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\title{Supercompactness and Level by Level Equivalence
are Compatible with Indestructibility for
Strong Compactness
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
indestructibility, Gitik iteration, Prikry forcing,
non-reflecting stationary set of ordinals,
level by level equivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grant
66489-00-35 and a CUNY
Collaborative Incentive Grant.}
\thanks{The author wishes to thank
the referee for helpful comments and
suggestions which have been incorporated
into this 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{January 24, 2006\\
(revised May 24, 2006)}
\begin{document}
\maketitle
\begin{abstract}
It is known that if
$\gk < \gl$ are such that
$\gk$ is indestructibly
supercompact and $\gl$ is
$2^\gl$ supercompact, then
level by level equivalence
between strong compactness and
supercompactness fails.
We prove a theorem which
points towards this result
being best possible.
Specifically, we show that
relative to the existence
of a supercompact cardinal,
there is a model for level by
level equivalence between strong
compactness and supercompactness
containing a supercompact cardinal
$\gk$ in which $\gk$'s strong
compactness is indestructible
under $\gk$-directed closed forcing.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{AH4}, it was shown
(see Theorem 5) that
if $\gk$ is indestructibly supercompact
and level by level equivalence between
strong compactness and supercompactness
holds, then no cardinal $\gl > \gk$ is
$2^\gl$ supercompact.
Thus, in any universe in which
level by level equivalence between
strong compactness and supercompactness
holds and there is an indestructibly
supercompact cardinal, there must of
necessity be a restricted number of
large cardinals.
The purpose of this paper is to establish
a theorem which points in the
direction of showing that Theorem 5 of
\cite{AH4} is best possible.
Specifically, we prove the following.
\begin{theorem}\label{t1}
Suppose $V \models
``$ZFC + There
is a supercompact cardinal''.
There is then a model $\ov V \models
``$ZFC +
There is a supercompact cardinal $\gk$ +
Level by level equivalence between strong
compactness and supercompactness holds''
in which the strong compactness of $\gk$
is indestructible
under $\gk$-directed closed forcing.
\end{theorem}
\noindent In $\ov V$, it will
be the case that
no cardinal is supercompact up to an
inaccessible cardinal. Consequently,
$\gk$ of
necessity must be the only supercompact
cardinal in $\ov V$,
and $\ov V$ does not contain a measurable
cardinal above $\gk$.
Note that the work of \cite{AH4}
shows that in a universe with
a restricted number of large
cardinals, it is possible to have
either a fully indestructible
supercompact cardinal together
with level by level equivalence
between strong compactness and
supercompactness holding ``almost
everywhere'' (in a sense made clear
in \cite{AH4}), or
a partially indestructible supercompact
cardinal (in a sense again made
clear in \cite{AH4}), together
with precise level by level equivalence
between strong compactness
and supercompactness.
These results are to be contrasted
with Theorem \ref{t1} of this paper,
where in a universe with a
restricted number of large cardinals,
there is precise level by level equivalence
between strong compactness and
supercompactness, together with
a supercompact cardinal $\gk$
whose {\it strong compactness}
(although not necessarily supercompactness)
is fully indestructible under $\gk$-directed
closed forcing.
This is much closer to the desired
optimal result, namely a universe
with a restricted number of large
cardinals in which there is
exact level by level equivalence
between strong compactness and
supercompactness, together with
a supercompact cardinal $\gk$ whose
{\it supercompactness} is fully
indestructible under $\gk$-directed
closed forcing.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{A06}.
When forcing, $q \ge p$ means that
$q$ is stronger than $p$.
For $\ga \le \gb$ ordinals,
$[\ga, \gb]$ is the usual closed
interval of ordinals between
$\ga$ and $\gb$.
