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\title{Some Applications of Sargsyan's Equiconsistency Method
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E45, 03E55.}
\thanks{Keywords: Indestructibility,
equiconsistency, measurable cardinal, tall cardinal,
strong cardinal, hyperstrong cardinal,
strongly compact cardinal, supercompact cardinal,
hypercompact cardinal, almost huge
cardinal, core model.}}
\author{Arthur W.~Apter\thanks{The
author's research was
partially supported by
PSC-CUNY grants.}
\thanks{The author wishes to thank the
referee for helpful comments, suggestions,
and corrections which have been incorporated
into the current version of the paper.}\\
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/apter\\
awapter@alum.mit.edu}
%\date{July 27, 2010}
\date{August 20, 2010\\
(revised December 29, 2011)}
\begin{document}
\maketitle
\begin{abstract}
We apply techniques due to Sargsyan to
reduce the consistency strength of the
assumptions used to establish
an indestructibility theorem for supercompactness.
We then show how these and additional techniques
due to Sargsyan
may be employed to establish an equiconsistency
for a related indestructibility theorem
for strongness.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with the following definitions.
\begin{definition}\label{d1}
{\bf (Stewart Baldwin \cite{Ba})}
$\gk$ is {\em 0-hyperstrong} iff
$\gk$ is strong.
For $\ga > 0$, $\gk$ is {\em $\ga$-hyperstrong} iff
for any ordinal $\gd > \gk$, there is an
elementary embedding $j : V \to M$
witnessing the $\gd$-strongness of $\gk$
(i.e., ${cp}(j) = \gk$,
$j(\gk) > \card{V_\gd}$, and
$V_\gd \subseteq M$)
generated by a $(\gk, \gl)$-extender for some ordinal $\gl$
such that
$M \models ``\gk$ is $\gb$-hyperstrong
for every $\gb < \ga$''.
$\gk$ is {\em hyperstrong} iff $\gk$ is $\ga$-hyperstrong
for every ordinal $\ga$.
\end{definition}
%\noindent Note that Definition \ref{d1} was first
%given by Stewart Baldwin in \cite{Ba}, who in \cite{Ba}
%constructed a canonical inner model for a
%hyperstrong cardinal.
\noindent Note that in \cite{Ba}, Baldwin
constructed a canonical inner model for a
hyperstrong cardinal. The author and Sargsyan
showed in \cite[Theorem 2]{AS10} that
ZFC + There exists a Woodin cardinal $\vdash$
Con(ZFC + There exists a proper class of
hyperstrong cardinals) and also used the
notion of hyperstrong cardinal in \cite{AS10}
to establish an equiconsistency for a weak
form of universal indestructibility
(see \cite{AS10} and \cite{AH} for the relevant terminology).
The next definition is the obvious generalization of
Definition \ref{d1} to supercompactness.
It will play a key role in the proof of Theorem \ref{t2}.
\begin{definition}\label{d2}
$\gk$ is {\em 0-hypercompact} iff
$\gk$ is supercompact.
For $\ga > 0$,
$\gk$ is {\em $\ga$-hypercompact} iff
for any cardinal $\gd \ge \gk$, there is an
elementary embedding $j : V \to M$
witnessing the $\gd$-supercompactness of $\gk$
(i.e., ${cp}(j) = \gk$,
$j(\gk) > \gd$, and
$M^\gd \subseteq M$)
generated by a supercompact ultrafilter over
$P_\gk(\gd)$ such that
$M \models ``\gk$ is $\gb$-hypercompact for every $\gb < \ga$''.
$\gk$ is {\em hypercompact} iff $\gk$ is $\ga$-hypercompact
for every ordinal $\ga$.
\end{definition}
Observe that hyperstrong and hypercompact cardinals
are quite large in size with respect to strong
and supercompact cardinals respectively.
In particular, Definitions \ref{d1} and \ref{d2}
imply that a hyperstrong cardinal is a strong limit
of strong cardinals, and a hypercompact cardinal
is a supercompact limit of supercompact cardinals.
We continue with the main narrative.
In \cite{A02}, the following theorem was proven,
where indestructibility is as in Laver's
sense of \cite{L}.
\begin{theorem}\label{t1}
{\bf (\cite[Theorem 3]{A02})}
Let $V \models ``$ZFC + GCH + $\gk$ is almost
huge''. There is then a cardinal $\gl > \gk$
and a partial ordering $\FP \in V_\gl$,
$\card{\FP} = \gk$ such that in
$V^\FP_\gl \models $ZFC,
the following hold.
\begin{enumerate}
\item $\gk$ is a supercompact
limit of supercompact cardinals.
\item The
strongly compact and supercompact cardinals
coincide except at measurable limit points.
\item Every supercompact cardinal $\gd$ is
indestructible under $\gd$-directed closed forcing.
\item Every non-supercompact strongly compact cardinal $\gd$
has both its strong compactness and degree of
supercompactness indestructible under
$\gd$-directed closed forcing.
\end{enumerate}
\end{theorem}
The assumption of an almost huge cardinal
used in the proof of Theorem \ref{t1}
is of course rather strong. Thus, one may
ask if it is possible to prove Theorem \ref{t1}
from weaker hypotheses.
The first goal of this paper is to show
that this is indeed the case.
We begin by establishing the following result,
whose conclusion is identical to that of Theorem \ref{t1}.
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + GCH + $\gk$ is a
hypercompact cardinal''. There is then a partial
ordering $\FP \in V$, $\card{\FP} = \gk$ such that
in $V^\FP$, the following hold.
\begin{enumerate}
\item $\gk$ is a supercompact
limit of supercompact cardinals.
\item The
strongly compact and supercompact cardinals
coincide except at measurable limit points.
\item Every supercompact cardinal $\gd$ is
indestructible under $\gd$-directed closed forcing.
\item Every non-supercompact strongly compact cardinal $\gd$
has both its strong compactness and degree of
supercompactness indestructible under
$\gd$-directed closed forcing.
\end{enumerate}
\end{theorem}
Note that by \cite[Theorem 5]{A08},
ZFC + GCH + There exists an almost huge cardinal $\vdash$
Con(ZFC + GCH + There exists a proper class of hypercompact
cardinals which are limits of hypercompact cardinals).
This means that the hypotheses used to prove
Theorem \ref{t2} represent a bona fide reduction
in consistency strength from those
used to prove Theorem \ref{t1}.\footnote{Note that
\cite[Theorem 5]{A08} actually shows that for
the notion of {\em enhanced supercompact cardinal}
defined in \cite{A08},
ZFC + GCH + There exists an almost huge cardinal $\vdash$
Con(ZFC + GCH + There exists a proper class of enhanced
supercompact cardinals which are
limits of enhanced supercompact cardinals).
However, since any enhanced supercompact cardinal
must be hypercompact,
%an enhanced supercompact cardinal
%is a stronger notion than a hypercompact cardinal
%(in the sense that any enhanced supercompact
%cardinal must be hypercompact),
the desired reduction in consistency strength follows.}
Such a weakening of hypotheses was unattainable prior to
the introduction of Sargsyan's techniques in \cite{AS10}.
As our methods will show, the proof of Theorem \ref{t2}
actually yields the following theorem.
\begin{theorem}\label{t3}
Let $V \models ``$ZFC + GCH + $\gk$ is a
hypercompact cardinal''. There is then a partial
ordering $\FP \in V$, $\card{\FP} = \gk$ such that
in $V^\FP$, the following hold.
\begin{enumerate}
\item $\gk$ is a supercompact
limit of supercompact cardinals.
\item The
strongly compact and supercompact cardinals
coincide except at measurable limit points.
\item Every supercompact cardinal $\gd$ is
indestructible under $\gd$-directed closed forcing.
\item Every measurable limit of supercompact cardinals $\gd$
has its degree of
supercompactness indestructible under
$\gd$-directed closed forcing.\footnote{Since
by Menas \cite{Me}, any measurable limit of
strongly compact cardinals is in fact strongly
compact, every measurable limit of supercompact
cardinals $\gd$ actually also has its strong compactness
indestructible under $\gd$-directed
closed forcing.}
\end{enumerate}
\end{theorem}
Thus, in the spirit of the equiconsistency proven
in \cite{AS10}, one can ask the following
\bigskip\noindent Question: Is it possible to establish
an equiconsistency if Theorem \ref{t3} is recast
in terms of strongness?
