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\title{Indestructibility, Strong Compactness,
and Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, indestructibility,
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 grants
and CUNY
Collaborative Incentive grants.
In addition, 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{August 21, 2007\\
(revised April 20, 2009)}
\begin{document}
\maketitle
\begin{abstract}
We show the relative consistency
of the existence of two
strongly compact cardinals
$\gk_1$ and $\gk_2$ which
exhibit indestructibility
properties for their
strong compactness, together
with level by level equivalence
between strong compactness and
supercompactness holding at
all measurable cardinals except
for $\gk_1$.
In the model constructed,
$\gk_1$'s strong compactness is
indestructible under arbitrary
$\gk_1$-directed closed forcing,
$\gk_1$ is a limit of measurable cardinals,
$\gk_2$'s strong compactness is
indestructible under $\gk_2$-directed
closed forcing which is also
$(\gk_2, \infty)$-distributive, and
$\gk_2$ is fully supercompact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin by mentioning that we
assume throughout 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.
In particular, we say that $\gk$ is supercompact
(or strongly compact)
up to a cardinal $\gl$ if
$\gk$ is $\gd$ supercompact
(or $\gd$ strongly compact)
for every $\gd < \gl$.
%(although $\gk$ need not be $\gl$ supercompact).
We continue with some key definitions.
Suppose $V$ is a model of ZFC
in which for all regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ 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 by the author
and Shelah in \cite{AS97a}.
In \cite{A04}, the author proved the following theorem.
\begin{theorem}\label{t1}
Assume that
$V \models\ $ZFC + GCH satsifies the properties:
\begin{enumerate}
\item $\K \neq \emptyset$
is the class of supercompact
cardinals.
\item Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal.
\item $\gk$ is the
least supercompact cardinal.
\end{enumerate}
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models\ $ZFC satisfies the properties:
\begin{enumerate}
\item $\gk$ is the least strongly compact cardinal.
\item $\gk$ is a limit of measurable cardinals
but is not supercompact.
\item $\K - \{\gk\}$ is the class of
supercompact cardinals.
\item Level by level
equivalence between strong
compactness and supercompactness
holds for every measurable
cardinal except for $\gk$.
\item $\gk$'s strong compactness is
indestructible under $\gk$-directed
closed forcing.
\end{enumerate}
\end{theorem}
Note that by construction,
in the model for
Theorem \ref{t1}, $\gk$ is the only
strongly compact cardinal exhibiting
any sort of nontrivial indestructibility
properties. Thus, one may wonder whether
it is possible to have a model
containing more than one %non-supercompact
strongly compact cardinal in which
each strongly compact cardinal exhibits
indestructibility properties for its
strong compactness and
level by level equivalence between strong
compactness and supercompactness holds nontrivially
at every measurable cardinal which is not
strongly compact.
The purpose of this paper is to provide
an affirmative answer to the above
question. Specifically, we prove the following theorem.
\begin{theorem}\label{t2}
Suppose that $V \models
``$ZFC + $\gk_1 < \gk_2$ are
supercompact''. There is then a model
$\ov V \models\ $ZFC satisfying the properties:
\begin{enumerate}
\item $\gk_1$ is a non-supercompact
strongly compact cardinal which is
a limit of measurable cardinals.
\item $\gk_2$ is supercompact.
\item No cardinal
less than or equal to $\gk_1$ is supercompact
up to an inaccessible cardinal.
\item No cardinal
is supercompact up to a
measurable cardinal.
\item Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
except for $\gk_1$.
\item $\gk_1$ and
$\gk_2$ are the first two
strongly compact cardinals.
\item $\gk_1$'s strong compactness
is indestructible under arbitrary
$\gk_1$-directed closed forcing.
\item $\gk_2$'s strong compactness is
indestructible under $\gk_2$-directed
closed forcing which is also
$(\gk_2, \infty)$-distributive.
\end{enumerate}
\end{theorem}
We take this opportunity to
make two remarks concerning
Theorem \ref{t2}. Note that
Theorem 5 of Apter-Hamkins \cite{AH4}
indicates that
if $\gk$ is indestructibly supercompact
(in Laver's sense of \cite{L}, i.e.,
$\gk$'s supercompactness is indestructible
under arbitrary $\gk$-directed closed forcing)
and level by level equivalence between
strong compactness and supercompactness
holds, then no cardinal $\gl > \gk$ is
$2^\gl$ supercompact.
This is in stark contrast to
Theorem \ref{t2}, which not only
tells us, as Theorem \ref{t1} does, that
level by level equivalence between strong
compactness and supercompactness can
hold nontrivially
both below and above
a cardinal $\gk_1$ which is
indestructibly strongly compact
(in the sense of Apter-Gitik \cite{AG}, i.e.,
$\gk_1$'s strong compactness is indestructible
under arbitrary $\gk_1$-directed closed forcing)
%an indestructibly strongly compact cardinal $\gk_1$,
but that there can
be in addition a supercompact cardinal
(namely $\gk_2$)
greater than $\gk_1$ whose strong compactness
is highly indestructible.
Also, since %we observe that
in $\ov V$, it is %will be
the case that
no cardinal
%greater than $\gk_1$
is supercompact up to a
measurable cardinal, %. Consequently,
$\gk_2$ of
necessity must be the only supercompact
cardinal in $\ov V$,
and $\ov V$ does not contain a measurable
cardinal greater than $\gk_2$.
