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\title{Strong Cardinals can be Fully Laver Indestructible
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, almost huge cardinal,
strong cardinal, indestructibility}}
\author{Arthur W.~Apter
\thanks{The author's research was partially
supported by PSC-CUNY Grant 61449-00-30.
In addition,
the author wishes to thank Joel Hamkins for
a helpful conversation on the subject
matter of this paper. Finally, the author
wishes to thank the referee, for helpful
comments and suggestions that have been
incorporated into this version of the
paper, as well as for pointing out an
error in the original proof of
Lemma \ref{l6}.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{August 17, 2001\\
(revised October 9, 2001)}
\begin{document}
\maketitle
\begin{abstract}
We prove three theorems which
show that it is relatively consistent
for any strong cardinal $\gk$ to be fully
Laver indestructible under $\gk$-directed
closed forcing.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
A well-known result of Gitik and Shelah \cite{GS}
is that any strong cardinal $\gk$ can be
forced to be indestructible under
$\gk$-weakly closed partial orderings
satisfying the Prikry property.
Although we leave it to readers to
consult \cite{GS} for the exact
definition of this sort of indestructibility,
we note that it includes indestructibility
under many different forcing notions.
Some examples are Prikry forcing done on a measurable
cardinal above $\gk$,
the forcing
which adds a non-reflecting stationary
set of ordinals of some fixed but
arbitrary cofinality $\gd < \gk$ to
any regular cardinal $\gg > \gk$,
and in general, as the discussion given
on page 633 of \cite{A01} immediately
following Definition 2.2 shows,
any $\gk$-strategically closed
forcing.
The sorts of partial orderings
not included, however, are those
which are $\gk$-directed closed,
such as the L\'evy collapse
${\rm Coll}(\gk, \gl)$ of
a regular cardinal $\gl > \gk$ to $\gk$.
This is no accident, since by the work of
\cite{MSS}, if it is possible to collapse
$\gk^+$ to $\gk$ and preserve
$\gk$'s measurability, then there must
be an inner model with a Woodin cardinal.
The purpose of this paper is to show
that if we are willing to use
strong enough assumptions, then it
is possible to obtain different models in
which a strong cardinal $\gk$ is
fully Laver indestructible (in the
sense of \cite{L}) under
$\gk$-directed closed forcing.
Specifically, we prove the following
three theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + $\gk$ is the
least supercompact cardinal''.
There is then a partial ordering
$\FP \in V$ so that
$V^\FP \models ``$ZFC + $\gk$ is
the least strong cardinal +
$\gk$ isn't $2^\gk$ supercompact +
$\gk$'s strongness is indestructible under
forcing with partial orderings which are
$\gk$-directed closed''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + There is a
proper class of supercompact cardinals +
There is no supercompact limit of supercompact
cardinals''.
There is then a partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC + There is a
proper class of strong cardinals +
No strong cardinal $\gk$ is $2^\gk$
supercompact + Every strong cardinal
$\gk$ has its strongness indestructible under
$\gk$-directed closed forcing''.
\end{theorem}
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$
is almost huge''.
%Assume
%$j : V \to M$ is an elementary
%embedding witnessing $\gk$'s
%hugeness, with $j(\gk) = \gl$.
There is then a partial ordering
$\FP \in V$ so that for
$V^*$ the universe $V^\FP$
truncated at $\gk$,
$V^* \models ``$ZFC + There is
a proper class of supercompact
limits of supercompact cardinals +
Every strong cardinal $\gk$ has
its strongness indestructible under
$\gk$-directed closed forcing +
Every supercompact cardinal $\gk$ has
its supercompactness indestructible under
$\gk$-directed closed forcing''.
\end{theorem}
We take this opportunity to make a
few remarks concerning
Theorems \ref{t1} - \ref{t3}.
We note that in Theorem 8 of
\cite{AH1}, a model is constructed
in which there is a strong cardinal
$\gk$, no cardinal $\gl > \gk$ is
weakly compact, and ``universal
indestructibility'' holds for every
cardinal $\gd$ which is strong,
partially strong, measurable,
Ramsey, and weakly compact.\footnote{The
property of universal indestructibility
holding for all cardinals $\gd$ which
are strong, partially strong, measurable,
Ramsey, and weakly compact
means that the relevant
property of all such $\gd$ is indestructible
under forcing with an arbitrary $\gd$-directed
closed partial ordering.}
This model, however, is constructed using
a high-jump cardinal (see \cite{AH1} for
the definition), something which implies the
consistency of a supercompact limit of
supercompact cardinals, and has
consistency strength
in the realm of almost hugeness.
This is in sharp contrast to
Theorems \ref{t1} and \ref{t2} of
this paper, which are proven using considerably
weaker hypotheses.
Further, by Theorems 11 and 12 of \cite{AH1}, since
universal indestructibility holds, the size of
the universe above $\gk$ must be severely restricted.
In particular, for the amount of universal
indestructibility given by Theorem 8 of \cite{AH1},
no cardinal above $\gk$ can be even weakly compact.
This again is in sharp contrast to the models
constructed in this paper, in which there are
essentially no restrictions placed on the
size of any of the universes witnessing the
conclusions of our theorems.
Before giving the proofs of our theorems,
we briefly mention some background 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 standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
Whenever $\FP$ is an iteration, $\FP_\ga$
as usual will be the $\ga^{\rm th}$ stage in
the definition of $\FP$.
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 $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a cardinal and $\FP$ is
a partial ordering,
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is 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 $p \in \FP$.
$\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even (including limit)
stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is
$\gk^+$-directed closed, then $\FP$ is
$\gk$-strategically closed.
We mention that we are assuming familiarity with the
large cardinal notions of measurability, strongness,
supercompactness, and almost hugeness.
Interested readers may consult \cite{K}
%or \cite{SRK}
for further details.
We mention only that unlike \cite{K}, we will say that
the cardinal $\gk$ is $\gl$ strong
for $\gl > \gk$ if there is
$j : V \to M$ an elementary embedding having
critical point $\gk$ so that
$j(\gk) > |V_\gl|$ and $V_\gl \subseteq M$.
As always, $\gk$ is strong if $\gk$ is $\gl$
strong for every $\gl > \gk$.
Also, $\gk$ is ${<}\gl$ strong if $\gk$ is
$\ga$ strong for every $\ga \in (\gk, \gl)$, and
$\gk$ is ${<}\gl$ supercompact if $\gk$ is
$\ga$ supercompact for every cardinal
$\ga \in [\gk, \gl)$.
