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\title{Universal Indestructibility for Degrees of
Supercompactness and Strongly Compact Cardinals
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Universal indestructibility,
indestructibility, measurable cardinal,
strongly compact cardinal, supercompact cardinal.}}
\author{Arthur W.~Apter\thanks{The
first author's research was
partially supported by
PSC-CUNY grants and
CUNY Collaborative
Incentive grants.}
\thanks{The first author
wishes to thank James Cummings for
helpful discussions on the subject
matter of this paper.
In addition, both authors wish to
thank the referee, for many helpful
comments and suggestions which were
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\\
\\
Grigor Sargsyan\\
Group in Logic and the Methodology of Science\\
University of California\\
Berkeley, California 94720 USA\\
http://math.berkeley.edu/$\sim$grigor\\
grigor@math.berkeley.edu}
\date{November 10, 2006\\
(revised March 16, 2008)}
\begin{document}
\maketitle
\begin{abstract}
We establish two theorems concerning
strongly compact cardinals and
universal indestructibility for
degrees of supercompactness.
In the first theorem,
we show that universal
indestructibility for
degrees of supercompactness
in the presence of a strongly compact
cardinal is consistent with the
existence of a proper class of
measurable cardinals.
In the second theorem,
we show that universal
indestructibility for
degrees of supercompactness
is consistent in the presence of
two non-supercompact
strongly compact cardinals,
each of which exhibits a significant
amount of indestructibility for
its strong compactness.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with a brief discussion of
the terminology we will be using.
%when discussing indestructibility,
{\em Full indestructibility}
(for measurability, strong compactness,
or supercompactness)
%for a cardinal $\gk$
means that
$\gd$ retains these properties
%any of its
%measurability, strong compactness, or supercompactness
after forcing with an
arbitrary $\gd$-directed closed
partial ordering. In addition,
if $\gd$ is either $\gl$ strongly
compact or $\gl$ supercompact,
this degree of strong compactness
or supercompactness is {\em fully indestructible}
if $\gd$ remains $\gl$ strongly
compact or $\gl$ supercompact after forcing with an
arbitrary $\gd$-directed closed
partial ordering.
We say that
{\it universal indestructibility for
supercompactness}
holds in a model $V$ for ZFC containing
a supercompact cardinal if, in $V$,
every supercompact and every
partially supercompact (including measurable)
cardinal $\gd$
has its degree of supercompactness
fully indestructible.
If, however, in the preceding definition,
$V$ does not contain a supercompact cardinal,
then we say that
{\it universal indestructibility for
degrees of supercompactness} holds.
Analogously,
%fully Laver indestructible \cite{L} under $\gd$-directed
%closed forcing. Analogously,
{\it universal indestructibility for
strong compactness}
holds in a model $V$ for ZFC if, in $V$,
every strongly compact and every
partially strongly compact (including measurable)
cardinal $\gd$
has its degree of strong compactness
fully indestructible.
%under $\gd$-directed closed forcing.
The concept of
{\rm universal indestructibility}
was originally
established and shown to be
relatively consistent by the first
author and Hamkins in \cite{AH}.
Most notably, several
theorems concerning universal
indestructibility and both
strongly compact and supercompact
cardinals were proven
in that paper, including the relative
consistency of universal
indestructibility for supercompactness
in a model containing a supercompact
cardinal (Theorems 5 and 6), the
relative consistency of universal
indestructibility for strong compactness
in a model containing a strongly compact
cardinal (Theorem 7), and the fact that
if there are
two supercompact cardinals, then universal
indestructibility for supercompactness
must of necessity fail (Theorem 10).
The investigation of universal indestructibility
in the presence of
strongly compact cardinals
was continued in \cite{A05}.
In particular, the relative consistency
of universal indestructibility for
both strong compactness and supercompactness
in the presence of two fully indestructible
strongly compact cardinals was shown.
Note, however, that in the model
constructed in \cite{A05} for
universal indestructibility for
supercompactness containing two
strongly compact cardinals,
the second of these cardinals
is supercompact (although the
first isn't).
This (together with the results
mentioned in the preceding paragraph
and implicitly the results of \cite{A02})
%what had been previously known)
left open the question
of whether universal indestructibility
for degrees of supercompactness in a model
containing two non-supercompact
strongly compact cardinals, together
with some form of indestructibility
for strong compactness for both
of the strongly compact
cardinals, is possible.
Also, in either of the models
constructed in \cite{A05}
(and indeed, in any other model
previously given for universal
indestructibility containing
a strongly compact cardinal, such as
%those found in \cite{AH}),
those mentioned in the preceding paragraph),
there are only
set many measurable cardinals.
This left open in particular
the question of whether universal
indestructibility for degrees of supercompactness
in a model containing a strongly
compact cardinal and a proper class of
measurable cardinals is possible.
The purpose of this paper is to
%further and expand upon the studies begun in
%the aforementioned two papers.
investigate the questions raised
in the previous paragraph.
Specifically, we prove the following two theorems,
which answer these questions completely.
\begin{theorem}\label{t1}
Universal indestructibility
for degrees of supercompactness
in the presence of a
fully indestructible
strongly compact cardinal
and a proper class of
measurable cardinals is consistent
relative to the existence of a
proper class of supercompact cardinals.
\end{theorem}
\noindent We point out explicitly that
since the statement of Theorem \ref{t1}
mentions universal indestructibility
only for degrees of supercompactness,
the model witnessing its conclusions
contains no supercompact cardinals.
\begin{theorem}\label{t2}
%If there is a transitive model
%containing two supercompact cardinals,
%then there is a transitive model containing
Universal indestructibility
for degrees of supercompactness
in the presence of two
non-supercompact
indestructible strongly
compact cardinals
$\gk_1$ and $\gk_2$ is consistent
relative to the existence of
two supercompact cardinals.