We now very briefly give some
preliminary information
concerning notation and terminology.
When forcing, $q \ge p$ means that
$q$ is stronger than $p$.
For $\ga < \gb$ ordinals,
$(\ga, \gb)$, $(\ga, \gb]$, and
$[\ga, \gb)$
%, and $[\ga, \gb]$
are as in standard interval notation.
For $\gk$ a cardinal, $\FP$ is
{\rm $\gk$-directed closed} if every directed
set of conditions of cardinality less
than $\gk$ has an upper bound.
$\FP$ is {\rm $\gk$-strategically closed} if
in the two person game in which the
players construct an increasing 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 {\rm ${\prec} \gk$-strategically closed} if
in the two person game in which the players
construct an increasing 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 {\rm $(\gk, \infty)$-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 $\gk$.
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 $\check x$,
especially when $x$ is in
$V$ or is a variant of $G$.
%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
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
A corollary of Hamkins' work on
gap forcing found in
\cite{H2} and \cite{H3}
will be employed in the
proof of Theorem \ref{t2}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2} and \cite{H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\rm admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{H3},
$\FP$ is {\rm mild}
with respect to a cardinal $\gk$
iff every set of ordinals $x$ in
$V^\FP$ of size less than $\gk$ has
a ``nice'' name $\tau$
in $V$ of size less than $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2}, \cite{H3}, and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\rm amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2} and \cite{H3}
we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let
$V \models ``$ZFC + $\gk_1 < \gk_2$ are
supercompact''.
By first using the folklore result
that GCH can be forced while
preserving all ground model
supercompact cardinals, then
forcing as in \cite{AS97a},
and then both taking the appropriate
submodel and renaming cardinals if necessary,
we slightly abuse notation
and also assume in addition that
$V \models\ $ZFC + GCH satisfies the properties:
\begin{enumerate}
\item Level by level
equivalence between strong compactness
and supercompactness holds.
\item No cardinal greater than $\gk_1$ is supercompact up
to a measurable cardinal.
\item $\gk_1$ and $\gk_2$ are both
the first two strongly compact
and supercompact cardinals.
\end{enumerate}
The partial ordering $\FP^*$ of
Theorem \ref{t1} may be defined with
respect to $\gk_1$
so as to have cardinality $\gk_1$
and so that after forcing with $\FP^*$,
no cardinal less than or equal to $\gk_1$
is supercompact up to an inaccessible cardinal.
Consequently, if we now force with
$\FP^*$ and once
again slightly abuse notation
and denote the resulting
generic extension by $V$, then by standard arguments
and the work of \cite{A04}
and L\'evy-Solovay \cite{LS}, we may assume that
$V \models\ $ZFC satisfies the properties:
\begin{enumerate}
\item $\gk_1$ is the least
strongly compact cardinal.
\item $\gk_1$ is
a limit of measurable cardinals but
is not supercompact.
\item $\gk_1$ is indestructibly
strongly compact.
\item Level by level equivalence between strong
compactness and supercompactness holds
for every measurable cardinal except
for $\gk_1$.
\item GCH holds for
every cardinal greater than or equal to $\gk_1$.
\item $\gk_2$ is both supercompact
and the second strongly compact cardinal.
\item No cardinal greater than
$\gk_1$ is supercompact up to a
measurable cardinal.
\item No cardinal
less than or equal to $\gk_1$ is supercompact
up to an inaccessible cardinal.
\end{enumerate}
The strategy in proving
Theorem \ref{t2} will be to
adjust in a suitable way our
proof of Theorem 1 of \cite{A07}.
More specifically, we will redefine
the partial ordering used to prove
the aforementioned theorem so as to
add non-reflecting stationary sets of
ordinals of high enough
cofinality (namely $\gk_1$) instead
of Prikry sequences. This will destroy measurable
cardinals witnessing failures of level by level equivalence
between strong compactness and supercompactness,
but will also allow us to preserve
(unlike when a Prikry iteration is used)
that $\gk_1$ remains a non-supercompact
indestructible strongly compact cardinal.
It will also allow us to employ appropriately
modified arguments
from \cite{A07} in the proofs of
Lemmas \ref{l1} -- \ref{l3}.
Whereas the proofs of Lemmas \ref{l4} and
\ref{l5} will not be very difficult,
the proof of Lemma \ref{l6} will pose the
greatest technical challenge. It is the
heart of the argument, and requires the
use of an ingenious method developed by Sargsyan.
With Lemma \ref{l7}, which shows that $\gk_1$
and $\gk_2$ are the first two strongly compact cardinals,
the proof of Theorem \ref{t2} will be complete.
In accordance with this plan of attack,
the partial ordering $\FP$ used in the
proof of Theorem \ref{t2} is a
reverse Easton iteration of length $\gk_2$,
which we will index as
$\la \la \FP_\ga, \dot \FQ_\ga \ra
\mid \ga \in [\gk_1, \gk_2) \ra$.
It is a modification of the
partial ordering used in \cite{A07}.
Specifically, $\FP_{\gk_1}$ is the partial
ordering for adding a Cohen subset
of the least inaccessible cardinal
greater than $\gk_1$.