\bigskip The second goal of this paper is to provide
an affirmative answer to the Question. Specifically,
we also establish the following result
%(which is the conjunction of Theorems
(which follows from Theorems
\ref{t4a} and \ref{t4b}, to be stated and proved
in Section \ref{s2}), where
for a cardinal $\gd$ exhibiting a nontrivial
degree of strongness, {\em weak indestructibility}
means indestructibility of $\gd$'s degree of
strongness under partial orderings
which are both ${<} \gd$-strategically closed and
$(\gd, \infty)$-distributive.
Once again, prior to the introduction of
Sargsyan's methods of \cite{AS10},
establishing this sort of equiconsistency would
have been impossible.
\setcounter{theorem}{+5}
\begin{theorem}\label{t4}
The theories ``ZFC + There is a hyperstrong cardinal'' and
``ZFC + $T_1$'', where $T_1$ is the theory composed of
the statements
``There is a strong limit of strong cardinals'',
``Every strong cardinal
has its strongness weakly indestructible'', and
``Every measurable
limit of strong cardinals has its degree of strongness
weakly indestructible'' are equiconsistent.
\end{theorem}
We conclude Section \ref{s1} with
some definitions and terminology
which will be found throughout the course of the paper.
We use standard interval notation for
intervals of ordinals.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
If $\FP \in V$ is a partial ordering
and $G \subseteq V$ is $V$-generic over $\FP$,
then we will abuse notation somewhat and use
$V[G]$ and $V^\FP$ interchangeably to denote
the generic extension.
We also abuse notation slightly by occasionally
writing $x$ when we mean $\dot x$ or $\check x$,
especially for ground model objects and variants
of the generic object.
If $\FP$ is a partial ordering
and $\gk$ is a 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
%$\la p_\ga : \ga < \gd < \gk \ra$ 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 : \ga < \delta \rangle $ of elements of $\FP$,
%(where $\langle p_\alpha : \alpha < \delta \rangle$
%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 $p \in \FP$.
$\FP$ is {\it $\gk$-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 stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ 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 stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is ${<}\gk$-{\em strategically closed}
if $\FP$ is $\gd$-strategically closed for
every cardinal $\gd < \gk$.
$\FP$ is $(\gk, \infty)$-{\em distributive}
if for every sequence
$\la D_\ga : \ga < \gk \ra$ of dense open
subsets of $\FP$, $\bigcap_{\ga < \gk} D_\ga$
is also a dense open subset of $\FP$.
Note that since forcing with a partial ordering
%which is both ${<}\gk$-strategically
%closed or both $\gk$-closed and
which is $(\gk, \infty)$-distributive
adds no new subsets of $\gk$,
the measurability of any measurable
cardinal $\gk$ (or equivalently,
its $\gk + 1$-strongness) is automatically
indestructible under such partial orderings.
\section{The Proofs of Theorems \ref{t2} and \ref{t4}}\label{s2}
We turn now to the proofs of Theorems \ref{t2} and \ref{t4}
and a brief discussion as to why the proof of
Theorem \ref{t2} also yields a proof of Theorem \ref{t3}.
Key to the proofs of these theorems
are techniques developed by Sargsyan, which were used
to prove the main theorem (Theorem 1) of \cite{AS10}.
We begin with the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is a
hypercompact cardinal''.
Without loss of generality, by truncating
the universe if necessary, we assume in addition that
$V \models ``$No cardinal $\gd > \gk$ is measurable''.
We start by giving a very slight variant of
the definition of
the partial ordering $\FP$ used in the
proof of \cite[Theorem 3]{A02}, quoting
verbatim from that article when appropriate.
Suppose $\gg < \gd < \gk$ are such that $\gg$ is
regular and $\gd$ is supercompact.
$\FP_{\gg, \gd}$ is defined to be a modification
of Laver's indestructibility partial ordering
of \cite{L}. More specifically, $\FP_{\gg, \gd}$ is
an Easton support
iteration of length $\gd$
defined in the style of \cite{L}
satisfying the following properties.
\begin{enumerate}
\item Every stage at which a nontrivial forcing
is done is a ground model measurable cardinal.
\item The least stage at which a nontrivial forcing
is done can be chosen to be an arbitrarily large
measurable cardinal in $(\gg, \gd)$.
\item At a stage
$\ga$ when a nontrivial
forcing $\FQ$ is done,
$\FQ = \FQ^0 \ast \dot \FQ^1$,
where $\FQ^0$ is $\ga$-directed closed,
and $\dot \FQ^1$ is a term for the
forcing adding
a non-reflecting stationary set of
ordinals of cofinality $\gg$
to some cardinal $\gb > \ga$.
\end{enumerate}
\noindent By its definition, $\FP_{\gg, \gd}$
is a $\gg$-directed closed partial ordering
of rank $\gd + 1$ with
$|\FP_{\gg, \gd}| = \gd$.
By \cite[Lemma 13, pages 2028 -- 2029]{ASh}
(see also the proof of the Theorem of \cite{A98}),
${V}^{\FP_{\gg, \gd}} \models
``$There are no strongly compact cardinals in
the interval $(\gg, \gd)$
since unboundedly many cardinals in
$(\gg, \gd)$ contain non-reflecting
stationary sets of ordinals of
cofinality $\gg$ + $\gd$ is an
indestructible supercompact cardinal''.
This has as a consequence that
${V}^{\FP_{\gg, \gd}} \models
``$Any partial ordering not adding bounded
subsets to $\gd$ preserves that there are
no strongly compact cardinals in the
interval $(\gg, \gd)$''.
Let $\la \gd_\ga : \ga < \gk \ra$
enumerate the $V$-supercompact cardinals
below $\gk$ together with their
measurable limits.
We define now an Easton support iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
of length $\gk$ as follows:
\begin{enumerate}
\item\label{e1} $\FP_1 = \FP_0 \ast \dot \FQ_0$,
where $\FP_0$ is the partial ordering for
adding a Cohen subset to $\omega$, and
$\dot \FQ_0$ is a term for
$\FP_{\ha_2, \gd_0}$.
\item\label{e2} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and
$\forces_{\FP_\ga} ``$There is a $\gd_\ga$-directed
closed partial ordering such that after forcing with it,
$\gd_\ga$ is not $\zeta$-supercompact
for $\zeta$ minimal'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for such
a partial ordering of minimal rank which
destroys the $\zeta$-supercompactness of
$\gd_\ga$.% for the minimal possible $\zeta$.
\item\label{e3} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and Case
\ref{e2} above does not hold
(which will mean that
$\forces_{\FP_\ga} ``\gd_\ga$ is a measurable
limit of supercompact cardinals whose
degree of supercompactness is
indestructible under
$\gd_\ga$-directed closed forcing
and whose strong
compactness is also indestructible under
$\gd_\ga$-directed closed forcing''), then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the
trivial partial ordering
$\{\emptyset\}$.
\item\label{e4} If $\gd_\ga$ is not a
measurable limit of supercompact cardinals,
$\ga = \gb + 1$, $\gd_\gb$ is a measurable
limit of supercompact cardinals,
and Case \ref{e2} above holds for $\gd_\gb$,
then inductively, since a direct limit must
be taken at stage $\gb$,
$|\FP_\gb| = \gd_\gb < \gd_{\gb + 1} = \gd_\ga$.
This means inductively $\FP_\gb$
has been defined so as
to have rank less than $\gd_\ga$,
so by \cite[Lemma 3.1]{A02}
and the succeeding remark,
$\dot \FQ_\gb$ can be
chosen to have rank less than $\gd_\ga$.
Also, by \cite[Lemma 3.1]{A02} and the succeeding remark,
$\zeta < \gd_\ga$ for $\zeta$ the least such that
${V}^{\FP_\gb \ast \dot \FQ_{\gb}} =
{V}^{\FP_\ga} \models ``\gd_\gb$
is not $\zeta$-supercompact''.
(Note that \cite[Lemma 3.1]{A02} and the succeeding remark
say that if $\FP^*$ is a partial ordering,
$\gk^*$ is supercompact, $|\FP^*| < \gk^*$, and
$\dot \FQ$ and $\gg$ are such that
$\forces_{\FP^* \ast \dot \FQ} ``\gk^*$ is not
$\gg$-supercompact'', then $\dot \FQ$ and $\gg$ can be chosen
so that the rank of $\dot \FQ$ is below $\gk^*$
and $\gg < \gk^*$.)