Finally, we note that when discussing indestructibility,
we will say that a strong or supercompact cardinal
$\gk$ has its strongness or supercompactness
(fully) indestructible under $\gk$-directed
closed forcing if after forcing with an
arbitrary $\gk$-directed closed partial ordering,
$\gk$ retains its relevant property.
%The strong cardinal $\gk$ will be
%(fully) indestructible under $\gk$-strategically
%closed forcing if after forcing with an
%arbitrary $\gk$-strategically closed partial ordering,
%$\gk$ retains its strongness.
\section{The Proofs of Theorems \ref{t1} and
\ref{t2}}\label{s2}
We begin Section \ref{s2} with the proof
of Theorem \ref{t1}.
\begin{proof}
For the benefit of readers, we start by
recalling the definition from Section 3
of \cite{H4} of the lottery sum of
a collection of partial orderings.
If ${\cal C}$ is such a collection,
then the lottery sum is the partial ordering
$\oplus {\cal C} = \{\la \FQ, q \ra :
\FQ \in {\cal C}$ and $q \in \FQ\} \cup
\{0\}$, ordered with $0$ below everything and
$\la \FQ, q \ra \le \la \FQ', q' \ra$ iff
$\FQ = \FQ'$ and $q \le q'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\cal C}$, then $G$ first selects
an element of ${\cal C}$
(or as Hamkins says in \cite{H4},
``holds a lottery among the posets of
${\cal C}$'') and then forces
with it.\footnote{The definition of
the lottery sum of a collection of
partial orderings has been around for
quite a while and predates Hamkins'
applications of \cite{H4}
to forcing indestructibility.
In particular, the lottery sum of
partial orderings has also been called a
``disjoint sum of posets'' and has been
referred to via the slogan
``choose which poset to force with
generically''.
It was used by Woodin in
unpublished work about making
hypermeasurability indestructible
under Cohen forcing.}
Assume
$V \models ``$ZFC + $\gk$ is the
least supercompact cardinal''.
We are now ready to give the definition
of the partial ordering $\FP$ used
in the proof of Theorem \ref{t1}.
$\FP$ is an Easton support iteration
of length $\gk$ which begins by adding
a Cohen subset of $\go$ and then
does a non-trivial
forcing only at those stages
which are strong cardinals in $V$.
At such a stage $\ga$, the forcing done
is the lottery sum of all partial
orderings $\FR$ having rank below the least
$V$-strong cardinal $\eta$ above $\ga$ which
are $\ga$-directed closed
and which have the
additional property that
forcing with $\FR$ destroys
the $\gg$ strongness of $\ga$ for some
$\gg < \eta$,
assuming this sort of $\FR$ exists.
If this is not the case, then the
forcing done at stage $\ga$ is the
trivial forcing $\{\emptyset\}$.
\begin{lemma}\label{l1}
$V^\FP \models ``$No cardinal $\gd < \gk$
is strong''.
\end{lemma}
\begin{proof}
Fix $\gd < \gk$ so that
$V^\FP \models ``\gd$ is strong''. Write
$\FP = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
In Hamkins' terminology of \cite{H1}, \cite{H2},
and \cite{H3}, $\FP$ ``admits a gap at $\go$''.
The Gap Forcing Theorem of \cite{H2} and
\cite{H3} states that if
$V[G]$ is a forcing extension obtained
by forcing which admits a gap at
$\gg < \gd$ and
$j : V[G] \to M[j(G)]$ is an embedding
with critical point $\gd$ so that
${M[j(G)]}^\gg \subseteq M[j(G)]$ in $V[G]$, then
$M = V \cap M[j(G)]$ (so $M \subseteq V$), and
the restricted embedding
$j \rest V : V \to M$ is definable in $V$.
This implies that any cardinal which is
strong in $V^\FP$ had to have been strong in
$V$. Thus, it must be the case that
$V \models ``\gd$ is strong'' as well.
In analogy to what was done in the preceding
paragraph, we can write
$\FP_\gd = \FP_0 \ast \dot \FQ^*$, where
$|\FP_0| = \go$ and
$\forces_{\FP_\gd} ``\dot \FQ^*$ is
$\ha_1$-strategically closed''.
However, by Corollary 5.4 of \cite{H4},
since $\FP_\gd$ therefore admits a gap at $\go$,
and since $\gd$, being below the least
supercompact cardinal, isn't supercompact,
there must be some cardinal $\gg > \gd$
so that after forcing over
$V^{\FP_\gd}$ with ${\rm Coll}(\gd, \gg)$,
$\gd$ is no longer measurable.
Note now that it is possible to infer that
the least such $\gg > \gd$ as just mentioned
must be below the least $V$-strong cardinal $\eta$ above
$\gd$. This is since if such a $\gg$ is chosen
to be above $\eta$, then we can choose
$\rho > \max(\gg, \eta)$ sufficiently large so that
for $j : V \to M$ an elementary embedding
witnessing the $\rho$ strongness of $\eta$,
$M \models ``$Forcing over
$V^{\FP_\gd}$ with ${\rm Coll}(\gd, \gg)$
destroys $\gd$'s measurability''.
Since the critical point of $j$ is $\eta$
and $\gd < \eta$, this means we can find
by reflection some cardinal $\gs < \eta$
so that forcing over
$V^{\FP_\gd}$ with ${\rm Coll}(\gd, \gs)$
destroys $\gd$'s measurability.
Therefore, the forcing done at stage $\gd$
in the definition of $\FP$ is non-trivial.
In particular, we know that
$V^{\FP_{\gd + 1}} \models ``\gd$ isn't
$\gs$ strong for some $\gs < \eta$''.
Since we can write
$\FP = \FP_{\gd + 1} \ast \dot
\FP^{\gd + 1}$, where
$\forces_{\FP_{\gd + 1}}
``\dot \FP^{\gd + 1}$ is both
$\gg$-strategically closed and
$\gg$-directed closed for every cardinal
$\gg < \eta$'', we have that
$V^{\FP_{\gd + 1} \ast \dot \FP^{\gd + 1}} = V^\FP \models
``\gd$ isn't a strong cardinal''.
This proves Lemma \ref{l1}.