In the model constructed,
$\gk_1$ and $\gk_2$ are the
only measurable cardinals and have
no nontrivial degrees of supercompactness,
$\gk_1$'s strong compactness is
fully indestructible, and
%under ${} \gk_1$-directed closed forcing, and
$\gk_2$'s strong compactness is
indestructible under
${} \gk_2$-directed closed forcing
which is also $(\gk_2, \infty)$-distributive.
\end{theorem}
We take this opportunity
to make several comments
concerning the above theorems.
Theorem 10 of \cite{AH}
also demonstrates that if there is a
supercompact cardinal
$\gk$ in the universe and
universal indestructiblity
holds, no cardinal above $\gk$
can be measurable.
As the work of
\cite{A05} (and implicitly
the work of \cite{A02}) show,
this is not true if
$\gk$ is strongly compact.
%However, in
%all of the previously known
%models for universal
%indestructibility for supercompactness
%containing at least one strongly compact
%cardinal there are only
%set many measurable cardinals.
%As our remarks prior to the statements of Theorems
%\ref{t1} and \ref{t2} indicate,
In fact, Theorem \ref{t1}
%provides the first example of a model
indicates that it is possible to have a model
containing a strongly compact
cardinal in which universal
indestructibility for degrees of supercompactness
holds and there is
also a proper class of
measurable cardinals.
%In fact, in all of the models previously
%constructed in which any form
%of universal indestructibility holds
%and there is a strongly compact cardinal,
%there are only set many measurable cardinals.
Theorem \ref{t2} is a generalization
of Theorem 1 of \cite{A02}.
In that result, there is a model
in which the first two strongly
compact cardinals $\gk_1$
and $\gk_2$ are the only
two measurable cardinals,
$\gk_1$'s strong compactness is
fully indestructible,
%under ${} \gk_1$-directed closed forcing,
and $\gk_2$'s measurability,
although not necessarily its
strong compactness, is fully
indestructible.
%under ${} \gk_2$-directed closed forcing.
Thus, although not stated
explicitly in \cite{A02},
universal indestructibility for degrees of
supercompactness holds in this model.
We will have all of these same
features in the model we construct
here which witnesses the conclusions
of Theorem \ref{t2}, with the
additional property that
$\gk_2$'s strong compactness is
indestructible under
${} \gk_2$-directed closed forcing
which is also $(\gk_2, \infty)$-distributive.
Further, Theorem \ref{t2} provides
the first example of an iteration of
strategically closed partial orderings
which forces any form of
indestructibility for a strongly compact
cardinal which isn't a limit of measurable cardinals.
(The partial ordering found in \cite{AG}
which forces both full indestructibility
for the least strongly compact
cardinal and also forces the least
strongly compact cardinal
to be the least measurable
cardinal is a Prikry iteration.)
We present now
very briefly some background information.
For anything left unexplained, readers
may consult \cite{A02} -- \cite{AS06}.
We will abuse notation
slightly and use both $V^\FP$ and
$V[G]$ to indicate the universe obtained
by forcing with the partial ordering $\FP$.
For $\gk$ a cardinal, $\FP$ is
{\em $\gk$-directed closed} if every directed
set of conditions of cardinality less
than $\gk$ has a common extension.
$\FP$ is {\em $\gk$-strategically closed} if
in the two person game in which the
players construct a sequence of conditions
$\la p_\ga \mid \ga \le \gk \ra$ such that
$\ga < \gb$ implies $p_\gb$ extends $p_\ga$,
where player I plays odd stages and
player II plays even and limit stages,
player II has a strategy ensuring the
game can always be continued.
$\FP$ is {\em ${\prec} \gk$-strategically closed} if
in the two person game in which the players
construct a sequence of conditions
$\la p_\ga \mid \ga < \gk \ra$ such that
$\ga < \gb$ implies $p_\gb$ extends $p_\ga$,
where player I plays odd stages and
player II plays even and limit stages,
player II has a strategy ensuring the
game can always be continued.
$\FP$ is {\em $(\gk, \infty)$-distributive} if
given a sequence
$\la D_\ga \mid \ga < \gk \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$ is
also a dense open subset of $\FP$.
Note that if $\FP$ is
$(\gk, \infty)$-distributive, then
forcing with $\FP$ adds no new
subsets of $\gk$.
Finally, we will say that
$\gk$ is {\em supercompact up to the
measurable cardinal $\gl$} if
$\gk$ is $\gd$ supercompact for
every $\gd < \gl$.
\section{The Proofs of Theorems \ref{t1}
and \ref{t2}}\label{s2}
We turn first to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + There is a
proper class of supercompact cardinals
$\la \gk_\ga \mid \ga \in {\rm Ord} \ra$''.
Without loss of generality,
we assume in addition that
$V \models {\rm GCH}$ and that
there are no inaccessible limits
of supercompact cardinals in $V$.
Let $\FP$ be the partial ordering
of Theorem 1 of \cite{AG}
defined with respect to $\gk_0$.
Since $V \models {\rm GCH}$,
the arguments of \cite{AG} show that
$V^\FP \models ``$ZFC + $\gk_0$ is
both the least strongly compact
and least measurable cardinal +
$\gk_0$'s strong compactness is
fully indestructible''.
%under ${} \gk_0$-directed closed forcing''.
Since $\FP$ may be defined so that
$\card{\FP} = \gk_0$, standard
arguments show that GCH holds at
all cardinals at and above
$\gk_0$ after forcing with $\FP$.
In addition, the L\'evy-Solovay
arguments \cite{LS} show that
$V^\FP \models ``$The $\gk_\ga$
for $\ga > 0$ form the
class of supercompact cardinals''.
Work now in $\ov V = V^\FP$.
The partial ordering
$\FQ$ we will use to complete
the proof of Theorem \ref{t1}
is, roughly speaking,
the partial ordering
of Theorem 2 of \cite{A02}
defined using $\gk_0$
instead of $\go$ and the
supercompact cardinals of $\ov V$.