Let $\dot \gg_\ga$ be such that
$\forces_{\FP_\ga} ``\dot \gg_\ga = \gd^+_\gb$'',
and let
$\sigma \in (\gd_\gb, \gd_\ga)$ be the least
measurable cardinal (in either $V$ or
$V^{\FP_\ga}$) such that
$\forces_{\FP_{\ga}} ``\sigma >
\max(\dot \gg_\ga, \dot \zeta,
{\rm rank}(\dot \FQ_\gb))$''. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$ defined such that
$\sigma$ is below the least stage at which,
in the definition of
$\FP_{\gg_\ga, \gd_\ga}$, a nontrivial forcing is done.
\item\label{e5} If $\gd_\ga$ is not a measurable limit
of supercompact cardinals and Case \ref{e4} does not hold,
then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where for
$\gg_\ga = {(\bigcup_{\gb < \ga} \gd_\gb)}^+$,
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$.
\end{enumerate}
We observe that \cite[Lemma 3.1]{A02}
and the succeeding remark
remain true if ``supercompact''
is replaced by ``strong''.
To see this, we use the notation found in
(\ref{e4}) above. Assume that
$\forces_{\FP^* \ast \dot \FQ} ``\gk^*$ is not
$\gg$-strong''. Let $\gl$ be sufficiently large with
$j : V \to M$ an elementary embedding witnessing the
$\gl$-strongness of $\gk^*$ such that in $M$,
$\forces_{\FP^* \ast \dot \FQ} ``\gk^*$ is not
$\gg$-strong''. By reflection, since ${\rm cp(j)} =
\gk^*$ and $|\FP^*| < \gk^*$, there must be
$\FQ^*$ having rank below $\gk^*$ and $\gg^* < \gk^*$
such that
$\forces_{\FP^* \ast \dot \FQ^*} ``\gk^*$ is not
$\gg^*$-strong''.
The arguments of \cite[Lemmas 4.1 -- 4.2]{A02} and the remark
immediately following the proof of Lemma 4.1 literally
unchanged show that in
$V^\FP$, the following hold.
\begin{enumerate}
\item $\gk$ is a
limit of supercompact cardinals.
\item The
strongly compact and supercompact cardinals below $\gk$
coincide except at measurable limit points.
\item Every supercompact cardinal $\gd < \gk$ is
indestructible under
$\gd$-directed closed forcing.
\item Every non-supercompact strongly compact cardinal $\gd < \gk$
has both its strong compactness and degree of
supercompactness indestructible under
$\gd$-directed closed forcing.
\end{enumerate}
%Thus, the proof of Theorem \ref{t2} will be
%completed by the following lemma.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is an indestructible
supercompact cardinal''.
\end{lemma}
\begin{proof}
We use ideas found in the proof of \cite[Lemma 1.4]{AS10}.
We proceed inductively, taking as our inductive hypothesis
that if $\ga \ge 0$ is an ordinal and $N \subseteq V$ is
%a transitive model of ZFC + GCH
such that either
$N = V$ or for some $\gl \ge \gk$, $N$ is the
transitive collapse of $V^{P_\gk(\gl)}/\U$ for
some supercompact ultrafilter $\U$ over $P_\gk(\gl)$
%$\FP \in N$, $V_\gk \in N$,
and $N \models ``\gk$ is $\ga$-hypercompact'',
then $N^\FP \models ``$The
$\gk^{+ \ga}$-supercompactness of $\gk$ is indestructible
under $\gk$-directed closed forcing''.
We assume the inductive hypothesis is true for $\gb < \ga$.
If it is false at $\ga$, then let $N$ and $\FQ' \in N^\FP$
which is $\gk$-directed closed and of minimal rank $\gd$
be such that $N^{\FP \ast \dot \FQ'} \models ``\gk$ is not
$\gk^{+ \ga}$-supercompact''. For the sake of simplicity,
we assume without loss of generality that $N = V$.
Choose $\gl$ to be sufficiently large, e.g.,
suppose $\gl$ is the least strong limit cardinal
greater than $\max(|{\rm TC}(\FP \ast \dot \FQ')|, \gd,
\gk^{+ \ga})$. Let
$j : V \to M$ be an elementary embedding witnessing the
$\gl$-supercompactness of $\gk$ generated by a supercompact
ultrafilter over $P_\gk(\gl)$ such that
$M \models ``\gk$ is $\gb$-hypercompact for every $\gb < \ga$''.
Because $M^\gl \subseteq M$, the definition of
$\FP$ implies that
$j(\FP) = \FP \ast \dot \FQ \ast \dot \FR$, where
$\FQ \in (V_{\gd} )^{M^\FP} = (V_{\gd} )^{V^\FP}$ and
$M^{\FP \ast \dot \FQ} \models
``\gk$ is not $\gk^{+ \ga}$-supercompact''.
Another appeal to the closure properties of $M$
yields that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is not $\gk^{+ \ga}$-supercompact'' as well.
We complete the proof of Lemma \ref{l1}
by showing that
$V^{\FP \ast \dot \FQ} \models ``\gk$ is
$\gk^{+ \ga}$-supercompact'', a contradiction.
To see that this is the case, let $G$ be $V$-generic
over $\FP$ and let $H$ be $V[G]$-generic over $\FQ$.
Observe that since
$M \models ``$No cardinal $\gd > \gk$ is measurable'',
the least ordinal at which $\dot \FR$ is forced
to do nontrivial forcing is well above $\gl$.
Therefore, standard arguments,
as mentioned, e.g., in \cite{L} yield that
$j$ lifts in $V[G][H][H'][H'']$ to
$j : V[G][H] \to M[G][H][H'][H'']$, where $H''$
contains a master condition for $j '' H$ and
$H' \ast H''$ is both $V[G][H]$ and $M[G][H]$-generic over
$\FR \ast j(\dot \FQ)$, a partial ordering which is
$\gl$-directed closed in both $V[G][H]$ and $M[G][H]$.
We consequently have that
${\cal U} \in V[G][H][H'][H'']$ given by
$x \in {\cal U}$ iff $\la j(\gg) : \gg < \gk^{+ \ga} \ra
\in j(x)$ is an ultrafilter over
$(P_\gk(\gk^{+ \ga}))^{V[G][H]}$ witnessing the
$\gk^{+ \ga}$-supercompactness of $\gk$ which,
by the closure properties of $\FR \ast j(\dot \FQ)$
in both $M[G][H]$ and $V[G][H]$, is a member of
$V[G][H]$ as well.
This contradiction completes the proof of Lemma \ref{l1}.
\end{proof}
Since $\FP$ may be defined so that $|\FP| = \gk$,
the L\'evy-Solovay results \cite{LS} show that
$V^\FP \models ``$No cardinal $\gd > \gk$ is measurable''.
Hence, Lemma \ref{l1} completes the proof of Theorem \ref{t2}.
\end{proof}
We know of course that in $V^\FP$, the strongly
compact and supercompact cardinals coincide
except at measurable limit points, every
non-supercompact strongly compact cardinal $\gd$ has
its degree of supercompactness indestructible under
$\gd$-directed closed forcing, and every
supercompact cardinal $\gd$ is indestructible under
$\gd$-directed closed forcing.
Also, Menas' result of \cite{Me}
tells us that every measurable limit of supercompact
cardinals is in fact strongly compact, and that
if $\ga < \gd$ and $\gd$ is the $\ga^{\rm th}$
measurable limit of supercompact cardinals, then
$\gd$ is strongly compact but is not supercompact.
Consequently, there are non-supercompact strongly
compact cardinals present in $V^\FP$, and the only
such cardinals are the measurable limits of supercompact
cardinals. Hence, $V^\FP \models ``$Every measurable
limit of supercompact cardinals has its degree of
supercompactness indestructible under
$\gd$-directed closed forcing'', i.e., $V^\FP$
is a model for the conclusions of Theorem \ref{t3}.
\begin{pf}
Having completed our discussion of Theorems
\ref{t2} and \ref{t3}, we turn our attention
to the proof of Theorem \ref{t4}.
As in \cite{AS10}, for clarity of exposition,
we split its presentation into two distinct components.
We begin with our forcing construction, i.e.,
we first prove the following result.
\setcounter{theorem}{+3}
\begin{theorem}\label{t4a}
Let $T_1$ be the theory composed of
the statements
``There is a strong limit of strong cardinals'',
``Every strong cardinal
has its strongness weakly indestructible'', and
``Every measurable
limit of strong cardinals has its degree of strongness
weakly indestructible''.
Then
Con(ZFC + There is a hyperstrong cardinal) $\implies$
Con(ZFC + $T_1$).
\end{theorem}
\begin{proof}
Let $V \models ``$ZFC + $\gk$ is a hyperstrong cardinal''.
