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\title{Indestructible Strong Compactness but not
Supercompactness
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal, indestructibility, Prikry forcing,
Prikry sequence, non-reflecting stationary set of ordinals,
lottery sum.}
\thanks{The authors wish to acknowledge helpful conversations on
the subject matter of this paper with James Cummings and
Joel Hamkins.}}
% The authors also wish to thank Grigor Sargsyan,
% to whom they owe a huge debt of gratitude.
% Without him, this paper would not be possible.}}
\author{Arthur W.~Apter\thanks{The
first author's research was partially
supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu
\\
\\
Moti Gitik\thanks{The second author's research
was partially supported by ISF Grant 234/08.}\\
Department of Mathematics\\
Tel Aviv University\\
69978 Tel Aviv, Israel\\
http://www.math.tau.ac.il/$\sim$gitik\\
gitik@post.tau.ac.il
\\
\\
Grigor Sargsyan\thanks{This material is
partially based upon work supported by the
National Science Foundation under Grant
No$.$ DMS-0902628.}\\
Department of Mathematics\\
Rutgers University\\
New Brunswick, New Jersey 08904\\
http://grigorsargis.weebly.com\\
grigor@math.rutgers.edu}
\date{October 3, 2011\\
(revised January 15, 2012)}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
Starting from a supercompact cardinal $\gk$,
we force and construct a model in which
$\gk$ is both the least strongly compact
and least supercompact cardinal and $\gk$'s
strong compactness, but not its supercompactness,
is indestructible under arbitrary $\gk$-directed
closed forcing.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
One of the most celebrated results in large
cardinals and forcing is due to Laver \cite{L},
who showed that any supercompact cardinal
$\gk$ can have its supercompactness forced to be
indestructible under arbitrary $\gk$-directed
closed forcing. This raises the following
\bigskip
\noindent Question: Is it possible to force a supercompact
cardinal $\gk$ to have its strong compactness, but not
its supercompactness, indestructible under
arbitrary $\gk$-directed closed forcing?
\bigskip
The purpose of this paper is to answer the
above question in the affirmative.
Specifically, we will prove the following theorem.
\begin{theorem}\label{t1}
%Suppose $V \models ``$ZFC + $\gk$ is supercompact''.
%There is then a partial ordering $\FP \subseteq V$ such that
%$V^\FP \models ``\gk$ is both the least strongly compact
%and least supercompact cardinal + $\gk$'s strong compactness,
%but not its supercompactness, is indestructible under
%arbitrary $\gk$-directed closed forcing''.
Let $V \models ``$ZFC +
$\gk$ is supercompact''.
There is then a partial ordering
$\FP \subseteq V$ %$\card{\FP} = \gk$
such that $V^\FP \models ``\gk$ is both
supercompact and the least strongly
compact cardinal''.
For any $\FQ \in V^\FP$ which is
$\gk$-directed closed,
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact''.
Further, there is $\FR \in V^\FP$ which is
$\gk$-directed closed and nontrivial such that
$V^{\FP \ast \dot \FR} \models ``\gk$
is not supercompact''.
Moreover, for this $\FR$,
$V^{\FP \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank''.
\end{theorem}
\noindent Forcing to obtain a model in which
the least strongly compact cardinal is the
same as the least supercompact cardinal was
of course first done by Magidor in \cite{Ma}.
Key to the proof of Theorem \ref{t1}
(specifically the fact that $\gk$'s supercompactness
is not indestructible)
is the following
result due %theorem due
to the second author, which we will prove as well.
\begin{proposition}\label{p1}
Suppose $\gk$ is a Mahlo cardinal and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga \le \gk \ra$
is an Easton support iteration of length $\gk + 1$ satisfying
the following properties.
\begin{enumerate}
\item $\FP_0 = \{\emptyset\}$.
\item For each $\ga < \gk$,
$\forces_{\FP_\ga} ``\card{\dot \FQ_\ga} < \gk$''.
\item $\forces_{\FP_\gk} ``\dot \FQ_\gk$ is
${<} \gk$-strategically closed''.
%\item For some $\ga < \gk$, $\forces_{\FP_\ga} ``\dot \FQ_\ga$
%is nontrivial''.
\item For some $\ga, \gd < \gk$,
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ adds a new subset of $\gd$''.
\item $\gk$ is Mahlo in $V^{\FP_{\gk + 1}} = V^\FP$.
\end{enumerate}
Then in $V^\FP$, %$V^{\FP_{\gk + 1}}$,
there are no fresh subsets of $\gk$.
\end{proposition}
We note that Proposition \ref{p1} is an %weak
analogue of results due to Hamkins
(see \cite{H3, H2, H03}).
Adopting the terminology of these papers,
Hamkins shows that for a suitably large
cardinal $\gk$ (measurable, supercompact, etc.)
and an iteration $\FP$ {\it admitting
a gap below $\gk$}, after forcing with $\FP$,
there are no fresh subsets of $\gk$.
The iterations we consider need not be
gap forcings, yet they retain
this crucial property vital to the proof
of Theorem \ref{t1}.
%the crucial property of adding no fresh subsets of $\gk$.
%for what he calls {\it gap iterations},
Before beginning the proofs of our theorems, we briefly mention
some preliminary information and terminology.
%Before beginning the proofs of
%Theorems \ref{t1} and
%\ref{p1}, we briefly
%mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in the usual interval notation.
If $\gk \ge \go$ is a regular cardinal, then
$\add(\gk, 1)$ is the standard partial ordering
for adding a single Cohen subset of $\gk$.
When forcing, $q \ge p$ will mean that
{\it $q$ is stronger than $p$}.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
The abuse of notation mentioned above will be compounded
by writing $x \in V^\FP$ instead of $\dot x \in V^\FP$.
Any term for trivial forcing will
always be taken as a term for the
partial ordering $\{\emptyset\}$.
If $\varphi$ is a formula in the forcing language
with respect to $\FP$ and $p \in \FP$, then
$p \decides \varphi$ means that
{\it $p$ decides $\varphi$}.
If $\FP$ is
an arbitrary partial ordering
and $\gk$ is a regular cardinal,
%$\FP$ is {\it $\gk$-distributive}
%if for every sequence
%$\la D_\ga : \ga < \gk \ra$
%of dense open subsets of $\FP$,
%$\bigcap_{\ga < \gk} D_\ga$
%is dense open.