For $\gk$ a cardinal, the
partial ordering $\FP$ is
$\gk$-directed closed if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is $\gk$-strategically closed
if in the two person game in which
the players construct an increasing sequence
$\la p_\ga : \ga \le \gk \ra$, where
player I plays odd stages and player II
plays even and limit stages (always
choosing the trivial condition at stage 0),
player II has a strategy which ensures
the game can always be continued.
%$\FP$ is ${<}\gk$-strategically closed
%if $\FP$ is $\gd$-strategically closed
%for every cardinal $\gd < \gk$.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
For $p \in \FP$ and $\varphi$
a formula in the forcing language
with respect to $\FP$,
$p \decides \varphi$ means that
$p$ decides $\varphi$.
We recall for the benefit of readers the
definition given by Hamkins in
Section 3 of \cite{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the lottery sum is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
Suppose $V$ is a model of ZFC
%containing supercompact cardinals
in which for all regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact.
%except possibly if $\gk$ is a measurable limit
%of cardinals $\gd$ which are $\gl$ supercompact.
Such a model will be said to
witness level by level
equivalence between strong
compactness and supercompactness.
We will also say that $\gk$
is a witness to level by
level equivalence between strong
compactness and supercompactness
iff for every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact
iff $\gk$ is $\gl$ supercompact.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
The partial ordering $\FP$ which
will be used in the proof of
Theorem \ref{t1} is a Gitik
iteration of Prikry-like forcings.
As such, we take this opportunity
to mention some of the basic
properties of this sort of iteration.
We begin our discussion
with a preliminary definition found in \cite{GS}.
Suppose $\gd$ is inaccessible.
A partial ordering $\FQ$ is
%$\la \FQ, \le, \le^* \ra$ is
$\gd$-weakly closed and satisfies the Prikry
property if it meets the following criteria.
%$\le^* \ \subseteq \ \le$ satisfies the following properties.
\begin{enumerate}
\item $\FQ$ has two partial orderings
$\le$ and $\le^*$, with
$\le^* \ \subseteq \ \le$.
\item For any condition $p \in \FQ$ and any formula
$\varphi$ in the forcing language with respect to
$\FQ$, there is some
$q \ge^* p$ such that $q \decides \varphi$.
\item For each $\gg < \gd$,
any $\le^*$ increasing chain of
elements of $\FQ$ of length
$\gg$ has an upper bound.
%For any increasing chain
%$p_0 \le^* p_1 \le^* \cdots \le^* p_\ga \le^* \cdots
%(\ga < \gg < \gd)$
%with respect to $\le^*$ of elements of $\FQ$,
%there is an upper bound $q$ for the whole chain.
\end{enumerate}
Given the above definition,
a Gitik iteration $\FP$
of Prikry-like forcings
having length $\gk$ is an
Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra
: \ga < \gk \ra$ in which at each
nontrivial stage $\gd$
(which for our purposes will always
be inaccessible), the forcing used is
$\gd$-weakly closed and satisfies
the Prikry property. In our situation,
each component $\dot \FQ_\gd$ of the
iteration used at a nontrivial
stage $\gd$ has the form
$\dot \FQ^0_\gd \ast \dot \FQ^1_\gd \ast \dot \FQ^2_\gd$,
where $\dot \FQ^0_\gd$ is a term for a
$\gd$-directed closed partial ordering,
$\dot \FQ^1_\gd$ is a term for either
trivial forcing or Prikry forcing, and
$\dot \FQ^2_\gd$ is a term for a
$\gd$-strategically closed partial ordering.
(That a $\gd$-strategically closed
partial ordering may be used as
a component in a Gitik iteration
of Prikry-like forcings was shown in \cite{A01}.)
The ordering $\le$ on $\FP$ is a modification of the usual
ordering used when an Easton support iteration is defined
except,
roughly speaking, the stems of
Prikry conditions are extended
nontrivially only finitely often.
Its definition is given in \cite{G},
\cite{GS}, and \cite{AG},
but for concreteness, we repeat it here.
