
then either $\gg \in \lim C_{\gb_i i}$ or
$\gg = \gb_i$, and in either case it is easy to see
that $i \in X_\gg$ and $C_{\gg i} = C_{\gb_i i} \cap \gg = C_{\ga i}
\cap \gg$.
Let $i, j \in X_\ga$ with $i < j$. By induction
\[
C_{\gb_i i} = \bigcup_{k < i} C_{\gb_k i} \subseteq \bigcup_{k < i}
C_{\gb_k j} \subseteq \bigcup_{k < j} C_{\gb_k j} = C_{\gb_j j}.
\]
Since $C_{\gb_j j}$ is club in $\gb_j$ and $C_{\gb_i i}$ is
cofinal in $\gb_i$, it follows that $\gb_i \in C_{\gb_j j}$.
Therefore by definition $C_{\ga i} \subseteq C_{\ga j}$.
Finally let $\gg \in S \cap \ga$, and observe that since $\gb \notin S$
we have $S \cap \ga = S \cap \gb$.
Find $i$ such that $\gg < \gb_i$, and then
$j \in X_\ga$ such that $i < j$ and $\gg \in \lim C_{\gb_i j}$.
Since $C_{\gb_j j} = \bigcup_{k < j} C_{\gb_k j}$,
$\gg \in \lim C_{\gb_j j}$.
This concludes the proof.
\end{proof}
\subsection{The principle $\square^B_\gk$}
The following version of $\square_\gk$ was studied by Baumgartner. Its main interest for us is that
unlike the original $\square_\gk$ principle it is consistent for $\gk$ supercompact.
\begin{definition}
Let $\gk$ be regular.
A $\square^B_\gk$-sequence is a sequence $\seq{C_\ga: \ga \in T}$ where
\begin{enumerate}
\item $T$ is a set of limit ordinals less than $\gk^+$.
\item $\setof{\ga < \gk^+}{\cf(\ga) = \gk} \subseteq T$.
\item For all $\ga \in T$, $C_\ga$ is a club subset of $\ga$ with $\ot{C_\ga} \le \gk$.
\item If $\ga \in T$ and $\gb \in \lim(C_\ga)$ then $\gb \in T$ and $C_\gb = C_\ga \cap \gb$.
\end{enumerate}
\end{definition}
As usual we say ``$\square^B_\gk$ holds'' if there is a $\square^B_\gk$-sequence.
\begin{fact}[Baumgartner \cite{JEB}]
\label{jimforcing}
Let $\gk$ be regular. Then there exists a forcing poset
$\FP=\FP(\gk)$ such that
\begin{enumerate}
\item $\FP$ is $\gk$-directed closed.
\item $\FP$ is strategically closed for the game of length $\gk+1$.
\item $\forces_\FP \hbox{``$\square^B_\gk$ holds''}$.
\end{enumerate}
\end{fact}
\begin{proof}
We force with the set of initial segments of successor length of such a sequence. More formally
let $\FP$ be the set of sequences $\seq{C_\ga: \ga \in s}$ where
\begin{enumerate}
\item $s$ is a bounded set of limit ordinals less than $\gk^+$, with a
maximal element $\gg$.
\item $\setof{\ga < \gg}{\cf(\ga) = \gk} \subseteq s$.
\item For all $\ga \in s$, $C_\ga$ is a club subset of $\ga$ with $\ot{C_\ga} \le \gk$.
\item If $\ga \in s$ and $\gb \in \lim(C_\ga)$ then $\gb \in s$ and $C_\gb = C_\ga \cap \gb$.
\end{enumerate}
$\FP$ is ordered as follows: if $p = \seq{C_\ga: \ga \in s}$ and $q = \seq{D_\ga: \ga \in t}$
then $p \le q$ iff $t = s \cap (\max(t) + 1)$ and $C_\ga = D_\ga$ for all $\ga \in t$.
Now we take the claims made in the theorem one by one. We adopt the
convention ``a sequence is a function is a set of ordered pairs'',
so that $\seq{C_\ga: \ga \in s} = \setof{ (\ga, C_\ga)}{ \ga \in s}$.
\begin{enumerate}
\item Since the partial ordering on $\FP$ is treelike it suffices to show that $\FP$ is $\gk$-closed.
Let $\seq{p_i: i < \gz}$ be a decreasing $\gz$-sequence of conditions for some $\gz < \gk$, and let $s_i = \dom(p_i)$
and $\gg_i = \max(s_i)$. Let $\gb^* = \sup_i \gg_i$, $\gb = \gb^* + \go$, $c = [\gb^*, \gb)$. Define
$p = \bigcup_i p_i \cup \{ (\gb, c) \}$. It is routine to check that $p$ is a condition, the key point being
that since $\cf(\gb^*) = \cf(\gz) < \gk$ we are under no obligation to put $\gb^*$ into the
domain of $p$.
\item We describe a winning strategy for player II in the game of
length $\gk+1$ played on $\FP$.
Let the move made at stage
$i$ be $p_i = \seq{C_\ga: \ga \in s_i}$, with $\max(s_i) = \gg_i$.
Player II will play to guarantee that if $i, j$ are even with $i < j$ then $\gg_i \in \lim(C_{\gg_j})$
and $C_{\gg_i} = C_{\gg_j} \cap i$.
Suppose we have reached an even stage $\ga$ and it is the turn of Player II.
\medskip
\noindent Case I. $\ga$ is a successor ordinal, say $\ga = \gb + 2$. We set $\gg_\ga = \gg_{\gb +1} + \go$,
$C_{\gg_\ga} = C_{\gg_\gb} \cup \{ \gg_\gb \} \cup [\gg_{\gb+1}, \gg_\ga)$,
$s_\ga = s_{\gb +1} \cup \{ \gg_\ga \}$,
and finally $p_\ga = p_{\gb + 1} \cup \{ (\gg_\ga, C_{\gg_\ga}) \}$.
\noindent Case II. $\ga$ is a limit ordinal with $\ga \le \gk$. We set $\gg_\ga = \sup_{j < \ga} \gg_j$,
$C_{\gg_\ga} = \setof{\gg_j}{j < \ga}$, and finally
$p_\ga = \bigcup_{\gg < \ga} p_\gg \bigcup \{ (\gg_\ga, C_{\gg_\ga}) \}$.
\medskip
It is routine to check that this is a successful strategy for player II.
The key points are that II always plays a set of order type less than or equal to $\gk$,
and that if $\ga$ is limit and $\gb \in \lim(C_{\gg_\ga})$ then $\gb = \gg_\gd$ for
limit $\gd < \ga$ and so
\[
C_{\gg_\gd} = \setof{\gg_j}{j < \gd} = C_{\gg_\ga} \cap \gg_\gd.
\]
\item It is easy to see that for all $p \in \FP$ and $\gb < \gk^+$ there exists $q \le p$
such that $\max(\dom(q)) > \gb$. The proof is a routine induction on $\gb$, using the same idea as in the
proof of strategic closure to get through stages of cofinality $\gk$.
By strategic closure, $\FP$ adds no $\gk$-sequence of ordinals so preserves
all cardinals and cofinalities up to $\gk^+$. It follows that if $G$ is $\FP$-generic then $\bigcup G$ is a
$\square^B_\gk$-sequence in $V[G]$.
\end{enumerate}
\end{proof}
We will need to know later that $\square^B_\gk$ is consistent with
$\gk$ being a large cardinal.
We will use the following basic facts, which originated in
Silver's work on Reverse Easton forcing.
\begin{fact} Let $M$ and $N$ be models of ZFC, let
$k: M \lra N$ be an elementary embedding.
Let $\FP \in M$ be a
forcing
poset and let $G$ and $H$ be such that
\begin{enumerate}
\item $G$ is $\FP$-generic over $M$.
\item $H$ is $k(\FP)$-generic over $N$.
\item $\forall p \in G \; k(p) \in H$.
\end{enumerate}
Then there is a unique elementary embedding $k^+: M[G] \lra N[H]$ such that
$k^+$ extends $k$ and $k^+(G) = H$.
\end{fact}
\begin{fact} Let $M$ and $N$ be models of ZFC with $M \subseteq
N$. Let $\gl$ be an $N$-cardinal. Let $\FP \in M$ be a forcing poset such that
\begin{enumerate}
\item $N \models \hbox{``$\FP$ is $\gl$-closed''}$.
\item $\setof{ A \in M}{\hbox{$A$ is a maximal antichain in $\FP$}}$
has cardinality $\gl$ in $N$.
\end{enumerate}
Then for every $p \in \FP$ there exists $H \in N$ such that $p \in
H$
and $H$ is $\FP$-generic over $M$.