%$\FP$ is {\it $\gk$-closed}
%if every increasing chain of elements
%$\la p_\ga : \ga < \gk \ra$ of $\FP$
%has an upper bound $p \in \FP$.
$\FP$ is {\it $\gk$-directed closed}
if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga \mid \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha \mid \alpha < \delta \rangle$
is {\it directed} if
every two elements
$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
%If in addition, any directed subset of
%$\FP$ of size $\gk$ has an upper bound,
%then $\FP$ is said to be
%{\it ${\le} \gk$-directed closed}.
$\FP$ is {\it $\gk$-strategically closed} if in the
two person game
of length $\gk + 1$
in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga \le \gk\rangle$, where player I plays odd
stages and player
II plays even stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\it ${<}\gk$-strategically closed} if
$\FP$ is $\gd$-strategically closed for every
$\gd < \gk$.
Note that if $\FP$ is
$\gk$-directed closed, then $\FP$ is
${<}\gk$-strategically closed
(so since $\add(\gk, 1)$ is $\gk$-directed
closed, $\add(\gk, 1)$ is ${<}\gk$-strategically
closed as well).
We adopt Hamkins' terminology of \cite{H3, H2, H03}
and say that {\it $x \subseteq \gk$
is a fresh subset of $\gk$
with respect to $\FP$} if
$\FP$ is nontrivial forcing,
$x \in V^\FP$, $x \not\in V$, yet
$x \cap \ga \in V$ for every $\ga < \gk$.
%Note that
%if $\FP$ is $\gk$-strategically closed,
%then $\FP$ is $\gk$-distributive.
%Also, if $\FP$ is
%$\gk$-strategically closed and
%$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$\FP$ is {\it ${\prec}\gk$-strategically closed} if in the
%two person game in which the players construct an increasing
%sequence
%$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
%stages and player
%II plays even and limit stages (again choosing the
%trivial condition at stage 0),
%then player II has a strategy which
%ensures the game can always be continued.
From time to time within the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik iterations}.
By this we will mean an Easton support iteration
as first given by the second author in \cite{G},
to which we refer readers for a discussion
of the basic properties of
and terminology associated with such an iteration.
For the purposes of this paper,
at any stage $\gd$ at which
a nontrivial forcing is done in a Gitik iteration,
we assume the partial ordering
$\FQ_\gd$ with which we force is
either $\gd$-directed closed or is
Prikry forcing defined with respect to
a normal measure over $\gd$
(although other types of partial orderings
may be used in the general case --- see
\cite{G} for additional details).
%By Lemmas 1.2 and 1.4 of \cite{G},
%if $\gd_0$ is the first stage in the
%definition of $\FP$ at which a nontrivial
%forcing is done, then forcing with
%$\FP$ adds no bounded subsets to $\gd_0$.
We recall for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\it lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
Finally, we mention that we are assuming
familiarity with the large cardinal
notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{J}
or \cite{SRK} for further details.
We do note, however, that the measurable
cardinal $\gk$ is said to have {\it trivial Mitchell rank} if
there is no elementary embedding
$j : V \to M$ generated by a
normal measure ${\cal U}$ over $\gk$ such that
$M \models ``\gk$ is a measurable cardinal''.
We explicitly observe that if $\gk$ has trivial
Mitchell rank, then $\gk$ is not supercompact
(and in fact, if $\gk$ has trivial Mitchell
rank, then $\gk$ is not even $2^\gk$ supercompact).
\section{The Proofs of Theorem \ref{t1} and
Proposition \ref{p1}}\label{s2}
We begin with the proof of Proposition \ref{p1}.
\begin{proof}
Suppose $\gk$ is a Mahlo cardinal,
$\FP = \FP_{\gk + 1} = \la \la \FP_\ga,
\dot \FQ_\ga \ra \mid \ga \le \gk \ra$ is an
Easton support iteration of length $\gk + 1$
with $\FP_0 = \{\emptyset\}$,
for each $\ga < \gk$,
$\forces_{\FP_\ga} ``\card{\dot \FQ_\ga} < \gk$'',
$\forces_{\FP_\gk} ``\dot \FQ_\gk$ is
${<}\gk$-strategically closed'',
and $\gk$ is Mahlo in $V^\FP$.
Assume without loss of generality that
forcing with $\FQ_0$ over $V$ adds a
new subset of $\go$
%$\dot \FQ_0$ is a term for a partial ordering
%adding a new subset of $\go$.
(so $\forces_{\FP_0} ``\dot \FQ_0$
adds a new subset of $\go$'').
Let $G = G_{\gk + 1} = G_\gk \ast G(\gk)$
be $V$-generic over $\FP = \FP_\gk \ast \dot \FQ_\gk$
(so $G_1$ is $V$-generic over $\FP_1 = \FP_0 \ast \dot \FQ_0$),
and let $\dot x$ be a term such that
$\forces_{\FP} ``\dot x$ is a fresh subset of $\gk$''.
Work for the time being in $V[G]$.
Because $\forces_{\FP} ``\dot x$ is a fresh
subset of $\gk$'', it is the case that
$\forces_{\FP} ``$For every $\eta < \gk$,
$\dot x \cap \eta \in \check V$''.
Therefore, for every inaccessible cardinal
$\eta < \gk$, we may choose a condition
$p(\eta) = \la \dot p_\nu \mid \nu \in \supp(p(\eta)) \ra
\in G$ deciding $\dot x \cap \eta$.
Since $\FP$ is an Easton support iteration,
$\supp(p(\eta)) \cap \eta$ is a bounded subset of $\eta$.
Now, using the fact that $\gk$ is Mahlo in $V[G]$ and
Fodor's Theorem applied to the function
$f(\eta) =$ The least $\xi$ such that
${\rm supp}(p(\eta)) \cap \eta \subseteq \xi$,
let $S \subseteq \gk$, $S \in V[G]$ be stationary
such that for $\eta_1, \eta_2 \in S$,
$\supp(p(\eta_1)) \cap \eta_1 = \supp(p(\eta_2)) \cap \eta_2$.
Since
%$\la {\rm supp}(p(\eta) \cap \eta) \mid \eta \in S \ra$
$\la p(\eta) \rest \eta \mid \eta \in S \ra$
forms a $\Delta$-system, working in $V[G]$, we can
find $\eta^* < \gk$ and $p^* \in \FP_{\eta^*}$
such that for $\gk$ many values of $\eta$, it is the case that
$p(\eta) \rest \eta = p^*$.