Specifically, let $p, q \in \FP$, $p = \la \dot p_\ga : \ga < \gk \ra$,
$q = \la \dot q_\ga : \ga < \gk \ra$.
Then $q \ge p$ iff
$q$ extends $p$ with respect to the
usual Easton support iteration
ordering, but in addition, for some finite
$A \subseteq {\hbox{\rm support}}(p)$ and all
$\gb \in {\hbox{\rm support}}(p) - A$,
for $\dot q_\gb = \la \dot r_\gb', \dot s_\gb', \dot t_\gb' \ra$,
$\dot p_\gb = \la \dot r_\gb, \dot s_\gb, \dot t_\gb \ra$,
$q \rest \gb \ast \dot r_\gb' \in \FP_\gb \ast \dot \FQ^0_\gb$,
$p \rest \gb \ast \dot r_\gb \in \FP_\gb \ast \dot \FQ^0_\gb$,
$q \rest \gb \ast \dot r_\gb' \forces_{\FP_\gb \ast \dot \FQ^0_\gb}
``$If $\dot s_\gb'$ and $\dot s_\gb$ are conditions
with respect to Prikry forcing,
then $| \dot s_\gb' - \dot s_\gb|
= 0$, where $|\ \ \ |$ is the distance function of
\cite{Ma}''. (Intuitively, $s_\gb' \ge s_\gb$
and $|s_\gb' - s_\gb| = 0$ means that
$s_\gb'$ is obtained from $s_\gb$
``by shrinking measure 1 sets''.)
Further, if $A = \emptyset$ in the above definition,
then $q$ is called an Easton extension of $p$.
By Lemma 1.4 of \cite{G},
for any $p \in \FP$ and any formula
$\varphi$ in the language of
forcing with respect to $\FP$,
there is some Easton extension $q$ of $p$
such that $q \decides \varphi$.
By Lemma 1.2 of \cite{G}, if $\gd$ is
the least cardinal in the iteration and
$\la p_\ga : \ga < \gg < \gd \ra$ is a
sequence of elements of $\FP$ such that
$\ga < \gb$ implies $p_\gb$ is
an Easton extension of $p_\ga$,
%$\le^*$ increasing chain of elements of $\FP$,
there is some condition
$p \in \FP$ which is an Easton extension
of each $p_\ga$.
Therefore, in analogy to Prikry forcing,
forcing with $\FP$ adds no new bounded
subsets of $\gd$. In addition, Lemmas 1.4
and 1.2 of \cite{G} show that $\FP$ is
$\gd$-weakly closed and satisfies the
Prikry property, with $\le^*$ defined as
Easton extension.
We assume familiarity with the
large cardinal notions of
measurability, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} and \cite{SRK} for further details.
We do wish to point out explicitly,
however, that an indestructibly
supercompact cardinal $\gk$ is
one as in \cite{L}, i.e.,
a supercompact cardinal which
remains supercompact after
$\gk$-directed closed forcing.
Also, we say that $\gk$ is supercompact
up to a cardinal $\gl$ if
$\gk$ is $\gd$ supercompact
for every $\gd < \gl$.
%(although $\gk$ need not be $\gl$ supercompact).
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + $\gk$ is
supercompact''.
By first forcing GCH, then
forcing as in \cite{AS97a},
and then taking the appropriate
submodel if necessary,
we slightly abuse notation
and also assume in addition that
$V \models ``$GCH + Level by level
equivalence between strong compactness
and supercompactness holds +
No cardinal is supercompact up
to an inaccessible cardinal''.
The partial ordering $\FP$ used in the
proof of Theorem \ref{t1} is a
Gitik iteration of Prikry-like
forcings
$\la \la \FP_\ga, \dot \FQ_\ga \ra
: \ga < \gk \ra$.
Specifically, $\FP_0$ is the partial
ordering for adding a Cohen subset of $\go$.
