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\title{Indestructibility and Strong Compactness
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, indestructibility,
non-reflecting stationary set of ordinals}}
\author{Arthur W.~Apter
\thanks{The contents of this paper were presented
in the Special Session in Set Theory
of Logic Colloquim 2003, held
August 14-20, 2003 in Helsinki, Finland.
The author wishes to thank the organizers
for having invited him to speak at and
participate in a very stimulating conference.
The author also wishes to thank Hy Line
Cruises, for having provided a very
relaxing setting under which the
main result of this paper was proven.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{November 3, 2003\\
(revised April 23, 2004)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model in which
the first two strongly compact
cardinals aren't supercompact
yet satisfy significant
indestructibility properties
for their strong compactness.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
The study of indestructibility for
non-supercompact strongly compact cardinals
is one which has been the subject of
a great deal of investigation over
the last few years, most notably
in the papers \cite{A98}, \cite{AG},
\cite{AH}, \cite{A02}, and \cite{H4}.
We refer readers to the introductory
section of \cite{A02} for a thorough
discussion of the relevant history.
We note, however, that in spite
of all of the work done, the
basic question of whether it
is consistent, relative to
anything, for the first two
strongly compact cardinals to
be non-supercompact yet to
satisfy significant indestructibility
properties for their strong
compactness had heretofore
been left unanswered.
This should be contrasted to the
relative ease with which Laver's
forcing of \cite{L} iterates,
to produce models such as the
one given in \cite{A98} in which
there is a proper class of
supercompact cardinals and
every supercompact cardinal
$\gk$ has its supercompactness
indestructible under $\gk$-directed
closed forcing.
The purpose of this paper is
to provide an affirmative answer
to the above question.
Specifically, we will prove
the following theorem.
\begin{theorem}\label{t1}
It is consistent, relative
to the existence of two
supercompact cardinals,
for the first two strongly
compact cardinals $\gk_1$
and $\gk_2$ to be
non-supercompact yet to
satisfy significant
indestructibility properties
for their strong compactness.
Specifically, in our
final model $V^\FP$,
$\gk_1$'s strong compactness
is indestructible under
arbitrary $\gk_1$-directed
closed forcing, and $\gk_2$'s
strong compactness is
indestructible under either
trivial forcing or $\gk_2$-directed
closed forcing that can be written
in the form
$\add(\gk_2, 1) \ast \dot \FQ'$.
\end{theorem}
We note that Theorem \ref{t1}
is a generalization of a sort of
Theorem 1 of \cite{A02}.
In the model constructed for
this theorem, the first
two strongly compact cardinals
$\gk_1$ and $\gk_2$ aren't
supercompact (and in fact,
are the first two measurable
cardinals), $\gk_1$'s
strong compactness is fully
indestructible under
$\gk_1$-directed closed forcing,
yet $\gk_2$'s measurability,
but not necessarily its
strong compactness, is indestructible
under arbitrary $\gk_2$-directed
closed forcing.\footnote{Theorem 1
of \cite{A02} is of course a
generalization of a celebrated
result of Magidor, who showed in
\cite{Ma} that it is consistent,
relative to the existence of a
strongly compact cardinal, for
the least strongly compact cardinal
to be the least measurable cardinal.
Readers are urged to consult \cite{Ma}
and \cite{AC1} for further details on
both this result and some generalizations.}
Also, for our Theorem \ref{t1},
as for Theorem 1 of \cite{A02},
it is impossible simply to iterate
the forcing conditions given in
\cite{AG}, where a model
in which the least strongly
compact cardinal $\gk$ is both
the least measurable cardinal
and has its strong compactness
fully indestructible under
$\gk$-directed closed forcing
is constructed.
The reason is that these forcing
conditions, being a Gitik-style
iteration of Prikry-like forcings
as introduced in \cite{G},
add bona fide Prikry sequences,
which in turn then add weak square
sequences.