\end{proof}
%We remark that the proof of Lemma \ref{l1}
%shows that the forcing done at any stage
%$\ga$ in the definition of $\FP$ for
%$\ga$ a strong cardinal is non-trivial.
\begin{lemma}\label{l2}
$V^\FP \models ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
\end{lemma}
\begin{proof}
If the conclusions of Lemma \ref{l2} are false,
let $\FQ \in V^\FP$ and $\gd > \gk$ be so that
$V^\FP \models ``\FQ$ is $\gk$-directed closed'' and
$V^{\FP \ast \dot \FQ} \models ``\gk$ isn't
$\gd$ strong''.
By Solovay's theorem of \cite{So}, let
$\gg > \max(\gd, |{\rm TC}(\dot \FQ)|)$
be a cardinal at which GCH and
$|{[\gg]}^{< \gk}| = \gg$ hold, e.g.,
a singular strong limit cardinal of
cofinality $\gk$. If
$j : V \to M$ is an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$,
$M^{\FP \ast \dot \FQ} \models ``\gk$
isn't $\gd$ strong''.
Since $\gg > 2^\gk$,
we know by Lemma 2.1 of \cite{AC2} and
the succeeding remark that
$M \models ``\gk$ is a strong cardinal''.
Further, as $j$ and $M$ may be chosen so that
$M \models ``\gk$ isn't $\gg$ supercompact'',
%the relevant portion of
the proof of
Lemma 2.4 of \cite{AC2} implies that
$M \models ``$No cardinal in the interval
$(\gk, \gg]$ is strong''.
Thus, at stage $\gk$ in $M$ in the definition
of $j(\FP)$, a non-trivial forcing is done, and
$\dot \FQ$ is a term for an allowable choice
in the lottery held at stage $\gk$ in $M$
in the definition of $j(\FP)$.
Also, the next non-trivial stage in $M$
in the definition of $j(\FP)$ takes place
well after $\gg$.
If we now write
$j(\FP) = \FP_{\gk + 1} \ast \dot \FR$
and take $G_0$ to be $V$-generic over
$\FP_\gk = \FP$ and $G_1$ to be
$V[G_0]$-generic over $\FQ$, in
$V[G_0][G_1]$, standard arguments
(see, e.g., the proofs of Corollary 4.6 and
Theorem 4.8 of \cite{H4}) show that
$j$ lifts to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
where $G_2$ and $G_3$ are suitably generic
objects constructed in $V[G_0][G_1]$, and
$G_3$ contains a master condition for $G_1$.
We can thus infer that
$V^{\FP \ast \dot \FQ} \models ``\gk$
is $\gg$ supercompact'', which, by the
choice of $\gg$, implies that
$V^{\FP \ast \dot \FQ} \models ``\gk$ is
$\gd$ strong''.
This contradiction completes the proof
of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``\gk$ isn't $2^\gk$ supercompact''.
\end{lemma}
\begin{proof}
By the preceding lemma,
$V^\FP \models ``\gk$ is a strong cardinal''.
Therefore, by Lemma 2.1 of \cite{AC2}, if $\gk$ were
$2^\gk$ supercompact, $\gk$ would be a limit of
strong cardinals.
By Lemma \ref{l1} of this paper, however,
this is not the case.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l1} - \ref{l3} complete the proof
of Theorem \ref{t1}.
\end{proof}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let
$V \models ``$ZFC +
$\la \gk_\ga : \ga \in {\rm Ord} \ra$
is an increasing enumeration of the
supercompact cardinals + There is
no supercompact limit of supercompact cardinals''.
By our assumptions on $V$,
it is the case that
$\bigcup_{\gb < \ga} \gk_\gb < \gk_\ga$.
We are now ready to give the definition
of the partial ordering $\FP$ used
in the proof of Theorem \ref{t2}.
$\FP$ is a proper class Easton support iteration
which begins by adding
a Cohen subset of $\go$ and then
does a non-trivial
forcing only at those stages
which are non-supercompact strong cardinals in $V$.
At such a stage $\ga$, the forcing done
is the lottery sum of all partial
orderings $\FR$ having rank below the least
$V$-strong cardinal $\eta$ above $\ga$ which
are $\ga$-directed closed
and which have the
additional property that
forcing with $\FR$ destroys
the $\gg$ strongness of $\ga$ for some
$\gg < \eta$.
By the proof of Lemma \ref{l1},
this sort of $\FR$ always exists
at a stage $\ga$ which is a non-supercompact
strong cardinal,
and the forcing at any such stage
is always non-trivial.
By the proof of Lemma \ref{l1}, all strong
cardinals in $V$ which aren't supercompact
are destroyed in $V^\FP$, and any cardinal
which is strong in $V^\FP$ had to have
been strong in $V$. For any
$\gk \in \la \gk_\ga : \ga \in {\rm Ord} \ra$,
if we write
$\FP = \FP_\gk \ast \dot \FP^\gk$, then since
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is
$\gk$-directed closed'', we will know that
$V^\FP \models ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''
if we know that
$V^{\FP_\gk} \models ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
This last fact, however, follows as in the
proof of Lemma \ref{l2}.
Thus, the only strong cardinals in $V^\FP$
are the elements of
$\la \gk_\ga : \ga \in {\rm Ord} \ra$.
Since by our choice of $V$ we know that
$\bigcup_{\gb < \ga} \gk_\gb < \gk_\ga$,
no strong cardinal in $V^\FP$ is a
limit of strong cardinals. Hence, as in
Lemma \ref{l3},
$V^\FP \models ``$No strong cardinal $\gk$
is $2^\gk$ supercompact''.
Since the usual arguments show that
$V^\FP \models {\rm ZFC}$,
this completes the proof of Theorem \ref{t2}.
\end{proof}
We remark that the proof of Theorem \ref{t2}
remains valid if we drop the restriction on
$V$ that there is no supercompact limit of
supercompact cardinals.
%$\bigcup_{\gb < \ga} \gk_\gb < \gk_\ga$.
This constraint on $V$ is imposed in order
to infer that
$V^\FP \models ``$No strong cardinal
$\gk$ is $2^\gk$ supercompact''.
In fact, without this limitation,
it conceivably could be the case that some
large supercompact cardinal $\gk$ remains
supercompact in $V^\FP$, although our
proof techniques do not seem to be able
to yield this fact.
This is the reason for Theorem \ref{t3},
whose proof we give below.
\section{The Proof of Theorem \ref{t3}}\label{s3}
We turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Let
$V \models ``$ZFC + $\gk$ is almost huge''. Take
$j : V \to M$ as an elementary embedding witnessing
$\gk$'s almost hugeness, with $j(\gk) = \gl$.
The partial ordering $\FP$ we use in the proof of
Theorem \ref{t3} is a modification of the
partial ordering defined in the proof
of Theorem \ref{t1}.