The only nontrivial stages of forcing
$\gd < \gk$ occur at $V$-measurable
cardinals. At such a stage $\gd$,
%$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$,
$\dot \FQ_\gd$ has the form
$\dot \FQ^0_\gd \ast \dot \FQ^1_\gd \ast \dot \FQ^2_\gd$,
where for $\gd'$
the least inaccessible in $V$
(and $V^{\FP_\gd}$ as well)
above $\gd$, $\dot \FQ^0_\gd$ is a
term for the lottery sum of all
$\gd$-directed closed partial orderings having
rank below $\gd'$. If
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``$Level by
level equivalence between strong compactness
and supercompactness fails at $\gd$'', then
$\dot \FQ^1_\gd$ is a term for Prikry forcing over
$\gd$ defined with respect to some normal measure;
otherwise, $\dot \FQ^1_\gd$ is a term for trivial forcing.
$\dot \FQ^2_\gd$ is then a term for the partial ordering
adding a non-reflecting stationary set of ordinals
of cofinality $\go$ to $\gd^*$, where
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd \ast \dot \FQ^1_\gd}
``\gd^*$ is the successor
of the least strong limit cardinal
(in $V^{\FP_\gd \ast \dot \FQ^0_\gd \ast \dot \FQ^1_\gd}$)
above $\gd$'s degree of
supercompactness in $V$''.\footnote{As our
proofs will indicate, other definitions
of $\gd^*$ are possible. The definition
of $\gd^*$ just given, however,
is sufficient for the purposes of this paper.}
%to allow the arguments that follow to be given successfully.}
(Note that a precise definition of the
partial ordering for adding a
non-reflecting stationary set of ordinals
of cofinality $\go$ to a successor
cardinal $\gg^+$
%a partial ordering which is $\gd$-strategically closed,
may found in, e.g., Section 1 of
\cite{AS97a}. A property of
this partial ordering we use here
is that it is $\gg$-strategically closed.)
\begin{lemma}\label{l1}
Suppose $\gd$ is measurable in $V$. Then
$\forces_{\FP_\gd} ``$Level by level equivalence
between strong compactness and supercompactness
holds at $\gd$''.
\end{lemma}
\begin{proof}
Since $V \models {\rm GCH}$, by Lemma 1.5 of
\cite{G},
$\forces_{\FP_\gd} ``\gd$ is a measurable
cardinal''.
We consequently assume inductively that for
every measurable cardinal $\gg < \gd$,
$\forces_{\FP_\gg} ``$Level
by level equivalence
between strong compactness and supercompactness
holds at $\gg$''.
Let $\gl > \gd$ be a regular cardinal in
$V^{\FP_\gd}$ such that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ strongly compact''.
Note that because $\gd$ is measurable in $V$,
$\FP_\gd$ is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
In addition,
$\FP_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FP_\gd}$,
since $\gd$ is
measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$
is a subordering of the
direct limit of
$\la \FP_\ga : \ga < \gd \ra$
as calculated in $V^{\FP_\gd}$.
Hence, by Theorem 2.1.5 of \cite{H}
(see also the proofs of
Lemma 3 of \cite{AC1} or
Lemma 8 of \cite{A97}),
every $\gd$-additive
uniform ultrafilter over a cardinal
$\gg \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gg$ in $V$.
Therefore, since the $\gl$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FP_\gd}$-regular cardinal
$\gg \in [\gd, \gl]$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FP_\gd}$,
and since the fact $\FP_\gd$
is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$
tells us the regular cardinals
at or above $\gd$ in
$V^{\FP_\gd}$ are the same
as those in $V$,
the preceding sentence implies
that every $V$-regular cardinal
$\gg \in [\gd, \gl]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gl$ strongly
compact in $V$.