These weak square sequences
can't exist above a strongly
compact cardinal, as
can be inferred from the proof
of Theorem 5.4 of \cite{CFM}.
%discussed in Theorem 5.4 of \cite{CFM}.
Before beginning the proof of
Theorem \ref{t1}, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
%If we also have that $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
%is an Easton support iteration of length $\gk$
%so that at stage $\ga$, a non-trivial forcing is done
%based on the ordinal $\gd_\ga$, then we will say that
%$\gd_\ga$ is in the field of $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a regular
cardinal and $\gl$ is
an ordinal, $\add(\gk, \gl)$
is the standard partial
ordering for adding
$\gl$ Cohen subsets of $\gk$.
If $\FP$ is
an arbitrary partial ordering,
%$\FP$ is {\it $\gk$-distributive}
%if for every sequence
%$\la D_\ga : \ga < \gk \ra$
%of dense open subsets of $\FP$,
%$\bigcap_{\ga < \gk} D_\ga$
%is dense open.
$\FP$ is {\it $\gk$-directed closed}
if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$
is {\it directed} if
every two elements
$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
$\FP$ is {\it $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is
$\gk^+$-directed closed, then $\FP$ is
$\gk$-strategically closed.
%Note that
%if $\FP$ is $\gk$-strategically closed,
%then $\FP$ is $\gk$-distributive.
Also, if $\FP$ is
$\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$\FP$ is {\it ${\prec}\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (again choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
In this paper, we will use non-reflecting
stationary set forcing $\FP_{\eta,\gl}$.
Specifically, if $\eta<\lambda$ are both regular
cardinals, then conditions in $\FP_{\eta,\gl}$
are bounded subsets $s\subseteq\gl$ consisting of
ordinals of cofinality $\eta$ such that for every
$\alpha<\lambda$, the initial segment
$s\intersect\alpha$ is non-stationary in
$\alpha$, ordered by end-extension. It is
well-known that if $G$ is $V$-generic over
$\FP_{\eta, \gl}$ (see \cite{Bu} or \cite{A01})
and GCH holds in $V$, then in
$V[G]$, the set $S=S[G]=\bigcup G \subseteq \gl$
is a non-reflecting stationary set of ordinals of
cofinality $\eta$, the bounded subsets of $\gl$
are the same as those in $V$, and cardinals,
cofinalities and GCH have been preserved. It
is virtually immediate that $\FP_{\eta, \gl}$ is
$\eta$-directed closed. It follows from
work of Solovay
(Theorem 4.8 of \cite{SRK} and the
succeeding remarks) that the existence of a
non-reflecting stationary subset of $\lambda$,
consisting of ordinals of confinality $\eta$,
implies that no cardinal
$\delta\in(\eta,\lambda]$ is $\lambda$ strongly
compact. Thus, iterations of this forcing provide
a way to destroy all strongly compact cardinals
in an interval.
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 {\it 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''.}
A result which will be used in the proof
of Theorem \ref{t1} is Hamkins'
Gap Forcing Theorem of \cite{H2} and \cite{H3}.
We therefore state this theorem now, along
with some associated terminology, quoting
freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| < \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\it admits a gap at $\gd$}.
Also, as in the terminology of
\cite{H2} and \cite{H3} (and elsewhere),
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The Gap Forcing Theorem is then
the following.
\begin{theorem}\label{t2}
{\bf (Hamkins' Gap Forcing Theorem)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a gap at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ 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 $V$.
\end{theorem}
Finally, we mention that we are assuming
familiarity with the large cardinal
notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{K}
or \cite{SRK} for further details.
We do wish to note, however,
that a measurable cardinal $\gk$ has
{\it trivial Mitchell rank} if
there is no embedding
$j : V \to M$ for which
${\rm cp}(j) = \gk$ and
$M \models ``\gk$ is measurable''.