\end{fact}
\begin{theorem}
\label{prep}
Let $\gk$ be $\gk^{+n}$-supercompact for some $n$ with
$0 < n < \go$, and let GCH
hold. Let $\FP_{\gk+1}$ be the Reverse Easton iteration
of length $\gk+1$ where at every inaccessible $\ga \le \gk$
we force with $\FP(\ga)_{V^{\FP_\ga}}$ (where $\FP(\ga)$ is
the forcing to add $\square^B_\ga$ defined in Fact
\ref{jimforcing}).
Then in $V^{\FP_{\gk+1}}$
\begin{enumerate}
\item $V$-cardinals greater than or equal to $\gk$ are preserved.
\item GCH holds.
\item $\gk$ is $\gk^{+n}$-supercompact.
\end{enumerate}
\end{theorem}
\begin{proof} $\FP_\gk$ is a $\gk$-c.c.~forcing of size $\gk$, so
forcing with $\FP_\gk$ preserves cardinals greater than or equal to
$\gk$ and GCH holds at and above $\gk$ in $V^{\FP_\gk}$.
We saw in the proof of Fact \ref{jimforcing} that
$\FP(\gk)$ adds no $\gk$-sequences of ordinals, so it preserves all
cardinals and cofinalities up to $\gk^+$. $2^\gk = \gk^+$ in
$V^{\FP_\gk}$ so $\FP(\gk)$
has cardinality $\gk^+$, hence it is $\gk^{++}$-c.c.~and preserves
all cardinals and cofinalities above $\gk^+$. Standard arguments show
that GCH still holds at and above $\gk$ in $V^{\FP_{\gk+1}}$, and a routine
induction using the above analysis shows that GCH holds below $\gk$
in $V^{\FP_{\gk+1}}$.
To show the preservation of supercompactness, suppose that $G$ is
$\FP_\gk$-generic over $V$ and $g$ is $\FP(\gk)$-generic over
$V[G]$. Fix $j : V \lra M$ which is the ultrapower map defined
from some supercompactness measure on $\powerset_\gk(\gk^{+n})$.
Notice that $\gk^{+n+1} = \gk^{+n+1}_M < j(\gk)$. In particular if
we let $\gee$ be the least $M$-inaccessible greater than $\gk$
then $\gee > \gk^{+n+1}$.
$\card{\powerset_\gk{\gk^{+n}}} =\gk^{+n}$, and it follows using GCH that
$j(\gk^{++}) < \gk^{+n+2}$.
$j(\FP_{\gk+1})$ is an iterated forcing in $M$, and the resemblance
between $V$ and $M$ implies that $\FP_{\gk+1}$ is an initial segment
of $j(\FP_{\gk+1})$. By an easy chain condition argument
$V[G][g] \models {}^{\gk^{+n}} M[G][g] \subseteq M[G][g]$.
Let $\FR$ be the forcing $j(\FP_\gk)/G * g$. Then in $M[G][g]$ the
forcing $\FR$ is $\gee$-closed and has $j(\gk)$ maximal antichains.
$\card{j(\gk)} = \gk^{+n+1}$, so working in
$V[G][g]$ we may build a descending $\gk^{+n+1}$-chain of conditions in $\FR$
to decide each antichain in $M[G][g]$. Thus we may build $H \in V[G][g]$ which is
$\FR$-generic over $M[G][g]$.
$M[G][g][H] \subseteq V[G][g]$, and $V[G][g] \models {}^{\gk^{+n}} M[G][g][H] \subseteq M[G][g][H]$.
Since $j`` G \subseteq G * g * H$, working in $V[G][g]$ we may lift $j$ to get a new
embedding (which we also denote by $j$, without risk of confusion)
$j: V[G] \lra M[G][g][H]$.
Now we use Silver's ``master condition'' argument. Consider
$p =_{\rm def} \bigcup j``g$. $p \in M[G][g][H]$ and $p$ is
a sequence $\seq{D_\gb : \gb \in A}$,
where $A$ is a bounded subset of $j(\gk^+)$
because
\[
M[G][g][H] \models \cf(\sup(j``\gk^+)) = \gk^+ < j(\gk^+).
\]
The only way in which $p$ falls short of being
a condition in $\FP(j(\gk))$ is that $A$ does not have a largest
element. Since $\sup(A) = \sup(j``\gk^{+})$ and $M \models
\cf(\sup(j``\gk^{+})) = \gk^+ < j(\gk)$, we are not obliged to define a
club at $\sup(A)$. Accordingly we let $\gg = \sup(A) + \go$, let
$D_\gg = \setof{\sup(A) + n}{n < \go}$ and $A^* = A \cup \{ \gg \}$.
Now $p^* =_{\rm def} \seq{D_\gb: \gb \in A^*}$ is a condition and
$\forall q \in g \; p^* \le j(q)$.
In $M[G][g][H]$, the forcing $\FP(j(\gk))$ is $\gk^{+n+1}$-closed and
has $j(\gk^{++})$ antichains.
Since $\card{j(\gk^{++})} = \gk^{+n+1}$ and
\[
V[G][g] \models {}^{\gk^{+n}}M[G][g][H] \subseteq M[G][g][H],
\]
we may build a
chain of conditions in $\FP(j(\gk))$ to decide each antichain
in $M[G][g][H]$, and so build $h \in V[G][g]$ which is
$\FP(j(\gk))$-generic over the model $M[G][g][H]$. Taking the first element
of that chain to be $p^*$ we may assume that $p^* \in h$ and hence
that $j `` g \subseteq h$.
So we may lift $j$ again to get $j: V[G][g] \lra M[G][g][H][h]$.
This map is defined in $V[G][g]$ and witnesses that
$\gk$ is $\gk^{+n}$-supercompact in $V[G][g]$.
\end{proof}
\subsection{Proof of Lemma \ref{squarelemma}}
We are now ready to prove Lemma \ref{squarelemma}.
\begin{proof}[Lemma \ref{squarelemma}]
As before, let
\[S = \setof{\ga < \gk^+}{\cf(\ga) < \gk}.\]
Let
\[
\seq{C_{\ga i}: \ga \in S, i \in X_\ga}
\]
be a good matrix and let $\seq{D_\ga: \ga \in T}$ witness the truth
of $\square^B_\gk$. We will define a $\square_{\gk, \gl}$-sequence
$\seq{ {\cal E}_\ga: \ga < \gk^+}$ in $W$.
Let $E \in W$ be a club in $\gk$ which has order type $\gl$ and
meets every $>\go$-club from $V$, and let
$E = \seq{\gk_i : i < \gl}$ with $\gk_i$ increasing.
\medskip
\noindent {\bf Case 1:} $V \models \cf(\ga) < \gk$, $\ga \notin T$.
Let ${\cal E}_\ga = \setof{ C_{\ga \gk_j} }{ \gk_j \in X_\ga}$.
\medskip
\noindent {\bf Case 2:} $V \models \cf(\ga) < \gk$, $\ga \in T$.
Let ${\cal E}_\ga = \setof{ C_{\ga \gk_j} }{ \gk_j \in X_\ga}
\cup \{ D_\ga \}$.
\medskip
\noindent {\bf Case 3:} $V \models \cf(\ga) = \gk$ (which implies
that $\ga \in T$).
Let ${\cal E}_\ga = \{ D_\ga \}$.
\medskip
We verify that this is a $\square_{\gk, \gl}$-sequence. Clearly
each ${\cal E}_\ga$ is a family of closed unbounded subsets of $\ga$
which have order type at most $\gk$, and
the cardinality of ${\cal E}_\ga$ is at most $\gl$.
Each ${\cal E}_\ga$ is non-empty by our assumption on the
club $E$.
Let $C \in {\cal E}_\ga$ and let $\gb \in \lim C$. There are
two possibilities.
\medskip
\noindent 1. $V \models \cf(\ga) < \gk$ and $C = C_{\ga \gk_j}$ for
some $j < \gl$ with $\gk_j \in X_\ga$. By the coherence properties
of a good matrix, we have $\gk_j \in X_\gb$ and
$C \cap \gb = C_{\gb \gk_j}$. The definition of ${\cal E}_\gb$
now tells us that $C \cap \gb \in {\cal E}_\gb$.
\medskip
\noindent 2. $C = D_\ga$ for some $\ga \in T$. By the coherence
properties of a $\square^B_\gk$-sequence, $\gb \in T$ and
$C \cap \gb = D_\gb$. Again the definition of ${\cal E}_\gb$
tells us that $C \cap \gb \in {\cal E}_\gb$.
\end{proof}
\begin{remark} If $\gl = \go$ (that is to say we are in the case of Fact
\ref{ernesto})
then we do not need the assumption that $\square^B_\gk$ holds in $V$.
In this case
we just set
\[
{\cal E}_\ga = \setof{C_{\ga \gk_j}}{\gk_j \in X_\ga}
\]
for all $\ga$ with $V \models \cf(\ga) < \gk$, and set ${\cal E}_\ga = \{ X_\ga \}$
for some $X_\ga$ cofinal in $\gk$ with order type $\go$ when
$V \models \cf(\ga) = \gk$.