By \cite[Theorem 3.12]{Ba}, it is also possible to
assume that $V \models {\rm GCH}$. As in the proof of
Theorem \ref{t2}, by truncating the universe if necessary,
we once again assume that $V \models ``$No cardinal
$\gd > \gk$ is measurable''.
The partial ordering $\FP$ used in the proof of
Theorem \ref{t4a} will be the
partial ordering used in the proof of Theorem \ref{t2}
recast in terms of strongness.
Suppose $\gg < \gd < \gk$ are such that
$\gg$ is regular and $\gd$ is strong. By the proof of
\cite[Theorem 4.10]{H4}, there is a ${<}\gg$-strategically
closed, $(\gg, \infty)$-distributive partial ordering
$\FP_{\gg, \gd} \in V$ of rank $\gd + 1$ with
$|\FP_{\gg, \gd}| = \gd$ such that
$V^{\FP_{\gg, \gd}} \models ``\gd$ is a weakly
indestructible strong cardinal''.
$\FP_{\gg, \gd}$
is a slight variant of
Hamkins' partial ordering of \cite[Theorem 4.10]{H4}.
As in \cite{H4}, it is
defined as an Easton support iteration of length $\gd$,
with the difference from
\cite{H4} that nontrivial forcing takes place only at
stages $\gs > \gg$ with component partial orderings which
are (at least) both ${<}\gs$-strategically closed and
$(\gs, \infty)$-distributive.
The lifting arguments used in the
proof of \cite[Theorem 4.10]{H4},
which will be given in the proof of Lemma \ref{l4}, then show
that $\FP_{\gg, \gd}$ is as desired.
Let $\la \gd_\ga : \ga < \gk \ra$
enumerate the $V$-strong cardinals
below $\gk$ together with their
measurable limits.
We define now an Easton support iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
of length $\gk$ as follows:
\begin{enumerate}
\item\label{f1} $\FP_1 = \FP_0 \ast \dot \FQ_0$,
where $\FP_0$ is the partial ordering for
adding a Cohen subset to $\omega$, and
$\dot \FQ_0$ is a term for
$\FP_{\ha_2, \gd_0}$.
\item\label{f2} If $\gd_\ga$ is a measurable
limit of strong cardinals
(meaning that $\gd_\ga = \sup_{\gb < \ga} \gd_\gb$) and
$\forces_{\FP_\ga} ``$There is a ${<} \gd_\ga$-strategically
closed, $(\gd_\ga, \infty)$-distributive
partial ordering such that after forcing with it,
$\gd_\ga$ is not $\zeta$-strong
for $\zeta$ minimal'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for such
a partial ordering of minimal rank which
destroys the $\zeta$-strongness of
$\gd_\ga$.% for the minimal possible $\zeta$.
\item\label{f3} If $\gd_\ga$ is a measurable
limit of strong cardinals and Case
\ref{f2} above does not hold
(which will mean that
$\forces_{\FP_\ga} ``\gd_\ga$ is a measurable
limit of strong cardinals whose
degree of strongness is
indestructible under
${<}\gd_\ga$-strategically closed,
$(\gd_\ga, \infty)$-distributive partial orderings''), then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the
trivial partial ordering
$\{\emptyset\}$.
\item\label{f4} If $\gd_\ga$ is not a
measurable limit of strong cardinals,
$\ga = \gb + 1$, $\gd_\gb$ is a measurable
limit of strong cardinals,
and Case \ref{f2} above holds for $\gd_\gb$,
then inductively, since a direct limit must
be taken at stage $\gb$,
$|\FP_\gb| = \gd_\gb < \gd_{\gb + 1} = \gd_\ga$.
This means inductively $\FP_\gb$
has been defined so as
to have rank less than $\gd_\ga$,
so by \cite[Lemma 3.1]{A02}
and the succeeding remark (which as we
have previously observed remain
valid if ``supercompact'' is replaced
by ``strong''), $\dot \FQ_\gb$ can be
chosen to have rank less than $\gd_\ga$.
Also, by \cite[Lemma 3.1]{A02} and the succeeding remark,
$\zeta < \gd_\ga$ for $\zeta$ the least such that
${V}^{\FP_\gb \ast \dot \FQ_{\gb}} =
{V}^{\FP_\ga} \models ``\gd_\gb$
is not $\zeta$-strong''.
%Let $\dot \gg_\ga$ be such that
%$\forces_{\FP_\ga} ``\dot \gg_\ga = \gd^+_\gb$'',
Let $\gg_\ga = \gd^+_\gb$,
and let
$\sigma \in (\gd_\gb, \gd_\ga)$ be the least
measurable cardinal (in either $V$ or
$V^{\FP_\ga}$) such that
$\forces_{\FP_{\ga}} ``\sigma >
\max(\gg_\ga, \dot \zeta,
{\rm rank}(\dot \FQ_\gb))$''.\footnote{As opposed
to the proof of Theorem \ref{t2},
it is possible to take $\gg_\ga = \gd^+_\gb$,
instead of just having
$\forces_{\FP_\ga} ``\dot \gg_\ga = \gd^+_\gb$''.
This is since
$\forces_{\FP_\gb} ``\dot \FQ_\gb$ is
$(\gd_\gb, \infty)$-distributive'', which means
that forcing with $\FP_\gb \ast \dot \FQ_\gb = \FP_{\gb + 1} =
\FP_{\ga}$ preserves $\gd^+_\gb$.}
Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$ defined such that
$\sigma$ is below the least stage at which,
in the definition of
$\FP_{\gg_\ga, \gd_\ga}$, a nontrivial forcing is done.
\item\label{f5} If $\gd_\ga$ is not a measurable limit
of strong cardinals and Case \ref{f4} does not hold,
then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where for
$\gg_\ga = {(\bigcup_{\gb < \ga} \gd_\gb)}^+$,
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$.
\end{enumerate}
\begin{lemma}\label{l2}
$V^\FP \models ``$There are $\gk$ many strong cardinals
$\gd < \gk$''. In addition,
$V^\FP \models ``$Every strong cardinal
$\gd < \gk$ which is not a limit of strong
cardinals has its strongness weakly indestructible''.
\end{lemma}
\begin{proof}
Suppose $V \models ``\gd < \gk$ is a strong cardinal
which is not a limit of strong cardinals''.
This means we can
let $\ga < \gk$ be such that $\gd = \gd_\ga$ and
$\gd \neq \sup_{\gb < \ga} \gd_\gb$.
%and write
%$\FP = \FP_{\ga + 1} \ast \dot \FQ_{\ga + 1} \ast \dot \FR =
%\FP_{\ga + 2} \ast \dot \FR$.
Write $\FP = \FP_\ga \ast \dot \FQ_\ga \ast \dot \FR =
\FP_{\ga + 1} \ast \dot \FR$.
By the
definition of $\FP$, $\forces_{\FP_{\ga + 1}} ``\gd$ is
a weakly indestructible strong cardinal and $\dot \FR$
is ${<}\gd$-strategically closed and
$(\gd, \infty)$-distributive'', from which it immediately follows that
$V^{\FP_{\ga + 1} \ast \dot \FR} =
V^\FP \models ``\gd$ is a weakly indestructible strong cardinal''.
Thus, the proof of Lemma \ref{l2} will be complete
once we have shown that $V^ \FP \models ``$Any
strong cardinal $\gd < \gk$ which is not a
limit of strong cardinals is such that
for some $\ga < \gk$, $\gd = \gd_\ga$ and
$\gd \neq \sup_{\gb < \ga} \gd_\gb$''.
To do this, suppose
$V^\FP \models ``\gd < \gk$ is a strong cardinal which is not
a limit of strong cardinals''.
Write $\FP = \FP' \ast \dot \FP''$, where
$|\FP'| = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
By Hamkins' Gap Forcing Theorem of \cite{H2, H3},
this factorization tells us
that $V \models ``\gd$ is a strong cardinal'',
from which we immediately infer that
$\gd = \gd_\ga$ for some $\ga < \gk$.
If $\gd = \sup_{\gb < \ga} \gd_\gb$, then we have that
$\gd = \sup_{\gb < \ga} \gd_{\gb + 1}$.
Since by the first paragraph of the proof of this lemma,
for any $\gb < \ga$, $V^\FP \models ``\gd_{\gb + 1}$
is a strong cardinal'',
$V^\FP \models ``\gd$ is a limit of
strong cardinals''.
This contradiction completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
${V}^\FP \models ``$If $\gd < \gk$ is a measurable
limit of strong cardinals, then
$\gd$'s degree of strongness is weakly
indestructible''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 4.1]{A02}.