More explicitly,
let $\la \gd_\ga \mid \ga \in {\rm Ord} \ra$ enumerate
$\{\gd \mid \gd$ is measurable but
%$\gd > \gk_0$,
$\gd \neq \gk_\ga$
for any $\ga \in {\rm Ord}\}$. As in
Theorem 2 of \cite{A02},
we define a reverse Easton
class iteration
$\FQ = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga \in {\rm Ord} \ra$ as follows:
\begin{enumerate}
%\item\label{d0} $\FP_{\gk_0} =
%\{\emptyset\}$, i.e., $\FP$ is
%defined as trivial forcing
%up through and including
%stage $\gk_0$.
\item\label{d1} $\FP_{0}$ is the partial ordering
for adding a Cohen subset of $\gk_0$.
%i.e., $\dot \FQ_{\gk_0}$ is a term
%for the partial ordering which adds a
%Cohen subset of $\gk_0$.
%\item\label{d2} If $\forces_{\FP_\ga} ``\gd_\ga$
%isn't measurable'', 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{d3} If $\forces_{\FP_\ga} ``\gd_\ga$
is measurable and there is a $\gd_\ga$-directed
closed partial ordering such that forcing with it
destroys $\gd_\ga$'s measurability'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for a
$\gd_\ga$-directed closed
partial ordering of least possible rank
destroying $\gd_\ga$'s measurability.
\item\label{d4} If neither Case \ref{d1}
nor Case \ref{d3} holds, then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
the trivial partial ordering $\{\emptyset\}$.
\end{enumerate}
By the arguments of Lemmas 3.1 -- 3.3
of \cite{A02},
$\ov V^\FQ \models ``$ZFC +
$\la \gk_\ga \mid \ga \ge 1 \ra$
are the measurable cardinals
above $\gk_0$ + Each measurable
cardinal $\gd$ above $\gk_0$
has its measurability fully
indestructible''.
%under ${} \gd$-directed closed forcing''.
Since by its definition,
$\FQ$ is a ${} \gk_0$-directed closed
reverse Easton class iteration
defined in $\ov V$ whose first
nontrivial stage adds a Cohen
subset of $\gk_0$ and whose
remaining stages don't change
the (size of the)
power set of $\gk_0$,
$\ov V^\FQ \models ``\gk_0$ is both
the least strongly compact and
least measurable cardinal + $\gk_0$'s
strong compactness is fully indestructible +
%under ${} \gk_0$-directed closed forcing +
$2^{\gk_0} = \gk^+_0$''.
In addition,
%by clause (\ref{d4}) above in the definition of $\FQ$,
by the proof of Lemma 3.2 of \cite{A02},
for $\ga > 0$, $V^{\FP_{\gk_\ga}} \models
``\gk_\ga$ is a measurable cardinal whose
measurability is fully indestructible''.
%under $\gk_\ga$-directed closed forcing''.
It then follows that the first nontrivial
stage of forcing in $\FQ$ after $\gk_\ga$
is (at least) $\gk^+_\ga$-directed closed. So,
by the fact $\card{\FP_{\gk_\ga}} = \gk_\ga$,
%by the definition of $\FQ$,
%we know that $\gd_\ga$-directed closed forcing
%where $\gd_\ga > \gk_\ga$
%is done at stage
%$\gk_{\ga + 1}$ when $\ga > 0$,
%and $\gk_\ga$ is supercompact,
%which means by the definition of $\FQ$ that
$\ov V^\FQ \models ``2^{\gk_\ga} =
\gk^+_\ga$ if $\ga > 0$''.
Further, $\ov V^\FQ \models ``$No measurable
cardinal $\gk$ is $2^\gk = \gk^+$
supercompact''.
This follows from the facts that
$\la \gk_\ga \mid \ga \in {\rm Ord} \ra$
%the sequence of $V$-supercompact cardinals,
enumerates the measurable cardinals of
$\ov V^\FQ$, and, since
in $V$, there are no inaccessible
limits of supercompact cardinals,
$\ov V^\FQ \models ``$For all ordinals $\ga$,
$2^{\gk_\ga} = \gk^+_\ga$''.
This means that in $\ov V^\FQ$,
$\gk_0$ is a strongly compact
cardinal, $\gk_0$'s strong compactness
and degree of supercompactness
(namely measurability) are
fully indestructible, and
%under ${} \gk_0$-directed closed forcing, and
any measurable cardinal $\gk > \gk_0$
has its degree of supercompactness
(once again only measurability)
fully indestructible.
%under ${} \gk$-directed closed forcing.
Consequently, $\ov V^\FQ$ is
a model containing a fully
indestructible strongly compact
cardinal and a proper class of
measurable cardinals in which
universal indestructibility for
degrees of supercompactness holds.
This completes the proof of
Theorem \ref{t1}.
\end{proof}
Having completed the proof of
Theorem \ref{t1}, we turn now
to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk_1 < \gk_2$
are supercompact''. Without loss of
generality, we assume in addition that
$V \models {\rm GCH}$ and that no
cardinal above $\gk_1$ is supercompact
up to a measurable cardinal in $V$.
In particular,
$V \models ``$No cardinal above
$\gk_2$ is measurable''.
As in the proof of Theorem \ref{t1},
let $\FP$ be the partial ordering
of Theorem 1 of \cite{AG}
defined with respect to $\gk_1$.
As before,
$V^\FP \models ``$ZFC + $\gk_1$ is
both the least strongly compact
and least measurable cardinal +
$\gk_1$'s strong compactness is
fully indestructible''.
%under ${} \gk_1$-directed closed forcing''.
Since $\FP$ may be defined so that
$\card{\FP} = \gk_1$, once again, standard
arguments show that GCH holds at
all cardinals at and above
$\gk_1$ after forcing with $\FP$.