\end{remark}
\section{Radin forcing}
We assume in this section that GCH holds and that
$\gk$ is a $\gk^{+5}$-supercompact cardinal.
We describe a partial ordering $\FP$ which will have the properties that
\begin{enumerate}
\item $\gk$ is $\gk^{+5}$-supercompact in $V^\FP$.
\item If $\gm < \gk$ is any limit cardinal in $V^\FP$ then $V \models \hbox{``$\gm$ is $\gm^+$-supercompact''}$.
\end{enumerate}
The forcing $\FP$ will be obtained by a minor modification in the construction of Foreman and
Woodin's model where GCH fails everywhere \cite{FW}.
In \cite{FW} Foreman and Woodin are working with a cardinal $\gk$ which is $\beth_\go(\gk)$-supercompact
and is such that $\beth_n(\gk)$ is weakly inaccessible for each $n$. They show (see Section 7 of
their paper) that it is possible to construct some forcing $\FP$ such that
\begin{enumerate}
\item $\FP$ preserves cardinals.
\item $\gk$ is $\beth_3(\gk)$-supercompact in $V^\FP$.
\item The forcing $\FP$ adds a club set $C \subseteq \gk$ such that if
$C$ is enumerated in increasing order as $\seq{\gk_i : i < \gk }$ then
\begin{enumerate}
\item Each $\gk_i$ is a large cardinal in $V$.
\item The forcing adds an ${\rm Add}(\beth_4(\gk_i), \gk_{i+1})$-generic object to
$V$ for each $i < \gk$.
\end{enumerate}
\end{enumerate}
Truncating $V^\FP$ at $\gk$ then gives a model in which GCH fails everywhere.
We will build $\FP$ in a very similar way so that
\begin{enumerate}
\item $\gk$ is $\gk^{+5}$-supercompact in $V^\FP$.
\item The forcing $\FP$ adds a club set $C \subseteq \gk$ such that if
$C$ is enumerated in increasing order as $\seq{\gk_i : i < \gk }$ then
\begin{enumerate}
\item $\gk_0 = \ha_0$, $\gk_i$ is a large cardinal in $V$ for $i > 0$.
\item The forcing adds a ${\rm Coll}(\gk_i^{+6}, < \gk_{i+1})$-generic object to
$V$ for each $i < \gk$.
\item Cardinals not lying in the intervals $(\gk_i^{+6}, \gk_{i+1})$ are preserved.
\end{enumerate}
\end{enumerate}
We will assume that the reader has a copy of \cite{FW} to hand. Rather than reproducing
the excellent exposition in that paper with minor changes, we will mostly just give
references to that paper and leave the reader to fill in the details. We attempt to
use notation which parallels that of \cite{FW}. In particular we define
\[
\powerset_\gg \gd = \setof{X \subseteq \gd}{\ot{X} < \gg, X \cap \gg \in \gg}.
\]
Given $x,y \in \powerset_\gg \gd$ we define a relation ``$x \prec y$'' by
\[
x \prec y \iff (x \subseteq y \wedge \ot{x} < y \cap \gg).
\]
This definition really depends on the values of $\gg$ and $\gd$ but these should
always be clear from the context.
We begin by setting $\gs = \gk^{+5}$, $\gr = \gk^{+4}$, $\gl^* = \gk^{+3}$.
For the rest of this paper
we fix $j$ such that
\begin{enumerate}
\item $\crit(j) = \gk$, $j(\gk) > \gs$, ${}^\gs M \subseteq M$.
\item $j$ arises as the ultrapower map associated with some normal fine ultrafilter $U$
on the set $\powerset_\gk \gs$.
\end{enumerate}
We will define a sequence $\vec M = \seq{M_\ga : \ga < \gl^*}$ such that $M_0 = j`` \gl^*$
and for $\ga > 0$ $M_\ga$ is a measure on
\[
Z = \powerset_\gk \gl^* \times V_\gk \times V_\gk.
\]
The definition of $\vec M$ will be recursive, and will go as follows: for $0 < \ga < \gl^*$ we will
define some function $g_\ga$ such that $\dom(g_\ga) = A_\ga \in M_\ga$ and
$g_\ga : A_\ga \rightarrow V_\gk$,
and will then set
\[
M_\ga = \setof{X \subseteq Z}{(M(0), \seq{M_\gb: 0 < \gb < \ga}, \seq{g_\gb: 0 < \gb < \ga}) \in j(X)}.
\]
It is routine to check that $(M(0), \seq{M_\gb: 0 < \gb < \ga}, \seq{g_\gb: 0 < \gb < \ga}) \in j(Z)$, so
$M_\ga$ is a $\gk$-complete measure on $Z$ for all $\ga < \gl^*$. The argument of
Lemma 3.1 from \cite{FW} shows that ${\rm Ult}(V, M_\ga)$ is closed under $\gl^*$-sequences.
We will need to know that $M_\ga$ concentrates on triples whose second entries are
constructed in a manner similar to that in which $\vec M$ is constructed; we prove
this by a reflection argument.
We define $j_0 : V \lra M_0^*$ to be the ultrapower map arising from the ultrafilter
\[
U_0 = \setof{x \in \powerset_\gk \gr}{j `` \gr \in j(x)}.
\]
Defining $F: \powerset_\gk \gs \lra \powerset_\gk \gr$ by $F(x) = x \cap \gr$ it is
clear that $X \in U_0 \iff F^{-1}[X] \in U$. It follows that $F$ induces a map from
$M_0^* = {\rm Ult}(V, U_0)$ to $M = {\rm Ult}(V, U)$ given by $k : [f]_{U_0} \longmapsto [f \circ F]_{U}$,
and it is routine to check that $k$ is an elementary embedding and that $k \circ j_0 = j$.
Since $\gr + 1 \subseteq \rge(k)$, $\crit(k) > \gr$. Since $\gr^+_{M_0^*} = \gr^+_M = \gs$
but $\gr^{++}_{M_0^*} < \gr^{++}_M$ it follows that $\crit(k) = \gr^{++}_{M_0^*}$.
We also know that $H_\gs^{M_0^*} = H_\gs^M = H_\gs$, and can argue in a standard way that $k \restriction H_\gs = {\rm id}$.
If we now define a sequence $\vec N$ in the same way as $\vec M$ is defined, save that $j$ is replaced
by $j_0$, then a routine inductive proof shows that $\vec N = \vec M$; the key point is that
for every $\ga \le \gl^*$ the sequence $\vec M \restriction \ga$ lies in
$M_0^*$ and is fixed by $k$.
By GCH we see that $U_0 \in M$. Let $i_0: M \lra N = {\rm Ult}(M, U_0)$ be the ultrapower map.
$i_0 \restriction H_\gs = j_0 \restriction H_\gs$ and so if $\vec N_0$ is defined
like $\vec M$ using $i_0$ in place of $j$ then $\vec N_0 = \vec N = \vec M$. It follows that
$M_\ga$ will concentrate on triples whose second entries are sequences of measures
constructed in the same manner as $\vec M$.
We defer for the moment the question of exactly how the function $g_\ga$ is defined; we will return
to this point after some general discussion of the construction of $\FP$.
The first step in the construction of $\FP$ will be to build some forcing $\FR$ which
adds to the universe sequences $\seq{x_i : i < \gk}$ and $\seq{G_i: i < \gk}$ such that
\begin{enumerate}
\item $x_i \in \powerset_\gk \gl^*$, $i 0$) will concentrate on
the set of triples $(u, v, w)$ such that
\begin{enumerate}
\item $u \in \powerset_\gk \gl^*$.
\item Defining $\gk_u = u \cap \gk$ and $\gl_u = \ot{u}$,
\begin{enumerate}
\item $\dom(v) = \dom(w) = [1, \gb)$ for some $\gb < \gl_u$.
\item $v(\ga)$ is a measure on $Z_u = \powerset_{\gk_u} \gl_u \times V_{\gk_u} \times V_{\gk_u}$.
\item $w(\ga)$ is a function such that $\dom(w(\ga)) \in v(\ga)$ and $\rge(w(\ga)) \subseteq V_{\gk_u}$.
\end{enumerate}
\end{enumerate}
The building blocks of the forcing $\FR$ will be triples of this general form.
A condition will determine a finite sequence of triples
$\seq{ (u_i, v_i, w_i) : i < n}$ such that $u_0 \prec u_1 \prec \ldots \prec u_{n-1}$.
If $\dom(v_i) = [1, \gb_i)$ then the condition will associate with the triple $(u_i, v_i, w_i)$
a sequence $\seq{A_i(j): 1 \le j < \gb_i}$ such that $A_i(j) \in v_i(j)$ for $j \in [1, \gb_i)$.
The idea is that the sets $A_i(j)$ will constrain the triples $(u, v, w)$ that can be interpolated
between $(u_{i-1}, v_{i-1}, w_{i-1})$ and $(u_i, v_i, w_i)$ when the condition is extended.