By the fact $V \models ``$Level by
level equivalence between strong compactness
and supercompactness holds and no cardinal
is supercompact up to an inaccessible cardinal'',
$V \models ``\gd$ is $\gl$
supercompact'', and $\gl$ is below
the least $V$-inaccessible cardinal
above $\gd$. The proof of Lemma \ref{l1}
will therefore be complete once we have shown
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
To do this, fix $j : V \to M$ an
elementary embedding witnessing the
$\gl$ supercompactness of $\gd$.
We note that since
$j(\gd) > \gl > \gd$, our inductive
assumptions in $V$ imply that in $M$,
$\forces_{\FP_\gd} ``\gd$ is a measurable
cardinal and level by level equivalence
between strong compactness and supercompactness
holds at $\gd$''.
Also,
%if we opt for trivial forcing
at stage $\gd$ in $M$ in the definition of
$j(\FP_\gd)$, $\dot \FQ_\gd =
{(\dot \FQ^0_\gd)}^M \ast
{(\dot \FQ^1_\gd)}^M \ast
{(\dot \FQ^2_\gd)}^M$, where
${(\dot \FQ^0_\gd)}^M$ is a term for the
stage $\gd$ lottery sum performed in
the definition of $j(\FP_\gd)$,
${(\dot \FQ^1_\gd)}^M$
is a term for either trivial forcing
or Prikry forcing, and
${(\dot \FQ^2_\gd)}^M$
is a term for a partial
ordering which adds a non-reflecting stationary
set of ordinals to an ordinal well above $\gl$.
Thus, if we opt for trivial forcing in the
stage $\gd$ lottery sum done
in $M$ in the definition of
$j(\FP_\gd)$, our inductive assumptions
allow us to take ${(\dot \FQ^1_\gd)}^M$
as a term for trivial forcing.
Consequently, above the appropriate condition
in $M$, $j(\FP_\gd)$ is forcing equivalent to
$\FP_\gd \ast \dot \FP^{*}$, where $\dot
\FP^{*}$ is a term for a Gitik iteration
of Prikry-like forcings whose first
nontrivial stage takes place well beyond $\gl$.
Since $V \models {\rm GCH}$, we may now apply the
argument found in Lemma 1.5 of \cite{G}
to show that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $\gl > \gd$ are regular cardinals and
$V \models ``\gd$ is $\gl$
supercompact'', then
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
Since $V \models ``\gd$ is
$\gl$ supercompact'', let
$j : V \to M$ be an elementary
embedding witnessing this fact.
Since $\gl > \gd$, by GCH in $V$,
$M \models ``\gd$ is measurable''.
Therefore,
by Lemma \ref{l1} applied in $M$,
$\forces_{\FP_\gd} ``\gd$ is
a measurable cardinal and level by
level equivalence between strong
compactness and supercompactness
holds at $\gd$''. Consequently,
the argument given in the last paragraph
of the proof of Lemma \ref{l1} now applies to
show that in $V$,
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
Since in Lemma \ref{l2}, $\gd$ and
$\gl$ can be arbitrary, it immediately
follows that in $V^\FP$, $\gk$ is
supercompact.
\begin{lemma}\label{l3}
If $V^\FP \models ``\gd$ is measurable'', then
$V \models ``\gd$ is measurable''.
\end{lemma}
\begin{proof}
Suppose $V \models ``\gd$ isn't measurable'' but
$V^\FP \models ``\gd$ is measurable''.
Since measurable cardinals are Mahlo and
forcing can't create a new Mahlo cardinal,
it must be the case that
$V \models ``\gd$ is Mahlo''.
Write $\FP = \FP_\gd \ast \dot \FP^\gd$.
Since $\gd$ isn't measurable in $V$,
the definition of $\FP$ implies that
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$
doesn't add subsets to the least
inaccessible cardinal above $\gd$''.
We may therefore infer that
$V^{\FP_\gd} \models ``\gd$ is measurable''.