In addition, the
arguments of \cite{LS} show that
$V^\FP \models ``\gk_2$ is
supercompact and no cardinal
above $\gk_1$ is supercompact
up to a measurable cardinal''.
Further, since
$V \models ``$No cardinal above
$\gk_2$ is measurable'',
$V^\FP \models ``$No cardinal above $\gk_2$
is measurable''.
Working now in $\ov V = V^\FP$, let
$ f : \gk_2 \to V_{\gk_2}$ be a Laver
function \cite{L} for $\gk_2$.
Without loss of generality,
we assume $f$ is defined nontrivially
only on measurable cardinals in
the open interval $(\gk_1, \gk_2)$. Let
$\la \gd_\ga \mid \ga < \gk_2 \ra$ enumerate the
measurable cardinals in the open interval
$(\gk_1, \gk_2)$.
Using $f$, we define a
reverse Easton iteration
$\FQ = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk_2 \ra$
having length $\gk_2$ as follows:
\begin{enumerate}
\item\label{c1} $\FP_0$ is the partial
ordering for adding a Cohen subset
of $\gk_1$.
%i.e., $\dot \FQ_0$ is a term for the partial ordering
%which adds a Cohen subset of $\gk_1$.
\item\label{c2} If $f(\gd_\ga)
= \la 0, \dot \FQ' \ra$,
$\dot \FQ'$ has rank below $\gd_{\ga + 1}$,
and $\forces_{\FP_\ga}
``\dot \FQ'$ is a
${} \gd_\ga$-directed closed
partial ordering of rank below $\gd_{\ga + 1}$ with
%the least measurable cardinal above $\gd_\ga$ with
the property
that forcing with it destroys $\gd_\ga$'s
measurability'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga = \dot \FQ'$.
%is a term for such a partial ordering.
\item\label{c3} If
$f(\gd_\ga) = \la 1, \dot \FQ' \ra$,
$\dot \FQ'$ has rank below $\gd_{\ga + 1}$, and
$\forces_{\FP_\ga} ``\dot \FQ'$ is a
${} \gd_\ga$-directed closed partial
ordering which is also
$(\gd_\ga, \infty)$-distributive
of rank below $\gd_{\ga + 1}$'', then
%the least measurable cardinal above $\gd_\ga$'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ' \ast
\dot \FR$, where $\dot \FR$ is a term
for the partial ordering which adds a
non-reflecting stationary set of ordinals
of cofinality $\gk_1$ to $\gd_\ga$.
\item\label{c4} If none of the above
cases holds, then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ is
the partial ordering which adds a non-reflecting
stationary set of ordinals of cofinality
$\gk_1$ to $\gd_\ga$''.
\end{enumerate}
\begin{lemma}\label{l1}
$\ov V^\FQ \models ``$No cardinal
$\gd \in (\gk_1, \gk_2)$ is measurable''.
\end{lemma}
\begin{proof}
By the definition of $\FQ$,
since adding a non-reflecting stationary
set of ordinals to a regular cardinal $\gd$
ensures that $\gd$ isn't weakly compact,
we know that for any $\ga < \gk_2$,
$\ov V^{\FP_{\ga + 1}}
\models ``\gd_\ga$ isn't measurable''.
Further, since
$\FQ = \FP_{\ga + 1} \ast \dot \FP^{\ga + 1}$ where
$\forces_{\FP_{\ga + 1}} ``\dot \FP^{\ga + 1}$ is
$\eta$-strategically closed for $\eta$ the
least inaccessible cardinal above $\gd_\ga$'',
$\ov V^\FQ \models ``$No $\ov V$-measurable cardinal
$\gd \in (\gk_1, \gk_2)$ is measurable''.
This means the proof of Lemma \ref{l1}
will be complete once we have shown that
forcing with $\FQ$ creates no new measurable
cardinals in the open interval $(\gk_1, \gk_2)$.
To do this, write
$\FQ = \FP_0 \ast \dot \FR$, where
$\FP_0$ is nontrivial,
$\card{\FP_0} = \gk_1$, and
$\forces_{\FP_0} ``\dot \FR$ is
$\gg$-strategically closed for
$\gg$ the least inaccessible
cardinal above $\gk_1$''.
In the terminology of \cite{H2}
and \cite{H3}, $\FQ$ ``admits a gap
at $\gk_1$'', so by the Gap Forcing Theorem
of \cite{H2} and \cite{H3}, any cardinal
measurable in $\ov V^\FQ$
in the open interval $(\gk_1, \gk_2)$
had to have been
measurable in $\ov V$.
From this, we immediately infer that
$\ov V^\FQ \models ``$No cardinal
$\gd \in (\gk_1, \gk_2)$ is measurable''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$\ov V^\FQ \models ``\gk_2$ is a measurable cardinal
whose measurability is fully indestructible''.
%under ${} \gk_2$-directed closed forcing''.
\end{lemma}
\begin{proof}
If this is not the case, let
$\FR \in \ov V^\FQ$ be such that
$\ov V^\FQ \models ``\FR$ is
${} \gk_2$-directed closed and
forcing with $\FR$ destroys
$\gk_2$'s measurability''.
Let $\dot \FR$ be a
canonical term for $\FR$, and let
$\gl > \max(\card{{\rm TC}(\dot \FR)}, 2^{\gk_2})$
be a regular cardinal. Since $f$ is a
Laver function for $\gk_2$, take
$j : \ov V \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness
of $\gk_2$ such that
$j(f)(\gk_2) = \la 0, \dot \FR \ra$.