Unfortunately the measure $v_i(j)$ concentrates on the set $Z_{u_i}$, while a candidate
for interpolation is a triple $(u, v, w)$ as above where $u \in \powerset_\gk \gl^*$
and $u \prec u_i$. While $v$ and $w$ are of the right form (because
$\gl_u < \gk_{u_i}$), $u$ is not literally a member of $\powerset_{\gk_{u_i}} \gl_{u_i}$.
We define some functions that will be used to deal with this problem.
If $u, u' \in \powerset_\gk \gl^*$ and $u \prec u'$ then (since $u \subseteq u'$) there is a
natural map $i_{u u'}: \gl_u \rightarrow \gl_{u'}$ induced by the inclusion map from $u$ to
$u'$. Notice that $i_{u u'} \restriction \gk_u = {\rm id}$.
Abusing notation, we also denote by $i_{u u'}$ the map from $Z_u$ to $Z_{u'}$ given by
\[
i_{u u'} (a , b, c) = (i_{u u'}(a), b , c).
\]
In the situation of the last paragraph, the appropriate question to ask will be whether
$i_{u u_i}(u , v, w) \in A_i(j)$ where $j = i_{u u_i}(\lh(v))$.
Similarly if $u \in \powerset_\gk \gl^*$ the inclusion map induces a natural map $i_u: \gl_u \rightarrow \gl^*$
with $i_u \restriction \gk_u = {\rm id}$.
As in the preceding paragraph we abuse notation slightly and define $i_u: Z_u \lra Z$ by
$i_u (a , b, c) = (i_u (a), b , c)$.
It is clear from the definitions that if $u \prec v$ then
$i_u = i_v \circ i_{uv}$, and similarly if $u \prec v \prec w$ then
$i_{uw} = i_{v w} \circ i_{u v}$.
Following \cite{FW} we will henceforth identify the triple
$(u, v, w)$ with the pair $(u \frown v, w)$, and will also regard $u \frown v$ as lying
in $\powerset_\gk \gl^*$.
Following the notation of \cite{FW} we
usually write such a pair as $(\vec u, \vec w)$, where with the conventions we are now using
\begin{enumerate}
\item $\vec u$ is a sequence whose domain is some ordinal $\beta$; we write
$\lh(\vec u) = \gb$ and sometimes with a mild abuse of notation
$\lh( (\vec u, \vec w) ) = \gb$.
$\dom(\vec w) = [1, \gb)$.
\item $u_0$ is an element of $\powerset_\gk \gl^*$. We will define
$\gk_{\vec u} = u_0 \cap \gk$, $\gl_{\vec u} = \ot{u_0}$. We also let
$\gk_{(\vec u, \vec w)} = \gk_{\vec u}$ and
$\gl_{(\vec u, \vec w)} = \gl_{\vec u}$.
\item For $0 < \ga < \gb$, $u_\ga$ is a measure on
$\powerset_{\gk_{\vec u}} \gl_{\vec u} \times V_{\gk_{\vec u}}$, and $w_\ga$ is a
function with domain lying in $u(\ga)$ and with range a subset
of $V_{\gk_{\vec u}}$.
\end{enumerate}
\begin{definition} $(\vec u, \vec w)$ is a {\em good pair} if conditions 1, 2, and 3
above hold.
If $(\vec u, \vec w)$ is a good pair and $u_0 \prec x$ then
$i_{u_0 x}(\vec u, \vec w) = (\vec u^*, \vec w)$ where $\vec u^*$ is the sequence obtained
from $\vec u$ by replacing $u_0$ with $i_{u_0 x}(u_0)$.
\end{definition}
The sequence of lemmas and definitions which follows runs exactly parallel to
the corresponding discussion in \cite{FW}.
We leave the proofs to the reader and just make a few
motivating remarks. We will state some results about the sequence $\vec M$
before defining the functions $g_\ga$, but this is not problematic because these
results do not depend on the exact definition of the $g$'s.
\begin{lemma} \label{blah}
Fix a well-ordering $\lhd$ of $H_{\gl^{+6}}$.
There exists a sequence of mutually disjoint sets $\seq{A_\ga : 0 < \ga < \gl^*}$
such that
if $(\vec u, \vec h) \in A_\gb$ then
\begin{enumerate}
\item $(\vec u, \vec h)$ is a good pair.
\item $\lh(\vec u) < \gl_{\vec u}$, $i_{\vec u}(\lh(\vec u)) = \gb$.
\item The structure $\seq{ H_{sup(u_0)^{+5}}, \in, (\vec u, \vec h), \lhd}$
is elementarily equivalent to the structure
$\seq{ H^M_{sup(j`` \gl^*)^{+5}}, \in, (\vec M \restriction \gb, \vec g \restriction \gb), \lhd}$.
\end{enumerate}
\end{lemma}
The point of this lemma is that $M_\ga$ concentrates on sequences which resemble
$(\vec M \restriction \ga, \vec g \restriction \ga)$. In particular if $(\vec u, \vec h)$
is in $A_\ga$ then since $U_0 \in H^M_{sup(j`` \gl^*)^{+5}}$
it will follow that there is some elementary embedding $i$
such that
\begin{enumerate}
\item $i$ witnesses that $\gk_{\vec u}$ is $\gk^{+4}_{\vec u}$-supercompact.
\item $\vec u$ and $\vec h$ are constructed from $i$ in the same way that $\vec M$ and
$\vec g$ are constructed from $j$.
\end{enumerate}
As in \cite{FW} we may define a class $U_\infty$ of pairs $(\vec u, \vec h)$ such that
if $(\vec u, \vec h) \in U_\infty$ then
\begin{enumerate}
\item $(\vec u, \vec h)$ has the properties from clauses 1, 2 and 3 of Lemma \ref{blah}.
\item Each measure in $\vec u$ concentrates on a subset of $U_\infty$.
\end{enumerate}
Henceforth we assume that all pairs $(\vec u, \vec h)$ which we consider are drawn from
$U_\infty$.
At this point we are finally ready to be more precise about the definition of the functions $g_\ga$
involved in the definition of $\vec M$. At the same time we will define a map $\gp$, which will
eventually be used to define the forcing $\FP$ as a projection of the forcing $\FR$.
$g_\ga$ will be chosen so that $\dom(g_\ga) \in M_\ga$ and
$g_\ga( \vec u, \vec h) \in {\rm Coll}(\gk_{\vec u}^{+6}, < \gk)$ for all $(\vec u, \vec h) \in \dom(g_\ga)$.
%\begin{lemma} Let $0 < \ga < \gl^*$. Let
%\[
% X = \setof{ h }{ \dom(h) \in M_\ga, \forall a \in \dom(h) \; h(a) \in {\rm Coll}(\gk^{+6}_a, <\gk) }
%\]
% Define a preordering $\le_{M_\ga}$ on $X$ by
%
%\[
% h_1 \le M_\ga h_0 \iff \setof{a}{h_1(a) \le h_0(a)} \in M_\ga.
%\]
% Any decreasing sequence in $(X, \le_{M_\ga})$ of length less than or equal to $\gl^*$
% has a lower bound.
%\end{lemma}
We define a map $\gp$ whose domain is $U_\infty$. $\gp(\vec u, \vec f)$ will be defined
by induction on $\gk_{\vec u}$ to be
$(\vec v, \vec {\cal G})$ where
\begin{enumerate}
\item $\lh(\vec v) = \lh(\vec u)$, $\dom(\vec{\cal G}) = \dom(\vec f)$.
\item $v_0 = \gk_{\vec u}$.
\item For $0 < \ga < \lh(\vec v)$
\begin{enumerate}
\item $v_\ga$ is the measure on $V_{v_0}$ defined by
\[
v_\ga = \setof{X \subseteq V_{v_0}}{ \gp^{-1}[X] \in u(\ga)}.
\]
\item ${\cal G}_\ga$ is a certain filter on the Boolean algebra given by the ultraproduct
$Q(\vec v, \ga) = \prod_a {\rm RO}({\rm Coll}(\gk_a^{+6}, < \gk))/v_\ga$. ${\cal G}_\ga$ is generated
by elements of the form $[b(\vec u, \vec f, \vec A, \ga)]_{v_\ga}$, where
we define
\[
b(\vec u, \vec f, \vec A, \ga)(c) = \bigvee \setof{ f_\ga(a) }{a \in A_\ga, \gp(a) = c}
\]
for each $c \in V_{v_0}$ and each sequence $\vec A$ such that $\lh(\vec A) = \lh(\vec v)$
and $A_\ga \in v_\ga$ for all $\ga$.
\end{enumerate}
\end{enumerate}
We define $U^\gp_\infty = \setof{\gp(a)}{a \in U_\infty}$.