An ultrafilter ${\cal U}$
generating this sort of
embedding will be said to
have trivial Mitchell rank as well.
Ultrafilters of trivial Mitchell rank
always exist for every measurable cardinal.
Also, if $\gk$ is $2^\gk$ supercompact,
then $\gk$ has non-trivial Mitchell rank.
This implies that $\gk$ is the
$\gk^{\rm th}$ measurable cardinal,
which further implies that the
least measurable cardinal $\gd$
isn't $2^\gd$ 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 + $\gk_1$ and
$\gk_2$ are supercompact''.
Without loss of generality,
by first forcing GCH and then
forcing with the
partial ordering given in
Theorem 1 of \cite{AG} (which
can be defined so as to
have cardinality $\gk_1$),
%followed by the $\gk_1$-directed
%closed partial ordering which
%forces GCH for all cardinals
%greater than or equal to $\gk_1$,
we assume that
$V^*$ has been generically
extended to a model
$V$ such that
$V \models ``\gk_1$
is both the least strongly
compact and least
measurable cardinal +
$\gk_1$'s strong compactness
is indestructible under
$\gk_1$-directed closed forcing +
GCH holds for all cardinals
greater than or equal to $\gk_1$ +
$\gk_2$ is supercompact''.
The partial ordering $\FP$
to be used in the proof of
Theorem \ref{t1} is now
defined as follows.
For any ordinal $\gd$, let
$\gd'$ be the least
$V$-strong cardinal above
$\gd$.
$\FP$ begins by adding a
Cohen subset of $\gk_1$.
The remainder of $\FP$ is the
reverse Easton iteration
having length $\gk_2$
which does a non-trivial forcing
only at those cardinals in
the open interval
$(\gk_1, \gk_2)$
%$\gd \in (\gk_1, \gk_2)$
which are in $V$ measurable
limits of strong cardinals.
At such a stage $\gd$, the
forcing done is
$\FQ^\gd \ast \dot \FP_{\gk_1, \gd'}$,
where
$\forces_{\FP_\gd} ``\dot \FQ^\gd$ is
the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$
which can be written in the form
$\add(\gd, 1) \ast \dot \FQ'$''.
The intuition behind the
above definition of
$\FP$ is as follows.
The non-reflecting
stationary set of ordinals
of cofinality $\gk_1$ added
at each non-trivial stage of
forcing is used to destroy
all strongly compact cardinals
in the open interval
$(\gk_1, \gk_2)$.
The lottery sum employed at each
non-trivial stage of forcing
is used to force indestructibility
for $\gk_2$.
The Cohen subset added at each
non-trivial stage of forcing is
used to ensure that $\gk_2$
becomes a non-supercompact
strongly compact cardinal.
The entire iteration $\FP$
is defined in a way so as to
be $\gk_1$-directed closed,
which means that after
forcing with $\FP$,
$\gk_1$ remains as both the
least strongly compact and
least measurable cardinal and
retains the indestructibility
of its strong compactness
under $\gk_1$-directed closed forcing.
The following lemmas show that
$\FP$ is as desired.
Throughout, we assume that
$G$ is $V$-generic over $\FP$.
\begin{lemma}\label{l1}
In $V^\FP$, there are no strongly
compact cardinals in the open
interval $(\gk_1, \gk_2)$.
\end{lemma}
\begin{proof}
By Lemma 2.1 of \cite{AC2},
the supercompactness
of $\gk_2$ in $V$ implies that in $V$,
there are unboundedly many in $\gk_2$
measurable limits
of strong cardinals in
the open interval $(\gk_1, \gk_2)$.
Therefore, by its definition, after forcing
with $\FP$, unboundedly many in $\gk_2$
cardinals in the open interval $(\gk_1, \gk_2)$
will contain non-reflecting
stationary sets of ordinals of
cofinality $\gk_1$.