In addition, $\FP_\gd$ is
forcing equivalent either to a partial
ordering having size below $\gd$ or to
the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
Hence,
by applying either the results of
\cite{LS} if the former holds
or the argument found in the last
paragraph of the proof of Lemma \ref{l1}
if the latter holds,
we may infer that
$V \models ``\gd$ is measurable'' iff
$V^{\FP_\gd} \models ``\gd$ is measurable''.
This contradiction completes the proof of
Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds below $\gk$''.
\end{lemma}
\begin{proof}
Suppose $\gd < \gk$ and
$V^\FP \models ``\gl > \gd$ is
a regular cardinal and
$\gd$ is $\gl$ strongly compact''.
By Lemma \ref{l3}, we know that
$\gd$ is measurable in $V$.
Hence, $\gd$ is a stage at
which a nontrivial forcing
takes place, so we may write
$\FP = (\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd) \ast
(\dot \FQ^2_\gd \ast \dot \FS') =
\FR \ast \dot \FS$.
By Lemmas 1.4
and 1.2 of \cite{G} and the
definition of $\FP$ as a Gitik
iteration of Prikry-like forcings,
$\forces_\FR ``$Forcing with
$\dot \FS$ doesn't add
bounded subsets of $\gd^*$ but
does add a non-reflecting
stationary set of ordinals of
cofinality $\go$ to $\gd^*$''.
Consequently, by Theorem 4.8 of
\cite{SRK} and the succeeding remarks,
$\forces_\FP ``\gd$ isn't $\gd^*$
strongly compact''. Thus, by
the preceding two sentences,
$\forces_{\FP} ``\gl < \gd^*$'', and
$\forces_\FR ``\gd$ is $\gl$
strongly compact''.
Therefore, since $\FR =
\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd$ and
$\dot \FQ^1_\gd$ must be a
term either for trivial
forcing or for Prikry forcing
over $\gd$, $\dot \FQ^1_\gd$
is a term for trivial forcing, and
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd}
``\gd$ is $\gl$ strongly compact''.
Hence, as $\dot \FQ^1_\gd$ isn't
a term for Prikry forcing over $\gd$,
by the definition of $\FP$,
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd}
``\gd$ is $\gl$ supercompact''.
The preceding then immediately implies that
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd \ast \dot \FS} ``\gd$
is $\gl$ supercompact'', i.e.,
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$No cardinal is supercompact
up to an inaccessible cardinal''.
\end{lemma}
\begin{proof}
Suppose $\gd < \gk$ and
$V^\FP \models ``\gl > \gd$
is a regular cardinal
and $\gd$ is $\gl$ supercompact''.
By the proof of Lemma \ref{l4},
$V^\FP \models ``\gl < \gd^*$''.
Since $\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd$ is forcing
equivalent to a partial ordering
having size below $\gd'$,
$V^\FP \models ``\gd^* < \gd'$''.
Thus, $V^\FP \models ``$No cardinal
below $\gk$ is supercompact up
to an inaccessible cardinal''.
Further, as the fact
$V \models ``\gk$ is supercompact
and no cardinal is supercompact
up to an inaccessible cardinal''
implies $V \models ``$No cardinal
above $\gk$ is inaccessible'',
$V^\FP \models ``$No cardinal above
$\gk$ is inaccessible'', i.e.,
$V^\FP \models ``$No cardinal at or
above $\gk$ is supercompact up to
an inaccessible cardinal''.
This completes the proof of
Lemma \ref{l5}.
\end{proof}
Lemmas \ref{l4} and \ref{l5} together
now immediately imply that
$V^\FP \models ``$Level by level equivalence
between strong compactness and
supercompactness holds''.
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$'s strong compactness
is indestructible under $\gk$-directed
closed forcing''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6} is
virtually identical to the proof of
Lemma 2.2 of \cite{A06}.
For the convenience of readers,
we repeat the proof here,
taking the liberty to quote
verbatim when necessary.
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is $\gk$-directed
closed''.