Exactly as in \cite{FW} we may now choose the sequence of functions $g_\ga$ in such a way that
$\gp(\vec M, \vec g) = (\vec w, \vec {\cal F})$ with each of the ${\cal F}_\ga$ an ultrafilter
on the appropriate Boolean algebra. To be more precise we use the closure of
${\rm Ult}(V, M_\ga)$ to choose the $g_\ga$ in such a way that for every $b \in Q(\vec w, \ga)$
there is $\vec A$ such that $[b(\vec M, \vec g, \vec A, \ga)]_{w_\ga}$
either refines or is incompatible with $b$.
A key point in the construction that follows is this: if $\vec k$ is a sequence of
functions with $B_\ga = \dom(k_\ga) \in w_\ga$, $B_\ga \subseteq \dom(g_\ga)$ and
$k_\ga(a) \le g_\ga(a)$ for all $a$ then $[b(\vec M, \vec k, \vec b, \ga)]_{w_\ga} \in {\cal F}_\ga$
for all $\ga$. This will be crucial in the proof that $\FP$ collapses only those cardinals
which it ought to.
We are now ready to define the forcing $\FR$. In fact we will define for each $a \in U_\infty$ a
forcing $\FR_a$. Our final $\FR$ will be $\FR_a$ where $a$ is a certain initial segment
of $(\vec M, \vec g)$.
Following \cite{FW} we say that $(\vec u, \vec f, \vec A, \vec k, s)$
is a {\em suitable quintuple} iff
EITHER
\begin{enumerate}
\item $(\vec u, \vec f) \in U_\infty$.
\item $\vec A$ is a sequence such that $A_\ga \in u_\ga$ for $0 < \ga < \lh(\vec u)$.
\item $\vec k$ is a sequence such that for $0 < \ga < \lh(\vec u)$
\begin{enumerate}
\item $\dom(k_\ga) = A_\ga$.
\item For all $a \in A_\ga$, $k_\ga(a) \in {\rm Coll}(\gk_a^{+6}, < \gk_{\vec u})$ and
$k_\ga(a) \le f_\ga(a)$.
\end{enumerate}
\item $s \in {\rm Coll}(\gk_{\vec u}^{+6} < \gk)$.
\end{enumerate}
OR
\begin{enumerate}
\item $\vec u = \langle \ha_0 \rangle$, where by convention we set $\gk_{\vec u} = \ha_0$.
\item $\vec f = \vec A = \vec k = 0$.
\item $s \in {\rm Coll}(\ha_6, <\gk)$.
\end{enumerate}
When $u_0 \prec x$ we let
$i_{u_0 x}(\vec u, \vec f, \vec A, \vec k, s) = (\vec u^*, \vec f, \vec A, \vec k, s)$,
where $u^*$ is the result of replacing $u_0$ by $i_{u_0 x}$ in $u$.
\begin{definition} Let $a = (\vec u, \vec f) \in U_\infty$. A condition in $\FR_a$ is
a sequence
\[
\seq{t_1, \ldots, t_n, (\vec u, \vec f, \vec A, \vec k)}
\]
such that
\begin{enumerate}
\item Each $t_i$ is a suitable quintuple
$t_i = (\vec u_i, \vec f_i, \vec A_i, \vec k_i, s_i)$.
\item $t_1 = (\langle \ha_0 \rangle, 0, 0, 0, s_1)$.
\item $(\vec u_i)_0 \prec (\vec u_{i+1})_0$ for $1 \le i < n$ and
$(\vec u_i)_0 \in \powerset_{\gk_{\vec u}} \gl_{\vec u}$ for $1 \le i \le n$.
\item $\vec A$ is a sequence of measure one sets for $\vec u$. $\dom(k_\gd) = A_\gd$
and $k_\gd(a) \in {\rm Coll}(\gk_a^{+6}, <\gk_{\vec u})$ for all $a \in A_\gd$.
\item $s_i \in {\rm Coll}( \gk_{\vec u_i}^{+6}, < \gk_{\vec u_{i+1}})$ for $1 \le i < n$
and $s_n \in {\rm Coll}( \gk_{\vec u_n}^{+6}, < \gk_{\vec u} )$.
\end{enumerate}
\end{definition}
Intuitively this condition is intended to give a certain amount of information about a
pair of sequences $\vec x$ and $\vec G$ where
\begin{enumerate}
\item $\vec x$ is a continuous and $\prec$-increasing chain of sets from $\powerset_{\gk_{\vec u}} \gl_{\vec u}$
with union $\gl_{\vec u}$.
\item $G_i$ is ${\rm Coll}(\gk_{x_i}^{+6}, < \gk_{x_{i+1}})$-generic.
\end{enumerate}
The ordering on $\FR_a$ will be defined accordingly in Definition \ref{Rdef}.
\begin{definition}
Let $q^0 = (\vec u^0, \vec f^0, \vec A^0, \vec k^0, s^0)$ and
$q^1 = (\vec u^1, \vec f^1, \vec A^1, \vec k^1, s^1)$ be suitable quintuples.
\begin{enumerate}
\item $q^1$ {\em shrinks} $q^0$ iff
\begin{enumerate}
\item $(\vec u^1, \vec f^1) = (\vec u^0, \vec f^0)$.
\item $A^1_\ga \subseteq A^0_\ga$ for all $\ga$.
\item $k^1_\ga(a) \le k^0_\ga(a)$ for all $a \in A^1_\ga$.
\item $s^1 \le s^0$.
\end{enumerate}
\item $q^1$ is {\em addable} to $q^0$ iff
\begin{enumerate}
\item $(\vec u^1)_0 \prec (\vec u^0)_0$ (so we may define a map $i = i_{(\vec u^1)_0 (\vec u^0)_0}$).
\item $i(\lh(\vec u^1)) < \lh(\vec u_0)$.
\item $i(\vec u^1, \vec f^1) \in A^0_{i(\lh(\vec u^1))}$.
\item $i`` A^1_\gd \subseteq A^0_{i(\gd)}$ for $0 < \gd < \lh(\vec u^1)$.
\item $k^1_\gd(a) \le k^0_{i(\gd)}(i(a))$ for all $a \in A^1_\gd$, $0 < \gd < \lh(\vec u^1)$.
\item $s^1 \le k^0_{i(\lh(\vec u^1))}(i(\vec u^1, \vec f^1))$.
\end{enumerate}
\end{enumerate}
\end{definition}
Notice that the definition of addability did not involve $s^0$, so that with a mild abuse of language we may
say that ``$(\vec u^1, \vec f^1, \vec A^1, \vec k^1, s^1)$ is addable to
$(\vec u^0, \vec f^0, \vec A^0, \vec k^0)$''.
\begin{definition}
\label{Rdef}
Let $a = (\vec u, \vec f) \in U_\infty$ and let
\begin{eqnarray*}
p^0 & = & \seq{t^0_1, \ldots, t^0_n, (\vec u, \vec f, \vec A^0, \vec k^0)} \cr
p^1 & = & \seq{t^1_1, \ldots, t^1_m, (\vec u, \vec f, \vec A^1, \vec k^1)} \cr
\end{eqnarray*}
be two conditions in $\FR_a$,
where $t^d_e = (\vec u^d_e, \vec f^d_e, \vec A^d_e, \vec k^d_e, s^d_e)$.
Then $p^1 \le p^0$ iff
\begin{enumerate}
\item $n \le m$.
\item $A^1_\ga \subseteq A^0_\ga$ and $a \in A^1_\ga \implies k^1_\ga(a) \le k^0_\ga(a)$
for all $0 < \ga < \lh(\vec u)$.
\item For every $i$ with $1 \le i \le n$ there is $j$ with $1 \le j \le m$
such that $t^1_j$ shrinks $t^0_i$.
\item For each $j$ with $1 \le j \le m$ one of the following statements holds:
\begin{enumerate}
\item $t^1_j$ is addable to $t^0_1$.
\item There exists $i$ such that $1 \le i \le n$ and $t^1_j$ shrinks $t^0_i$.
\item There exists $i$ such that $1 \le i < n$, $(\vec u^0_i)_0 \prec (\vec u^1_j)_0 \prec (\vec u^0_{i+1})_0$
and $t^1_j$ is addable to $t^0_{i+1}$.
\item $(\vec u^0_n)_0 \prec (\vec u^1_j)_0$ and $i_{(\vec u^1_j)_0}(t^1_j)$ is addable to $(\vec u, \vec f, \vec A^0, \vec k^0)$.
\end{enumerate}
\end{enumerate}
\end{definition}
We are now able to define the projected forcing $\FP$. A {\em suitable quintuple for $\FP$} is
$(\vec v, \vec {\cal G}, \vec B, \vec b, s)$ where
EITHER
\begin{enumerate}
\item $(\vec v, \vec {\cal G}) \in U^\gp_\infty$.
\item $\vec B$ is a sequence of measure one sets for $\vec v$.