$\FP$ is an Easton support iteration
of length $\gk$ which begins by adding
a Cohen subset of $\go$ and then
does a non-trivial
forcing only at those stages $\ga < \gk$
which are ${<}\gk$ strong cardinals in $V$.
At such a stage, the forcing done
is the lottery sum of all partial
orderings $\FR$ having rank below the least
$V$-${<}\gk$ strong cardinal $\eta$ above $\ga$ which
are $\ga$-directed closed
and which have the
additional property that
forcing with $\FR$ destroys the
$\gg$ strongness of $\ga$ for some $\gg < \eta$,
assuming this sort of $\FR$ exists.
If this sort of $\FR$ does not exist, then the
forcing done at stage $\ga$ is the
lottery sum of all partial orderings
$\FS$ having rank below the least
$V$-${<}\gk$ strong cardinal $\eta$ above $\ga$
which are $\ga$-directed closed and
which have the additional property that
forcing with $\FS$ destroys the $\gg$
supercompactness of $\ga$ for some
$\gg < \eta$, again assuming this kind of
$\FS$ exists.
If this kind of $\FS$ does not exist, then the
forcing done at stage $\ga$ is the
trivial forcing $\{\emptyset\}$.
Work now in $V_\gk$.
We observe that the only difference
between the definition of $\FP$
just given and the definition of
$\FP$ given in the proof of
Theorem \ref{t2} is that there
are stages $\ga$, namely
those $\ga$ which are
supercompact and which become
indestructibly strong cardinals,
at which an additional forcing is done.
This forcing, however, is $\ga$-directed closed,
meaning that it combined with all further
forcing will preserve the fact that
$\ga$ has become an indestructibly strong cardinal.
Therefore, using the proof of
Theorem \ref{t2} combined with the
succeeding remark, we are able to infer
the following.
\begin{lemma}\label{n1}
$V^\FP_\gk \models ``$ZFC + There is a
proper class of strong cardinals + Every
strong cardinal $\gd$ has its strongness
indestructible under $\gd$-directed
closed forcing''.
\end{lemma}
In addition,
by analyzing the proof of Theorem \ref{t2},
we are also able to infer the following.
\begin{lemma}\label{n2}
For any cardinal $\ga < \gk$ which is
supercompact in $V_\gk$, after forcing with
$\FP_\ga$, $\ga$ has become a
${<}\gk$ strong cardinal whose
${<}\gk$ strongness is indestructible when
forcing with any $\ga$-directed closed partial
ordering having rank below $\gk$.
\end{lemma}
\begin{lemma}\label{l5}
$V^\FP_\gl \models ``\gk$ is a fully
indestructible supercompact cardinal''.
\end{lemma}
\begin{proof}
If the conclusions of Lemma \ref{l5} are false,
let $\FQ \in V^\FP_\gl$ and $\rho < \gl$ be so that
$V^\FP \models ``\FQ$ is $\gk$-directed closed'' and
$V^{\FP \ast \dot \FQ} \models ``\gk$ isn't
$\rho$ supercompact''.
By using an argument similar to
one found in the proof of Lemma \ref{l1},
we can show that for
$\gd > \gk$ the least ${<}\gl$ strong
cardinal in $V$
(which exists since $\gk$ is
almost huge), $\rho$ can be chosen to be
below $\gd$, and $\dot \FQ$ can be
chosen to have rank below $\gd$.
To see this, if either $\rho$ or
the rank of $\dot \FQ$ is chosen to
be above $\gd$, then by choosing
$\gg \in (\gd, \gl)$ to be sufficiently large and
$i : V \to N$ as an elementary embedding
witnessing the $\gg$ strongness of $\gd$,
$N \models ``$After forcing with
$\FP \ast \dot \FQ$, $\gk$ is no longer
$\rho$ supercompact''.
Since $\gk$ is below the critical point of
$i$, it then follows by reflection that both
$\rho$ and the rank of $\dot \FQ$ can be
chosen to have rank below $\gd$.
Also, as
$\FP = {j(\FP)}_\gk$,
elementarity and the remarks immediately
preceding the proof of this lemma
imply that in $M^\FP$,
$\gk$ is a ${<}\gl$ strong cardinal whose
${<}\gl$ strongness is indestructible when
forcing with any $\gk$-directed closed
partial ordering having rank below $\gl$.
This means since $M^{{<}\gl} \subseteq M$
that $\FQ$ is an allowable
choice in the lottery held at stage $\gk$ in
$M^\FP$.
%The remainder of the proof of Lemma \ref{l5} is very
%similar to the argument given in Lemma \ref{l2}.
Let now
$\gb > \max(\rho, |{\rm TC}(\dot \FQ)|)$,
$\gb < \gd$ be sufficiently large, e.g.,
as in the proof of Lemma \ref{l2},
the least singular strong limit cardinal
of cofinality $\gk$
above $\max(\rho, |{\rm TC}(\dot \FQ)|)$.
%the least inaccessible above
%$\max(\rho, |{\rm TC}(\dot \FQ)|)$.
Let
$k : V \to M_*$ be the elementary embedding
witnessing the $\gb$ supercompactness of $\gk$
which is the restriction of $j$
to $\gb$, i.e., $k$ is generated
by the ultrafilter ${\cal U}$ defined by
$x \in {\cal U}$ iff $\la j(\ga) : \ga < \gb \ra
\in x$.
Take $\ell : M_* \to M$ as the
canonical elementary embedding so that
$\ell \circ k = j$.
It is the case that
$M \models ``$After forcing with the
iteration defined through stage $\gk$,
there is a non-trivial lottery sum used
in the definition of the stage
$\gk + 1$ iteration for which $\FQ$
is an allowable choice,
and the next non-trivial stage in the
iteration is above $\gb$''.
Therefore, since the critical point of
$\ell$ is above $\gb$, by elementarity,
$M_* \models ``$After forcing with the
iteration defined through stage $\gk$,
there is a non-trivial lottery sum used
in the definition of the stage
$\gk + 1$ iteration for which $\FQ$
is an allowable choice,
and the next non-trivial stage in the
iteration is above $\gb$''.
%Further, by the choice of $\gb$,
%$M_* \models ``$No cardinal in the interval
%$(\gk, \gb]$ is strong''.
%the next non-trivial stage in $M_*$
%in the definition of $k(\FP)$ takes place
%after $\gb$.