This means we can choose a condition
$q^{**} \in G$, $q^{**} \ge p^*$ such that
$q^{**} \forces_{\FP} ``$For $\gk$ many ordinals $\eta < \gk$,
there is $p(\eta) \in \dot G$ such that
$p(\eta) \rest \eta = q^{**} \rest \eta^*$ and
$p(\eta) \decides \dot x \cap \eta$''.
For the remainder of the proof of Proposition \ref{p1},
let $q^{**} \rest \eta^* = p^{**}$.
Work now in $V$.
Using the properties of $q^{**}$ given
in the last sentence of the preceding paragraph,
we define by induction a %an increasing
sequence $\la p^t \mid t \in 2^{< \go} \ra$
of conditions in $\FP$,
a sequence of ordinals
$\la \eta^t \mid t \in 2^{< \go} \ra$,
and a sequence
$\la z^t \mid t \in 2^{< \go} \ra$ such that the following hold.
\begin{enumerate}
\item\label{c1}
If $t$ extends $s$, then $\gk > \eta^t > \eta^s$.
\item\label{c2} $p^t \rest \eta^t = p^{**}$.
\item\label{c3} If $t$ extends $s$, then $\sup(\supp(p^s) \cap \gk)
< \eta^t$.
\item\label{c4} If $t$ extends $s$, then $p^t \ge p^s - \eta^t$.
\item\label{c5} $p^t \forces ``\dot x \cap \eta^t = z^t$''.
\item\label{c6} If $p^t$ and $p^s$ are incompatible, then for some
$\ga \le \max(\sup(z^t), \sup(z^s))$, it is the case that
$z^t \cap \ga \neq z^s \cap \ga$.
\end{enumerate}
We begin the induction by picking $p^{\la \ra}$ to be a
condition such that for $\eta < \gk$ and some
$z \subseteq \eta$, $p^{\la \ra} \rest \eta = p^{**}$ and
$p^{\la \ra} \forces ``\dot x \cap \eta = z$''.
Set $\eta^{\la \ra} = \eta$ and $z^{\la \ra} = z$.
Suppose now that $p^t$, $\eta^t$, and $z^t$ are all defined.
We define
$p^{t^\frown 0}$, $\eta^{t^\frown 0}$, $z^{t^\frown 0}$,
$p^{t^\frown 1}$, $\eta^{t^\frown 1}$, and $z^{t^\frown 1}$.
Note that there must be $\eta > \eta^t$,
$z, y \subseteq \eta$ with $z \neq y$, and $p, q \in \FP$ such that
$p \rest \eta = p^{**} = q \rest \eta$,
$p, q \ge p^t - \eta$,
$p \forces ``\dot x \cap \eta = z$'', and
$q \forces ``\dot x \cap \eta = y$''.
If not, then it is possible to decide $\dot x$ completely in
$V$, contradicting our hypothesis that
$\forces_{\FP} ``\dot x$ is a fresh subset of $\gk$''.
%We consequently choose such
%$\eta$, $p$, $q$, $z$, and $y$ and define
%$p^{t^\frown 0}$, $\eta^{t^\frown 0}$, $z^{t^\frown 0}$,
%$p^{t^\frown 1}$, $\eta^{t^\frown 1}$, and $z^{t^\frown 1}$
%accordingly, using a fixed strategy ${\cal S}$
%to pick $p$ and $q$.
We consequently choose such
$\eta$, $p$, $q$, $z$, and $y$ and define
$p^{t^\frown 0} = p$, $\eta^{t^\frown 0} = \eta$, $z^{t^\frown 0} = z$,
$p^{t^\frown 1} = q$, $\eta^{t^\frown 1} = \eta$,
and $z^{t^\frown 1} = y$,
going under the assumption that we have fixed
at the beginning of the construction a term $\dot {\cal S}$
such that
$\forces_{\FP_\gk} ``\dot {\cal S}$ is a strategy for
$\dot \FQ_\gk$ which has been used to choose
$\dot p_\gk$ and $\dot q_\gk$''.
%using a fixed strategy ${\cal S}$
%to pick $p$ and $q$.
This completes our induction.
Note that for every $V[G_1]$-branch $f : \go \to 2$
through $2^{< \go}$, there is an upper bound
$p^f \in \FP / G_1$ for the sequence
$\la p^{f \rest n} \mid n < \go \ra$. This is since
conditions %(\ref{c1}),
(\ref{c2}) and (\ref{c3}) ensure that
if $t$ extends $s$, then
$p^t \rest \eta^t = p^s \rest \eta^s
= p^{**}$ and $\supp(p^s) \cap \gk < \eta^t$.
Thus, the only common elements of the supports of
$p^t$ and $p^s$ below $\gk$ are those of $p^{**}$.
By condition (\ref{c2}), these agree on
$\supp(p^{**})$.
By condition (\ref{c4}),
$p^t$ extends $p^s$ on $\gk$,
the only common coordinate where conditions extend.
%This means that
%the only common coordinate where conditions extend is $\gk$.
Hence, because %by the fact that
$\forces_{\FP_\gk} ``\dot \FQ_\gk$ is
${<}\gk$-strategically closed and
$\la \dot p^{f \rest n}_\gk \mid n < \go \ra$
was constructed using $\dot {\cal S}$'',
it is possible to define an upper bound $p^f$
in $V[G_1]$ roughly speaking by putting together
$p^{**}$, everything in the union of the supports
of each $p^{f \rest n}$ below $\gk$, and
an upper bound to those conditions occurring
at coordinate $\gk$.
Because $(p^{f \rest n})_0 = p^{\la \ra} \rest 1 \in G_1$
for each $n < \go$,
$p^f$ is a well-defined condition in $\FP/G_1$.
%this definition makes sense.
%Even though this definition appears as though
%it is being given in $V[G_1]$, since we are
%working with an Easton support iteration,
%it can actually be given in $V$.