By the factorization of $\FP$ given in
the second paragraph of the
proof of Lemma \ref{l2} and the
results of \cite{H2, H3},
any strong cardinal in ${V}^\FP$
had to have been strong in $V$, and
$\gd$ must be in $V$ a measurable limit
of strong cardinals.
This means that $\gd = \gd_\ga$ for some
limit ordinal $\ga < \gk$
and $\gd = \sup_{\gb < \ga} \gd_\gb$.
If
${V}^\FP \models ``\gd_\ga$'s degree of
strongness is not weakly indestructible'', then let
$\gz$ smallest and
$\FQ \in {V}^\FP$ of minimal rank
be such that
${V}^\FP \models ``\FQ$ is
${<}\gd_\ga$-strategically closed and
$(\gd_\ga, \infty)$-distributive'',
$V^\FP \models ``\gd_\ga$ is $\zeta$-strong'', yet
${V}^{\FP \ast \dot \FQ} \models
``\gd_\ga$ is not $\zeta$-strong''.
Write $\FP = \FP_\ga \ast \dot \FQ_\ga \ast
\dot \FQ_{\ga + 1} \ast \dot \FR =
\FP_{\ga + 1} \ast \dot \FQ_{\ga + 1} \ast
\dot \FR = \FP_{\ga + 1} \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FR = \FP_{\ga + 2} \ast \dot \FR$.
As in the first paragraph of the
proof of Lemma \ref{l2},
the definition of $\FP$ ensures that
$\forces_{\FP_{\ga + 2}} ``\gd_{\ga + 1}$ is a
weakly indestructible strong cardinal and
$\dot \FR$ is
${<}\gd_{\ga + 1}$-strategically closed and
$(\gd_{\ga + 1}, \infty)$-distributive''. Hence,
${V}^{\FP_{\ga + 2} \ast
%\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FR} =
{V}^\FP \models ``\gd_{\ga + 1}$ is a weakly
indestructible strong cardinal'', so
by \cite[Lemma 3.1]{A02}
and the succeeding remark (applied to strong cardinals),
$\FQ, \zeta \in (V_{\gd_{\ga + 1}})^{V^\FP}$.
Therefore, the preceding tells us
$\FQ \in {V}^{\FP_{\ga + 2}} =
{V}^{\FP_\ga \ast \dot \FQ_\ga \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}}}$ and
${V}^{\FP_\ga \ast \dot \FQ_\ga \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FQ} =
{V}^{\FP_\ga \ast \dot \FQ^*} \models
``\gd_\ga$ is not $\zeta$-strong''.
Since $\forces_{\FP_\ga}
``\dot \FQ^*$ is ${<}\gd_\ga$-strategically
closed and $(\gd_\ga, \infty)$-distributive'',
we must be in Case \ref{f4} at stage
$\ga + 2$ of the definition of $\FP$.
This means that for some $\zeta' \le \zeta$,
${V}^{\FP_\ga \ast \dot \FQ_\ga} =
{V}^{\FP_{\ga + 1}} \models
``\gd_\ga$ is not $\zeta'$-strong'',
and consequently, ${V}^\FP \models ``\gd_\ga$ is not
$\zeta'$-strong''as well. Since
${V}^\FP \models ``\gd_\ga$ is
$\zeta$-strong'', this is a
contradiction.
This proves Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``\gk$ is a weakly indestructible strong cardinal''.
\end{lemma}
\begin{proof}
We follow the proofs of Lemma \ref{l1} and \cite[Lemma 1.4]{AS10},
quoting verbatim as appropriate.
We proceed inductively,
taking as our inductive hypothesis that
if $\ga \ge 1$ is an ordinal and
$N \subseteq V$ is such that either $N = V$ or
for some $\gl$, $N$ is the transitive collapse of
${\rm Ult}(V, {\cal E})$ where ${\cal E}$ is a
$(\gk, \gl)$-extender and
%an arbitrary
%transitive model of ZFC + GCH such that
%$\FP \in N$, $V_\gk \in N$, and
%$N \models ``$GCH + $\gk$ is $\gb$-hyperstrong for
%every $\gb < \ga$'', then
$N \models ``\gk$ is $\ga$-hyperstrong'', then
$N^\FP \models ``$The $\gk + \ga$-strongness of $\gk$ is
weakly indestructible''.
For $\ga = 1$,
this amounts to showing that if
%$\FP \in N$, $V_\gk \in N$, $N \subseteq V$, and
%and $N \models ``$GCH + $\gk$ is $0$-hyperstrong'', i.e., if
$N \models ``\gk$ is $1$-hyperstrong'',
then $N^\FP \models ``$The $\gk + 1$-strongness
of $\gk$, i.e., the measurability of $\gk$,
is weakly indestructible''.
To see that this is indeed the case, let
$\mu \in N$ be a normal measure over $\gk$
such that for $j_\mu : N \to M_\mu$ the
ultrapower embedding via $\mu$,
$M_\mu \models ``\gk$ is not measurable''.
Note that
by the fact $V_\gk \in N$ and the definition of $\FP$,
%$N \models ``\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra :
%\gd < \gk \ra$ is a reverse Easton iteration
%such that at nontrivial stages $\gd$,
%$\forces_{\FP_\gd} `\dot \FQ_\gd$ is
%${<}\gd$-strategically closed' ''.
%The preceding two sentences yield that
$j_\mu(\FP) = \FP \ast \dot \FQ'$, where the
first ordinal at which $\dot \FQ'$ is forced
to do nontrivial forcing is above $\gk^+$, and
$\forces_{\FP} ``\dot \FQ'$ is ${\prec} \gk^+$-strategically
closed''. Since $N \models {\rm GCH}$,
standard arguments
%as given, e.g., in \cite[Theorem 3.5]{H4},
yield that
$j_\mu$ lifts in $N$ to
$j_\mu : N^\FP \to M^{j_\mu(\FP)}_\mu$.
(An
outline of these arguments is as follows.
Let $G$ be $N$-generic over $\FP$.
Since $\FP$ is $\gk$-c.c., $M_\mu[G]$ remains
$\gk$-closed with respect to $N[G]$.
Because $N \models {\rm GCH}$ and $M_\mu$
is given by an ultrapower embedding, we may let
$\la D_\gb : \gb < \gk^+ \ra \in N[G]$
enumerate the dense open subsets of $\FQ'$
present in $M_\mu[G]$. As in the
construction of the generic object
$H'$ given later in the proof
of this lemma, it is possible to use the
${\prec}\gk^+$-strategic closure of
$\FQ'$ in both $M_\mu[G]$ and $N[G]$ to
build in $N[G]$ an $M_\mu[G]$-generic
object $G'$ over $\FQ'$. Since
$j_\mu '' G \subseteq G \ast G'$,
$j_\mu$ lifts to $j_\mu : N[G] \to M_\mu[G][G']$.)
From this, it follows that
$N^\FP \models ``\gk$ is measurable''.
Since the measurability
of $\gk$ is weakly indestructible,
we have established the base case of our induction.
We now assume that $\ga > 1$ is an arbitrary
(successor or limit) ordinal.
If our inductive hypothesis is false at $\ga$, then
let $N$ and $\FQ' \in N^\FP$ of minimal rank $\gd$ which is
${<}\gk$-strategically closed and $(\gk, \infty)$-distributive
be such that
$N^{\FP \ast \dot \FQ} \models ``\gk$
is not $\gk + \ga$-strong''.
For the sake of simplicity, we assume without
loss of generality that $N = V$. Choose
$\gl$ to be sufficiently large, e.g.,
suppose $\gl$ is the least strong limit cardinal above
$\max(\card{{\rm TC}(\FP \ast \dot \FQ)}, \gd, \gk + \ga)$
having cofinality $\gk$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$-strongness of $\gk$
generated by a $(\gk, \gl)$-extender such that
$M \models ``\gk$ is $\gb$-hyperstrong
for every $\gb < \ga$''.
By the choice of $j$ and $M$, $\FQ' \in M^\FP$.
Because $V_\gl \subseteq M$, the definition of
$\FP$ implies that
$j(\FP) = \FP \ast \dot \FQ \ast \dot \FR$, where
$\FQ \in (V_{\gd} )^{M^\FP} = (V_{\gd} )^{V^\FP}$ and
$M^{\FP \ast \dot \FQ} \models
``\gk$ is not $\gk + \ga$-strong''.
Another appeal to the fact $V_\gl \subseteq M$
yields that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is not $\gk + \ga$-strong'' as well.