If we now write
$k(\FP) = \FP_{\gk + 1} \ast \dot \FR$
and take $G_0$ to be $V$-generic over
$\FP_\gk$ and $G_1$ to be
$V[G_0]$-generic over $\FQ$, as before, in
$V[G_0][G_1]$, standard arguments show that
$k$ lifts to
$k : V[G_0][G_1] \to M_*[G_0][G_1][G_2][G_3]$,
where $G_2$ and $G_3$ are suitably generic
objects constructed in $V[G_0][G_1]$, and
$G_3$ contains a master condition for $G_1$.
We can thus infer that
$V^{\FP \ast \dot \FQ} \models ``\gk$
is $\gb$ supercompact'', which, by the
choice of $\gb$, implies that
$V^{\FP \ast \dot \FQ} \models ``\gk$
is $\rho$ supercompact''.
This contradiction completes the proof
of Lemma \ref{l5}.
\end{proof}
By the preceding lemma, we now know that
in $V$,
$\forces_\FP ``\gk$ is a ${<}\gl$
supercompact cardinal whose
${<}\gl$ supercompactness is indestructible
when forcing with
$\gk$-directed closed partial orderings
having rank below $\gl$''.
We abbreviate this as
$\forces_{\FP} ``\varphi(\gk, \gl)$'', where
$\varphi(x, y)$ is a formula in the
language of set theory containing the
free variables $x$ and $y$.
Therefore, since $M^{{<}\gl} \subseteq M$,
in $M$, it is also the case that
$\forces_\FP ``\varphi(\gk, \gl)$''.
Hence, by reflection, in $V$,
there are unboundedly in $\gk$
many $\gd < \gk$ so that
$\forces_{\FP_\gd} ``\varphi(\gd, \gk)$''.
For any such $\gd$, $\FP$ factors as
$\FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\gd$-directed closed''. This means that
by elementarity, we can infer that
in $M$,
$\forces_\FP ``\varphi(\gk, \gl)$ and
$\gk$ is a limit of cardinals $\gd$
so that $\varphi(\gd, \gl)$''.
Therefore, again by reflection, we can
infer that in $V$, there are unboundedly
in $\gk$ many $\gd < \gk$ so that
$\forces_{\FP} ``\varphi(\gd, \gk)$ and
$\gd$ is a limit of cardinals $\gg$ so that
$\varphi(\gg, \gk)$''.
\begin{lemma}\label{l6}
$V^\FP_\gk \models ``$Any
supercompact cardinal
$\gd < \gk$ has its supercompactness
fully indestructible''.
\end{lemma}
\begin{proof}
By the paragraph immediately preceding
this lemma, we know that
$V^\FP_\gk \models ``$There is a
proper class of supercompact cardinals''.
We therefore fix $\gd < \gk$ so that
$V^\FP_\gk \models ``\gd$ is supercompact''.
Since an analysis such as
the one given in Lemma \ref{l1}
yields that $\FP$ admits a
gap at $\go$, the Gap Forcing Theorem of
\cite{H2} and \cite{H3} shows that
$V_\gk \models ``\gd$ is supercompact''.
Let
$\FS = \FP_\gd$.
We show that
$V^\FS_\gk \models ``\gd$ is a
supercompact cardinal whose supercompactness
is indestructible under $\gd$-directed
closed forcing''. This suffices, since
as above, the factorization
$\FP = \FP_\gd \ast \dot \FP^\gd$ yields that
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\gd$-directed closed''.
If it is not the case that
$V^\FS_\gk \models ``\gd$ is a
supercompact cardinal whose supercompactness
is indestructible under $\gd$-directed
closed forcing'',
let $\FQ \in V^\FS_\gk$ and $\rho^* < \gk$ be so that
$V^\FS_\gk \models ``\FQ$ is $\gd$-directed closed'' and
$V^{\FS \ast \dot \FQ}_\gk \models ``\gd$ isn't
$\rho^*$ supercompact''. By the same argument
as given in the first paragraph of the proof
of Lemma \ref{l5}, it is possible to choose
both $\rho^*$ and the rank of
$\dot \FQ$ to be below $\eta$, the least
cardinal in $V$ above $\gd$
which is ${<}\gk$ strong in $V$.
Therefore, since by Lemma \ref{n2},
%our remarks in the sentence
%immediately preceding the proof of
%Lemma \ref{l5},
$V^\FS \models ``\gd$ is a
${<}\gk$ strong cardinal whose
${<}\gk$ strongness is indestructible
when forcing with $\gd$-directed
closed partial orderings having
rank below $\gk$'', the definition
of $\FP$ tells us that
$V^{\FP_{\gd + 1}} \models ``$For some
$\rho < \eta$,
$\gd$ isn't $\rho$ supercompact''.
Write
$\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$.
By the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\FP^{\gd + 1}$ is
both $\gg$-strategically closed and
$\gg$-directed closed for every cardinal
$\gg < \eta$''.
By our work in the preceding paragraph, it
then immediately follows that
$V^{\FP_{\gd + 1} \ast \dot \FP^{\gd + 1}} =
V^\FP \models ``$For some
$\rho < \eta$,
$\gd$ isn't $\rho$ supercompact'', a
contradiction to our assumption that
$V^\FP_\gk \models ``\gd$ is supercompact''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{n1} - \ref{l6},
along with the intervening remarks,
complete the proof of Theorem \ref{t3}.
\end{proof}
We wish to emphasize that implicit in
Corollary 5.4 of \cite{H4}
%and Lemma \ref{l1} of this paper
is the
necessity of supercompactness in the
proofs of all three of the theorems of this paper.
In fact, Corollary 5.4 of \cite{H4}
explicitly states that if the measurability of
a particular cardinal $\gd$ has been made
indestructible via ${\rm Coll}(\gd, \gg)$
for any cardinal
$\gg > \gd$ via a partial ordering admitting
a gap at $\go$, then $\gd$ had to have
been supercompact in the ground model.
Given this, and given the results of \cite{MSS},
we conjecture that the hypotheses used in the
proofs of Theorems \ref{t1} and \ref{t2}
are exactly what is needed to obtain the
conclusions of these theorems.
The situation is somewhat different, however,
with Theorem \ref{t3}, which
should be contrasted with
Theorems 5 and 6 of \cite{A00a}.
In these theorems, starting with just
a class $\K$ of supercompact cardinals,
models are constructed in which the
strong and supercompact cardinals all
exhibit a uniform degree of indestructibility.