Working now in $V[G_1]$
(or $V[H_1]$ where $H_1 \subseteq \FP_1$ is
$V$-generic over $\FP_1$ with $p^{\la \ra} \rest 1 \in H_1$),
define $\eta^f = \bigcup_{n < \go} \eta^{f \rest n}$ and
$z^f = \bigcup_{n < \go} z^{f \rest n}$. Since
$V[G_1] \models ``\gk$ is a Mahlo cardinal'',
$\eta^f < \gk$. Consequently, because
$p^f \forces_{\FP/G_1} ``\dot x \cap \eta^f = z^f$'' and
$\forces_{\FP} ``$For every $\eta < \gk$,
$\dot x \cap \eta \in \check V$'', $z^f \in V$.
In addition, by condition (\ref{c6}), if
$g : \go \to 2$, $g \in V[G_1]$, $g \neq f$ is
another branch through $2^{< \go}$ such that $g \neq f$,
then $z^g \neq z^f$.
%$g_2 : \go \to 2$,
%$g_1, g_2 \in V[G_1]$ are two branches through
%$2^{< \go}$ such that $g_1 \neq g_2$, then $z^{g_1} \neq z^{g_2}$.
Moreover, for every $s \in 2^{< \go}$, $z^s$ is an
initial segment of $z^f$ iff $s$ is an initial segment of $f$.
Let $H_1$ be $V$-generic over $\FP_1$
such that $p^{\la \ra} \rest 1 \in H_1$,
and let $r : \go \to 2$
be a new real added.
Let $H$ be a $V$-generic subset of $\FP$
extending $H_1$ such that $p^r \in H$.
Consider $z^r$. By our observations in
the preceding paragraph, $z^r \in V$, and
$r$ can be reconstructed from $z^r$ as the set
$\bigcup \{s \in 2^{< \go} \mid z^s$ is an
initial segment of $z^r\}$.
It thus immediately follows that $r \in V$.
%Since for each $n < \go$, the only
%member of the $n^{\rm th}$ level of the
%inductive construction that $p^r$ can extend is
%$p^{r \rest n}$,
%$r$ is definable in $V$ as
%$\bigcup \{s \in 2^{< \go} \mid p^r \ge p^s\}$.
This contradiction to the fact that
$(\dot x)_H$ is a fresh subset of
$\gk$ with respect to $\FP$ completes
the proof of Proposition \ref{p1}.
\end{proof}
Suppose that for some
$\ga, \gd < \gk$ with $\gd \neq \go$,
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ adds a new subset of
the cardinal $\gd$''.
Assume without loss of generality that
$\ga = 0$ and $\gd$ is the least cardinal to
which a new subset is added.
We explicitly observe that since
$\forces_{\FP_\gk} ``\dot \FQ_\gk$
is ${<}\gk$-strategically closed'',
the same inductive construction as given above
remains valid, with every occurrence of $\go$
replaced by an occurrence of $\gd$.
Having completed the proof of Proposition \ref{p1},
we turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + $\gk$ is supercompact''.
Without loss of generality, we assume that
$V \models {\rm GCH}$ as well.
For any ordinal $\gd$, let
$\gd'$ be the least
$V$-strong cardinal above
$\gd$.
The partial ordering
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk \ra$
to be used in the proof of
Theorem \ref{t1}
is the Gitik iteration of length $\gk$
which has the following properties.
%which is defined as follows.
\begin{enumerate}
\item $\FP$ begins by forcing with $\add(\go, 1)$, i.e.,
$\FP_0 = \{\emptyset\}$ and
$\forces_{\FP_0} ``\dot \FQ_0 = \dot \add(\go, 1)$''.
\item The only stages at which
$\FP$ (possibly) does nontrivial forcing are
those ordinals $\gd$ which are, in $V$,
Mahlo limits of strong cardinals.
At such a stage $\gd$,
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FL_\gd
\ast \dot \FR_\gd$, where $\dot \FL_\gd$
is a term for the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$.
\item If either $V^{\FP_\gd \ast \dot \FL_\gd} =
V^{\FP_\gd}$, i.e., the lottery selects
trivial forcing at stage $\gd$, or
$V^{\FP_\gd \ast \dot \FL_\gd} \models
``\gd$ is not measurable'', then
$\dot \FR_\gd$ is a term for trivial forcing.
\item If $V^{\FP_\gd \ast \dot \FL_\gd} \models
``\gd$ is measurable'' and
$V^{\FP_\gd \ast \dot \FL_\gd} \neq
V^{\FP_\gd}$, i.e., the lottery selects
nontrivial forcing at stage $\gd$, then
$\dot \FR_\gd$ is a term for Prikry forcing
defined with respect to some normal measure
over $\gd$.
\end{enumerate}
The intuition behind the
above definition of
$\FP$ is as follows.
The fact that nothing is
done at stage $\gd$ when the
lottery selects trivial forcing,
i.e., that no Prikry sequence is added,
ensures that
$V^\FP \models ``\gk$ is supercompact''.
Since a Prikry sequence is added when
a nontrivial forcing at stage $\gd$
preserves the measurability of $\gd$,
there will be a partial ordering
$\FR \in V^\FP$ such that
$V^{\FP \ast \dot \FR} \models ``\gk$
is not supercompact''.
%$\gk$'s supercompactness will be destroyed.
The lottery sum at stage $\gd$, in
conjunction with the Prikry forcing, will
allow us to show that in $V^\FP$,
$\gk$'s strong compactness is preserved
by nontrivial forcing.
Because unboundedly many in $\gk$
Prikry sequences will have been added
by $\FP$, $V^\FP \models ``$No cardinal
below $\gk$ is strongly compact'', i.e.,
$V^\FP \models ``\gk$ is the least strongly
compact cardinal''.
The following lemmas show that
$\FP$ is as desired.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A06a}.
Let $\gl \ge \gk^+ = 2^\gk$
be any regular cardinal.
Take $j : V \to M$ as an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk$
such that $M \models ``\gk$
is not $\gl$ supercompact''.
By \cite[Lemma 2.1]{AC2},
in $M$, $\gk$ is a
Mahlo limit of
strong cardinals. This means
by the definition of $\FP$ that
it is possible to opt for trivial forcing in
the stage $\gk$ lottery
held in $M$ in the
definition of $j(\FP)$.
Further, $M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is
strong''. This is since
otherwise, in $M$,
$\gk$ is supercompact
up to a strong cardinal,
so by the proof of \cite[Lemma 2.4]{AC2},
$\gk$ is supercompact in $M$.
Consequently, in $M$, above the appropriate condition,
$j({\FP})$ is forcing equivalent to ${\FP} \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gl$.
We may now apply the argument of \cite[Lemma 1.5]{G}.