The only nontrivial stages of forcing
$\gd \in (\gk_1, \gk_2)$ 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 measurable cardinal in $V$
(and $V^{\FP_\gd}$ as well)
greater than $\gd$, $\dot \FQ^0_\gd$ is a
term for the lottery sum of all
$\gd$-directed closed partial orderings
which are also $(\gd, \infty)$-distributive
having rank less than $\gd'$. 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 the partial
ordering adding a non-reflecting stationary
set of ordinals of cofinality $\gk_1$ to $\gd$;
otherwise, $\dot \FQ^1_\gd$ is a term for trivial forcing.
(Note that a precise definition of the
partial ordering for adding a
non-reflecting stationary set of ordinals
of cofinality $\gk_1$ to an
inaccessible 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 will use in what follows
is that it is both $\gk_1$-directed closed
and ${\prec} \gg$-strategically closed.)
\begin{lemma}\label{l1}
Suppose $\gd \in (\gk_1, \gk_2)$
is measurable in $V$. Then
$\forces_{\FP_\gd} ``$Level by level equivalence
between strong compactness and supercompactness
holds at $\gd$''.
\end{lemma}
\begin{proof}
We modify the proof of Lemma 2.1 of \cite{A07}.
Since $V \models ``$GCH holds for
every cardinal greater than or equal to $\gk_1$'',
by standard arguments
(see, e.g., the
proof of Lemma 8.1 of \cite{AH4}),
$\forces_{\FP_\gd} ``\gd$ is a measurable
cardinal''.
We consequently assume inductively that for
every measurable cardinal $\gg < \gd$,
$\gg \in (\gk_1, \gk_2)$,
$\forces_{\FP_\gg} ``$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''.
Since by its definition, $\FP_\gd$
is forcing equivalent to a
partial ordering which
admits a gap below $\gd$ and
is mild with respect to $\gd$,
by Theorem \ref{t3},
$V \models ``\gd$ is $\gl$ strongly compact''.
By our assumptions on $V$ (including level by
level equivalence between strong compactness
and supercompactness for every measurable
cardinal except for $\gk_1$), $V \models ``\gd$
is $\gl$ supercompact and $\gl < \gd'$
(so $\gl$ is non-measurable)''.
The proof of Lemma \ref{l1}
will therefore be complete once we have shown that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
To do this, fix $j : V \to M$ an
elementary embedding witnessing the
$\gl$ supercompactness of $\gd$.
We note that since
$j(\gd) > \gl > \gd > \gk_1$, our inductive
assumptions in $V$ together
with the fact that
$V \models ``$GCH holds for
every cardinal greater than or equal to $\gk_1$''
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,
at stage $\gd$ in $M$ in the definition of
$j(\FP_\gd)$, ${(\dot \FQ_\gd)}^M =
{(\dot \FQ^0_\gd)}^M \ast
{(\dot \FQ^1_\gd)}^M$, where
${(\dot \FQ^0_\gd)}^M$ is a term for the
stage $\gd$ lottery sum performed in
the definition of $j(\FP_\gd)$ and
${(\dot \FQ^1_\gd)}^M$
is a term for either trivial forcing
or the forcing adding a non-reflecting
stationary set of ordinals of
cofinality $j(\gk_1) = \gk_1$ to $\gd$.
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
also 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
reverse Easton iteration of
suitably closed partial orderings whose first
nontrivial stage takes place well beyond $\gl$.
Since $V \models ``$GCH holds for
every cardinal greater than or equal to $\gk_1$'',
we may once again employ the
arguments used in the
proof of Lemma 8.1 of \cite{AH4}
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 $\gl > \gd > \gk_1$ are regular cardinals and
%with $\gd, \gl \in (\gk_1, \gk_2)$ and
$V \models ``\gd$ is $\gl$
supercompact'', then
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of Lemma 2.2 of \cite{A07}.
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 $V$
for cardinals
greater than or equal to $\gk_1$,
$M \models ``\gd$ 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 cardinals
greater than $\gk_1$,
as in \cite{A07},
it immediately
follows that in $V^\FP$, $\gk_2$ is
supercompact.
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds for every measurable
cardinal $\gd \in (\gk_1, \gk_2)$''.
\end{lemma}
\begin{proof}
%We follow the proof of Lemma 2.4 of \cite{A07}.
Suppose $\gd, \gl \in (\gk_1, \gk_2)$ and
$V^\FP \models ``\gl > \gd$ is
a regular cardinal and
$\gd$ is $\gl$ strongly compact''.
As in Lemma \ref{l1},
by Theorem \ref{t3}, the
definition of $\FP$, and level by
level equivalence between
strong compactness and supercompactness
in $V$ for every measurable cardinal
except for $\gk_1$, we have that
$V \models ``\gd$ is $\gl$ supercompact
and $\gl < \gd'$ (so $\gl$ is
non-measurable)''.
Thus, if we write
$\FP = \FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd \ast \dot \FS = \FR \ast \dot \FS$,
the definition of $\FP$ tells us that
$\forces_{\FR} ``$Forcing with $\dot \FS$
adds no bounded subsets of $\gd'$''.
