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Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
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\title{Strong Compactness and a
Global Version of a Theorem of Ben-David and Magidor}
\date{July 22, 1999}
\author{Arthur W.~Apter
\thanks{This research was partially supported
by PSC-CUNY Grants 665337 and 667379. In addition, the author wishes to
thank James Cummings and Moti Gitik for helpful conversations
on the subject matter of this paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\}
\begin{document}
\maketitle
\begin{abstract}
Starting with a model in which
$\gk$ is the least inaccessible limit of cardinals $\gd$
which are $\gd^+$ strongly compact, we force and construct
a model in which $\gk$ remains inaccessible and in which,
for every cardinal $\gg < \gk$, $\Box_{\gg^{+ \go}}$ fails but
$\Box_{\gg^{+ \go}, \go}$ holds. This generalizes a result of
Ben-David and Magidor and provides an analogue in the context
of strong compactness to a result of the author and Cummings
in the context of supercompactness.
\end{abstract}
\baselineskip=24pt
In \cite{BDM}, Ben-David and Magidor proved the theorem
Con(ZFC + GCH + $\gk$ is $\gk^+$ supercompact)
$\implies$
Con(ZFC + $\Box^*_{\ha_\omega}$ + $\neg \Box_{\ha_\omega}$).
In \cite{AH}, the author and Henle were able to reduce
the original hypotheses of Ben-David and Magidor and
proved the theorem
Con(ZFC + GCH + $\gk$ is $\gk^+$ strongly compact)
$\implies$
Con(ZFC + $\Box^*_{\ha_\omega}$ + $\neg \Box_{\ha_\omega}$).
Then, in \cite{AC}, the author and Cummings proved a global
version of the result of \cite{BDM} by proving
Con(ZFC + GCH + $\gk$ is $\gk^{+5}$ supercompact)
$\implies$
Con(ZFC + $\gk$ is $\gk^{+5}$ supercompact + For every
singular cardinal $\gl < \gk$, there exists a stationary
$S \subseteq \gl^+$ such that if
$\vec S = \la S_\ga : \ga < \gb < {\rm cof}(\gl) \ra$
is a sequence of stationary subsets of $S$, then
$\vec S$ reflects simultaneously to cofinality $\gd$ for
unboundedly many $\gd < \gl$ + For every singular
cardinal $\gl < \gk$, $\Box_{\gl, {\rm cof}(\gl)}$).
The proof of this result uses the generalized form of
Radin forcing given by Foreman and Woodin in \cite{FW}.
The purpose of this paper is to show that a weaker global
version of the result of \cite{BDM} than the one just stated
can be proven using hypotheses involving strong compactness
instead of supercompactness. Specifically, we prove the following.
\begin{theorem}\label{Theorem}
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{theorem}
The proof of Theorem \ref{Theorem} uses a Gitik style iteration
as given in \cite{G1} and \cite{G2}. Note that since
no degree of strong compactness implies the amount of
closure needed in order to define any sort of Radin
forcing, unless there is some (as yet unknown) way of
producing degrees of supercompactness from degrees of
strong compactness, the above result is the best sort
one can hope to obtain starting from hypotheses
involving just strong compactness.
Before beginning the proof of Theorem \ref{Theorem},
we very briefly give some preliminary information
concerning definitions and notation.
When forcing, $q \ge p$ will mean that
$q$ is stronger than $p$.
Also, if
$\FP$ is our forcing partial ordering, then for
$G \subseteq \FP$ $V$-generic, $V^\FP$ and
$V[G]$ will mean the same thing, and for
$\varphi$ a formula in the forcing language
with respect to $\FP$, $p \in \FP$,
$p \decides \varphi$ will mean that $p$
decides $\varphi$. In addition, we may
from time to time confuse a term with
the set it denotes, especially when the
term denotes a fixed ground model set.
The cardinal $\gk$ will be
$\gk^+$ strongly compact if
$P_\gk(\gk^+) = \{x \subseteq \gk^+ : |x| < \gk\}$
carries a $\gk$-complete, fine measure.