We now show that the embedding $j$
lifts in $V^{\FP \ast \dot \FQ}$ to
$j : V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
%that the embedding $j$ lifts
The methods for doing this
are quite similar to those given in the proof of
\cite[Theorem 4.10]{H4} (as well as elsewhere).
For the benefit of readers, we give
the argument here as well, taking the
liberty to quote freely from \cite[Theorem 4.10]{H4}
and \cite[Lemma 1.4]{AS10}.
Since $j$ is an extender embedding, we have that
$M = \{j(f)(a) : a \in {[\gl]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
Because $V_\gl \subseteq M$ and $V \models
``$No cardinal $\gd > \gk$ is measurable'',
we may write
$j(\FP)$ as $\FP \ast \dot \FQ \ast \dot \FR$,
where the first ordinal at which
$\dot \FR$ is forced to do nontrivial forcing
is above $\gl$.
Since $\gl$ has been chosen to have cofinality $\gk$,
we may assume that $M^\gk \subseteq M$.
This means that if
$G$ is $V$-generic over $\FP$ and
$H$ is $V[G]$-generic over $\FQ$,
$\FR$ is ${\prec} \gk^+$-strategically closed in both
$V[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$.
As in \cite{H4} and \cite{AS10},
by using a suitable coding that allows us
to identify finite subsets of $\gl$
with elements of $\gl$,
by the definition of $M$,
there must be some
$\ga_0 < \gl$ and function $g$ such that
$\dot \FQ = j(g)(\ga_0)$.
%(assuming that $\dot \FQ$ has been chosen reasonably).
Let $N^* = \{i_{G \ast H}(\dot z) : \dot z =
j(f)(\gk, \ga_0, \gl)$ for some function $f \in V\}$.
It is easy to verify that
$N^* \prec M[G][H]$, that $N^*$ is closed under
$\gk$ sequences in $V[G][H]$, and that
$\gk$, $\ga_0$, $\gl$, $\FQ$, and $\FR$ are all
elements of $N^*$.
Further, since
%$\FP \subseteq V_\gk$ and
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : [\gk]^3 \to V_\gk$ in $V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N^*$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G][H]$ and $V[G][H]$, we can
%use the argument for the construction of the
%generic object $G_1$ to be given in Lemma \ref{l5} to
build $H' \subseteq \FR$
in $V[G][H]$ as follows. Let
%an $N$-generic object $H'$ over $\FR$ as follows. Let
$\la D_\gs : \gs < \gk^+ \ra$ enumerate in
$V[G][H]$ the dense open subsets of
$\FR$ present in $N^*$ so that
every dense open subset of $\FR$
occurring in $N^*$ appears at an
odd stage at least once in the
enumeration.
If $\gs$ is an odd ordinal,
$\gs = \tau + 1$ for some $\tau$.
Player I picks
$p_\gs \in D_\gs$ extending $q_\tau$
%$\sup(\la q_\gb : \gb < \gs \ra)$
(initially, $q_{0}$ is the empty condition),
and player II responds by picking
$q_\gs \ge p_\gs$
according to a fixed strategy ${\cal S}$
(so $q_\gs \in D_\gs$).
If $\gs$ is a limit ordinal, player II
uses ${\cal S}$ to pick
$q_\gs$ extending each
$q \in \la q_\gg : \gg < \gs \ra$.
%$\sup(\la q_\gb : \gb < \gs \ra)$.
By the ${\prec} \gk^+$-strategic closure of
$\FR$ in $V[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs : \gs < \gk^+ \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gk^+$.
Let
$H' = \{p \in \FR : \exists \gs <
\gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $H'$ is actually
$M[G][H]$-generic over $\FR$.
If $D$ is a dense open subset of
$\FR$ in $M[G][H]$, then
$D = i_{G \ast H}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$,
where $\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that
$j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of
$\FR$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$ and therefore $(\gl, \infty)$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is
definable without the parameters
$\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N^*$.
Thus, since $H'$ meets every dense open
subset of $\FR$ present in $N^*$,
$H' \cap i_{G \ast H}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$H' \cap i_{G \ast H}(\dot D) \neq \emptyset$.
This means $H'$ is
$M[G][H]$-generic over $\FR$, so in
$V[G][H]$, as
$j''G \subseteq G \ast H \ast H'$,
$j$ lifts to
$j : V[G] \to M[G][H][H']$ via the definition
$j(i_G(\tau)) = i_{G \ast H \ast H'}(j(\tau))$.
It remains to lift $j$ through the forcing $\FQ$
while working in $V[G][H]$.
To do this, it suffices to show that
$j''H \subseteq j(\FQ)$ generates an
$M[G][H][H']$-generic object $H''$ over $j(\FQ)$.
Given a dense open subset
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = i_{G \ast H \ast H'}(\dot D)$ for some name
$\dot D = j(\vec D)(a)$, where
$a \in {[\gl]}^{< \go}$ and
$\vec D = \la D_\gs : \gs \in {[\gk]}^{|a|} \ra$
is a function.
%{[\gk]}^{< \go} \ra$.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $(\gk, \infty)$-distributive, it follows that
%$D' = \bigcap_{\gs \in {[\gk]}^{< \go}} D_\gs$ is
$D' = \bigcap_{\gs \in {[\gk]}^{|a|}} D_\gs$ is
also a dense open subset of $\FQ$. As
$j(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$j''H \cap D \neq \emptyset$. Thus,
$H'' = \{p \in j(\FQ) : \exists q \in j''H
[q \ge p]\}$ is our desired generic object, and
$j$ lifts in $V[G][H]$ to
$j : V[G][H] \to M[G][H][H'][H'']$.
This final lifted version of $j$ is
$\gl$-strong since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$.
Therefore, since
$V[G][H] \models ``\gl > \gk + \ga$ is a
strong limit cardinal'',
$V[G][H] \models ``\gk$ is $\gk + \ga$-strong''.
This contradiction completes our induction
and the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l2} -- \ref{l4} imply that
in $V^\FP$, the following hold.
\begin{enumerate}
\item $\gk$ is a strong limit
of strong cardinals.
\item Every strong cardinal
$\gd \le \gk$ has its strongness weakly
indestructible
\item Every measurable limit of
strong cardinals $\gd \le \gk$ has its
degree of strongness weakly indestructible.
\end{enumerate}
\noindent Since $\FP$ may be defined so that $|\FP| = \gk$,
as before, the results of \cite{LS} show that
$V^\FP \models ``$No cardinal $\gd > \gk$ is measurable''.
This means that $V^\FP$ witnesses the conclusions of
Theorem \ref{t4a}.
\end{proof}
%Hence, Lemmas \ref{l2} -- \ref{l4} complete the proof of
%Theorem \ref{t4a}.
%\end{proof}
Having completed the proof of Theorem \ref{t4a},
we present now the inner model portion
of our argument. Specifically, we establish the
following result.
\begin{theorem}\label{t4b}
Let $T_2$ be the theory composed of the statements
``There is a strong limit of strong cardinals'' and
``Every measurable limit of strong cardinals has
its degree of strongness weakly indestructible''. Then
Con(ZFC + $T_2$) $\implies$
Con(ZFC + There is a hyperstrong cardinal).
%Let $T_1$ be the theory composed of
%the statements
%``There is a strong limit of strong cardinals'',
%``Every strong cardinal
%has its strongness weakly indestructible'', and
%``Every measurable
%limit of strong cardinals has its degree of strongness
%weakly indestructible''.
%Then
%Con(ZFC + $T_1$) $\implies$
%Con(ZFC + There is a hyperstrong cardinal).
\end{theorem}
\begin{proof}
We follow very closely the proof of
\cite[Theorem 4]{AS10}, frequently quoting verbatim
when appropriate.
We argue using standard core model techniques
exposited in \cite{St} and \cite{Ze}.
We are done if there is an inner model with a
hyperstrong cardinal, so we assume without
loss of generality that this is not the case.
Suppose that
$V$ is a model of ZFC in which the following hold.
\begin{enumerate}
\item $\gk$ is a strong limit of
strong cardinals.
\item Every strong cardinal has
its strongness weakly indestructible.
\item Every measurable limit of strong
cardinals has its degree of strongness weakly indestructible.
\end{enumerate}
\noindent Let $\gl > \gk$
%, e.g., $\gl = \gk^+ + \go$,
be an arbitrary strong limit cardinal.
%ordinal which is sufficiently large, e.g.,
%$\gl$ is of size at least $\gk + \go$.