The specific amount of indestructibility is that
if $\gk$ is strong,
$\gk$'s strongness is indestructible under
$\gk$-directed closed forcings which are
also $\gk$-strategically closed, and if
$\gk$ is supercompact, $\gk$'s supercompactness
is indestructible under $\gk$-directed closed
forcings which are also $\gk$-strategically closed.
Thus, since this uniform degree of indestructibility
can be obtained by using hypotheses which are
much weaker than almost hugeness,
we conclude this paper by asking if it is
possible to weaken significantly the
hypotheses used to prove Theorem \ref{t3}.
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\begin{graveyard}
Suppose that
$\gk < \gl$ are regular cardinals.
A partial ordering $\FP_{\gk, \gl}$ that will be used
in this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP_{\gk, \gl} =
\{s : s$ is a bounded subset of
$\gl$ consisting of ordinals of cofinality
$\gk$ so that for every $\ga < \gl$,
$s \cap \ga$ is non-stationary in $\ga\}$,
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%\{ p$ : For some
%$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
%function of $S_p$, a subset of $\ga$ not stationary at its
%supremum nor having any initial segment which is stationary
%at its supremum, so that $\gb \in S_p$ implies
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%ordered by $q \ge p$ iff $q \supseteq p$,
%and $S_p = S_q \cap \sup (S_p)$,
%i.e., $S_q$ is an end extension of $S_p$.
It is well-known
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\cite{Bu}, \cite{A01}, or \cite{KM}), in $V[G]$,
if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G \} \subseteq \gl$
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bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH
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\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
\bibitem{So} R.~Solovay, ``Strongly Compact Cardinals
and the GCH'', in:
{\it Proceedings of the Tarski Symposium},
{\bf Proceedings of Symposia in Pure Mathematics 25},
American Mathematical Society, Providence,
1974, 365--372.
i.e., if $\FP$ is $\gk$-directed
closed, then forcing with $\FP$
preserves the fact that $\gk$
is strong.
Note that if $\FP$ is
$\gk^+$-directed closed, then $\FP$ is
$\gk$-strategically closed.
Also, if $\FP$ is
$\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$\FP$ is ${\prec}\gk$-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.
If we also have that $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an Easton support iteration of length $\gk$
so that at stage $\ga$, a nontrivial forcing is done
based on the ordinal $\gk_\ga$, then we will say that
$\gk_\ga$ is in the field of $\FP$.
Further, if $V^\FP \models ``\gk$ is huge'',
by Exercise 24.12, page 334 of \cite{K},
$V^\FP_\gk \models ``$There is a proper
class of supercompact limits of
supercompact cardinals''.
For the forward direction, we note that since
$V \models ``\gk$ is ${<\gl}$ strong'',
$V \models ``\gk$ is ${<}\gl$ supercompact'',
so by the fact $M^\gl \subseteq M$,
$M \models ``\gk$ is ${<}\gl$ supercompact'' as well.
By elementariness,
$M \models ``\gl$ is measurable'', so
$M \models ``\gk$ is ${<\gl}$ strong''.
Further, by elementariness, since
$M \models ``\gl$ is $j(\gl)$ supercompact'',
if $\gk < \gl$ is so that
$M \models ``\gk$ is ${<\gl}$ strong'',
then Magidor's theorem
implies that
$M \models ``\gk$ is $j(\gl)$ supercompact''.
Also, since $j(\gl) > \gl$, it clearly is
the case that if
$M \models ``\gk$ is $j(\gl)$ supercompact'', then
$M \models ``\gk$ is ${<\gl}$ strong''.
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$ is huge''.
\end{lemma}
\begin{proof}
We begin by observing that
for $\gk < \gl$,
$V \models ``\gk$ is ${<}\gl$ strong'' iff
$M \models ``\gk$ is ${<}j(\gl)$ strong''.
The forward direction follows by elementarity.
The reverse direction follows since
$M^\gl \subseteq M$.
This means that $\FP_\gk$, the partial
ordering $\FP$ through stage $\gk$, is
an initial segment of $j(\FP)$, and $\FP$ is an
initial segment of $j(\FP)$.
Let $G$ be $V$-generic over $\FP$,
where $G = G_0 \ast G_1$ is so that
$G_0$ is $V$-generic over $\FP_\gk$,
and $G_1$ is $V$-generic over
$\FP^\gk$, the portion of $\FP$
defined between stages $\gk$ and $\gl$.
Since $\FP_\gk$ is an Easton support
iteration and $\gk$ is measurable,
this means that
$j''G_0 \subseteq G_0 \ast G_1$.
Hence, in $V[G]$, $j$ extends to
$j_0 : V[G_0] \to M[G_0][G_1]$.
Further, as $\gl$ is measurable in both
$V$ and $M$ and $\FP$ is an Easton
support iteration, $\FP$ is $\gl$-c.c.
Thus, standard arguments show that
$M[G]$ remains $\gl$ closed with
respect to $V[G]$. Therefore,
working in $V[G]$, since
$A = \{j''p : p \in G_1\}$ is a compatible
set of conditions of size $\gl$
in $j(\FP^\gk)$,
a partial ordering which is
$\gl^+$-directed closed in both
$M[G]$ and $V[G]$, we can let $q$ be
a master condition for $A$.
Since we may assume that $j$ is generated
by a huge ultrafilter ${\cal U}$ over
$P^\gk(\gl)$, we can infer by GCH in $V$ that
$|j(\gl)| =
|\{f : f : P^\gk(\gl) \to \gl\}| =
|\{f : f : \gl \to \gl\}| = 2^\gl = \gl^+$.
And, since $j(\FP^\gk)$ is $j(\gl)$-c.c$.$
in $M[G]$, in $M[G]$,
$|\{{\cal A} : {\cal A}$ is a maximal antichain of
$j(\FP^\gk)\}| =
|{[j(\gl)]}^{< j(\gl)}| = j(\gl)$.
This means that in
$V[G]$, this set has cardinality $\gl^+$.
Thus, if
$\la {\cal A}_\ga : \ga < \gl^+ \ra$ is
an enumeration in $V[G]$ of these maximal antichains,
since $j(\FP^\gk)$ is $\gl^+$-directed closed in $V[G]$,
we can let $p_0$ be a common extension of $q$ and
the element of ${\cal A}_0$ with which it is
compatible, take $p_\gb$ for $\gb < \gl^+$ a
limit ordinal as an upper bound for
$\la p_\ga : \ga < \gb \ra$, and take
$p_{\ga + 1}$ for $\ga < \gl^+$ as a
common extension of $p_\ga$ and the element of
${\cal A}_{\ga}$ with which it is compatible.