Since $V^\FP = V^{\FR \ast \dot \FS} \models
``\gd$ is $\gl$ strongly compact'', we may hence infer that
$\forces_{\FR} ``\gd$ is $\gl$ strongly compact''
and that $\dot \FQ^1_\gd$ is a term
for trivial forcing (since otherwise,
$V^\FP \models ``\gd$ contains a non-reflecting
stationary subset of ordinals of
cofinality $\gk_1$ and $\gd$ is
weakly compact'', a contradiction).
As a consequence, another appeal to the
definition of $\FP$ indicates that
not only is it the case that
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``\gd$
is $\gl$ strongly compact'',
but it is additionally true that
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``\gd$
is $\gl$ supercompact'' as well.
As we now know that $\FP$ is forcing equivalent to
a partial ordering of the form
$\FP_\gd \ast \dot \FQ^0_\gd \ast \dot \FT$ where
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``$Forcing
with $\dot \FT$ adds no bounded subsets of $\gd'$'',
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$No cardinal is supercompact
up to a measurable cardinal''. In fact,
$V^\FP \models ``$No cardinal less than or equal to
$\gk_1$ is supercompact up to an inaccessible cardinal''.
\end{lemma}
\begin{proof}
Since $V \models ``$No cardinal
is supercompact up to a measurable cardinal''
and $\FP$ admits a sufficiently
small gap above $\gk_1$,
by Theorem \ref{t3},
$V^\FP \models ``$No cardinal
greater than $\gk_1$ is supercompact
up to a measurable cardinal''.
Since the first nontrivial stage
of forcing in $\FP$ occurs at the
least $V$-inaccessible cardinal
greater than $\gk_1$ and
$V \models ``$No cardinal less than or equal to $\gk_1$
is supercompact up to an inaccessible cardinal'',
$V^\FP \models ``$No cardinal less than or equal to $\gk_1$
is supercompact up to an inaccessible cardinal''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$Level by level equivalence
between strong compactness and
supercompactness holds for
every measurable cardinal except
for $\gk_1$''.
\end{lemma}
\begin{proof}
Since $V^\FP \models ``\gk_2$ is supercompact'',
by Lemma \ref{l4},
$V^\FP \models ``$There are no measurable
cardinals greater than $\gk_2$''. Thus, by Lemma \ref{l3},
$V^\FP \models ``$Level by level equivalence between
strong compactness and supercompactness holds for
every measurable cardinal greater than $\gk_1$''.
In addition,
as forcing with $\FP$ adds no bounded
subsets of the least $V$-inaccessible
cardinal greater than $\gk_1$ and
$V \models ``$Level by level equivalence between
strong compactness and supercompactness holds
for every measurable cardinal except for
$\gk_1$ and no cardinal less than or equal to $\gk_1$
is supercompact up to an inaccessible cardinal'',
$V^\FP \models ``$Level by level equivalence
holds for every measurable cardinal less than $\gk_1$
and $\gk_1$ is not supercompact up to an
inaccessible cardinal''.
Finally, because $V \models ``\FP$ is
$\gk_1$-directed closed and $\gk_1$
is indestructibly strongly compact'',
$V^\FP \models ``\gk_1$ is a non-supercompact
indestructibly strongly compact cardinal''.
We therefore now infer that
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
fails at $\gk_1$''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``\gk_2$'s strong compactness
is indestructible under $\gk_2$-directed
closed forcing which is also
$(\gk_2, \infty)$-distributive''.
\end{lemma}
\begin{proof}
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is $\gk_2$-directed
closed and $(\gk_2, \infty)$-distributive''.
%Let $\FQ \in {V}^\FP$ be such that
%${V}^\FP \models ``\FQ$ is both
%${} \gk_2$-directed closed and
%$(\gk_2, \infty)$-distributive''.
Let $\dot \FQ$ be a
canonical term for $\FQ$, and let
$\gl > \max(\card{{\rm TC}(\dot \FQ)}, 2^{\gk_2})$
be an arbitrary regular cardinal.
Take $j : {V} \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness of $\gk_2$.
By the choice of $\gl$ and the fact that
$V \models ``$No cardinal greater than
$\gk_2$ is measurable'',
$M \models ``\gk_2$ is measurable and there
are no measurable cardinals in the interval
$(\gk_2, \gl]$''. Thus,
$\FQ$ is allowed in the stage
$\gk_2$ lottery sum held in $M^{\FP}$
in the definition of $j(\FP)$.
We therefore assume without loss of generality
that we are forcing above a condition which
picks $\FQ$ in the stage
$\gk_2$ lottery sum held in $M^{\FP}$
in the definition of $j(\FP)$.
We will consequently in what follows
slightly abuse notation and replace
any term for the stage $\gk_2$ lottery sum
held in $M^{\FP}$ in the definition of
$j(\FP)$ with $\dot \FQ$.
We consider below two cases.
\medskip\noindent Case 1: In $M$,
$\forces_{\FP \ast \dot \FQ} ``$Level
by level equivalence between strong
compactness and supercompactness holds
at $\gk_2$''.
Under these circumstances, since
$\dot \FQ^1_{\gk_2}$ is a term for trivial
forcing (our slight abuse of notation
allows us to consider the terms
$\dot \FQ$ and $\dot \FQ^0_{\gk_2}$
as being the same), in $M$,
$j(\FP \ast \dot \FQ)$ is forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$,
where the first nontrivial stage in
the forcing denoted by $\dot \FR$
takes place well after $\gl$.