If $\gk < \gl$ are cardinals, $\gk$ is regular,
and $\gl$ is inaccessible, then
${\rm Coll}(\gk, < \gl)$ will be the usual
L\'evy collapse ordering for collapsing all
cardinals $\gd$, $\gk < \gd < \gl$, to $\gk$.
The following definition was given in \cite{AC}.
For convenience, we repeat it now.
\begin{definition}
Let $\gk$ be an infinite cardinal, and let
$\gl \le \gk$ be a cardinal. Then a
$\Box_{\gk, \gl}$ sequence is a sequence
$\la {\cal C}_\ga : \lim(\ga), \ga < \gk^+ \ra$ such that
\begin{enumerate}
\item $1 \le |{\cal C}_\ga| \le \gl$.
\item For every $C \in {\cal C}_\ga$
\begin{enumerate}
\item $C$ is club in $\ga$.
\item $o.t.(C) \le \gk$.
\item For all $\gb \in \lim(C)$, $C \cap \gb \in {\cal C}_\gb$.
\end{enumerate}
\end{enumerate}
We say that $``\Box_{\gk, \gl}$ holds'' iff there exists a
$\Box_{\gk, \gl}$ sequence.
\end{definition}
The principle $\Box_{\gk, \gl}$ was defined by
Schimmerling \cite{Sch} in his work on the core
model for one Woodin cardinal. It is a common
generalization of two principles studied by
Jensen \cite{Je}, the principles
$\Box_\gk = \Box_{\gk, 1}$ and
$\Box^*_\gk = \Box_{\gk, \gk}$.
The following definitions, which were also used in \cite{AC},
are used in this paper as well. We
repeat them here.
\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 ${\rm cof}(\ga) > \go$ 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 ${\rm cof}(\ga) = \gm$ and $S_i \cap \ga$ is
stationary for all
$i < \gb$.
\end{definition}
We turn now to the proof of Theorem \ref{Theorem}.
\begin{proof}
Let
$V \models ``$ZFC + GCH +
$\la \gk_\ga : \ga < \gk \ra$ is a sequence of
cardinals so that each $\gk_\ga$ is
$\gk^+_\ga$ strongly compact and $\gk$ is inaccessible''.
Without loss of generality, we assume that $\gk$ is
the least such cardinal.
We also define $\gd_0 = \go_2$, and for
$0 < \ga < \gk$,
$\gd_\ga = {(\cup_{\gb < \ga} \gk_\gb)}^{++}$.
We are now in a position to define the partial ordering
$\FP$ which will be used in the proof of Theorem \ref{Theorem}.
Since the iteration given in \cite{G1} and \cite{G2}
has Easton supports, with only the order relation defined
slightly differently from usual, we can write
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$,
where we start by letting $\FP_0 = \{ \emptyset \}$.
We take as an inductive hypothesis that
$|\FP_\ga| < \gk_\ga$, and we write
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$ for a certain
term $\dot \FQ_\ga$. Because $\gk$ has been chosen as
the least inaccessible limit of cardinals $\gd$ which
are $\gd^+$ strongly compact, the inductive hypothesis is
preserved at limit stages.
In order to describe $\dot \FQ_\ga$, we first note that
by the L\'evy-Solovay results \cite{LS},
$\forces_{\FP_\ga} ``\gk_\ga$ is $\gk^+_\ga$
strongly compact and any ultrafilter witnessing the
$\gk^+_\ga$ strong compactness or measurability of
$\gk_\ga$ is composed of sets containing ground model
measure 1 sets''. Thus, as in \cite{AH}, we can let
${\cal U} \in V^{\FP_\ga}$ be a $\gk^+_\ga$
strongly compact ultrafilter over $P_{\gk_\ga}(\gk^+_\ga)$
so that the Rudin-Keisler projection ${\cal U}_*$ of
${\cal U}$ to an ultrafilter over $\gk_\ga$ is a
normal measure over $\gk_\ga$ and ${\cal U}$ is
weakly normal in the sense that if
$\forces_{\FP_\ga} ``\dot f : \dot P_{\gk_\ga}(\gk^+_\ga)
\to \gk_\ga$ is so that $\dot f(p) \in p$ on a
${\cal U}$ measure 1 set, then $\dot f$ is
constant on a ${\cal U}$ measure 1 set''.