As mentioned in Section \ref{s1},
by work of Solovay (Theorem
4.8 of \cite{SRK} and the
succeeding remarks), this
means that in $V^\FP$,
no cardinal $\gd \in (\gk_1, \gk_2)$
is strongly compact.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
In $V^\FP$, $\gk_2$ is
strongly compact.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} uses
a technique of Magidor
for preserving strong compactness,
which is unpublished by him
but is used, e.g., in \cite{A01},
%under an iteration of forcing to add
%non-reflecting stationary sets of ordinals
%and is similar to proofs for the preservation
%of strong compactness given in
\cite{AC1}, \cite{AC2}, and \cite{AH3}.
Let $\gl > 2^{\gk_2}$ be a
regular cardinal, and let
$k_1 : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of ${\gk_2}$ such that
$M \models
``{\gk_2}$ isn't $\gl$ supercompact''.
%By the choice of $\gl$,
%$M \models ``{\gk_2}$ is $\gg$ supercompact for every
%$\gg < \gl$''. Also,
$\gl$ is large enough so that we
may assume by selecting a normal ultrafilter of
trivial Mitchell rank over ${\gk_2}$ that
$k_2 : M \to N$ is an embedding witnessing the
measurability of ${\gk_2}$ definable in $M$ such that
$N \models ``{\gk_2}$ isn't measurable''.
In addition, as $\gl > 2^{\gk_2}$,
Lemma 2.1 of \cite{AC2} and the succeeding
remarks imply that in both $V$ and $M$,
${\gk_2}$ is a
measurable cardinal which is a limit of
strong cardinals.
This means by the definition of $\FP$
it is possible to opt for
${\rm Add}(\gk_2, 1)$ in
the stage $\gk_2$ lottery
held in $M$ in the
definition of $k_1(\FP)$.
We therefore assume that
this has been done, meaning that in $M$,
%We may therefore assume that in $M$,
above the appropriate condition,
$k_1(\FP)$ is forcing equivalent to
$\FP \ast \dot \FS \ast \dot \FT$,
where $\dot \FS$ is a term for
$\add(\gk_2, 1)$, and
$\dot \FT$ is a term for the
rest of $k_1(\FP)$.
We consequently assume for the
rest of the proof of Lemma \ref{l2}
that we are forcing above
such a condition.
Since as in Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in (\gk_2, \gl]$ is strong''
(this is since otherwise, $\gk_2$
would be supercompact up to
a strong cardinal and hence be
fully supercompact),
the first ordinal in the realization
of $\dot \FT$ occurs above $\gl$.
%strong cardinal which is a
%limit of strong cardinals,
%and in fact, in both $V$ and $M$, ${\gk_2}$
%carries a normal measure concentrating on
%strong cardinals.
Suppose that
$k : V \to N$ is an elementary embedding
definable in $V$ with
critical point ${\gk_2}$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
such that $x \subseteq y$ and
$N \models ``|y| < k({\gk_2})$''.
By Theorem 22.17 of \cite{K},
$k$ witnesses the $\gl$
strong compactness of ${\gk_2}$
in the sense that the existence
of this sort of embedding implies that
${\gk_2}$ is $\gl$ strongly compact.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of ${\gk_2}$.
We show that $j$ lifts to
$j : V^{\FP} \to N^{j(\FP)}$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of ${\gk_2}$ in
$V^{\FP}$ and $\gl > 2^{\gk_2}$
is an arbitrary regular
cardinal, this proves Lemma \ref{l2}.
To do this,
write $j(\FP)$ as
$\FP \ast \dot \FQ \ast \dot \FR$, where
$\dot \FQ$ is a term for the portion of
$j(\FP)$ between ${\gk_2}$ and $k_2({\gk_2})$ and
$\dot \FR$ is a term for the rest of
$j(\FP)$, i.e., the part above $k_2({\gk_2})$.
Note that since
$N \models ``{\gk_2}$ isn't measurable'',
the realization of $\dot \FQ$
is trivial at stage $\gk_2$.