Standard arguments, as given, e.g., in the
proof of Lemma 8.3 of \cite{AH4}
now show that $j$ lifts in
$V^{\FP \ast \dot \FQ}$ to
$j : V^{\FP \ast \dot \FQ} \to
M^{j(\FP \ast \dot \FQ)}$, i.e.,
$V^{\FP \ast \dot \FQ} \models ``\gk_2$ is
$\gl$ supercompact''.
This completes our discussion of Case 1.
\hfill$\square$
\medskip\noindent Case 2: In $M$,
$\forces_{\FP \ast \dot \FQ} ``$Level
by level equivalence between strong
compactness and supercompactness fails
at $\gk_2$''.
Under these circumstances,
the proof of Lemma \ref{l6} is
virtually identical to the proof of
Lemma 2.3 of Apter-Sargsyan \cite{AS06}.
%For the convenience of readers,
%we repeat the proof here,
%taking the liberty to quote
%verbatim when necessary.
Specifically, because $\gl$ has been chosen large enough,
we may assume by taking %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
$N \models ``\gk_2$ is not measurable''.
It is the case that if $i : {V} \to N$
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 $\gk_2$.
To complete our discussion of Case 2,
we show that $i$ lifts in
${V}^{\FP \ast \dot \FQ}$ to
$i : {V}^{\FP \ast \dot \FQ} \to
N^{i(\FP \ast \dot \FQ)}$.
%Since this lifted embedding witnesses
%the $\gl$ strong compactness of $\gk_2$
%in ${V}$, and since $\gl$ is arbitrary,
%this proves Lemma \ref{l6}.
To do this, we use a modification
of an argument originally due to Magidor,
unpublished by him but found in,
among other places,
Lemma 8 of Apter-Hamkins \cite{AH03}.
The modification is due to Sargsyan.
Note that throughout the course
of the remainder of the proof of Lemma \ref{l6},
we will refer readers to the construction
given in %these lemmas
Lemma 8 of \cite{AH03} when relevant,
and omit details already presented therein.
Let $G_0$ be ${V}$-generic over $\FP$,
and let $H$ be ${V}[G_0]$-generic
over $\FQ$.
Write
$i(\FP) = \FP \ast \dot \FP^1 \ast
\dot \FP^2 \ast \dot \FP^3$,
where $\dot \FP^1$ is a term for the portion of
the forcing
%starting at stage $\gk_2$
defined from stage $\gk_2$ to stage $k(\gk_2)$,
$\dot \FP^2$ is a term for the
forcing done at stage $k(\gk_2)$, and
$\dot \FP^3$ is a term for the remainder
of the forcing, i.e., the portion done
after stage $k(\gk_2)$.
We will build in ${V}[G_0][H]$
generic objects for the different
portions of $i(\FP)$.
We begin by constructing an
$N[G_0]$-generic object $G_1$ for $\FP^1$.
The argument used is essentially the
same as the one given in the construction
of the generic object $G_1$ found in
Lemma 8 of \cite{AH03}
(and will therefore be carried out in
$M[G_0] \subseteq {V}[G_0] \subseteq
{V}[G_0][H]$).
Specifically, since
$N \models ``\gk_2$ is not measurable'',
only trivial forcing is done at
stage $\gk_2$ in $N$, which means that
$\dot \FP^1$ is forced to act nontrivially
on ordinals in the open interval $(\gk_2, k(\gk_2))$.
In addition, since GCH holds in $N$
for cardinals
greater than or equal to $\gk_1$
(as it does in ${V}$ and $M$),
standard counting arguments
%found in Lemma 2.3 of \cite{A02} and Lemma 8 of \cite{AH03}
show that
$N[G_0] \models ``\card{\FP^1} = k(\gk_2)$ and
$\card{\wp(\FP^1)} = 2^{k(\gk_2)} = k(\gk^+_2)$''.
Consequently, since GCH
for cardinals
greater than or equal to $\gk_1$ 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 ${V}[G_0]$ or
$M[G_0]$ of the dense open subsets of $\FP^1$
present in $N[G_0]$. We then build in
%either ${V}[G_0]$ or
$M[G_0]$ an $N[G_0]$-generic object $G_1$
for $\FP^1$ by meeting in turn each member of
$\la D_\ga \mid \ga < \gk^+_2 \ra$, using the fact that
$\FP^1$ is ${\prec} \gk^+_2$-strategically
closed in $N[G_0]$ and $M[G_0]$.
%and ${V}[G_0]$,
This strategic closure property
of $\FP_1$ follows
from the fact that standard arguments
show $N[G_0]$ remains $\gk_2$-closed with
respect to (both $V[G_0]$ and) ${M}[G_0]$.
We next analyze the exact nature of $\dot \FP^2$.
By the definition of $\FP$,
the closure properties
of $M$, and the fact that we are
in Case 2, we may write
$j(\FP \ast \dot \FQ) = \FP \ast \dot \FQ \ast \dot \FQ'
\ast \dot \FR \ast j(\dot \FQ)$, where
$\dot \FQ \ast \dot \FQ'$ is a term for
the forcing taking place at stage
$\gk_2$ in $M$ and
$\dot \FQ'$ 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 $\dot \FP^2$ is a
term for the forcing which takes place
at stage $k(\gk_2)$ in $N$, we may write
$\dot \FP^2 = k(\dot \FQ) \ast k(\dot \FQ')$.