In analogy to \cite{AH},
in $V^{\FP_\ga}$,
the results of \cite{LS} and the fact $|\FP_\ga| < \gk_\ga$ imply
$D = \{p \in P_{\gk_\ga}(\gk^+_\ga) : p \cap \gk_\ga$
is inaccessible$\} \in {\cal U}$.
$\dot \FQ_\ga$ is a term for $\FQ_\ga \in V^{\FP_\ga}$,
the set of all finite tuples of the form
$\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
satisfying the following properties.
\begin{enumerate}
\item For $1 \le i \le n$, $p_i \in D$ and
$p_i \cap \gk_\ga < p_{i + 1} \cap \gk_\ga$.
\item For $\gk_i = p_i \cap \gk_\ga$,
$f_0 \in {\rm Coll}(\gd_\ga, < \gk_1)$,
$f_i \in {\rm Coll}(\gk^+_i, < \gk_{i + 1})$
for $1 \le i \le n - 1$, and
$f_n \in {\rm Coll}(\gk^+_n, < \gk_\ga)$.
\item We adopt as our notation, for any
$S \subseteq P_{\gk_\ga}(\gk^+_\ga)$,
${[S]}^{< \go} = \{ \la q_1, \ldots, q_m \ra :
m \in \go$ and $q_1 \cap \gk_\ga < q_2 \cap \gk_\ga
< \cdots < q_m \cap \gk_\ga\}$.
We can then write, without ambiguity, that
$A : {[D]}^{< \go} \to {\cal U}$
is a function.
\item If $q \in A(\la p_1, \ldots, p_n \ra)$,
$p_n \cap \gk_\ga < q \cap \gk_\ga$, then
$f_n \in {\rm Coll}(\gk^+_n, < q \cap \gk_\ga)$.
\item Let $D_{p_n} = \{q \in D : p_n \cap \gk_\ga <
q \cap \gk_\ga \}$. $F$ is a function with domain
${[D_{p_n}]}^{< \go}$ so that:
\begin{enumerate}
\item If $q \in A(\la p_1, \ldots, p_n,
q_1, \ldots, q_m \ra)$,
$q_m \cap \gk_\ga < q \cap \gk_\ga$, then
$F(\la q_1, \ldots, q_m, q \ra) \in
{\rm Coll}({(q \cap \gk_\ga)}^+, < \gk_\ga)$.
\item If $q \in A(\la p_1, \ldots, p_n,
q_1, \ldots, q_m \ra)$,
$q_m \cap \gk_\ga < q \cap \gk_\ga < p \cap \gk_\ga$, then
$F(\la q_1, \ldots, q_m, q \ra) \in
{\rm Coll}({(q \cap \gk_\ga)}^+, < p \cap \gk_\ga)$.
\end{enumerate}
\end{enumerate}
In the above, the condition $\pi$ is said to have length $n$,
written $\lh(\pi) = n$.
For $\pi, \pi' \in \FQ_\ga$,
$\pi = \la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$ and
$\pi' = \la q_1, \ldots, q_\ell, g_0, \ldots, g_\ell, B, G \ra$,
$\pi' \ge \pi$ iff the following hold.
\begin{enumerate}
\item $n \le \ell$ and $q_i = p_i$ for $ 1 \le i \le n$.
\item $f_i \subseteq g_i$ for $0 \le i \le n$.
\item $q_i \in A(\la p_1, \ldots, p_n, q_{n + 1},
\ldots, q_{i - 1} \ra)$ and
$F(\la q_{n + 1}, \ldots, q_i \ra) \subseteq g_i$ for
$n < i \le \ell$.