%doesn't add a Cohen subset to $\gk_2$.
Thus, the ordinals at which the realization of
$\dot \FQ$ does a non-trivial forcing
%is composed of all $N$-measurable limits of strong cardinals
lie in the interval
$({\gk_2}, k_2({\gk_2})]$ (the endpoint
$k_2({\gk_2})$ is included since by elementarity,
$k_2({\gk_2})$ is a measurable cardinal
which is a limit of
strong cardinals in $N$),
and the ordinals at which the realization
of $\dot \FR$ does a non-trivial forcing
lie in the interval
$(k_2({\gk_2}), k_2(k_1({\gk_2})))$.
Since we have assumed that
we have opted for $\add(\gk_2, 1)$
at stage $\gk_2$ in $M$, we may
infer that the forcing done
at stage $k_2(\gk_2)$ in $N$ is
$\add(k_2(\gk_2), 1)$.
%Let ${G}$ be $V$-generic over $\FP$.
We construct in $V[{G}]$ an
$N[{G}]$-generic object $G_1$ over
$\FQ$ and an
$N[{G}][G_1]$-generic object $G_2$ over
$\FR$. Since $\FP$ is a reverse
Easton iteration of length ${\gk_2}$,
a direct limit is taken at stage ${\gk_2}$,
and no forcing is done at stage ${\gk_2}$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' {G} \subseteq {G} \ast G_1 \ast G_2$.
This means that
$j : V \to N$ lifts to
$j : V[{G}] \to N[{G}][G_1][G_2]$ in
$V[{G}]$.
To build $G_1$, note that since $k_2$
is generated by an
ultrafilter
%${\cal U}$
over ${\gk_2}$ and
since in both $V$ and $M$, $2^{\gk_2} = \gk^+_2$,
$|k_2(\gk^+_2)| = |k_2(2^{\gk_2})| =
|\{ f : f : {\gk_2} \to \gk^+_2$ is a function$\}| =
|{[\gk^+_2]}^{\gk_2}| = \gk^+_2$. Thus, as
$N[{G}] \models ``|\wp(\FQ)| = k_2(2^{\gk_2})$'', we can let
$\la D_\ga : \ga < \gk^+_2 \ra$ enumerate in
$V[{G}]$ the dense open subsets of
$\FQ$ present in $N[{G}]$.
For the purpose of the construction of
$G_1$ to be given below, we further
assume without loss of generality that
for every dense open subset
$D \subseteq \FQ$ found in $N[{G}]$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$.
Since the ${\gk_2}$ closure of $N$ with respect to either
$M$ or $V$ implies the least ordinal
at which $\FQ$ does a non-trivial forcing
is above $\gk^+_2$, the definition of $\FQ$ implies that
%as the Easton support iteration which adds
%a non-reflecting stationary set of ordinals of
%cofinality $\xi$ to each $N[{G}]$-measurable limit
%of strong cardinals in the interval
%$({\gk_2}, k_2({\gk_2})]$ implies that
$N[{G}] \models ``\FQ$ is
${\prec} \gk^+_2$-strategically closed''.
By the fact the standard arguments show that
forcing with the ${\gk_2}$-c.c$.$ partial ordering
$\FP$ preserves that $N[{G}]$ remains
${\gk_2}$-closed with respect to either
$M[{G}]$ or $V[{G}]$,
$\FQ$ is ${\prec} \gk^+_2$-strategically closed
in both $M[{G}]$ and $V[{G}]$ as well.
We can now construct $G_1$ in either
$M[{G}]$ or $V[{G}]$ as follows.
Players I and II play a game of length
$\gk^+_2$. The initial pair of moves is
generated by player II choosing the
trivial condition $q_0$ and player
I responding by choosing
$q_1 \in D_1$.