Using GCH in $V$, let
$\gl > \max(\gk, \card{{\rm TC}(\dot \FQ)})$
be an arbitrary regular cardinal large
enough so that
${(2^{[\gl]^{< \gk}})}^V = \rho =
{(2^{[\gl]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \rho^+ =
2^\rho = 2^{[\rho]^{< \gk}}$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gk$.
%such that $M \models ``\gk$ isn't $\gs$ supercompact''.
Since $V \models ``$No cardinal $\gd > \gk$
is inaccessible'', by the closure properties of $M$,
$M \models ``\card{{\rm TC}(\dot \FQ)}$ is
below the least inaccessible cardinal above $\gk$''.
%This has as an immediate consequence that
Thus, as $M \models ``\gk$ is measurable'',
$\dot \FQ$ is a term for an allowable choice
in the stage $\gk$ lottery sum performed in
$M$ in the definition of $j(\FP)$.
We hence
%assume without loss of generality
for the remainder of this proof force in $M$
above a condition that opts
for $\FQ$ in $M^\FP$, so that
$j(\FP)$ is forcing equivalent to
$\FP_\gk \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast (\dot \FQ^2_\gk \ast \dot \FR') =
\FP \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast \dot \FR$.
The remainder of the proof of Lemma \ref{l6},
as with the proof of Lemma 2.2 of \cite{A06},
is as in the proof of Lemma 2 of \cite{AG}.
%For concreteness, we provide a sketch of the proof, and
We therefore refer readers to the
proof of Lemma 2 of \cite{AG} for any
missing details.
By the last sentence of the
preceding paragraph, in $M$,
$j(\FP \ast \dot \FQ)$ is
forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FQ^1_\gk$
is a term for either Prikry forcing
or trivial forcing''. Further, since
$M \models ``\gk$
is $\rho$ supercompact'',
%There are no inaccessible cardinals in the interval $(\gk, \gs]$'',
the first nontrivial
stage in the definition of
$\dot \FR$ after $\gk$ takes place
well above $\gs$.
Consequently, as in Lemma 2 of \cite{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga : \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of
${(P_\gk(\gl))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
$\FP \ast \dot \FQ \ast \dot \FQ^1_\gk$
names of elements of $\FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ^1_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) : \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\gl \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ^1_\gk}
``\dot p_\gl$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb : \gb < \gl \ra$''.
If we then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gl]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gl)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FQ^1_\gk$
either the trivial condition
(if $\FQ^1_\gk$ is trivial forcing)
or of the
form $\la \emptyset, B \ra$
(if $\FQ^1_\gk$ is Prikry forcing), in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) : \gb < \gl \ra \in
\dot X$'' for some name $\dot X$ of $X$,
then as in Lemma 2 of \cite{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gl$ strongly compact''.
Since $\gl$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l6}.
\end{proof}
By taking $\ov V = V^\FP$,
Lemmas \ref{l1} - \ref{l6} and the
intervening remarks complete
the proof of Theorem \ref{t1}.
\end{proof}
The techniques of this paper
are applicable when forcing
over models in which the
large cardinal structure
of the universe is slightly
more complicated above our
supercompact cardinal $\gk$.
For instance, if we start with
a model in which $\gk$ is
supercompact and no cardinal
is supercompact up to a
Mahlo cardinal, then it is
possible to force and
construct a model in which
$\gk$ is supercompact, no
cardinal is supercompact up
to a Mahlo cardinal,
and the further conclusions
of Theorem \ref{t1} hold.
(We simply change the definition
of $\gd'$ to be the least Mahlo
cardinal in $V$ above $\gd$
and essentially argue as before.)
Such a model, of course,
can contain inaccessible
cardinals above $\gk$.
We still do not know, however,
if it is possible to force
and construct a model in which
$\gk$ is indestructibly
supercompact and level by
level equivalence between
strong compactness and
supercompactness holds.
This is the question with which
we conclude.
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\end{document}