\item $B \subseteq^* A$, i.e., for every
$\la r_1, \ldots, r_m \ra \in {[D_{p_n}]}^{< \go}$,
$B(\la p_1, \ldots, p_n, r_1, \ldots, r_m \ra) \subseteq
A(\la p_1, \ldots, p_n, r_1, \ldots, r_m \ra)$ and
$B(\la p_1, \ldots, p_n \ra) \subseteq
A(\la p_1, \ldots, p_n \ra)$.
\item For all $\la r_1, \ldots, r_m \ra \in
{[D_{q_\ell}]}^{< \go}$ such that for
$1 \le i \le m$,
$r_i \in B(\la q_1, \ldots, q_\ell,\break r_1, \ldots, r_{i - 1} \ra)$,
$\{r : F(\la q_{n + 1}, \ldots, q_\ell, r_1, \ldots, r_m, r \ra)
\subseteq
G(\la r_1, \ldots, r_m, r \ra) \} \in {\cal U}$ and
$\{r : F(\la q_{n + 1}, \ldots, q_\ell, r \ra) \subseteq
G(\la r \ra) \} \in {\cal U}$.
\end{enumerate}
Note that $\FQ_\ga$ is defined in exactly the same manner as the
partial ordering $\FP$ of \cite{AH}, except that
$\go_2$ in the definition of $\FP$ of \cite{AH}
is replaced by $\gd_\ga$. Note also that by our previous
remarks, $\dot \FQ_\ga$ can be taken as a set of terms of
the form
$\la p_1, \ldots, p_n, \dot f_0, \ldots, \dot f_n,
A, \dot F \ra$,
where $\dot f_0, \ldots, \dot f_n$ and $\dot F$
are terms for the appropriate functions but
$p_1, \ldots, p_n$ and $A$ are actual elements of
${(P_{\gk_\ga}(\gk^+_\ga))}^V$ and ${\cal U}$.
Finally, the definition of $\dot \FQ_\ga$
ensures the inductive hypothesis is preserved at
successor stages.
As mentioned earlier, the partial ordering $\le$ on
$\FP$ is a modification of the usual Easton support
ordering. Although its definition can be found in
\cite{G1} and \cite{G2}, for concreteness, we give it
here as well. Specifically, let
$p, q \in \FP$,
$p = \la \dot p_\ga : \ga < \gk \ra$,
$q = \la \dot q_\ga : \ga < \gk \ra$.
Then $q \ge p$ iff $q \ge p$ with respect to the
usual Easton support iteration ordering, but in
addition, for some finite $A \subseteq {\rm support}(p)$
and all $\gb \in {\rm support}(p) - A$,
$q \rest \gb \forces ``\dot q_\gb \ge \dot p_\gb$
and $\lh(\dot q_\gb) = \lh(\dot p_\gb)$''.
Further, if $A = \emptyset$ in the above definition,
then $q$ is called an Easton or direct extension of $p$.
Let $G$ be $V$-generic over $P$. The full generic extension
$V[G]$, in which too many cardinals are collapsed, is not
our desired final model $V'$. In order to define $V'$,
we first define, for each $\ga < \gk$, a subsequence
$H_\ga \subseteq G_\ga$, where $G_\ga$ is the projection of
$G$ onto $\FQ_\ga$. $V'$ will then be taken as
$V[\la H_\ga : \ga < \gk \ra]$.
Let now $\la \gk^*_i : i < \go \ra$ be so that
$\gk^*_i = p_i \cap \gk_\ga$, where $p_i$ is so that
$\exists \pi \in G_\ga[\pi =
\la p_1, \ldots, p_i, \ldots, p_n, f_0, \ldots, f_n,
A, F \ra]$. Let $\la I_i : i < \go \ra$ be so that
$I_i = \{f : \exists \pi \in G_\ga [\pi =
\la p_1, \ldots, p_n, f_0, \ldots, f_{i - 1}, f,
f_{i + 1}, \ldots, f_n, A, F \ra] \}$.
By genericity, both $\la \gk^*_i : i < \go \ra$ and
$\la I_i : i < \go \ra$ are well-defined.