(To avoid redundancy, we let $p_1 = p_0$.)
If we then define
$G_2 = \{p \in j(\FP^\gk) : p$ is extended
by an element of the sequence
$\la p_\ga : \ga < \gl^+ \ra\}$, then
$G_2$ is an $M[G]$-generic object over
$j(\FP^\gk)$ constructed in $V[G]$ so that
$j''G = j''(G_0 \ast G_1) \subseteq
G_0 \ast G_1 \ast G_2$.
Thus, in $V[G]$, $j$ extends to
$\ov j : V[G] \to M[G][G_2]$, i.e.,
$V[G] \models ``\gk$ is huge''.
This proves Lemma \ref{l6}.
\end{proof}
Work in $V_* = V_\gl$ until otherwise specified.
We observe now that the only difference
between the definition of $\FP$
just given and the definition of
$\FP$ given in the proof of
Theorem \ref{t2} is that there
are stages $\ga$, namely
those $\ga$ which are
supercompact and which are
indestructibly strong cardinals,
at which an additional forcing is done.
This forcing, however, is $\ga$-directed closed,
meaning that it combined with all further
forcing will preserve the fact that
$\ga$ is an indestructibly strong cardinal.
Therefore, using the proof of
Theorem \ref{t2} combined with the
succeeding remark, we are able to infer that
$V^\FP_* \models ``$ZFC + There is a
proper class of strong cardinals + Every
strong cardinal $\gk$ has its strongness
indestructible under $\gk$-directed
closed forcing''.
\begin{lemma}\label{l6}
$V^\FP_* \models ``$Any supercompact
cardinal $\gk < \gk$ has its
supercompactness indestructible
under $\gk$-directed closed forcing''.
\end{lemma}
\begin{proof}
Fix $\gk < \gk$ so that
$V^\FP_* \models ``\gk$ is
supercompact''.
Such a $\gk$ exists, as by Lemma \ref{l6},
$V^\FP \models ``\gk$ is huge''.
Since an analysis such as
the one given in Lemma \ref{l1}
yields that $\FP$ admits a
gap at $\go$, the results of
\cite{H1}, \cite{H2}, and \cite{H3}
show that
$V_* \models ``\gk$ is supercompact''.
Let
$\FP = \FP_\gk$.
We show that
$V^\FP_* \models ``\gk$ is a
supercompact cardinal whose supercompactness
is indestructible under $\gk$-directed
closed forcing''. This suffices, since the factorization
$\FP = \FP_\gk \ast \dot \FP^\gk$ yields that
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is
$\gk$-directed closed''.
The remainder of the proof of Lemma \ref{l6} is very
similar to the argument given in Lemma \ref{l2}.
If the conclusions of Lemma \ref{l6} are false,
let $\FQ \in V^\FP_*$ and $\rho$ be so that
$V^\FP_* \models ``\FQ$ is $\gk$-directed closed'' and
$V^{\FP \ast \dot \FQ}_* \models ``\gk$ isn't
$\rho$ supercompact''. Let
$\gg > \max(\rho, |{\rm TC}(\dot \FQ)|)$
be a cardinal at which GCH holds, e.g.,
a singular strong limit cardinal of
cofinality $\gk$. If
$k : V_* \to M_*$ is an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$,
$M^{\FP \ast \dot \FQ}_* \models ``\gk$
isn't $\rho$ supercompact''.
Since $\gg > 2^\gk$,
we know by Lemma 2.1 of \cite{AC2} and
the succeeding remark that
$M_* \models ``\gk$ is a strong cardinal''.
In addition, since
$V_* \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing'',
by elementarity,
$M_* \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing''.
In particular,
$M_* \models ``\gk$'s strongness is indestructible
under $\gk$-directed closed forcing''.
Further, as $k$ and $M_*$ may be chosen so that
$M_* \models ``\gk$ isn't $\rho$ supercompact'',
Lemma 2.4 of \cite{AC2} implies that
$M \models ``$No cardinal in the interval
$(\gk, \gg]$ is strong''.
Thus, at stage $\gk$ in $M_*$ in the definition
of $k(\FP)$, a non-trivial forcing is done, and
$\dot \FQ$ is a term for an allowable choice
in the lottery held at stage $\gk$ in $M_*$
in the definition of $k(\FP)$.
Also, the next non-trivial stage in $M_*$
in the definition of $k(\FP)$ takes place
well after $\gg$.
If we now write
$k(\FP) = \FP_{\gk + 1} \ast \dot \FR$
and take $G_0$ to be $V_*$-generic over
$\FP_\gk$ and $G_1$ to be
$V[G_0]$-generic over $\FQ$, in
$V_*[G_0][G_1]$, standard arguments show that
$k$ lifts to
$k : V_*[G_0][G_1] \to M_*[G_0][G_1][G_2][G_3]$,
where $G_2$ and $G_3$ are suitably generic
objects constructed in $V_*[G_0][G_1]$, and
$G_3$ contains a master condition for $G_1$.
We can thus infer that
$V^{\FP \ast \dot \FQ}_* \models ``\gk$
is $\gg$ supercompact'', which, by the
choice of $\gg$, implies that
$V^{\FP \ast \dot \FQ} \models ``\gk$
is $\rho$ supercompact''.
This contradiction completes the proof
of Lemma \ref{l6}.
\end{proof}
Since
$V^\FP \models ``\gk$ is huge'',
by Exercise 24.12, page 334 of \cite{K},
$V^\FP_\gl = V^* \models ``$There is a proper
class of supercompact limits of
supercompact cardinals''.
This, together with Lemmas \ref{l6} and \ref{l6},
completes the proof of Theorem \ref{t3}.
\end{proof}
We begin by observing that
for $\gd < \gl$,
$V \models ``\gd$ is ${<}\gl$ strong'' iff
$M \models ``\gd$ is ${<}j(\gl)$ strong''.
The forward direction follows by elementarity.
The reverse direction follows since
$M^{{<}\gl} \subseteq M$.
This means that $\FP$ is
an initial segment of $j(\FP)$, and in $M$,
$\FP = {j(\FP)}_\gk$.
Since $M^{{<}\gl} \subseteq M$,
it must also be the case that
$M^\FP \models ``\FQ$ is $\gk$-directed closed'' and
$M^{\FP \ast \dot \FQ} \models ``\gk$ isn't
$\rho$ supercompact''.