Since $\ov V \models ``$No cardinal above
$\gk_2$ is measurable'', the closure properties
of $M$ imply that in $M$, $\gd_{\gk_2} = \gk_2$,
${(j(\FQ))}_{\gk_2} = \FQ$, and
$\forces_{\FQ} ``\dot \FR$ is a
${} \gk_2$-directed closed partial ordering
of rank below $\gd_{\gk_2 + 1}$
%the least measurable above $\gk_2$
with the property that forcing with it
destroys $\gk_2$'s measurability''.
By the definition of $\FQ$, this means that
$\dot \FR$ is a term for the partial
ordering used at stage $\gk_2$ in $M$
in the definition of $j(\FQ)$. Thus,
$j(\FQ \ast \dot \FR) = \FQ \ast \dot \FR \ast \dot \FS
\ast j(\dot \FR)$, where the first
nontrivial stage in the definition of
$\dot \FS$ is forced to occur
well above $\gl$.
Standard arguments (which are given
in the second paragraph of the proof of
Lemma 2.2 of \cite{A02}) then show
$j$ lifts to
$j^* : \ov V^{\FQ \ast \dot \FR} \to M^{j(\FQ \ast \dot \FR)}$
and that for $\mu$ the normal measure
over $\gk_2$ given by
$x \in \mu$ iff $\gk_2 \in j^*(x)$,
$\mu \in \ov V^{\FQ \ast \dot \FR}$. Thus,
$\ov V^{\FQ \ast \dot \FR} \models ``\gk_2$
is a measurable cardinal'', a contradiction.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$\ov V^\FQ \models ``\gk_2$ is a strongly compact cardinal
whose strong compactness
is indestructible under
${} \gk_2$-directed closed forcing
which is also $(\gk_2, \infty)$-distributive''.
\end{lemma}
\begin{proof}
Let
$\FR \in \ov V^\FQ$ be such that
$\ov V^\FQ \models ``\FR$ is both
${} \gk_2$-directed closed and
$(\gk_2, \infty)$-distributive''.
As in the proof of Lemma \ref{l2},
let $\dot \FR$ be a
canonical term for $\FR$, and let
$\gl > \max(\card{{\rm TC}(\dot \FR)}, 2^{\gk_2})$
be a regular cardinal. Since $f$ is a
Laver function for $\gk_2$, take
$j : \ov V \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness of $\gk_2$ such that
$j(f)(\gk_2) = \la 1, \dot \FR \ra$.
Because $\gl$ has been chosen large enough,
we may assume by choosing %in $M$
a normal measure over $\gk_2$
having trivial Mitchell
rank that $k : M \to N$ is an elementary
embedding witnessing the measurability of $\gk_2$
definable in $M$ such that
$N \models ``\gk_2$ isn't measurable''.
It is the case that if $i : \ov V \to N$
is an elementary embedding having critical point
$\gk_2$ and for any $x \subseteq N$ with
$\card{x} \le \gl$, there is some $y \in N$
such that $x \subseteq y$ and
$N \models ``\card{y} < i(\gk_2)$'',
then $i$ witnesses the $\gl$ strong
compactness of $\gk_2$.
Using this fact, it is easily verifiable that
$i = k \circ j$ is an elementary embedding
witnessing the $\gl$ strong
compactness of $\gk_2$.
We show that $i$ lifts in
$\ov V^{\FQ \ast \dot \FR}$ to
$i : \ov V^{\FQ \ast \dot \FR} \to
N^{i(\FQ \ast \dot \FR)}$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of $\gk_2$
in $\ov V^{\FQ \ast \dot \FR}$,
the facts that $\gl$ is arbitrary and
trivial forcing is both $\gk_2$-directed
closed and $(\gk_2, \infty)$-distributive
complete the proof of Lemma \ref{l3}.
Let $G_0$ be $\ov V$-generic over $\FQ$,
and let $H$ be $\ov V[G_0]$-generic
over $\FR$.
Since $N \models ``\gk_2$ isn't measurable'',
only trivial forcing is done at stage
$\gk_2$ in $N^\FQ$ in the definition
of $i(\FQ)$. Thus, we may write
$i(\FQ) = \FQ \ast \dot \FQ^1 \ast
\dot \FQ^2 \ast \dot \FQ^3$,
where $\dot \FQ^1$ is a term for the portion of
the forcing defined from stage
$\gk_2$ to stage $k(\gk_2)$,
$\dot \FQ^2$ is a term for the
forcing done at stage $k(\gk_2)$, and
$\dot \FQ^3$ is a term for the remainder
of the forcing, i.e., the portion done
between stages $k(\gk_2)$ and
$k(j(\gk_2)) = i(\gk_2)$
(inclusive of the term $i(\dot \FR)$ for the
forcing done at stage $i(\gk_2)$).
We will build in $\ov V[G_0][H]$
generic objects for the different
portions of $i(\FQ)$.
To do this, we use a modification
of an argument initially due to Magidor,
unpublished by him but found in,
among other places, Lemma 2.3 of \cite{A02}
and Lemma 8 of \cite{AH03}.
The modification is due to the second author.
In particular, in Magidor's original proof,
$i(\FQ) = \FQ \ast \dot \FQ^1 \ast
\dot \FQ^2$,
where $\dot \FQ^1$ is a term for the portion of
the forcing defined from stage
$\gk_2$ up to and including stage $k(\gk_2)$
and $\dot \FQ^2$ is a term for the remainder
of the forcing, i.e., the portion done
between stages $k(\gk_2)$ and
$k(j(\gk_2)) = i(\gk_2)$ (with trivial
forcing alone done at stage $i(\gk_2)$).
It is then only necessary to build an
$N[G_0]$-generic object $G_1$ for $\FQ_1$ and
an $N[G_0][G_1]$-generic object
$G_2$ for $\FQ_2$.
Because of the additional forcing $\FR$ done
at stage $\gk_2$ in $V^\FQ$, in what we are about
to present, we must take care of the forcing
done at stage $k(\gk_2)$ in $N^{\FQ \ast \dot \FQ_1}$
separately, and not as part of
an earlier partial ordering.