Specifically, let $G$ be $V$-generic over ${\FP}$.
%By the definition of ${\FP}$,
%$j '' G = G$.
Since GCH in $V$ implies that
$V \models ``2^{\gl} = \gl^+$'', we may let
$\la \dot x_\ga \mid \ga < \gl^+ \ra$ be an
enumeration in $V$ of all of the
canonical ${\FP}$-names of subsets of
$P_\gk(\gl)$.
By \cite[Lemmas 1.4 and 1.2]{G} and the
fact that $M^{\gl} \subseteq M$, we may
define an increasing sequence
$\la p_\ga \mid \ga < \gl^+ \ra$
of elements of $j({\FP})/G$
such that if $\ga < \gb < \gl^+$,
$p_\gb$ is an
Easton extension of $p_\ga$,\footnote{Roughly
speaking, this means that
$p_\gb$ extends $p_\ga$ as in a usual
reverse Easton iteration, except that
at coordinates at which Prikry forcing occurs in $p_\ga$,
measure 1 sets are shrunk and stems are not
extended. For a more precise definition,
readers are urged to consult \cite{G}.}
every initial segment of
the sequence is in $M$, and for every
$\ga < \gl^+$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x_\ga)$''.
The remainder of the argument of
\cite[Lemma 1.5]{G} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_\gk(\gl))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_\gk(\gl))}^{V[G]}$ and
for some $\ga < \gl^+$ and some
${\FP}$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j({\FP})/G}
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gk(\gl)$.
Thus, $\forces_{{\FP}} ``\gk$ is
$\gl$ supercompact''.
Since $\gl$ was arbitrary,
%$V^\FP \models ``\gk$ is supercompact''.
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk$-directed
closed. Then
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.2]{A06}.
Suppose $\FQ \in V^\FP$
is $\gk$-directed closed.
Let $\gl > \max(\gk, |{\rm TC}(\dot \FQ)|)$ be an
arbitrary regular cardinal large enough so that
$(2^{[\gl]^{< \gk}})^V = \gr =
(2^{[\gl]^{< \gk}})^{V^{\FP \ast \dot \FQ}}$ and
$\gr$ is regular in both $V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \gr^+ = 2^\gr$.
%Let $\gs > \gl > \max(|{\rm TC}(\dot \FQ)|, \gk)$
%be sufficiently large regular cardinals, and let
%Let $\gs > \gl$ be a sufficiently large
%regular cardinal, and take
%be sufficiently large regular cardinals, and let
%be a sufficiently large regular cardinal.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
As in Lemma \ref{l1}, by \cite[Lemma 2.1]{AC2},
$\gk$ is a Mahlo limit of strong cardinals in $M$.
Consequently, by the choice of $\gs$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j(\FP)$.
Further, as in Lemma \ref{l1}, since
$M \models ``$No cardinal
$\gd \in (\gk, \gs]$ is strong'',
the next nontrivial
forcing in the definition of
$j(\FP)$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
$j(\FP \ast \dot \FQ)$ is forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either Prikry forcing
or trivial forcing''.
The remainder of the proof of Lemma \ref{l2}
is as in the proof of \cite[Lemma 2]{AG}.
As in the proof of Lemma \ref{l1},
we outline the argument,
%For concreteness, we provide a sketch
%of the proof,
and refer readers to the
proof of \cite[Lemma 2]{AG} for any
missing details.
%By the last sentence of the
%preceding paragraph, in $M$,
%$j(\FP \ast \dot \FQ)$ is
%forcing equivalent to
%$\FP \ast \dot \FQ \ast \dot \FS_\gk
%\ast \dot \FR \ast j(\dot \FQ)$, where
%$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
%is a term for either Prikry forcing
%or trivial forcing''. Further, since
%$M \models ``$There are no Mahlo
%cardinals in the interval
%$(\gk, \gs]$'', the next nontrivial
%stage in the definition of
%$j(\FP)$ after $\gk$ takes place
%well above $\gs$. Consequently,
By the last two sentences of the preceding paragraph,
as in \cite[Lemma 2]{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of
${(P_\gk(\gl))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
$\FP \ast \dot \FQ \ast \dot \FS_\gk$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga \mid \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
If we then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gl]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gl)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FS_\gk$
either the trivial condition
(if $\FS_\gk$ is trivial forcing)
or of the
form $\la \emptyset, B \ra$
(if $\FS_\gk$ is Prikry forcing), in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) \mid \gb < \gl \ra \in
\dot X$'' for some name $\dot X$ of $X$,
then as in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gl$ strongly compact''.
Since $\gl$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l2}.
%Since $\gl$ was arbitrary,
%$V^{\FP \ast \dot \FQ} \models ``\gk$ is
%strongly compact''.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
Let $\gl = \gk^{+ \go}$.
Take $j : V \to M$ as an elementary embedding witnessing
the $\gl$ supercompactness of $\gk$.
Suppose $\FQ \in V^\FP$ is Add$(\gk, 1)$.
%i.e., the partial ordering for adding
%one Cohen subset of $\gk$.
By Lemma \ref{l2}, $V^{\FP \ast \dot \FQ} \models
``\gk$ is measurable'' (since
$V^{\FP \ast \dot \FQ} \models ``\gk$ is strongly compact'').
Because $\gl$ has been chosen large enough,
it therefore follows that
$M^{\FP \ast \dot \FQ} \models ``\gk$ is measurable''.
In addition, as in Lemma \ref{l2}, it is possible to opt
for $\FQ$ in the stage $\gk$ lottery held in $M$ in
the definition of $j(\FP)$. Therefore, by the
definition of $\FP$, above the appropriate condition,
$(j(\FP \ast \dot \FQ))_{\gk + 1} =
\FP_\gk \ast \dot \FQ_\gk = \FP_{\gk + 1}$ is forcing
equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS_\gk$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$ is
Prikry forcing defined over $\gk$''.
%This means that above the appropriate condition,
%$\forces_{\FP_\gk \ast \dot \FQ_\gk} ``\gk$ contains
%a Prikry sequence''.
This means that in $M$,
$\forces_{\FP_\gk} ``$By forcing above a condition
$\dot p^*_\gk$ ensuring that $\add(\gk, 1)$ is chosen in the
stage $\gk$ lottery
held in the definition
of $j(\FP)$, $\dot \FQ_\gk$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gk$''.