We now have
that if $j : V \to M$ is an elementary
embedding witnessing the $\gl$-strongness of
$\gk$ generated by a
$(\gk, \gl)$-extender ${\cal E}$, then
$M \models ``\gk$ is $(\gk + 2)$-strong''.
Since ${\rm cp}(j) = \gk$, for any
$\gg < \gk$ such that $V \models ``\gg$
is a strong cardinal'', $M \models ``j(\gg) = \gg$
is a strong cardinal''.
The previous two sentences therefore immediately
imply that $M \models ``\gk$ is a measurable
limit of strong cardinals''.
Hence, by elementarity,
$M \models ``$The $(\gk + 2)$-strongness of
$\gk$ is weakly indestructible''.
Consequently, for $\gd > \gk^+$ an arbitrary cardinal
and $\FP_\gd = ({\rm Coll}(\gk^+, \gd))^M$,
$M^{\FP_\gd} \models ``\gk$ is $(\gk + 2)$-strong''.
Since any subset of $\gd$ may now be coded
by a subset of $\gk^+$,
this means that %every subset of $\gd$ is present in
%there is actually an extender
%${\cal F} \in M^{\FP_\gd}$ witnessing
%enough strongness to show that all subsets
%of $\gd$ are captured in the relevant target model.
there is actually a $(\gk, \gk^{++})$-extender
${\cal F} \in M^{\FP_\gd}$
such that all subsets
of $\gd$ are captured in ${\rm Ult}(M^{\FP_\gd}, {\cal F})$.
By downwards absoluteness to the core model
${(K)}^{M^{\FP_\gd}}$, this last fact is true in
${(K)}^{M^{\FP_\gd}}$ as well, i.e., in
${(K)}^{M^{\FP_\gd}}$,
for some $\gg$,
there is a $(\gk, \gg)$-extender
${\cal F}^* = {\cal F} \rest {(K)}^{M^{\FP_\gd}}$
such that all subsets of
$\gd$ are captured in ${\rm Ult}({(K)}^{M^{\FP_\gd}}, {\cal F}^*)$.
By the absoluteness of the core model under set forcing,
in the core model ${(K)}^M = {(K)}^{M^{\FP_\gd}}$,
%there is some strongness extender
%${\cal F}^{**} = {\cal F}^* \rest {(K)}^M$
${\cal F}^*$ is a $(\gk, \gg)$-extender
witnessing that all subsets of
$\gd$ are captured in the relevant target model.
Since $\gd > \gk$ was arbitrary, this just means that
${(K)}^M \models ``\gk$ is a strong cardinal''.
Let $K = {(K)}^V$.
We show that $K \models ``\gk$
is $1$-hyperstrong''. To do this,
take once again $\gl$, ${\cal E}$, $j$, and $M$
as in the preceding paragraph.
Let ${\cal E}^* = {\cal E} \rest K$, with
$i : K \to N$ the $\gl$-strongness
embedding generated by ${\cal E}^*$ and
$\ell : N \to {(K)}^M$ the associated factor
elementary embedding whose critical point
is greater than $\gk$.
It is then the case that
$N \models ``\gk$ is a strong cardinal'', since
by elementarity,
$N \models ``\gk$ is a strong cardinal'' iff
${(K)}^M \models ``\ell(\gk)$ is a strong cardinal'',
i.e., iff
${(K)}^M \models ``\gk$ is a strong cardinal''.
Thus, for any $\gl > \gk$ which is a strong limit cardinal,
there is an elementary embedding witnessing the
$\gl$-strongness of $\gk$ in $K$ generated by a
$(\gk, \gl)$-extender such that in
the target model, $\gk$ is a strong cardinal.
Now that we know that
$K \models ``\gk$ is $1$-hyperstrong'',
we are able to proceed inductively.
Specifically, we assume that for
$\gl > \gk$ having been chosen to be a strong limit cardinal,
$j : V \to M$ an elementary
embedding witnessing the $\gl$-strongness of
$\gk$ generated by a
$(\gk, \gl)$-extender ${\cal E}$, and $\gr$ either
a successor or limit ordinal,
${(K)}^M \models ``\gk$ is $\ga$-hyperstrong
for every $\ga < \gr$''.
The proof given in the preceding
paragraph, with
``$\gk$ is $\ga$-hyperstrong
for every $\ga < \gr$''
replacing ``$\gk$ is a strong cardinal''
then shows that
$K \models ``\gk$ is $\gr$-hyperstrong''.
As $\gr$ was arbitrary,
this completes the proof of Theorem \ref{t4b}.
\end{proof}
Since Con(ZFC + $T_1$) $\implies$
Con(ZFC + $T_2$),
the proofs of Theorems \ref{t4a} and \ref{t4b}
complete the proof of Theorem \ref{t4}.
\end{pf}
%We conclude Section \ref{s2} by observing that
%the proof of Theorem \ref{t4b} actually yields
%the following slightly stronger result.
%\setcounter{theorem}{+6}
%\begin{theorem}\label{t4c}
%Let $T_2$ be the theory composed of the statements
%``There is a strong limit of strong cardinals'' and
%``Every measurable limit of strong cardinals has
%its degree of strongness weakly indestructible''. Then
%Con(ZFC + $T_2$) $\implies$
%Con(ZFC + There is a hyperstrong cardinal).
%\end{theorem}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we make several remarks.
We begin by conjecturing that, in analogy to
Theorem \ref{t4}, the conclusions of
Theorems \ref{t2} and \ref{t3} are actually equiconsistent
with the existence of a hypercompact cardinal.
Of course, since inner model theory for supercompactness
is still in its infancy, an attempt at establishing
this conjecture is not yet in sight.
We also ask whether it is possible to prove a version
of Theorem \ref{t4} for the kind of indestructibility
first described by Gitik and Shelah in \cite{GS}.
The proofs of Lemmas \ref{l2} and \ref{l3}
seem to suggest the use of a version of the
Gap Forcing Theorem for Prikry iterations,
something which has yet to be demonstrated.
Finally, we mention that in \cite{H5}, Hamkins
introduced the concept of {\em tall cardinal},
whose definition we now recall.
\begin{definition}\label{d3}
$\gk$ is {\em $\ga$-tall} iff there is an elementary
embedding $j : V \to M$ with ${cp}(j) = \gk$
such that $j(\gk) > \ga$ and $M^\gk \subseteq M$.
$\gk$ is {\em tall} iff $\gk$ is $\ga$-tall for every
ordinal $\ga$.
\end{definition}
\noindent Hamkins also presented in \cite{H5}
the thesis that ``tall is to strong as strongly
compact is to supercompact''. In light of this,
we finish by asking whether the theories ``ZFC + There is a
hyperstrong cardinal'' and ``ZFC + $T_3$'',
where $T_3$ is the theory composed of the statements
``There is a
strong limit of strong cardinals'', ``The strong
and tall cardinals coincide except at measurable
limit points'', ``Every strong cardinal is weakly
indestructible'', and ``Every non-strong tall cardinal
has its degree of strongness weakly indestructible''
%are equiconsistent.\footnote{Hamkins
%has shown in \cite{H5} that any measurable limit
%of tall cardinals is itself tall.
%Thus, if the strong and tall cardinals coincide
%except at measurable limit points, the only
%non-strong tall cardinals are measurable limits
%of strong cardinals. Consequently, since
%we have already observed in the proof of
%Lemma \ref{l4} that measurability is weakly indestructible,
%and since under the scenario just presented any strong
%cardinal is weakly indestructible, any non-strong
%tall cardinal trivially has its tallness weakly indestructible.}
are equiconsistent.\footnote{Hamkins has shown in
\cite{H5} that a tall cardinal $\gd$
must have its tallness indestructible
under $(\gd, \infty)$-distributive forcing.
This means that weak indestructibility for tallness,
in the sense of a tall cardinal $\gd$
having its tallness indestructible under
partial orderings which are both ${<}\gd$-strategically
closed and $(\gd, \infty)$-distributive, is always true.}
Although the technology for dealing with tall
cardinals is also still in its early stages
of development, we
conjecture that there is an affirmative
answer to this question.
\begin{thebibliography}{99}
\bibitem{A02} A.~Apter, ``Aspects of Strong
Compactness, Measurability, and Indestructibility'',
{\it Archive for Mathematical Logic 41}, 2002, 705--719.
\bibitem{A98} A.~Apter, ``Laver Indestructibility and the
Class of Compact Cardinals'', {\it Journal of Symbolic Logic 63},
1998, 149--157.