In addition, it is also necessary to
handle the issue of a ``master condition'' for
$j(\dot \FR)$ and $k(j(\dot \FR)) = i(\dot \FR)$.
The ideas for how to resolve these two issues
are due to the second author.
%We will, therefore, throughout the course
%of the remainder of the proof of Lemma \ref{l3},
%refer readers to the construction
%given in these lemmas when relevant,
%and omit details already presented therein.
We begin by constructing an
$N[G_0]$-generic object $G_1$ for $\FQ^1$.
The argument used is essentially the
same as the ones given in the construction
of the generic object $G_1$ found in
Lemma 2.3 of \cite{A02} and Lemma 8 of \cite{AH03}
(and will therefore be carried out in
$M[G_0] \subseteq \ov V[G_0] \subseteq
\ov V[G_0][H]$).
Specifically, since
$N \models ``\gk_2$ isn't measurable'',
only trivial forcing is done at
stage $\gk_2$ in $N$, which means that
$\dot \FQ^1$ is forced to act nontrivially
on ordinals in the open interval $(\gk_2, k(\gk_2))$.
In addition, since GCH holds in $N$
at and above $\gk_1$
(as it does in $\ov V$ and $M$),
standard counting arguments
%found in Lemma 2.3 of \cite{A02} and Lemma 8 of \cite{AH03}
show that
$N[G_0] \models ``\card{\FQ^1} = k(\gk_2)$ and
$\card{\wp(\FQ^1)} = 2^{k(\gk_2)} = k(\gk^+_2)$''.
Consequently, since GCH
at and above $\gk_1$ also yields that
$M \models ``\card{k(\gk^+_2)} = \gk^+_2$'',
we may let
$\la D_\ga \mid \ga < \gk^+_2 \ra$
be an enumeration in
%either $\ov V[G_0]$ or
$M[G_0]$ of the dense open subsets of $\FQ^1$
present in $N[G_0]$. We then build in
%either $\ov V[G_0]$ or
$M[G_0]$ an $N[G_0]$-generic object $G_1$
for $\FQ^1$ by meeting in turn each member of
$\la D_\ga \mid \ga < \gk^+_2 \ra$, using the fact
$\FQ^1$ is ${\prec} \gk^+_2$-strategically
closed in $N[G_0]$ and $M[G_0]$.
%and $\ov V[G_0]$,
This follows
from the fact that standard arguments
show $N[G_0]$ remains $\gk_2$-closed with
respect to $\ov V[G_0]$.
We next analyze the exact nature of $\dot \FQ^2$.
By the definition of $\FQ$
and the closure properties
of $M$, we may write
$j(\FQ \ast \dot \FR) = \FQ \ast \dot \FR \ast \dot \FR'
\ast \dot \FS \ast j(\dot \FR)$, where
$\dot \FR \ast \dot \FR'$ is a term for
the forcing taking place at stage
$\gk_2$ in $M$ and
$\dot \FR'$ is a
term for the partial ordering which adds
a non-reflecting stationary set of ordinals
of cofinality $\gk_1$ to $\gk_2$.
By elementarity, since $\dot \FQ^2$ is a
term for the forcing which takes place
at stage $k(\gk_2)$ in $N$, we may write
$\dot \FQ^2 = k(\dot \FR) \ast k(\dot \FR')$.
We will construct in $M[G_0][H]$
generic objects for %(the denotations of)
$k(\FR)$ and $k(\FR')$.
For $k(\FR)$, we use an argument containing
ideas due to Woodin, also presented in
Theorem 4.10 of \cite{H00}, Lemma 4.2 of
\cite{A03}, and Lemma 3.4 of \cite{AS06}.
First, note that since $N$ is given
by an ultrapower,
$N = \{k(h)(\gk_2) \mid h : \gk_2 \to M$
is a function in $M\}$.
Further, since by the definition of $G_1$,
$k '' G_0 \subseteq G_0 \ast G_1$,
$k$ lifts in both $M[G_0]$ and $M[G_0][H]$ to
$k : M[G_0] \to N[G_0][G_1]$.
From these facts, we may now show that
$k '' H \subseteq k(\FR)$ generates an
$N[G_0][G_1]$-generic object $G_2$ over
$k(\FR)$. Specifically, given a dense
open subset $D \subseteq k(\FR)$,
$D \in N[G_0][G_1]$, $D = i_{G_0 \ast G_1}(\dot D)$
for some $N$-name $\dot D = k(\vec D)(\gk_2)$, where
$\vec D = \la D_\ga \mid \ga < \gk_2 \ra$
is a function in $M$. We may assume that every
$D_\ga$ is a dense open subset of $\FR$.
Since $\FR$ is %${} \gk_2$-directed closed,
$(\gk_2, \infty)$-distributive,
it follows that
$D' = \bigcap_{\ga < \gk_2} D_\ga$ is also a
dense open subset of $\FR$. As
$k(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$k '' H \cap D \neq \emptyset$. Thus,
$G_2 = \{p \in k(\FR) \mid \exists
q \in k '' H [q$ extends $p]\}$,
which is definable in
$M[G_0][H]$, is our desired
$N[G_0][G_1]$-generic object over $k(\FR)$.
Then, since $k(\FR')$ is in
$N[G_0][G_1][G_2]$ the partial ordering
which adds a non-reflecting stationary
set of ordinals of cofinality $k(\gk_1)$
to $k(\gk_2)$, we know that
$N[G_0][G_1][G_2] \models
``\card{k(\FR')} = k(\gk_2)$ and
$\card{\wp(k(\FR'))} = 2^{k(\gk_2)} = k(\gk^+_2)$''.