We will construct in $M[G_0][H]$
generic objects for %(the denotations of)
$k(\FQ)$ and $k(\FQ')$.
For $k(\FQ)$, we use an argument containing
ideas due to Woodin, also presented in
Theorem 4.10 of \cite{H4}, Lemma 4.2 of
Apter \cite{A03}, Lemma 3.4 of Apter-Sargsyan \cite{AS06a},
and Lemma 2.3 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
$k '' H \subseteq k(\FQ)$ generates an
$N[G_0][G_1]$-generic object $G_2$ over
$k(\FQ)$. Specifically, given a dense
open subset $D \subseteq k(\FQ)$,
$D \in N[G_0][G_1]$, $D = i_{G_0 \ast G_1}(\dot D)$
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 $\FQ$.
Since $\FQ$ 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 $\FQ$. As
$k(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$k '' H \cap D \neq \emptyset$. Thus,
$G_2 = \{p \in k(\FQ) \mid \exists
q \in k '' H [q \ge p]\}$,
which is definable in
$M[G_0][H]$, is our desired
$N[G_0][G_1]$-generic object over $k(\FQ)$.
Then, since $k(\FQ')$ 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
$N[G_0][G_1][G_2] \models
``\card{k(\FQ')} = k(\gk_2)$ and
$\card{\wp(k(\FQ'))} = 2^{k(\gk_2)} = k(\gk^+_2)$''
Hence, since $N[G_0][G_1][G_2]$
remains $\gk_2$-closed with respect to $M[G_0][H]$,
which means $k(\FQ')$ is
${\prec} \gk^+_2$-strategically closed in
$N[G_0][G_1][G_2]$ and $M[G_0][H]$,
%and ${V}[G_0][H]$,
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 $k(\FQ')$.
We construct now (in ${V}[G_0][H]$) an
$N[G_0][G_1][G_2][G_3]$-generic object for $\FP^3$.
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 $j(\dot \FQ)$, i.e., we show the
existence of a term $\gt \in M$
in the language of forcing
with respect to $j(\FP)$ such that in $M$,
$\forces_{j(\FP)} ``\gt \in j(\dot \FQ)$
extends every $j(\dot q)$
for $\dot q \in \dot H$''.
We first note that since $\FP$ is
$\gk_2$-c.c$.$ in both ${V}$ and $M$, as
$\forces_{\FP} ``\dot \FQ$ is
${} \gk_2$-directed closed
%and $(\gk_2, \infty)$-distributive
and $\card{\dot \FQ} < \gl$'', the usual
arguments show $M[G_0][H]$ remains
$\gl$-closed with respect to ${V}[G_0][H]$.
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 $M$,
$\forces_{j(\FP)} ``\card{\dot T} < \gl < j(\gk_2)$,
any two elements of $\dot T$ are compatible, and
$\dot T$ is a subset of a
partial ordering (namely $j(\dot \FQ))$ which is
${} j(\gk_2)$-directed closed''. Thus, since
$M^\gl \subseteq M$,
$\forces_{j(\FP)} ``$There is an upper bound for
%a condition in $j(\dot \FQ)$ extending each element of
$\dot T$''.
A term $\gt$ for this upper bound %common extension
is as desired.
We work for the time being in $M$.
Consider the ``term forcing'' partial ordering
$\FR^*$ (see Foreman \cite{F} for the first
published account of term forcing or
Cummings \cite{C}, Section 1.2.5, page 8 --- the
notion is originally due to Laver)
associated with $\dot \FR \ast
j(\dot \FQ)$, i.e.,
$\gs \in \FR^*$ iff $\gs$ is a term
in the forcing language with respect to
$\FP \ast \dot \FQ \ast \dot \FQ'$ and
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ'}
``\gs \in \dot \FR \ast
j(\dot \FQ)$'', ordered by
$\gs_1 \ge \gs_0$ iff
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ'}
``\gs_1 \ge \gs_0$''.
Note that $\gt'$ defined as the term
in the language of forcing with respect to
$\FP \ast \dot \FQ \ast \dot \FQ'$ 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 $\FR^*$.
Clearly, $\FR^* \in M$. In addition, since
${V} \models ``$No cardinal greater than
$\gk_2$ is measurable'',
as in Case 1,
%by the fact that $M^\gl \subseteq M$, %closure properties of $M$,
$M \models ``$The first stage at which
$\dot \FR \ast j(\dot \FQ)$
is forced to do nontrivial forcing
is greater than $\gl$''. Thus,
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ'}
``\dot \FR \ast j(\dot \FQ)$
is ${\prec} \gl^+$-strategically
closed'', which, since $M^\gl \subseteq M$,
immediately implies that
$\FR^*$ itself is ${\prec} \gl^+$-strategically
closed in both ${V}$ and $M$. Further, since
${V}^\FP \models ``\card{\FQ} < \gl$'',
in $M$,
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ'}
``\card{\dot \FR \ast j(\dot \FQ)} < j(\gl)$''.