Also, as before, let ${\cal U}$ be the strongly compact ultrafilter
over $P_{\gk_\ga}(\gk^+_\ga)$ used to define $\FQ_\ga$.
As in \cite{AH}, for
$\pi = \la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$,
$\pi' = \la p_1', \ldots, p_\ell', f_0', \ldots, f_\ell', A', F' \ra$,
define an equivalence relation $\sim$ by saying that
$F \sim F'$ iff
$\exists B \in {\cal U} \forall \la q_1, \ldots, q_m \ra
\in {[B]}^{< \go}[F(\la q_1, \ldots, q_m \ra) =
F'(\la q_1, \ldots, q_m \ra)]$. Let
$\la F_i : i < \go \ra$ be so that
$F_i = \{ {[F]}_\sim : \exists \pi \in G_\ga [
F \in \pi$ and $\pi$ has length $n]\}$.
We can finally define
$H_\ga = \la \la \gk^*_i : i < \go \ra,
\la I_i : i < \go \ra, \la F_i : i < \go \ra \ra$.
By Lemma 3 of \cite{AH}, if
$\pi = \la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra
\in \FQ_\ga$
and $\varphi$ is a formula in the forcing language
with respect to $\FQ_\ga$, then there is $\pi' \ge \pi$,
$\pi' = \la p_1, \ldots, p_n, f_0', \ldots, f_n', A', F' \ra$
so that $\pi' \decides \varphi$. Since
$f_0 \in {\rm Coll}(\gd_\ga, < p_1 \cap \gk_\ga)$,
the proof of Lemma 3 of \cite{AH} implies that
$V[\la G_\gb : \gb < \ga \ra]$ and
$V[\la G_\gb : \gb \le \ga \ra]$
have the same subsets of
${(\cup_{\gb < \ga} \gk_\gb)}^+$,
so
$V[\la H_\gb : \gb < \ga \ra]$ and
$V[\la H_\gb : \gb \le \ga \ra]$ have the same subsets of
${(\cup_{\gb < \ga} \gk_\gb)}^+$.
Thus, by the definition of $\FQ_\gb$ for
$\gb \ge \ga$ and Lemmas 1.4 and 1.2 of
\cite{G1}, $V[G]$ and
$V[\la G_\gb : \gb < \ga \ra]$
contain the same subsets of
${(\cup_{\gb < \ga} \gk_\gb)}^+$, so
$V'$ and
$V[\la H_\gb : \gb < \ga \ra]$ contain the same subsets of
${(\cup_{\gb < \ga} \gk_\gb)}^+$.
\begin{lemma}
\label{l1}
If $V' \models ``\gd < \gk$ is a cardinal'', then for some
$\ga < \gk$, $V' \models ``\gd^{+ \go} = \gk_\ga$''.
\end{lemma}
\begin{proof}
Let $\ga$ be so that
$\cup_{\gb < \ga} \gk_\gb \le \gd < \gk_\ga$.
By Lemma 3 of \cite{AH} and the succeeding arguments,
$V[\la G_\gb : \gb < \ga \ra, H_\ga]$ and
$V[\la H_\gb : \gb \le \ga \ra]$ both satisfy
$``\gd^{+ \go} = \gk_\ga$''.
By our observations above, this is true in
$V'$ as well. This proves Lemma \ref{l1}.
\end{proof}
\begin{lemma}
\label{l2}
$V' \models ``$For every $\ga < \gk$,
$\Box_{\gk_\ga, \go}$''.
\end{lemma}
\begin{proof}
By the argument contained on pages 516-517 of \cite{AH}
after the proof of Lemma 3,
$V[\la G_\gb : \gb < \ga \ra, H_\ga]$ and
$V[\la H_\gb : \gb \le \ga \ra]$ both satisfy
$``\gk_\ga$ and ${(\gk^+_\ga)}^V$ are cardinals,
$\gk_\ga$ is inaccessible in $V$, and
${\rm cof}(\gk_\ga) = \go$''.