\item $b_\gg$ is a function with domain $B_\gg$ such that
$a \in B_\gg \implies b_\gg(a) \in {\rm RO}({\rm Coll}(\gk_a^{+6}, < \gk_{\vec v}))$.
\item $[b_\gg]_{v_\gg} \in {\cal G}_\gg$.
\item $s \in {\rm Coll}(\gk_{\vec v}^{+6}, < \gk)$.
\end{enumerate}
OR
\begin{enumerate}
\item $\vec v = \langle \ha_0 \rangle$, where by convention $\gk_{\vec v} = \ha_0$.
\item $\vec{\cal G} = \vec B = \vec b = 0$.
\item $s \in {\rm Coll}(\ha_6, < \gk)$.
\end{enumerate}
In the natural way we may define a map $\gp$ which takes suitable quintuples for $\FR$ to suitable quintuples
for $\FP$. We let $\gp(\vec u, \vec f, \vec A, \vec k, s) = (\vec v, \vec {\cal G}, \vec B, \vec b, s)$
where
\begin{enumerate}
\item $\gp(\vec u, \vec f) = (\vec v, \vec {\cal G})$.
\item $B_\gg = \gp``A_\gg$.
\item $b_\gg = b(\vec u, \vec k, \vec A, \gg)$.
\end{enumerate}
For the special case of a quintuple of form $q = (\langle \ha_0 \rangle, 0,0,0,s)$ we set
$\gp(q) = q$.
Now we define $\FP_c$ for every $c \in U^\gp_\infty$.
\begin{definition} Let $c = (\vec v, \vec G) \in U^\gp_\infty$.
A condition in $\FP_c$ is
a sequence
\[
\seq{t_1, \ldots, t_n, (\vec v, \vec {\cal G}, \vec B, \vec b)}
\]
such that
\begin{enumerate}
\item Each $t_i$ is a suitable quintuple for $\FP$, say
$t_i = (\vec v_i, \vec {\cal G}_i, \vec B_i, \vec b_i, s_i)$.
\item $t_1 = (\langle \ha_0 \rangle, 0 ,0 ,0 ,s_1)$.
\item $\gk_{\vec v_i} < \gk_{\vec v_{i+1}}$ for $1 \le i < n$ and
$\gk_{\vec v_n} < \gk_{\vec v}$.
\item $\vec B$ is a sequence of measure one sets for $\vec v$.
$b_\gg$ is a function such that $\dom(b_\gg) = B_\gg$,
$b_\gg(a) \in {\rm RO}({\rm Coll}(\gk_a^{+6}, < \gk_{\vec v}))$ for all $a \in B_\gg$,
and
$[b_\gg]_{v_\gg} \in {\cal G}_\gg$.
\item $s_i \in {\rm Coll}( \gk_{\vec v_i}^{+6}, < \gk_{\vec v_{i+1}})$ for $1 \le i < n$
and $s_n \in {\rm Coll}( \gk_{\vec v_n}^{+6}, < \gk_{\vec v} )$.
\end{enumerate}
\end{definition}
If $p = \seq{t_1, \ldots, t_n, (\vec v, \vec {\cal G}, \vec B, \vec b)} \in \FP_c$ then
we will divide $p$ into a ``lower part'' and an ``upper part'', to be more precise
we let $\lp(p) = \seq{t_1, \ldots, t_n}$
and $\up(p) = (\vec v, \vec {\cal G}, \vec B, \vec b)$.
\begin{definition}
Let $q^0 = (\vec v^0, \vec {\cal G}^0, \vec B^0, \vec b^0, s^0)$
and $q^1 = (\vec v^1, \vec {\cal G}^1, \vec B^1, \vec b^1, s^1)$
be suitable quintuples for $\FP$.
\begin{enumerate}
\item $q^1$ {\em shrinks} $q^0$ iff
\begin{enumerate}
\item $(\vec v^1, \vec {\cal G}^1) = (\vec v^0, \vec {\cal G}^0)$.
\item $B^1_\ga \subseteq B^0_\ga$ for all $\ga$.
\item $b^1_\ga(a) \le b^0_\ga(a)$ for all $a \in B^1_\ga$.
\item $s^1 \le s^0$.
\end{enumerate}
\item $q^1$ is {\em addable} to $q^0$ iff
\begin{enumerate}
\item There is $\gg < \lh(\vec v^0)$ such that
$(\vec v^1, \vec {\cal G}^1) \in B_\gg$ and
$s^1 \le b^0_\gg(\vec v^1, \vec {\cal G}^1)$.
\item There is an increasing $e: \lh(\vec v^1) \longrightarrow \lh(\vec v^0)$ such that
$B^1_\ga \subseteq B^0_{e(\ga)}$ and
$b^1_\ga(a) \le b^0_{e(\ga)}(a)$ for all $0 < \ga < \lh(\vec v^1)$ and all
$a \in B^1_\ga$.
\end{enumerate}
\end{enumerate}
\end{definition}
\begin{definition}
Let $c = (\vec v, \vec {\cal G}) \in U^\gp_\infty$ and let
\begin{eqnarray*}
p^0 & = & \seq{t^0_1, \ldots, t^0_n, (\vec v, \vec {\cal G}, \vec B^0, \vec b^0)} \cr
p^1 & = & \seq{t^1_1, \ldots, t^1_m, (\vec v, \vec {\cal G}, \vec B^1, \vec b^1)} \cr
\end{eqnarray*}
be two conditions in $\FP_c$,
where
$t^d_e = (\vec v^d_e, \vec {\cal G}^d_e, \vec B^d_e, \vec b^d_e, s^d_e)$.
Then $p^1 \le p^0$ iff
\begin{enumerate}
\item $n \le m$.
\item $B^1_\ga \subseteq B^0_\ga$ and $a \in B^1_\ga \implies b^1_\ga(a) \le b^0_\ga(a)$
for all $0 < \ga < \lh(\vec v)$.
\item For every $i$ with $1 \le i \le n$ there is $j$ with $1 \le j \le m$
such that $t^1_j$ shrinks $t^0_i$.
\item For each $j$ with $1 \le j \le m$ one of the following statements holds:
\begin{enumerate}
\item $t^1_j$ is addable to $t^0_1$.
\item There exists $i$ such that $1 \le i \le n$ and $t^1_j$ shrinks $t^0_i$.
\item There exists $i$ such that $1 \le i < n$, $(\vec v^0_i)_0 < (\vec v^1_j)_0 < (\vec v^0_{i+1})_0$
and $t^1_j$ is addable to $t^0_{i+1}$.
\item $(\vec v^0_n)_0 < (\vec v^1_j)_0$ and $t^1_j$ is addable to $(\vec v, \vec {\cal G}, \vec B^0, \vec b^0)$.
\end{enumerate}
\end{enumerate}
\end{definition}
At this point we are finally ready to define the forcings $\FP$ and $\FR$. To do this we observe that
by GCH there are only $2^{2^\gk} = \gk^{++}$ many measures on $V_\gk$, so that if
$(\vec w, \vec {\cal F}) = \gp(\vec M, \vec g)$ then the sequence $\vec w$ must
contain a repetition. Accordingly we define $\gl_1 < \gl^*$ to be minimal such that
for some $\gl_0 < \gl_1$ we have $w_{\gl_0} = w_{\gl_1}$. We then set
$\FR = \FR_{(\vec M \restriction \gl_1, \vec g \restriction \gl_1)}$
and $\FP = \FP_{(\vec w \restriction \gl_1, \vec {\cal F} \restriction \gl_1)}$.
The following properties are then proved by easy adaptations of proofs
in \cite{FW}.
\begin{lemma} $\FP_a$ has the $\gk_a^+$-chain condition.
\end{lemma}
\begin{lemma}
\label{basic_P}
$\FP_a$ adds to the universe a sequence
$\vec a$ of members of $U^\gp_\infty$ such that if $\gk_i = \gk_{a_i}$
then $\gk_0 = \ha_0$ and $\vec \gk$ is continuous, increasing and cofinal
in $\gk_a$. $\lh(a_i) > 1$ exactly when $i$ is a limit ordinal.
$\FP_a$ also adds a
sequence $\vec G$ such that $G_i$ is ${\rm Coll}(\gk_i^{+6}, < \gk_{i+1})$-generic
for all $i < \lh(\vec \gk) = \lh(\vec G)$.
\end{lemma}
\begin{lemma} \label{newlemma}
The generic club added by $\FP_a$ is eventually
contained in every set $X \subseteq \gk_a$ such that
$\setof{v}{\gk_v \in X}$ is measure one for all the measures
appearing in $a$. In particular the generic club is eventually
contained in every $>\go$-club subset of $\gk_a$.