Then, at an even stage $\ga + 2$,
player II picks
$q_{\ga + 2} \ge q_{\ga + 1}$ by
using some fixed strategy
${\cal S}$, where $q_{\ga + 1}$
was chosen by player I to be such that
$q_{\ga + 1} \in D_{\ga + 1}$ and
$q_{\ga + 1} \ge q_\ga$.
If $\ga$ is a limit ordinal, player II uses
${\cal S}$ to pick $q_\ga$ extending each
$q_\gb$ for $\gb < \ga$.
By the ${\prec} \gk^+_2$-strategic closure of
$\FQ$ in both $M[{G}]$ and $V[{G}]$,
the sequence
$\la q_\ga : \ga < \gk^+_2 \ra$
as just described exists.
By construction,
$G_1 = \{p \in \FQ : \exists \ga <
\gk^+_2 [q_\ga \ge p]\}$ is our
$N[{G}]$-generic object over $\FQ$.
It remains to construct in $V[{G}]$ the
desired $N[{G}][G_1]$-generic object
$G_2$ over $\FR$.
To do this,
we recall that by the forcing
equivalence in $M$ given in the first
paragraph of the proof of
this lemma of $k_1(\FP)$
with $\FP \ast \dot \FS \ast \dot \FT$,
the ordinals at which the
realization of $\dot \FT$
does a non-trivial forcing
lie in the interval
$(\gl, k_1({\gk_2}))$.
This implies that in $M$,
$\forces_{\FP \ast \dot \FS}
``\dot \FT$ is ${\prec} \gl^+$-strategically
closed''. Further,
since $\gl$ is a regular cardinal
and GCH holds in $V$ above $\gk_2$,
$|{[\gl]}^{< {\gk_2}}| = \gl$, and
$2^\gl = \gl^+$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter
%${\cal U}$
over $P_{\gk_2}(\gl)$,
$|k_1(\gk^+_2)| = |k_1(2^{\gk_2})| =
|2^{k_1({\gk_2})}| =
|\{ f : f : P_{\gk_2}(\gl) \to \gk^+_2$ is a function$\}| =
|{[\gk^+_2]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(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
$\dot \FT$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP \ast \dot \FS$ and
$\forces_{\FP \ast \dot \FS} ``\tau \in
\dot \FT$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP \ast \dot \FS} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT$ to be a set,
it is possible to restrict the terms
appearing in it to a sufficiently large
set-sized collection.
%with the additional
%crucial property that any term $\tau$
%forced to be in $\dot \FT$ is also forced
%to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M \models ``|\FT^*| = k_1({\gk_2})$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP \ast \dot \FS} ``\dot \FT$ is
${\prec}\gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V$ as well.
To show that
we may restrict the number of terms
so that
$M \models ``|\FT^*| = k_1({\gk_2})$'',
%we recall that in the official definition of $\FT^*$,
%the basic idea is to include only the canonical terms.
we observe that since
$\forces_{\FP \ast \dot \FS}
``|\dot \FT| = k_1({\gk_2})$'',
there is a set
$\{\tau_\alpha : \alpha 2^{\gk_2}$ was an arbitrary
%regular cardinal,
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
In $V^\FP$, $\gk_2$
isn't supercompact.
In fact, in $V^\FP$, $\gk_2$
has trivial Mitchell rank.
\end{lemma}
\begin{proof}
The proof uses an argument
due to Hamkins, which is
given in Lemma 2.4 of \cite{AH3}.
%Let $G$ be $V$-generic over $\FP$.
If $V[G]
\models ``{\gk_2}$ does not have trivial Mitchell
rank'', then let $j : V[G] \to M[j(G)]$ be an
embedding generated by a normal measure over
${\gk_2}$ in $V[G]$ witnessing this fact.
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \gk_1$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\gk^+_1$-strategically closed''.
$\FP$ thus admits a gap at $\gk_1$,
so by Theorem \ref{t2}
(the Gap Forcing Theorem of
\cite{H2} and \cite{H3}),
%(see also the generalization given in \cite{H5}),
$j$ must lift an embedding $j : V \to
M$ that is definable in $V$.