Therefore, by Fact 1.8 of \cite{AC}
and its proof (see the
proof of Lemma 2.1 of \cite{AC},
the remarks immediately following, and
also \cite{CS}),
%remarks following
%Lemma 2.1 of \cite{AC} and also \cite{CS}),
%(the proof of which is discussed at the end of the
%proof of Lemma 2.1 of \cite{AC}; see also
%\cite{CS}),
$\Box_{\gk_\ga, \go}$ is true in
both of the aforementioned models. By the observation
made prior to Lemma \ref{l1},
$\Box_{\gk_\ga, \go}$ is true in $V'$ as well.
This proves Lemma \ref{l2}.
\end{proof}
\begin{lemma}
\label{l3}
$V' \models ``$For every $\ga < \gk$, there exists a
stationary $S \subseteq \gk^+_\ga$ such that if
$S' \subseteq S$ is stationary, then $S'$ reflects at
$\gd$ for unboundedly many $\gd < \gk^+_\ga$''.
\end{lemma}
\begin{proof}
Take $V[\la G_\gb : \gb < \ga \ra]$ as our ground model.
By the automorphism argument found on pages 517-518 of
\cite{AH}, the filter
${\cal F} = \{ x \subseteq {(\gk^+_\ga)}^V : x \in
V[\la H_\gb : \gb \le \ga \ra]$ and
$\exists m[\la \gd_i : m \le i < \go \ra \subseteq x]\}$, where
$\la \gd_i : i < \go \ra = \la \sup(p_i) : i < \go \ra$
is the generic collapsing sequence through
${(\gk^+_\ga)}^V$ added by $G_\ga$, is an element of both
$V[\la G_\gb : \gb \le \ga \ra]$ and
$V[\la G_\gb : \gb < \ga \ra, H_\ga]$. Thus, since
$V[\la H_\gb : \gb < \ga \ra]$ is a definable submodel of
$V[\la G_\gb : \gb < \ga \ra]$, meaning an element of
$V[\la H_\gb : \gb \le \ga \ra]$ can be represented by
a term in the forcing language with respect to $\FQ_\ga$ in
$V[\la G_\gb : \gb < \ga \ra]$, the same automorphism argument shows
${\cal F} \in V[\la H_\gb : \gb \le \ga \ra]$.
But then, as in \cite{AH}, ${\cal F}$ extends in
$V[\la H_\gb : \gb \le \ga \ra]$ to an ultrafilter
${\cal U}^*$ which is $\gd$-indecomposable for any
$V[\la H_\gb : \gb \le \ga \ra]$-regular cardinal $\gd$ with
$\go < \gd < \gk_\ga$. By Lemma 2.2 of \cite{BDM},
as in both \cite{AH} and \cite{BDM}, this means
$V[\la H_\gb : \gb \le \ga \ra]$ satisfies the conclusions
of this lemma, i.e.,
by Lemmas 2.1 and 2.2 of \cite{BDM},
$V[\la H_\gb : \gb \le \ga] \models ``\neg
\Box_{\gk_\ga}$''.
By the observation made immediately prior to the proof of
Lemma \ref{l1}, both of these facts are true in $V'$ as well.
This proves Lemma \ref{l3}.
\end{proof}
By the observation made immediately prior to the proof of
Lemma \ref{l1}, if $\gd < \gk$ and $x \subseteq \gd$,
$x \in V[G]$, then
$x \in V[\la G_\gb : \gb < \ga \ra]$ for some fixed
$\ga < \gk$ that can be calculated from $\gd$.
Thus, since $\gk$ is inaccessible in $V$, this means that
$\gk$ is a strong limit cardinal in both
$V'$ and $V[G]$. If
$V \models ``\gk$ is Mahlo'', then since $\FP$ is an
Easton support iteration,
$V \models ``\FP$ is $\gk$-c.c.'', i.e., in both
$V'$ and $V[G]$, $\gk$ is regular and so is inaccessible.