\bibitem{A08} A.~Apter, ``Reducing the Consistency
Strength of an Indestructibility Theorem'',
{\it Mathematical Logic Quarterly 54}, 2008, 288--293.
%\bibitem{A05} A.~Apter,
%`` Universal Indestructibility is
%Consistent with Two Strongly Compact
%Cardinals'', {\it Bulletin of the
%Polish Academy of Sciences 53}, 2005, 131--135.
%\bibitem{AG} A.~Apter, M.~Gitik,
%``The Least Measurable can be
%Strongly Compact and Indestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 1404--1412.
\bibitem{AH} A.~Apter, J.~D.~Hamkins,
``Universal Indestructibility'',
{\it Kobe Journal of Mathematics 16},
1999, 119--130.
\bibitem{AS10} A.~Apter, G.~Sargsyan,
``An Equiconsistency for Universal Indestructibility'',
{\it Journal of Symbolic Logic 75}, 2010, 314--322.
%\bibitem{AS07} A.~Apter, G.~Sargsyan,
%``A Reduction in Consistency Strength for
%Universal Indestructibility'', {\it Bulletin
%of the Polish Academy of Sciences (Mathematics)
%55}, 2007, 1--6.
\bibitem{ASh} A.~Apter, S.~Shelah, ``Menas' Result is Best Possible'',
{\it Transactions of the American Mathematical Society 349},
1997, 2007--2034.
\bibitem{Ba} S.~Baldwin, ``Between Strong
and Superstrong'', {\it Journal of Symbolic
Logic 51}, 1986, 547--559.
%\bibitem{G} M.~Gitik, ``Changing Cofinalities
%and the Nonstationary Ideal'',
%{\it Israel Journal of Mathematics ??},
%1986, ??--??.
\bibitem{GS} M.~Gitik, S.~Shelah,
``On Certain Indestructibility of
Strong Cardinals and a Question of Hajnal'',
{\it Archive for Mathematical Logic 28}, 1989, 35--42.
%\bibitem{H4} J.~D.~Hamkins, ``Extensions
%with the Approximation and Cover
%Properties Have No New Large Cardinals'',
%{\it Fundamenta Mathematicae 180}, 2003, 257--277.
\bibitem{H2} J.~D.~Hamkins,
``Gap Forcing'', {\it Israel
Journal of Mathematics 125}, 2001, 237--252.
\bibitem{H3} J.~D.~Hamkins,
``Gap Forcing: Generalizing the L\'evy-Solovay
Theorem'', {\it Bulletin of Symbolic Logic
5}, 1999, 264--272.
\bibitem{H5} J.~D.~Hamkins, ``Tall Cardinals'',
{\it Mathematical Logic Quarterly 55}, 2009, 68--86.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery
Preparation'', {\it Annals of Pure and
Applied Logic 101}, 2000, 103--146.
%\bibitem{HW} J.~D.~Hamkins, W.~H.~Woodin,
%\bibitem{J} T.~Jech, {\it Set Theory:
%The Third Millennium Edition,
%Revised and Expanded},
%Springer-Verlag, Berlin and New York, 2003.
%\bibitem{K} A.~Kanamori, {\it The
%Higher Infinite}, Springer-Verlag,
%Berlin and New York, 1994.
\bibitem{L} R.~Laver, ``Making the Supercompactness of
$\gk$ Indestructible under $\gk$-Directed Closed
Forcing'', {\it Israel Journal of Mathematics 29},
1978, 385--388.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974/75, 327--359.
%\bibitem{MiSt} W.~Mitchell, J.~Steel,
%{\it Fine Structure and Iteration Trees},
%{\bf Lecture Notes in Logic 3},
%Springer-Verlag, Berlin and New York, 1994.
%\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
%``Strong Axioms of Infinity and Elementary Embeddings'',
%{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\bibitem{St} J.~Steel, {\it The Core Model
Iterability Problem}, {\bf Lecture Notes in
Logic 8}, Springer-Verlag, Berlin and New York, 1996.
\bibitem{Ze} M.~Zeman, {\it Inner Models and
Large Cardinals}, {\bf de Gruyter Series in
Logic and its Applications 5}, Walter de Gruyter
and Co., Berlin, 2002.
\end{thebibliography}
\end{document}
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is an indestructible
supercompact cardinal''.
\end{lemma}
\begin{proof}
We use ideas found in the proof of \cite[Lemma 1.4]{AS10}.
We proceed inductively, taking as our inductive hypothesis
that if $\ga \ge 0$ is an ordinal and
$\gz \le \gk$ is such that
$V \models ``\gz$ is
$\ga$ hypercompact'', then $V^{{\FP_\gz}} \models ``$The
$\gz^{+ \ga}$ supercompactness of $\gz$ is indestructible
under $\gz$-directed closed forcing''.
We assume the inductive hypothesis is true for $\gb < \ga$.
If it is false at $\ga$, then let
$\gz \le \gk$ and $\FQ' \in V^{{\FP_\gz}}$
which is $\gz$-directed closed and of minimal rank $\gd$
be such that $\gz$ is minimal and
$V^{{\FP_\gz} \ast \dot \FQ'} \models ``\gz$ is not
$\gz^{+ \ga}$ supercompact''.
Choose $\gl$ to be sufficiently large, e.g.,
suppose $\gl$ is the least strong limit cardinal
greater than $\max(|{\rm TC}({\FP_\gz} \ast \dot \FQ')|, \gd,
\gz^{+ \ga})$. Let
$j : V \to M$ be an elementary embedding witnessing the
$\gl$ supercompactness of $\gz$ generated by a supercompact
ultrafilter over $P_\gz(\gl)$ such that
$M \models ``\gz$ is $\gb$ hypercompact for every $\gb < \ga$''.
Because $M^\gl \subseteq M$
and $\gz < j(\gz) < j(\gk)$, the definition of
${\FP_\gz}$ and the inductive hypothesis
imply that
$j({\FP_\gz}) = {\FP_\gz} \ast \dot \FQ \ast \dot \FR$, where
$\FQ \in (V_{\gd} )^{M^{\FP_\gz}} = (V_{\gd} )^{V^{\FP_\gz}}$ and
$M^{{\FP_\gz} \ast \dot \FQ} \models
``\gz$ is not $\gz^{+ \ga}$ supercompact''.
Another appeal to the closure properties of $M$
yields that
$V^{{\FP_\gz} \ast \dot \FQ} \models
``\gz$ is not $\gz^{+ \ga}$ supercompact'' as well.
We complete the proof of Lemma \ref{l1}
by showing that
$V^{{\FP_\gz} \ast \dot \FQ} \models ``\gz$ is $\gz^{+ \ga}$
supercompact'', a contradiction.
To see that this is the case, let $G$ be $V$-generic
over ${\FP_\gz}$ and let $H$ be $V[G]$-generic over $\FQ$.
Observe that since
$M \models
``\gl$ is the least strong limit cardinal
greater than $\max(|{\rm TC}({\FP_\gz} \ast \dot \FQ')|, \gd,
\gz^{+ \ga})$'',
the least ordinal at which $\dot \FR$ is forced
to do nontrivial forcing is well above $\gl$.
Therefore, standard arguments,
as mentioned, e.g., in \cite{L} yield that
$j$ lifts in $V[G][H][H'][H'']$ to
$j : V[G][H] \to M[G][H][H'][H'']$, where $H''$
contains a master condition for $j '' H$ and
$H' \ast H''$ is both $V[G][H]$ and $M[G][H]$-generic over
$\FR \ast j(\dot \FQ)$, a partial ordering which is
$\gl$-directed closed in both $V[G][H]$ and $M[G][H]$.
We consequently have that
${\cal U} \in V[G][H][H'][H'']$ given by
$x \in {\cal U}$ iff $\la j(\gg) : \gg < \gz^{+ \ga} \ra
\in j(x)$ is an ultrafilter over
$(P_\gz(\gz^{+ \ga}))^{V[G][H]}$ witnessing the
$\gz^{+ \ga}$ supercompactness of $\gz$ which,
by the closure properties of $\FR \ast j(\dot \FQ)$
in both $M[G][H]$ and $V[G][H]$, is a member of
$V[G][H]$ as well.
This contradiction completes the proof of Lemma \ref{l1}.
\end{proof}
Since $\FP$ may be defined so that $|\FP| = \gk$,
the L\'evy-Solovay results \cite{LS} show that
$V^\FP \models ``$No cardinal $\gd > \gk$ is measurable''.
Hence, Lemma \ref{l1} completes the proof of Theorem \ref{t2}.
\end{proof}