Consequently, by reflection, for unboundedly
many $\gd < \gk$,
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that $\add(\gd, 1)$ is chosen in the
stage $\gd$ lottery
held in the definition
of $\FP$, $\dot \FQ_\gd$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gd$''.
%the forcing has taken place
%above a condition yielding that
%$\forces_{\FP_\gd \ast \dot \FQ_\gd} ``\gd$ contains
%a Prikry sequence''.
It now follows that
$\forces_{\FP} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''. To see this,
let $\gg < \gk$ be fixed but arbitrary.
Choose $p = \la \dot p_\ga \mid \ga < \gk \ra \in \FP$.
Since $\FP$ is an Easton support iteration,
let $\gr > \gg$ be such that for every
$\ga \ge \gr$,
$\forces_{\FP_\ga} ``\dot p_\ga$ is a term
for the trivial condition''.
We may now find $\gd > \gr > \gg$ such that
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that $\add(\gd, 1)$ is chosen in the
stage $\gd$ lottery
held in the definition
of $\FP$, $\dot \FQ_\gd$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gd$''.
This means that we may find $q \ge p$ such that
$q \forces ``\gd$ contains a Prikry sequence''.
%It then immediately follows that
%$V^\FP \models ``$Unboundedly many $\gd < \gk$
%contain Prikry sequences''.
Thus, $\forces_{\FP} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''.
Hence, by \cite[Theorem 11.1]{CFM},
%$\forces_{\FP} ``$Unboundedly many
$V^\FP \models ``$Unboundedly many
$\gd < \gk$ (i.e., the successors of those
cardinals having Prikry sequences) contain
non-reflecting stationary sets of ordinals
of cofinality $\go$''.
By \cite[Theorem 4.8]{SRK} and the succeeding remarks,
it thus follows that
%$\forces_{\FP} ``$No
$V^\FP \models ``$No
cardinal $\gd < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
For $\FR = ({\rm Add}(\gk, 1))^{V^\FP}$,
$V^{\FP \ast \dot \FR} \models ``\gk$
is not supercompact''.
In fact, in $V^{\FP \ast \dot \FR}$, $\gk$
has trivial Mitchell rank.
%, i.e., there is no normal measure $\mu$
%over $\gk$ in $V^{\FP \ast \dot \FR}$
%such that for
%$j : V^{\FP \ast \dot \FR} \to
%M^{j(\FP \ast \dot \FR)}$ the elementary
%embedding generated by the
%ultrapower via $\mu$,
%$M^{j(\FP \ast \dot \FR)} \models ``\gk$ is measurable''.
\end{lemma}
\begin{proof}
Let $G \ast H$ be $V$-generic over
$\FP \ast \dot \FR$. If $V[G \ast H]
\models ``{\gk}$ does not have trivial Mitchell
rank'', then let $j : V[G \ast H] \to M[j(G \ast H)]$ be an
elementary embedding generated by a normal measure
${\cal U} \in V[G \ast H]$ over $\gk$ such that
$M[j(G \ast H)] \models ``\gk$ is measurable''.
Note that since $M = \bigcup_{{\ga \in {\rm Ord}}} j(V_\ga)$,
$j \rest V : V \to M$ is elementary.
Therefore, because $j \rest \gk = {\rm id}$, we may infer
that $(V_\gk)^V = (V_\gk)^M$.
%Without fear of ambiguity, we will thus write $V_\gk$.
However, by Proposition \ref{p1}, we may further infer
that $(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$.
To see this, let $x \subseteq \gk$, $x \in M$.
Since $M \subseteq M[j(G \ast H)] \subseteq V[G \ast H]$,
$x \in V[G \ast H]$.
In addition, because $(V_\gk)^V = (V_\gk)^M$,
we know that $x \cap \ga \in V$ for every $\ga < \gk$.
This means that if $x \not\in V$, then $x$ is a
fresh subset of $\gk$ with respect to $\FP \ast \dot \FR$.
Since by Lemma \ref{l2},
$\gk$ is strongly compact and hence both measurable and Mahlo in
$V[G \ast H]$, this contradicts Proposition \ref{p1}.
Thus, $x \in V$, so
$(\wp(\gk))^M \subseteq (\wp(\gk))^V$.
From this, it of course immediately follows that
$(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$.
Let $I = j(G \ast H)$.
Note that if $V \models ``\gd < \gk$ is a strong cardinal'',
then $M \models ``j(\gd) = \gd$ is a strong cardinal''.
Also, $M \models ``\gk$ is a Mahlo limit of strong
cardinals'', since
$M[j(G \ast H)] \models ``\gk$ is a Mahlo cardinal'', and
forcing can't create a new Mahlo cardinal.
Hence, by the results of the preceding paragraph,
it follows as well that
%and hence that
$j(\FP) \rest \gk = \FP_\gk = \FP$ and
$I_\gk = G$.
Further, as
$V[G \ast H] \models
``M[I]^\gk \subseteq M[I]$'',
$H \in M[I]$.
%It cannot be the case that
%$H \in M[G_\gd]$ for
%any $\gd < \gk$, since $H$ codes the generic
%added at stage $\gd$ for unboundedly many $\gd < \gk$.
We know in addition
that in $M$, $\forces_{\FP_\gk \ast \dot \FQ_\gk}
``$The forcing beyond stage $\gk$
adds no new subsets of $2^\gk$'' and $\gk$ is
a stage at which nontrivial forcing
in $j(\FP)$ can take place.
%a nontrivial stage of forcing at stage
%$\gk$ in the definition of $j(\FP)$.
Consequently, $H \in M[I_{\gk + 1}] = M[G][I(\gk)]$, and
$M[I_{\gk + 1}] \models ``\gk$ is measurable''.
Note that since $\FP$ is defined by taking Easton supports,
$\FP$ is $\gk$-c.c$.$ in both $V$ and $M$.
Because $\FP$ is a Gitik iteration of
suitably directed closed partial orderings
together with Prikry forcing
and $(V_\gk)^V = (V_\gk)^M$,
$(V_\gk)^{V[G]} = (V_\gk)^{M[G]}$.
%$V_\gk[G]$ is the same when calculated in either
%$V[G]$ or $M[G]$.
It must therefore be the case that
$(\add(\gk, 1))^{V[G]} = (\add(\gk, 1))^{M[G]}$.