The following argument, which was essentially told to the
author by Gitik in August 1995,
shows that the assumption of $\gk$ being Mahlo can be
reduced to $\gk$ being inaccessible.
\begin{lemma}
\label{l4}
$V[G] \models ``\gk$ is regular''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l4} is a modification of the proof of
Lemma 1.3 of \cite{G2}. Let $g = \support(p)$, and let
$p \in \FP$, $p = \la \dot p_\ga : \ga \in g \ra$,
$\gd$, and $\dot f$ be so that
$p \forces ``\dot f : \gd \to \gk$''.
We show that for some direct extension $q \ge p$,
$q \forces ``\dot f$ is bounded in $\gk$''.
We follow closely the proof of Lemma 1.3 of \cite{G2}. Assume
$\la \dot p^*_\gb : \gb < \ga \in g \ra$ has been defined
and that no such $q$ exists. We show how to define
$\dot p^*_\ga$ so that
$\la \dot p^*_\gb : \gb < \ga \ra \forces_{\FP_\ga}
``\dot p^*_\ga \in \dot \FQ_\ga$ is a direct extension of
$\dot p_\ga$''. To do this, first fix $\gg < \gd$, and let
$H \subseteq \FP_{\ga + 1}$ be a $V$-generic set so that
$\la \la \dot p^*_\gb : \gb < \ga \ra, \dot p_\ga \ra
\in H$. As in \cite{G2}, let
$p - (\ga + 1) = \la \dot p_\gb : \gb \in g -
(\ga + 1) \ra$ be the element of $\FP/H$ corresponding to
$p$. This allows us to define
$p^* \in \FP/H$ by $p^*$ is any direct extension
$q \ge p - (\ga + 1)$ which forces
$``\dot f(\gg) < \dot \rho < \gk$'' if such a
$q$ and $\dot \rho$ exist, and
$p^* = p - (\ga + 1)$ otherwise. If we then let
$p^* = \la \dot q^*_\gb : \gb \in g^* \ra$,
it follows that
$g^* \in V[H]$. However, since $\FP_{\ga + 1}$
is of small cardinality, $g^*$ can easily be replaced
by a set in $V$. If we assume that this has already been done,
then as in \cite{G2},
$\la \la \dot p^*_\gb : \gb < \ga \ra, \dot p_\ga,
\dot p^* \ra \in \FP$ is a direct extension of $p$.
We show now there is $\dot q^*_\ga$ so that
$\la \dot p^*_\gb : \gb < \ga \ra \forces_{\FP_\ga}
``\dot q^*_\ga \ge \dot p_\ga$ is a direct extension and
$\la \la \dot p^*_\gb : \gb < \ga \ra, \dot q^*_\ga \ra
\forces_{\FP_{\ga + 1}} `\dot p^* =
\dot p - (\ga + 1)$' ''. To see this, let
$H' \subseteq \FP_\ga$ be $V$-generic, and work in
$V[H']$. Let $\dot q^*_\ga$ be a direct extension of
$p_\ga$ deciding the statement
$``\dot p^* = \dot p - (\ga + 1)$''.
If the decision is negative, then
$q^*_\ga \forces_{\FQ_\ga} ``\dot p^* \forces
`\dot f(\gg) < \dot \rho$' '' for some term
$\dot \rho$ denoting an ordinal $< \gk$. For every
$r \ge \la \la \dot p^*_\gb : \gb < \ga \ra,
\dot q^*_\ga \ra$, $r \in \FP_{\ga + 1}$
for which it is possible, let $\rho_r$
be a value decided by $r$ for $\dot \rho$.
Since $|\FP_{\ga + 1}| < \gk$ and $\gk$ is
regular in $V$,
$\rho = \sup_{r \in \FP_{\ga + 1}, r \ge
\la \la \dot p^*_\gb : \gb < \ga \ra, \dot q^*_\ga \ra}
\rho_r < \gk$ is so that
$\la \la \dot p^*_\gb : \gb < \ga \ra,
\dot q^*_\ga, \dot p^* \ra$ is a direct extension of $p$
forcing $``\dot f(\gg) < \rho$''.