$M^{\FP \ast \dot \FQ}_* \models ``\gk$
isn't $\rho$ supercompact''.
Since $\gg > 2^\gk$,
we know by Lemma 2.1 of \cite{AC2} and
the succeeding remark that
$M_* \models ``\gk$ is a strong cardinal''.
In addition, since
$V_* \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing'',
by elementarity,
$M_* \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing''.
In particular,
$M_* \models ``\gk$'s strongness is indestructible
under $\gk$-directed closed forcing''.
Further, as $k$ and $M_*$ may be chosen so that
$M_* \models ``\gk$ isn't $\rho$ supercompact'',
Lemma 2.4 of \cite{AC2} implies that
Thus, at stage $\gk$ in $M_*$ in the definition
of $k(\FP)$, a non-trivial forcing is done, and
$\dot \FQ$ is a term for an allowable choice
in the lottery held at stage $\gk$ in $M_*$
in the definition of $k(\FP)$.
Without loss of generality, by
truncating the universe at the least
strong limit of supercompact cardinals
if necessary,
By the preceding lemma, we now know that
in $V$,
$\forces_\FP ``\gk$ is a ${<}\gl$
supercompact cardinal whose
${<}\gl$ supercompactness is indestructible
when forcing with
$\gk$-directed closed partial orderings
having rank below $\gl$''.
Therefore, since $M^{{<}\gl} \subseteq M$,
in $M$, it is also the case that
$\forces_\FP ``\gk$ is a ${<}\gl$
supercompact cardinal whose
${<}\gl$ supercompactness is indestructible
when forcing with
$\gk$-directed closed partial orderings
having rank below $\gl$''.
Hence, by reflection, in $V$,
there are unboundedly in $\gk$
many $\gd < \gk$ so that
$\forces_{\FP_\gd} ``\gd$ is a ${<}\gk$
supercompact cardinal whose
${<}\gk$ supercompactness is indestructible
when forcing with
$\gd$-directed closed partial orderings
having rank below $\gk$''.
For any such $\gd$, $\FP$ factors as
$\FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\gd$-directed closed''. This means that
by elementarity, we can infer that
in $M$,
$\forces_\FP ``\gk$ is a ${<}\gl$
supercompact cardinal whose
${<}\gl$ supercompactness is indestructible
when forcing with
$\gk$-directed closed partial orderings
having rank below $\gl$
which is a limit of cardinals $\gd$
which are ${<}\gl$ supercompact and whose
${<}\gl$ supercompactness is indestructible
when forcing with
$\gd$-directed closed partial orderings
having rank below $\gl$''.
Therefore, again by reflection, we can
infer that in $V$, there are unboundedly
in $\gk$ many $\gd < \gk$ so that
$\gd$ is a ${<}\gk$ supercompact cardinal whose
${<}\gk$ supercompactness is indestructible
when forcing with
$\gd$-directed closed partial orderings
having rank below $\gk$
which is a limit of cardinals $\gg$
which are ${<}\gk$ supercompact and whose
${<}\gk$ supercompactness is indestructible
when forcing with
$\gg$-directed closed partial orderings
having rank below $\gk$.
\footnote{Readers are urged to
consult the discussion given on page
633 of \cite{A01} immediately following
Definition 2.2 for an explanation as to
why any $\gk$-strategically closed
partial ordering is $\gk$-weakly
closed and satisfies the Prikry property.}
The remainder of the proof of Lemma \ref{l6} is very
similar to the argument given in Lemma \ref{l2}.
There are also similarities to the argument given
at the end of the proof of Lemma \ref{l5}.
If the conclusions of Lemma \ref{l6} are false,
let $\FQ \in V^\FS_\gk$ and $\rho < \gk$ be so that
$V^\FS_\gk \models ``\FQ$ is $\gd$-directed closed'' and
$V^{\FS \ast \dot \FQ}_\gk \models ``\gd$ isn't
$\rho$ supercompact''. Let
$\gg > \max(\rho, |{\rm TC}(\dot \FQ)|)$,
$\gg < \gk$ be inaccessible. If
$k : V_\gk \to M_*$ is an elementary embedding
witnessing the $\gg$ supercompactness of $\gd$,
$M^{\FS \ast \dot \FQ}_* \models ``\gd$
isn't $\rho$ supercompact''.
Since $\gg > 2^\gd$,
we know by Lemma 2.1 of \cite{AC2} and
the succeeding remark that
$M_* \models ``\gd$ is a strong cardinal''.
In addition, since
$V_\gk \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing'',
by elementarity,
$M_* \models ``$All strong cardinals $\gg$
have their strongness indestructible under
$\gg$-directed closed forcing''.
In particular,
$M_* \models ``\gd$'s strongness is indestructible
under $\gd$-directed closed forcing''.
Further, as $k$ and $M_*$ may be chosen so that
$M_* \models ``\gd$ isn't $\gg$ supercompact'',
Lemma 2.4 of \cite{AC2} implies that
$M_* \models ``$No cardinal in the interval
$(\gd, \gg]$ is strong''.
Thus, at stage $\gd$ in $M_*$ in the definition
of $k(\FS)$, a non-trivial forcing is done, and
$\dot \FQ$ is a term for an allowable choice
in the lottery held at stage $\gd$ in $M_*$
in the definition of $k(\FS)$.
We can now argue as we did in the proofs of
Lemmas \ref{l2} and \ref{l5} to obtain
the contradiction that
%Also, the next non-trivial stage in $M_*$
%in the definition of $k(\FS)$ takes place
%well after $\gg$.
%If we now write
%$k(\FS) = \FS_{\gd + 1} \ast \dot \FR$
%and take $G_0$ to be $V_\gk$-generic over
%$\FS_\gd$ and $G_1$ to be
%$V[G_0]$-generic over $\FQ$, in
%$V_\gk[G_0][G_1]$, standard arguments show that
%$k$ lifts to
%$k : V_\gk[G_0][G_1] \to M_*[G_0][G_1][G_2][G_3]$,
%where $G_2$ and $G_3$ are suitably generic
%objects constructed in $V_\gk[G_0][G_1]$, and
%$G_3$ contains a master condition for $G_1$.
%We can thus infer that
%$V^{\FS \ast \dot \FQ}_\gk \models ``\gd$
%is $\gg$ supercompact'', which, by the
%choice of $\gg$, implies that
$V^{\FS \ast \dot \FQ}_\gk \models ``\gd$
is $\rho$ supercompact''.
\end{graveyard}