\end{lemma}
\begin{lemma}
\label{factorlemma}
Let $a = (\vec v, \vec {\cal G}) \in U^\gp_\infty$, let
\[
p = \seq{t_1, \ldots, t_n, (\vec v, \vec {\cal G}, \vec B, \vec b)} \in \FP_a
\]
with $t_i = (\vec v_i, \vec {\cal G}_i, \vec B_i, \vec b_i, s_i)$.
\begin{enumerate}
\item If $\lh(v_i) > 1$ then
\[
\FP_a/p \simeq \FP_{(\vec v_i, \vec {\cal G}_i)}/q \times \FP_a/r
\]
where
\[
q = \seq{t_1, \ldots, t_{i-1}, (\vec v_i, \vec {\cal G}_i, \vec B_i, \vec b_i)}
\]
and
\[
r = \seq{(\seq{\gk_{\vec v_i}}, \seq{}, \seq{}, \seq{}, s_i), t_{i+1}, \ldots, t_n,
(\vec v, \vec {\cal G}, \vec B, \vec b)}.
\]
\item If also $\lh(\vec v_{i+1}) = 1$ then
$\FP_a/p \simeq \FP_{(v_i, {\cal G}_i)}/q \times {\rm Coll}(\gk_{\vec v_i}^{+6}, < \gk_{\vec v_{i+1}})/s_i
\times \FP_a/r^*$ where
\[
r^* = \seq{ t_{i+1}, \ldots, t_n, (\vec v, \vec {\cal G}, \vec B, \vec b)}.
\]
\end{enumerate}
\end{lemma}
\begin{lemma}
Let $F$ be $\FP_a$-generic with $p \in F$.
Let $\vec a$ be the sequence from $U^\gp_\infty$ added
by $F$, and let $\vec G$ be the corresponding sequence
of generic collapses.
If $\lh(a_i) > 1$ then $V[F] = V[F_1 \times F_2]$ where
\begin{enumerate}
\item $F_1$ is $\FP_{a_i}$-generic.
\item $F_2$ is $\FP_a$-generic.
\item $F_1$ adds $\vec a \restriction i$ and $\vec G \restriction i$.
\item $F_2$ adds $\vec a^*$, $\vec G^*$ where $a^*_j = a_{i+ j}$ and
$G^*_j = G_{i+j}$.
\end{enumerate}
\end{lemma}
Abusing notation, we will denote $F_1$ by ``$F \restriction i$''.
If $p, q$ are conditions in $\FP_a$ we say that $p$ is a
{\em direct extension} of $q$ and write $p \le ^* q$ when $p$ and $q$
are sequences of the same length. To put it another way,
$p$ is obtained from $q$ by merely shrinking the quintuples which are
present in $p$ and the relevant parts of the final quadruple.
We write $p \le^* q$ when $p$ is a direct extension of $q$.
\begin{lemma}
\label{strongfactorlemma}
With the same hypotheses and notation as Lemma \ref{factorlemma},
if $b \in {\rm RO}(\FP_a)$ is a boolean value then there exist
a maximal antichain $A \subseteq \FP_{(\vec v_i, \vec {\cal G}_i)}/q$ and
a condition $r^* \le^* r$ such that $(s, r^*)$ decides $b$ for all
$s \in A$.
\end{lemma}
\begin{lemma}
\label{powersetlemma}
Let $F$ be $\FP_a$-generic and let $\vec \gk$, $\vec G$
be as in Lemma \ref{basic_P}. Let $\gl < \gk_a$ be a cardinal and let
$i$ be the largest limit ordinal such that $\gk_i \le \gl$.
Let $n < \go$ be minimal such that $\gl < \gk_{i+n}^{+6}$.
Let $G^* = F \restriction i \times \prod_{j < n} G_{i+j}$. Then
$G^*$ is
$\FP_{a_i} \times \prod_{j < n} {\rm Coll}(\gk_{i+j}^{+6}, < \gk_{i+j+1})$-generic and
$(\powerset \gl)_{V[F]} = (\powerset \gl)_{V[G^*]}$.
In particular, if $\gl < \gk$ and does not lie in an interval
$(\gk_i^{+6}, \gk_{i+1})$ then $\gl$ is preserved.
\end{lemma}
\begin{lemma} Let $G$ be $\FP$-generic. Then $\gk$ is $\gk^{+5}$-supercompact
in $V[G]$.
\end{lemma}
One more remark is in order before we begin to exploit the forcing
$\FP$. If $A \subseteq \gk$ and $\gk \in j(A)$ then every measure
$w_\ga$ will concentrate on the set of $a$ with $\gk_a \in A$.
Accordingly by forcing below an appropriate condition we may ensure
that the generic sequence $\vec \gk$ added by $\FP$ consists
of points from $A$.
\section{Proof of the main theorem}
\label{reflectionsection}
We are now in a position to prove Theorem \ref{mainthm}.
\begin{proof}[Theorem \ref{mainthm}]
We begin with a model ($V_0$ say) in which GCH holds and $\gk$ is
$\gk^{+5}$-supercompact. We do a Reverse Easton iteration
of the sort described in Theorem \ref{prep} and obtain a model
$V$ in which
\begin{enumerate}
\item GCH holds.
\item $\gk$ is $\gk^{+5}$-supercompact.
\item $\square^B_\gk$ holds.
\end{enumerate}
Let
\[
A = \setof{\ga < \gk}{\hbox{$\ga$ is $\ga^+$-supercompact and $\square^B_\ga$ holds}}.
\]
It is easy to see that $\gk \in j(A)$.
Working in $V$, we now use $j$ to build a forcing notion $\FP$ as in the preceding section.
Let $G_\FP$ be $\FP$-generic over $V$ and let $\vec a$, $\vec \gk$, $\vec G$ be as in
Lemma \ref{basic_P}. Forcing below a suitable condition we may assume that $\gk_i \in A$
for all $i > 0$.
We know by the work of the last section that
\begin{enumerate}
\item $\gk$ is $\gk^{+5}$-supercompact in $V[G_\FP]$.
\item If a cardinal $\gl$ is in $(\gk_i^{+6}, \gk_{i+1})$ for some $i$
then $\gl$ is collapsed to have cardinality $\gk_i^{+6}$ in $V[G_\FP]$.
Otherwise $\gl$ is preserved.
\end{enumerate}
The following claim will establish Theorem \ref{mainthm}.
\begin{claim} Let $\gl<\gk$ be a singular cardinal of
$V[G_\FP]$. Then in $V[G_\FP]$
\begin{enumerate}
\item $\gl = \gk_i$ for some limit $i < \gk$.
\item The combinatorial principle $\square_{\gl, \cf(\gl)}$ holds.
\item If $S = \setof{\ga < \gl^+}{\cf(\ga) < \gl}_V$ then $S$ is stationary.
Moreover
if $\vec S$ is a sequence of stationary subsets of $S$ and
$\lh(\vec S) < \cf(\gl)$ then $\vec S$ reflects to cofinality
$\gm$ for unboundedly many $\gm < \gl$.
\end{enumerate}
\end{claim}
\begin{proof}
We take the various claims in turn.
\begin{enumerate}
\item Since $\gl$ is singular, $\gl$ is a limit cardinal. We have collapsed all but finitely
many cardinals in each interval $[\gk_j, \gk_{j+1})$ so the only possibility is that
$\gl = \gk_i$ for some limit $i$. Notice that we must have $\lh(a_i) > 1$ because
sequences of length $1$ correspond to successor points on the sequence $\vec \gk$.
\item Since $\gl = \gk_i$, $\gl \in A$ and so $V \models \square^B_\gl$.
$\gl^+_V = \gl^+_{V[G_\FP]}$
by Lemma \ref{powersetlemma},
so
$V[G_\FP] \models \square_{\gl, \cf(\gl)}$ by
Lemma \ref{newlemma} and Lemma \ref{squarelemma}.
\item We know $\gl = \gk_i$ where $i$ is limit and $\lh(a_i) > 1$.
By Lemma \ref{powersetlemma} we know
$(\powerset \gl^+)_{V[G_\FP]} = (\powerset \gl^+)_{V[G_\FP \restriction \gl]}$.
Since $\FP_{a_i}$ is $\gl^+$-c.c.~the set $S$ is still stationary in
$V[G_\FP \restriction i]$, and so is stationary in $V[G_\FP]$.
It suffices to prove the desired reflection statement in
$V[G_\FP \restriction \gl]$. We will do this using the fact that $\gl \in A$, so that
$\gl$ is $\gl^+$-supercompact in $V$. Let $G^*_j = G_\FP \restriction j$ for
$j \le \lh(\vec a)$.
Let $\gm = \cf_{V[G_\FP]}(\gl) = \cf_{V[G^*_i]} (\gl)$.
Until further notice we work in $V[G^*_i]$.
Fix a sequence
$\seq{\gt_j: j < \gm}$ which is increasing and cofinal in $i$.
Then
$\seq{\gk_{\gt_j}: j < \gm}$ is increasing and cofinal in $\gk_i = \gl$.