This contradicts our assumption that no such $q$ exists. Thus,
$\la \la \dot p^*_\gb : \gb < \ga \ra, \dot q^*_\ga \ra \forces_
{\FP_{\ga + 1}}
``\dot p^* = \dot p - (\ga + 1)$'', and we take
$\dot p^*_\ga$ as this $\dot q^*_\ga$.
Let $q = \la \dot p^*_\ga : \ga \in g \ra$.
By definition, $q \ge p$ is a direct extension of
$p$. If $r \ge q$, $r \in \FP$ decides a value
$\rho$ for $\dot f(\gg)$, then by the definition of
extension, there is a maximal $\ga \in g$ so that
$r \rest \ga \forces_{\FP_\ga} ``\dot r_\ga$ is not
a direct extension of $\dot q_\ga$''.
This, however, contradicts the definition of
$q$, since $r - (\ga + 1)$ is a direct extension of
$q - (\ga + 1)$, $q - (\ga + 1)$ is a direct extension of
$p - (\ga + 1)$, so $r - (\ga + 1)$ is a direct extension of
$p - (\ga + 1)$ forcing
$``\dot f(\gg) < \rho + 1$''. Thus, there is a direct extension
$q$ of $p$ and an ordinal $\rho < \gk$ so that
$q \forces ``\dot f(\gg) < \rho$''.
And, since $|\FP_\ga| < \gk$ for any $\ga < \gk$,
$V[\la G_\gb : \gb < \ga \ra] \models
``\gk$ is regular''. Hence, without loss of generality,
we can assume that we are forcing over a model
$V[\la G_\gb : \gb < \ga \ra]$ so that for $\pi =
\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
a condition in the first partial ordering of our iteration,
$f_0$ is an element of a L\'evy collapse ordering
that is at least $\gd^+$-closed.
This means by the regularity of $\gk$ in $V$ and
Lemmas 1.4 and 1.2 of \cite{G1} that we can define in
$V$ a direct extension $q \ge p$ so that for some
ordinal $\rho < \gk$ and every $\gg < \gd$,
$q \forces ``\dot f(\gg) < \rho$''. This $q$ is
such that
$q \forces ``\dot f$ is bounded in $\gk$''.
This proves Lemma \ref{l4}.
\end{proof}
Since $V[G] \models ``\gk$ is regular'' and
$V' \subseteq V[G]$,
$V' \models ``\gk$ is regular'' as well.
Thus, Lemmas \ref{l1} - \ref{l4} complete
the proof of Theorem \ref{Theorem}.
\end{proof}
In conclusion, we remark that it would be
desirable, for the
$S \subseteq \gk^+_\ga$ which reflects,
to be able to prove that if $\vec S$
is a finite sequence of stationary subsets of
$S$, then $\vec S$ reflects simultaneously to cofinality
$\gd$ for unboundedly many $\gd < \gk_\ga$.
Whereas a modification of the proof given in
\cite{AC} will work if $\gk_\ga$ is
$\gk^+_\ga$ supercompact, we encounter problems if
$\gk_\ga$ is only $\gk^+_\ga$ strongly compact.
This is since the proof given in \cite{AC} relies
heavily on Solovay's unpublished fact that if
$\gk$ is $\gk^+$ supercompact and $\vec S$ is
a sequence of stationary subsets of
$\{ \ga < \gk^+ : {\rm cof}(\ga) < \gk \}$ such that
$\lh(\vec S) < \gk$, then there are unboundedly many
$\mu < \gk$ so that $\mu$ is the successor of an
inaccessible and $\vec S$ reflects simultaneously to
cofinality $\mu$. As Cummings has pointed out,
we can prove an analogous fact if
$\gk$ is $\gk^+$ strongly compact. We can't
necessarily infer under these circumstances, however, that
$\mu$ is the successor of an inaccessible.
Not being able to ascertain the exact nature of
$\mu$ blocks the proof of \cite{AC} from working
and means that the question of simultaneous
stationary reflection for $\vec S$ a finite
sequence of stationary subsets of $S$ remains open.
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