In addition, since $(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$,
the fact
$\FP$ is $\gk$-c.c$.$ in $M$ yields that %(as well as in $V$),
$(V_{\gk + 1})^{M[G]} \subseteq (V_{\gk + 1})^{V[G]}$.
This means that $H$ is $M[G]$-generic over $(\add(\gk, 1))^{M[G]}$,
since $H$ is $V[G]$-generic over
$(\add(\gk, 1))^{V[G]} = (\add(\gk, 1))^{M[G]}$,
and a dense open subset of
$(\add(\gk, 1))^{M[G]}$ in $M[G]$ is a member of
$(V_{\gk + 1})^{M[G]}$.
Hence, $H$ must be added by the stage $\gk$
forcing done in $M[G] = M[I_\gk]$, i.e., the stage
$\gk$ lottery held in $M[I_\gk]$
must opt for some nontrivial forcing.
%It also follows that
%$M[I_{\gk + 1}] \models ``\gk$ is measurable''.
By the definition of $\FP$ and
$j(\FP)$, we must then have that
$M[I_{\gk + 1}] \models ``\gk$ contains a Prikry
sequence''. %and hence has cofinality $\go$''.
This contradiction to the fact that
$M[I_{\gk + 1}] \models ``\gk$ is measurable''
completes the proof of Lemma \ref{l4}.
\end{proof}
We note that in general, if $j : V \to M$
is an elementary embedding having critical
point $\gk$, then %it is the case that
$(V_{\gk + 1})^V \subseteq (V_{\gk + 1})^M$.
To see this, we begin by observing that as in the proof
of Lemma \ref{l4}, since $j \rest \gk = {\rm id}$,
$(V_\gk)^V = (V_\gk)^M$. Without fear of ambiguity,
we therefore write $V_\gk$.
However, since for every $x \subseteq \gk$, $x \in V$,
$j(x) \in M$, $x$ is definable in $M$ as
$j(x) \cap V_\gk$.
In particular, this means that in Lemma \ref{l4},
it is actually true that
$(V_{\gk + 1})^V = (V_{\gk + 1})^M$.
%In general, though,
It could be the case, though, that
$j : V \to M$ is an elementary embedding with
critical point $\gk$, yet $(V_{\gk + 1})^V$ is a
proper subset of $(V_{\gk + 1})^M$.
We briefly outline two examples of this phenomenon,
which are as follows:
\begin{enumerate}
\item Let $\gk < \gl$ be such that
$\gk$ is a measurable cardinal and $\gl$ is a Woodin cardinal.
Suppose $\FP$ is the stationary tower forcing having
critical point $\gk^+$ which changes the cofinality of $\gk^+$
to $\go$ (see \cite{La} for a discussion of this partial
ordering). Since $V^\FP \models ``\gk$ is a measurable
cardinal'', let $j: V^\FP \to M^{j(\FP)}$ be an elementary
embedding witnessing $\gk$'s measurability.
Consider $j \rest V : V \to M$.
It will then be the case that $(V_{\gk + 1})^M$
contains a subset of $\gk \times \gk$ coding a well-ordering
of $(\gk^+)^V$ of order type $\gk$.
\item Let $\gk$ be a measurable cardinal.
Take $L[\mu]$ as our ground model. Force over $L[\mu]$
with the reverse
Easton iteration of length $\gk$ which adds
a Cohen subset of $\gk$ to each inaccessible cardinal
$\gd < \gk$. Let $V$ be the resulting generic extension.
Suppose $C$ is now $V$-generic over $(\add(\gk, 1))^V$.
Standard arguments show that
$V[C] \models ``\gk$ is a measurable cardinal''.
We can therefore once again let
$j : V[C] \to M[j(C)]$ be an elementary embedding
witnessing $\gk$'s measurability and consider
$j \rest V : V \to M$. It will then be the case that
$C \in (V_{\gk + 1})^M$.
\end{enumerate}
On the other hand, if there is no
inner model with a strong cardinal,
$V$ is the core model, and
$j \rest V : V \to M$ is the restriction of
$j : V^\FP \to M^{j(\FP)}$ for $\FP \in V$
(i.e., $\FP$ is set forcing over $V$), then
$j \rest V$ must be an elementary embedding
which is given by an iterated ultrapower
of $V$ starting with an extender over $\gk$
(see \cite{Z}).
This implies that $(V_{\gk + 1})^V = (V_{\gk + 1})^M$.
We note that $M$ here need not be a class in $V$
or even be contained in $V$, unless $\gk$ is not a
limit of measurable cardinals.
(If $\gk$ is not a limit of measurable cardinals,
then $o(\gk) = 1$, and $M$ is a finite iterate of
$V$ via the unique normal measure over $\gk$
present in $V$.)
To see this, suppose that $\gk$ is a limit
of measurable cardinals. Fix $x \in V$,
$x \subseteq \gk$ an unbounded subset of measurable cardinals, with
$x = \la \gl_\ga \mid \ga < \gk \ra$.
Suppose $r$ is $V$-generic for $\add(\go, 1)$.
View $r$ as a characteristic function.
Work in $V[r]$. By the L\'evy-Solovay
results \cite{LS}, each $\gl_\ga$ and $\gk$
remain measurable in $V[r]$.
Split $x$ into blocks $x_\ga = \{\gl_{\ga + n} \mid n < \go\}$,
where $\ga$ is either $0$ or a limit ordinal below $\gk$.
Now let $\FP^*$ be
Magidor's iteration of Prikry forcing
as first defined in \cite{Ma} which,
for only those $n$ such that
$r(n) = 0$, changes the cofinality of each
$\gl_{\ga + n}$ to $\go$.
%Now using Magidor's iteration of Prikry forcing
%as first defined in \cite{Ma}, for only those $n$ such that
%$r(n) = 0$, force to change the cofinality of each
%$\gl_{\ga + n}$ to $\go$.
By the work of \cite{Ma},
$\gk$ remains measurable in $(V[r])^{\FP^*}$.
Let $j : (V[r])^{\FP^*} \to (M[r])^{j(\FP^*)}$ be an elementary
embedding generated by some normal measure
$\mu \in (V[r])^{\FP^*}$ over $\gk$.
Consider $j \rest V : V \to M$.
%, which maps $V$ to an iterated ultrapower $M$.
We claim that $M \not \subseteq V$.
%To see this, let $\dot \FP^* \in V$ be a term for $\FP^*$.