If the reflection claim fails we can find $p \in G^*_i$, $\seq{\dot S_k : k < \gn}$
for some $\gn < \gm$, and $\gs < \gl$ such that $p \forces \dot S_k \subseteq \hat S$ and
\[
p \forces^V_{\FP_{a_i}} \hbox{``$\seq{\dot S_k : k < \gn}$ reflects simultaneously
at no cofinality $>\gs$''.}
\]
Let $S_k$ be the realisation of the term $\dot S_k$ by the generic $G^*_i$.
Then $S_k = \bigcup_{j < \gm} S^j_k$ where
\[
S^j_k = \setof{\ga}{\exists q \in G^*_i \; \lp(q) \in G^*_{\gt_j}, q \forces \ga \in \dot S_k}.
\]
Notice that $S^j_k$ increases with $j$.
For each $k<\gn$ there exists $j<\gm$ such that $S^j_k$ is stationary. Increasing $j$ further
we may also arrange that $S^j_k \cap X_j$ is stationary where
\[
X_j = \setof{\ga < \gk^+}{\cf(\ga) < \gk_{\gt_j}}_V.
\]
Since $\gn < \gm = \cf(\gm)$
we may find $j$ so large that
\begin{enumerate}
\item $S^j_k \cap X_j$ is stationary for all $k$.
\item $\lp(p) \in G^*_{\gt_j}$.
\end{enumerate}
We will now work in $V[G^*_{\gt_j}]$ until further notice. Let
\[
T_k = \setof{\ga}{\exists q \in \FP_{a_i} \; \lp(q) \in G^*_{\gt_j},
q \ \forces \ga \in \dot S_k \cap X_j}.
\]
$T_k \in V[G^*_{\gt_j}] \subseteq V[G^*_i]$, $S^j_k \cap X_j \subseteq T_k$
and $S^j_k \cap X_j$ is stationary in $V[G^*_i]$,
so clearly $T_k$ is stationary in $V[G^*_{\gt_j}]$.
Now $\gl$ is still $\gl^+$-supercompact in $V[G^*_{\gt_j}]$ because $\card{\FP_{a_{\gt_j}}} < \gl$.
$T_k = \bigcup_{p \in G^*_{\gt_j}} \setof{\ga}{p \forces \ga \in \dot T_k}$, and so there is
$p_k \in G^*_{\gt_j}$ such that $Y_k = \setof{\ga}{p_k \forces \ga \in \dot T_k}$ is
stationary.
That is to say, for every $k$ we may find $Y_k \subseteq T_k$ such that $Y_k$ is stationary
and $Y_k \in V$. $\seq{Y_k : k < \gn}$ is not necessarily in $V$
because $p_k$ depends on $k$ and $G^*_{\gt_j}$, but since
$\FP_{a_{\gt_j}}$ is $\gl$-c.c.~we may
find a family $X \in V$ of stationary subsets
of $X_j$ such that $\card{X} < \gl$ and $\setof{Y_k}{k < \gn} \subseteq X$.
We now work in $V$.
Appealing to Fact \ref{solovayfact} we may find $\gg$ such that
all the $Y_k$ reflect at $\gg$, and $\cf(\gg)$ is the successor of an inaccessible
with $\max \{ \gs, \gk_{\gt_j}^{+6} \} < \cf(\gg) < \gl$.
Choose $D \subseteq \gg$ club such that $\ot{D} = \cf(\gg)$.
Using the ``strong factorisation''
property from Lemma \ref{strongfactorlemma}, we work in $V[G^*_{\gt_j}]$ and
choose for each $k < \gn$ and each $\gb \in Y_k \cap D$
an upper part $q(k, \gb)$ such that
\[
\exists r \in G^*_{\gt_j} \; r \frown q(k, \gb) \forces \beta \in \dot S_k.
\]
Since $\max \{ \card{\FP_{a_{\gt_j}}}, \gn, \cf(\gg) \} < \gl$ we may find a set $Y$ of upper parts for
$\FP_{a_i}$ such that $Y \in V$, $\card{Y} < \gl$ and
$\setof{q(k, \gb)}{k< \gn, \gb \in Y_k \cap D} \subseteq Y$. The filters which
appear in $a_i$ are all $\gl$-complete
so we may find a single
upper part $q$ such that $q$ shrinks all the $q(k, \gb)$ as well
as the upper part of $p$.
Shrinking further if necessary we can guarantee that the condition $\seq{q}$
forces that the generic club added by $\FP_{a_i}$ has minimal entry greater than
$\cf(\gg)$.
We force below $\seq{q}$ to get an $H$ which is
$\FP_{a_i}$-generic over $V[G^*_{\gt_j}]$.
Let $\gm^*$ be the least entry on the
generic club added by $H$. We force $h$ which is ${\rm Coll}(\gk_{\gt_j}^{+6}, <\gm^*)_V$-generic
over $V[G^*_{\gt_j}][H]$.
By the second part of Lemma \ref{factorlemma}, $G^*_{\gt_j} \times h \times H$ can be rearranged
as a generic object $G^\dag$ for $\FP_{a_i}$. What is more we have arranged that
$p \in G^\dag$ and that $Y_k \cap D \subseteq \dot S_k^{G^\dag}$ for all $k$.
We will reach a contradiction by showing that $Y_k \cap D$ is stationary
in $V[G^\dag]$.
Since $\ot{D} = \cf_V(\gg)$ we may collapse $Y_k \cap D$ to get $Y'_k$ a stationary
subset of $\cf_V(\gg)$. Since $\gm^* > \cf_V(\gg)$ it will suffice to show
that $Y'_k$ is stationary in $V[G^*_{\gt_j} \times h]$.
$Y'_k$ consists of ordinals of cofinality less than $\gk_{\gt_j}$.
$\cf(\gg)$ is the successor of an inaccessible ($\gth$ say), so $\square^*_\gth$ holds.
So ${\rm Coll}(\gk_{\gt_j}^{+6}, <\gm^*)$
will preserve (see \cite{Komjath}) the stationarity of $Y'_k$. $\card{\FP_{\gt_j}} = \gk_{\gt_j}^+$
so $Y'_k$ (and thus $Y_k \cap D$) will be stationary in $V[G^*_{\gt_j} \times h]$ as required.
We now have a contradiction, so the simultaneous reflection property holds.
\end{enumerate}
\end{proof}
This concludes the proof of Theorem \ref{mainthm}.
\end{proof}
\begin{thebibliography}{99}
\bibitem{SCBDM} A.~Apter, ``Strong compactness and a global version of
a theorem of Ben-David and Magidor'', to appear in the
Mathematical Logic Quarterly.
\bibitem{AH} A.~Apter and J.~Henle, ``On box, weak box and strong compactness'',
Bulletin of the London Mathematical Society {\bf 24} (1992), 513--518.
\bibitem{JEB} J.~Baumgartner, unpublished.
\bibitem{BDM} S.~Ben-David and M.~Magidor, ``The weak $\square^*$ is really weaker
than the full $\square$'', Journal of Symbolic Logic {\bf 51} (1986), 1029-1033.
\bibitem{CFM} J.~Cummings, M.~Foreman and M.~Magidor,
``Scales, squares and stationary reflection'',
submitted to the Journal of Mathematical Logic.
\bibitem{CuSch} J.~Cummings and E.~Schimmerling, ``Indexed squares'',
submitted to the Israel Journal of Mathematics.
\bibitem{FW} M.~Foreman and W.~H.~Woodin, ``The generalized continuum hypothesis can fail everywhere'',
Annals of Mathematics {\bf 133} (1991), 1--35.
\bibitem{JeFS} R.~Jensen, ``The fine structure of the constructible hierarchy'',
Annals of Mathematical Logic {\bf 4} (1972), 229--308.
\bibitem{Komjath} P.~Komjath, ``Stationary reflection for uncountable cofinality'',
Journal of Symbolic Logic {\bf 51} (1986), 147--151.
\bibitem{SCH1} M.~Magidor, ``On the singular cardinals problem I'', Israel Journal of Mathematics
{\bf 28} (1977), 1--31.
\bibitem{Magidor} M.~Magidor, ``Changing cofinality of cardinals'',
Fundamenta Mathematicae {\bf 99} (1978), 61--71.
\bibitem{Prikry} K.~Prikry, ``Changing measurable into accessible cardinals'',
Dissertationes Mathematicae (Rozprawy Matematyczne) {\bf 68}
(1970), 5--52.
\bibitem{Radin} L.~Radin, ``Adding closed cofinal sequences to large cardinals'',
Annals of Mathematical Logic {\bf 22} (1982), 243--261.
\bibitem{Sch} E.~Schimmerling, ``Combinatorial principles in the core model for one Woodin cardinal'',
Annals of Pure and Applied Logic {\bf 74} (1995), 153--201.
\end{thebibliography}
\end{document}