Also, by GCH
for cardinals
greater than or equal to $\gk_1$ in both
${V}$ and $M$
and the fact that $j$ 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
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ'}
``\card{\wp(\dot \FR \ast j(\dot \FQ))} <
2^{j(\gl)} = j(\gl^+)$''.
Therefore, since as in the footnote
given in the proof of Lemma 8 of
\cite{AH03}, we may assume that
$\FR^*$ has cardinality less than
$j(\gl)$ in $M$,
we may let
$\la D_\ga \mid \ga < \gl^+ \ra \in {V}$
be an enumeration of the dense open subsets
of $\FR^*$ present in $M$.
It is then possible
using the ${\prec} \gl^+$-strategic closure
of $\FR^*$ in ${V}$ and the argument
employed in the construction of $G_1$
to build in ${V}$ an $M$-generic object
$G^*_4$ for $\FR^*$ 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 $k(\FR^*)$
containing $k(\gt')$.
By elementarity,
%since $\FP = \FP_{\gk_2}$,
$k(\FR^*)$ is the
term forcing in $N$ defined with respect to
$k(j(\FP_{})_{\gk_2 + 1}) =
\FP \ast \dot \FP^1 \ast \dot \FP^2$.
Therefore, since
$i(\FP \ast \dot \FQ) = k(j(\FP \ast \dot \FQ)) =
\FP \ast \dot \FP^1 \ast \dot \FP^2 \ast \dot \FP^3$,
$G^{**}_4$ is $N$-generic over $k(\FR^*)$, and
$G_0 \ast G_1 \ast G_2 \ast G_3$ is
$k(\FP \ast \dot \FQ)$-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 $\FP^3$.
In addition, since
the definition of $\gt$ tells us that
in $M$, the statement
``$\la p, \dot q \ra
\in j(\FP \ast \dot \FQ)$ implies that
$\la p, \dot q \ra \forces_{j(\FP \ast \dot \FQ)}
`\gt \ge \dot q$' ''
is true, by elementarity, in $N$, the statement
``$\la p, \dot q \ra
\in k(j(\FP \ast \dot \FQ))$ implies that
$\la p, \dot q \ra \forces_{k(j(\FP \ast \dot \FQ))}
`k(\gt) \ge \dot q$' ''
%``$p \in k(j(\FP))$ implies that
%$p \forces_{k(j(\FP))} ``k(\gt)$ extends $\check p$''
is true. In other words,
since $k \circ j = i$, in $N$, the statement
``$\la p, \dot q \ra \in i(\FP \ast \dot \FQ)$ implies that
$\la p, \dot q \ra \forces_{i(\FP \ast \dot \FQ)}
`k(\gt) \ge \dot q$' '' is true.
Thus, in $N$, $k(\gt)$ functions as a
term for a ``master condition'' for $i(\dot \FQ)$,
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 ${V}^{\FP \ast \dot \FQ}$ to
$i : {V}^{\FP \ast \dot \FQ} \to
N^{i(\FP \ast \dot \FQ)}$, i.e.
$V^{\FP \ast \dot \FQ} \models
``\gk_2$ is $\gl$ strongly compact''.
This completes our discussion of Case 2.
\hfill$\square$
\medskip We now see that regardless if
Case 1 or Case 2 holds,
$V^{\FP \ast \dot \FQ} \models
``\gk_2$ is $\gl$ strongly compact''.
Since $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``\gk_1$ and $\gk_2$
are the first two strongly compact cardinals''.
%In addition, $\ov V^\FP \models ``\gk_1$
\end{lemma}
\begin{proof}
It follows by Ketonen's characterization
of strong compactness given in
\cite{Ke} that any cardinal
which is strongly compact up to a
strongly compact cardinal is itself strongly compact.
Thus, since $V \models ``\gk_1$ is the
first strongly compact cardinal'',
it must be the case that for any $\gd < \gk_1$,
$V \models ``\gd$ is not strongly compact
up to $\gk_1$''. In addition,
as we observed in the proof of Lemma \ref{l5},
$\FP$ is $\gk_1$-directed closed,
and so cannot force a new degree
of strong compactness for any $\gd < \gk_1$.
Therefore, since $V \models ``\gk_1$
is both indestructibly strongly compact and
the first strongly compact cardinal'',
$V^\FP \models ``\gk_1$ is both indestructibly
strongly compact and the first
strongly compact cardinal''.
Next, if $\gd \in (\gk_1, \gk_2)$ is such that
$V^\FP \models ``\gd$ is measurable'',
then because $\FP$ admits a
sufficiently small gap above
$\gk_1$, by Theorem \ref{t3},
$V \models ``\gd$ is measurable''.
Hence, because by its definition,
$\FP$ is mild with respect to $\gd$,
Theorem \ref{t3} also tells us that
forcing with $\FP$ cannot create
any new degrees of strong compactness of $\gd$. As
$V \models ``$No cardinal $\gd \in (\gk_1, \gk_2)$
is strongly compact'',
we may consequently infer that
$V^\FP \models ``$No cardinal $\gd \in (\gk_1, \gk_2)$
is strongly compact''.
Since we have already seen that
$V^\FP \models ``\gk_2$ is supercompact'',
$V^\FP \models ``\gk_2$ is the second
strongly compact cardinal''.
This completes the proof of Lemma \ref{l7}.