%We observe that in $M$, $j(\dot \FP^*)$
To see this,
let $\ga$ be a limit ordinal such that $\gk < \ga < j(\gk)$.
We observe that in $M[r]$,
$j(\FP^*)$ is defined up to $j(\gk)$ and changes
the cofinality of $\gl_{\ga + n}$ to $\go$
only when $r(n) = 0$.
Thus, in $(M[r])^{j(\FP^*)}$, $\gl_{\ga + n}$
has cofinality $\go$ iff $r(n) = 0$.
Further, $(M[r])^{j(\FP^*)}$ is $\go$-closed with respect to
$(V[r])^{\FP^*}$. Hence, the only way for $\gl_{\ga + n}$
to have cofinality $\go$ in $(M[r])^{j(\FP^*)}$ is for
the measure over $\gl_{\ga + n}$ used to change
its cofinality to $\go$ to have been iterated $\go$ many
times in the construction of the iterated ultrapower $M$,
and for the sequence of critical points to form a
Prikry sequence through $\gl_{\ga + n}$.
In addition, when $r(n) = 1$, no normal measure over
$\gl_{\ga + n}$ is used in the construction of $M$.
(See \cite{BN} for a more detailed explanation of these facts.)
Consequently, if $M \subseteq V$, $r$ can easily
be recovered in $V$ via the definition of the
function $r^* : \go \to 2$ as
$r^*(n) = 0$ iff
%the measure over $\gl_{\gk + n}$ used
%by $j(\FP^*)$ to change its cofinality to $\go$
some measure over $\gl_{\gk + \go + n}$ has been iterated
$\go$ many times in the construction of the
iterated ultrapower $M$.
This, of course, contradicts the fact that
$r$ is $V$-generic for $\add(\go, 1)$.
Theorem \ref{t1} now follows from
Lemmas \ref{l1} -- \ref{l4}.
By Lemma \ref{l1}, $\gk$ is supercompact
in $V^\FP$, and by Lemma \ref{l2}, in $V^\FP$,
$\gk$'s strong compactness is indestructible
under arbitrary $\gk$-directed closed forcing.
By Lemma \ref{l3}, $V^\FP \models ``\gk$ is the
least strongly compact cardinal''.
By Lemma \ref{l4}, there is a nontrivial
$\gk$-directed closed forcing
$\FR \in V^\FP$ such that
$V^{\FP \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank and hence is not supercompact''.
This completes the proof
of Theorem \ref{t1}.
\end{proof}
In conclusion to this paper,
we ask if it is possible to get a model
witnessing the conclusions of Theorem \ref{t1}
in which $\gk$ is not the least strongly
compact cardinal.
Since Prikry forcing above a strongly
compact cardinal destroys strong compactness,
an answer to this question would require a different sort of
iteration from the one used in the
proof of Theorem \ref{t1}.
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\end{document}
On the other hand, if there is no
inner model with a strong cardinal,
$V$ is the core model, and
$j \rest V : V \to M$ is the restriction of
$j : V^\FP \to M^{j(\FP)}$ for $\FP \in V$
(i.e., $\FP$ is set forcing over $V$), then
$j \rest V$ must be an elementary embedding
which is given by an iterated ultrapower
of $V$ starting with an extender over $\gk$
(see \cite{Z}).
This implies that $(V_{\gk + 1})^V = (V_{\gk + 1})^M$.
We note that $M$ here need not be a class in $V$
or even be contained in $V$, unless $\gk$ is not a
limit of measurable cardinals.
To see this, suppose that $\gk$ is a limit
of measurable cardinals. Fix $x \in V$,
$x \subseteq \gk$ an unbounded subset of measurable cardinals, with
$x = \la \gl_\ga \mid \ga < \gk \ra$.
Suppose $r$ is $V$-generic for $\add(\go, 1)$.
View $r$ as a characteristic function.
Work in $\bar V = V[r]$. By the L\'evy-Solovay
results \cite{LS}, each $\gl_\ga$ and $\gk$
remain measurable in $\bar V$.
Split $x$ into blocks $x_\ga = \{\gl_{\ga + n} \mid n < \go\}$,
where $\ga$ is either $0$ or a limit ordinal below $\gk$.
Now let $\FP^*$ be
Magidor's iteration of Prikry forcing
as first defined in \cite{Ma} which,
for only those $n$ such that
$r(n) = 0$, changes the cofinality of each
$\gl_{\ga + n}$ to $\go$.
%Now using Magidor's iteration of Prikry forcing
%as first defined in \cite{Ma}, for only those $n$ such that
%$r(n) = 0$, force to change the cofinality of each
%$\gl_{\ga + n}$ to $\go$.
By the work of \cite{Ma},
$\gk$ remains measurable in $\bar V^{\FP^*}$.
Let $j : \bar V^{\FP^*} \to M^*$ be an elementary
embedding generated by some normal measure
$\mu \in \bar V^{\FP^*}$ over $\gk$.
Consider $j \rest V$, which maps $V$ to
an iterated ultrapower $M$.
We claim that $M \not \subseteq V$.
%To see this, let $\dot \FP^* \in V$ be a term for $\FP^*$.
%We observe that in $M$, $j(\dot \FP^*)$
To see this,
let $\ga$ be a limit ordinal such that $\gk < \ga < j(\gk)$.
We observe that in $M$,
$j(\FP^*)$ is defined up to $j(\gk)$ and changes
the cofinality of $\gl_{\ga + n}$ to $\go$
only when $r(n) = 0$.
By the work of \cite{BN}, for such an $n$,
the measure over $\gl_{\ga + n}$ used to change
its cofinality to $\go$ will be applied $\go$ many
times in the construction of the iterated ultrapower $M$.
In addition, when $r(n) = 1$, no normal measure over
$\gl_{\ga + n}$ is used in the construction of $M$.
Consequently, if $M \subseteq V$, $r$ can easily
be recovered in $V$ via the definition of the
function $r^* : \go \to 2$ as
$r^*(n) = 1$ iff
%the measure over $\gl_{\gk + n}$ used
%by $j(\FP^*)$ to change its cofinality to $\go$
some measure over $\gl_{\gk + \go + n}$ has been used
$\go$ many times in the construction of the
iterated ultrapower $M$.
This, of course, contradicts the fact that
$r$ is $V$-generic for $\add(\go, 1)$.