Hence, since $N[G_0][G_1][G_2]$
remains $\gk_2$-closed with respect to $M[G_0][H]$,
which means $k(\FR')$ is
${\prec} \gk^+_2$-strategically closed in
$N[G_0][G_1][G_2]$ and $M[G_0][H]$,
%and $\ov V[G_0][H]$,
the same argument used in the construction of
$G_1$ allows us to build in
$M[G_0][H]$ an
$N[G_0][G_1][G_2]$-generic object
$G_3$ for $k(\FR')$.
We construct now (in $\ov V[G_0][H]$) an
$N[G_0][G_1][G_2][G_3]$-generic object for $\FQ^3$.
We do this by combining the term forcing argument
found in Lemma 2.3 of \cite{A02} and
Lemma 8 of \cite{AH03} with the argument for
the creation of a ``master condition'' found in
Lemma 2 of \cite{AG}. Specifically, we
begin by showing the
existence of a term $\gt \in M$ for a
``master condition'' for $j(\dot \FR)$, i.e., we show the
existence of a term $\gt \in M$
in the language of forcing
with respect to $j(\FQ)$ such that in $M$,
$\forces_{j(\FQ)} ``\gt \in j(\dot \FR)$
extends every $j(\dot q)$
for $\dot q \in \dot H$''.
We first note that since $\FQ$ is
$\gk_2$-c.c$.$ in both $\ov V$ and $M$, as
$\forces_{\FQ} ``\dot \FR$ is
${} \gk_2$-directed closed
%and $(\gk_2, \infty)$-distributive
and $\card{\FR} < \gl$'', the usual
arguments show $M[G_0][H]$ remains
$\gl$-closed with respect to $\ov V[G_0][H]$.
This means $T = \{j(\dot q) \mid \exists
r \in G_0 [\la r, q \ra \in G_0 \ast H]\} \in
M[G_0][H]$ has a name $\dot T \in M$
such that in $M$,
$\forces_{j(\FQ)} ``\card{\dot T} < \gl < j(\gk_2)$,
any two elements of $\dot T$ are compatible, and
$\dot T$ is a subset of a
partial ordering (namely $j(\dot \FR))$ which is
${} j(\gk_2)$-directed closed''. Thus,
%since $M^\gl \subseteq M$,
$\forces_{j(\FQ)} ``$There is a condition
in $j(\dot \FR)$ extending
each element of $\dot T$''.
A term $\gt$ for this common extension
is as desired.
We work for the time being in $M$.
Consider the ``term forcing'' partial ordering
$\FS^*$ (see \cite{F} for the first
published account of term forcing or
\cite{C}, Section 1.2.5, page 8 --- the
notion is originally due to Laver)
associated with $\dot \FS \ast
j(\dot \FR)$, i.e.,
$\gs \in \FS^*$ iff $\gs$ is a term
in the forcing language with respect to
$\FQ \ast \dot \FR \ast \dot \FR'$ and
$\forces_{\FQ \ast \dot \FR \ast \dot \FR'}
``\gs \in \dot \FS \ast
j(\dot \FR)$'', ordered by
$\gs_1$ extends $\gs_0$ iff
$\forces_{\FQ \ast \dot \FR \ast \dot \FR'}
``\gs_1$ extends $\gs_0$''.
Note that $\gt'$ defined as the term
in the language of forcing with respect to
$\FQ \ast \dot \FR \ast \dot \FR'$ composed
of the tuple
all of whose members are forced to be
the trivial condition, with the exception
of the last member, which is $\gt$,
is an element of $\FS^*$.
Clearly, $\FS^* \in M$. In addition, since
$\ov V \models ``$No cardinal above
$\gk_2$ is measurable'',
by the closure properties of $M$,
$M \models ``$The first stage at which
$\dot \FS \ast j(\dot \FR)$
is forced to do nontrivial forcing
is above $\gl$''. Thus,
$\forces_{\FQ \ast \dot \FR \ast \dot \FR'}
``\dot \FS \ast j(\dot \FR)$
is ${\prec} \gl^+$-strategically
closed'', which, since $M^\gl \subseteq M$,
immediately implies that
$\FS^*$ itself is ${\prec} \gl^+$-strategically
closed in both $\ov V$ and $M$. Further, since
$\ov V^\FQ \models ``\card{\FR} < \gl$'',
in $M$,
$\forces_{\FQ \ast \dot \FR \ast \dot \FR'}
``\card{\dot \FS \ast j(\dot \FR)} < j(\gl)$''.
Also, by GCH
at and above $\gk_1$ in both
$\ov V$ and $M$
and the fact $j$ may be assumed
to be given via an ultrapower embedding
by a normal measure over $P_{\gk_2}(\gl)$,
$\card{j(\gl^+)} =
\card{\{f \mid f : P_{\gk_2}(\gl) \to \gl^+} =
%is a function$\}} =
\card{{[\gl^+]}^{\gl}}
= \gl^+$
and
$\forces_{\FQ \ast \dot \FR \ast \dot \FR'}
``\card{\wp(\dot \FS \ast j(\dot \FR))} <
2^{j(\gl)} = j(\gl^+)$''.
Therefore, since as in the footnote
given in the proof of Lemma 8 of
\cite{AH03}, we may assume that
$\FS^*$ has cardinality below
$j(\gl)$ in $M$,
we may let
$\la D_\ga \mid \ga < \gl^+ \ra \in \ov V$
be an enumeration of the dense open subsets
of $\FS^*$ present in $M$.
It is then possible
using the ${\prec} \gl^+$-strategic closure
of $\FS^*$ in $\ov V$ and the argument
employed in the construction of $G_1$
to build in $\ov V$ an $M$-generic object
$G^*_4$ for $\FS^*$ containing $\gt'$.
Note now that since $N$ is given by
an ultrapower of $M$ via a normal
measure over $\gk_2$, Fact 2 of
Section 1.2.2 of \cite{C} tells us that
$k '' G^*_4$ generates an $N$-generic
object $G^{**}_4$ over $k(\FS^*)$
containing $k(\gt')$.