\end{proof}
Since $V \models ``\gk_1$ is
a limit of measurable cardinals and
$\FP$ is $\gk_1$-directed closed'',
$V^\FP \models ``\gk_1$ is a limit of measurable cardinals''.
Therefore, by taking $\ov V = V^\FP$,
Lemmas \ref{l1} - \ref{l7}, their
proofs, and the intervening remarks complete
the proof of Theorem \ref{t2}.
\end{proof}
Theorem \ref{t2} leaves open
some interesting questions,
which we pose in conclusion
to this paper. In particular,
is it possible to prove an
analogue of Theorem \ref{t2}
for two strongly compact cardinals in
which the large cardinal structure of
the model constructed has fewer restrictions?
Is it possible to prove an analogue of
Theorem \ref{t2} for two strongly compact cardinals
in which
$\gk_2$'s strong compactness is indestructible
under arbitrary $\gk_2$-directed closed forcing?
Is it possible to prove an analogue of
Theorem \ref{t2} for two strongly compact cardinals
in which
$\gk_2$ is indestructibly supercompact?
Is it possible to prove an analogue of
Theorem \ref{t2} in which the
model constructed contains more than
two strongly compact cardinals?
\begin{thebibliography}{99}
%\bibitem{A02} A.~Apter,
%``Aspects of Strong Compactness,
%Measurability, and Indestructibility'',
%{\it Archive for Mathematical Logic 41},
%2002, 705--719.
\bibitem{A04} A.~Apter, ``Level by Level
Equivalence and Strong Compactness'',
{\it Mathematical Logic Quarterly 50}, 2004, 51--64.
\bibitem{A03} A.~Apter,
``Some Remarks on Indestructibility and
Hamkins' Lottery Preparation'',
{\it Archive for Mathematical Logic 42}, 2003, 717--735.
%\bibitem{A97} A.~Apter, ``Patterns of
%Compact Cardinals'', {\it Annals of
%Pure and Applied Logic 89}, 1997, 101--115.
\bibitem{A07} A.~Apter, ``Supercompactness and
Level by Level Equivalence are Compatible with
Indestructibility for Strong Compactness'',
{\it Archive for Mathematical Logic 46},
2007, 155--163.
%\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness'',
%{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
\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{AH03} A.~Apter, J.~D.~Hamkins,
``Exactly Controlling the Non-Supercompact
Strongly Compact Cardinals'',
{\it Journal of Symbolic Logic 68}, 2003, 669--688.
\bibitem{AH4} A.~Apter, J.~D.~Hamkins,
``Indestructibility and the Level-by-Level Agreement
between Strong Compactness and Supercompactness'',
{\it Journal of Symbolic Logic 67}, 2002, 820--840.
\bibitem{AS06a} A.~Apter, G.~Sargsyan,
``Identity Crises and Strong Compactness III:
Woodin Cardinals'', {\it Archive for Mathematical
Logic 45}, 2006, 307--322.
\bibitem{AS06} A.~Apter, G.~Sargsyan,
``Universal Indestructibility for Degrees of
Supercompactness and Strongly Compact Cardinals'',
{\it Archive for Mathematical Logic 47}, 2008, 133--142.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong
Equality between Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349},
1997, 103--128.
\bibitem{C} J.~Cummings, ``A Model in which GCH Holds at
Successors but Fails at Limits'', {\it Transactions of
the American Mathematical Society 329}, 1992, 1--39.
%\bibitem{DH} C.~DiPrisco, J.~Henle,
%``On the Compactness of $\ha_1$ and
%$\ha_2$'', {\it Journal of Symbolic Logic 43},
%1978, 394--401.
\bibitem{F} M.~Foreman, ``More Saturated Ideals'', in:
{\it Cabal Seminar 79-81}, {\bf Lecture Notes in
Mathematics 1019}, Springer-Verlag, Berlin and
New York, 1983, 1--27.
%\bibitem{G} M.~Gitik, ``Changing Cofinalities
%and the Nonstationary Ideal'',
%{\it Israel Journal of Mathematics 56},
%1986, 280--314.
%\bibitem{H1} J.~D.~Hamkins,
%``Destruction or Preservation As You
%Like It'',
%{\it Annals of Pure and Applied Logic 91},
%1998, 191--229.
\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{H} J.~D.~Hamkins, {\it Lifting and
%Extending Measures; Fragile Measurability},
%Doctoral Dissertation, University of California,
%Berkeley, 1994.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
{\it Annals of Pure and Applied Logic 101},
2000, 103--146.
%\bibitem{H5} J.~D.~Hamkins, ``Small Forcing Makes
%Any Cardinal Superdestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 51--58.
\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{Ke} J.~Ketonen, ``Strong Compactness and
Other Cardinal Sins'', {\it Annals of Mathematical
Logic 5}, 1972, 47--76.
%\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
%between the Concepts of Compactness and Supercompactness'',
%circulated manuscript.
\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{Ma} M.~Magidor, ``How Large is the First
%Strongly Compact Cardinal?'', {\it Annals of
%Mathematical Logic 10}, 1976, 33--57.
%\bibitem{Me} T.~Menas, ``On Strong Compactness and
%Supercompactness'', {\it Annals of Mathematical Logic 7},
%1974, 327--359.
%\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
%``Strong Axioms of Infinity and Elementary Embeddings'',
%{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}