By elementarity,
%since $\FQ = \FQ_{\gk_2}$,
$k(\FS^*)$ is the
term forcing in $N$ defined with respect to
$k(j(\FQ_{})_{\gk_2 + 1}) =
\FQ \ast \dot \FQ^1 \ast \dot \FQ^2$.
Therefore, since
$i(\FQ \ast \dot \FR) = k(j(\FQ \ast \dot \FR)) =
\FQ \ast \dot \FQ^1 \ast \dot \FQ^2 \ast \dot \FQ^3$,
$G^{**}_4$ is $N$-generic over $k(\FS^*)$, and
$G_0 \ast G_1 \ast G_2 \ast G_3$ is
$k(\FQ \ast \dot \FR)$-generic over $N$,
Fact 1 of Section 1.2.5 of \cite{C}
(see also \cite{F})
tells us that for
$G_4 = \{i_{G_0 \ast G_1 \ast G_2
\ast G_3}(\gs) \mid \gs \in G^{**}_4\}$, $G_4$ is
$N[G_0][G_1][G_2][G_3]$-generic over $\FQ^3$.
In addition, since
the definition of $\gt$ tells us that
in $M$, the statement
``$\la p, \dot q \ra
\in j(\FQ \ast \dot \FR)$ implies that
$\la p, \dot q \ra \forces_{j(\FQ \ast \dot \FR)}
`\gt$ extends $\dot q$' ''
is true, by elementarity, in $N$, the statement
``$\la p, \dot q \ra
\in k(j(\FQ \ast \dot \FR))$ implies that
$\la p, \dot q \ra \forces_{k(j(\FQ \ast \dot \FR))}
`k(\gt)$ extends $\dot q$' ''
%``$p \in k(j(\FQ))$ implies that
%$p \forces_{k(j(\FQ))} ``k(\gt)$ extends $\check p$''
is true. In other words,
since $k \circ j = i$, in $N$, the statement
``$\la p, \dot q \ra \in i(\FQ \ast \dot \FR)$ implies that
$\la p, \dot q \ra \forces_{i(\FQ \ast \dot \FR)}
`k(\gt)$ extends $\dot q$' '' is true.
Thus, in $N$, $k(\gt)$ functions as a
term for a ``master condition'' for $i(\dot \FR)$,
so since $G^{**}_4$ contains $k(\gt')$,
the construction of all of the above generic objects
immediately yields that
$i '' (G_0 \ast H) \subseteq G_0
\ast G_1 \ast G_2 \ast G_3 \ast G_4$.
This means that $i$ lifts
in $\ov V^{\FQ \ast \dot \FR}$ to
$i : \ov V^{\FQ \ast \dot \FR} \to
N^{i(\FQ \ast \dot \FR)}$.
This completes the proof of Lemma \ref{l3}.
\end{proof}
By its definition,
$\ov V \models ``\FQ$ is a
$\gk_1$-directed closed partial ordering
%such that $\card{\FQ} = \gk_2$
whose first nontrivial stage adds a
Cohen subset of $\gk_1$,
$\FQ$'s remaining stages don't
change the (size of the) power
set of $\gk_1$, and
$\card{\FQ} = \gk_2$''. Therefore,
as in the proof of Theorem \ref{t1},
$\ov V^\FQ \models ``\gk_1$ is
both the least strongly compact
and least measurable cardinal
and each of $\gk_1$'s strong
compactness and degree of supercompactness
(namely measurability) is fully
indestructible''.
%under $\gk_1$-directed closed forcing''.
In additon, since
$\ov V^\FQ \models ``2^{\gk_2} = \gk^+_2$'',
Lemmas \ref{l1} -- \ref{l3},
our earlier arguments, and the
results of \cite{LS} show that
$\ov V^\FQ \models ``\gk_2$ is the
second measurable cardinal, $\gk_2$'s
degree of supercompactness (namely
measurability) is fully indestructible,
%under $\gk_2$-directed closed forcing,
$\gk_2$ is a strongly compact cardinal whose
strong compactness is indestructible under
$\gk_2$-directed closed forcing which is also
$(\gk_2, \infty)$-distributive, and no
cardinal $\gd > \gk_2$ is measurable''. Hence,
$\ov V^\FQ$ is a model for universal
indestructibility for degrees of supercompactness
containing exactly two strongly compact cardinals
satisfying the remaining conclusions of Theorem
\ref{t2}.
This completes the proof of Theorem \ref{t2}.
\end{proof}
Theorems \ref{t1} and \ref{t2}
and what was previously known
in \cite{A02}, \cite{A05}, and \cite{AH}
leave open a number of interesting problems
related to the issues raised in this paper.
For instance:
\begin{enumerate}
\item Is it possible to have a model for
universal indestructibility for degrees of
supercompactness containing more than
one strongly compact cardinal and a
proper class of measurable cardinals? In
particular, could such a model also witness
the conclusions of Theorem \ref{t2}?
\item Is it possible to have a model analogous
to the one given in Theorem \ref{t2}
in which the second strongly compact
cardinal $\gk_2$ has its strong
compactness fully indestructible?
%under $\gk_2$-directed closed forcing?
\item Are there models analogous to the one constructed
for Theorem \ref{t2} containing more than
two strongly compact cardinals, e.g.,
a model containing more than two
strongly compact cardinals in which
universal indestructibility for degrees of supercompactness
holds and in which each strongly compact
cardinal satisfies some sort of indestructibility
property for its strong compactness?
\item Is there a model analogous to the one constructed
for Theorem \ref{t2} having unboundedly many
measurable cardinals between $\gk_1$ and
$\gk_2$?
%, or even class many measurable cardinals above $\gk_2$?
\end{enumerate}
\noindent These are the questions with which we
conclude this paper.
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\end{document}