Also by the Gap Forcing Theorem applied
in $M$,
${\gk_2}$ is measurable in $M$,
and by Lemma 2.1 of \cite{AC2},
in $V$, $\gk_2$ is a limit
of strong cardinals.
Since ${\rm cp}(j) = \gk_2$, if
$V \models ``\gd < \gk_2$ is
a strong cardinal'',
$M \models ``j(\gd) = \gd$ is
a strong cardinal''.
Therefore, in $M$,
%since as in Lemma \ref{l2},
${\gk_2}$ is a measurable
cardinal which is a
limit of strong cardinals,
so it follows that there is a non-trivial
forcing done at stage ${\gk_2}$ in $M$.
This means
$j(G) = G \ast S \ast H$, where $S$ is a
Cohen subset of ${\gk_2}$ added
by forcing over $M[G]$ with
${{\rm Add}({\gk_2}, 1)}^{M[G]}$
at stage ${\gk_2}$ in $M[G]$,
and $H$ is
$M[G][S]$-generic for the rest of the forcing
$j(\FP)$. Since
$V_{{\gk_2}+1}^V \subseteq M \subseteq
V$, it follows that $V_{{\gk_2} + 1}^{V[G]} = V_{{\gk_2}
+ 1}^{M[G]}$. From this it follows that
${{\rm Add}({\gk_2}, 1)}^{M[G]} =
{{\rm Add}({\gk_2}, 1)}^{V[G]}$,
and the dense open subsets of
what we can now unambiguously write as
${\rm Add}({\gk_2}, 1)$ are the same in both $M[G]$ and
$V[G]$. Thus, the set $S$, which is an element of
$V[G]$, is $V[G]$-generic over ${\rm Add}({\gk_2}, 1)$,
a contradiction. This completes
the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk_2$-directed
closed and can be written in the form
${\rm Add}(\gk_2, 1) \ast \dot \FQ'$. Then
%adds a Cohen subset to $\gk_2$. Then
$V^{\FP \ast \dot \FQ} \models ``\gk_2$
is strongly compact''.
In fact,
$V^{\FP \ast \dot \FQ} \models ``\gk_2$
is supercompact''.
\end{lemma}
\begin{proof}
Suppose $\FQ \in V^\FP$
is such a partial ordering. Let
$\gl > \max(|{\rm TC}(\dot \FQ)|, 2^{\gk_2})$
be an arbitrary regular cardinal, and let
$\gg = |2^{[\gl]^{< \gk_2}}|$. Let
$j : V \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness
of $\gk_2$ such that
$M \models ``\gk_2$ isn't $\gg$ supercompact''.
As in Lemma \ref{l2},
$M \models ``$No cardinal
$\gd \in (\gk_2, \gg]$ is strong''.
Therefore, by the choice of $\gg$, it is
possible to opt for $\FQ$ in the stage
$\gk_2$ lottery held in $M$ in the
definition of $j(\FP)$.
In addition,
the next non-trivial
forcing in the definition of
$j(\FP)$ takes place well beyond $\gg$.
Thus, above the appropriate condition,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast
\dot \FR \ast j(\dot \FQ)$.
We therefore assume we are forcing
above such a condition.
We are now in a position to be able to use
the standard reverse Easton techniques to
show that
$V^{\FP \ast \dot \FQ} \models ``\gk_2$ is
$\gl$ supercompact''.
We argue in analogy to, e.g., the
proof found in the last paragraph
of page 679 of \cite{AH3}.
Specifically, let
${G}$ be $V$-generic over $\FP$ and
$G_1$ be $V[{G}]$-generic over $\FQ$.
The usual arguments show that
$M[{G}][G_1]$ remains $\gg$ closed with
respect to $V[{G}][G_1]$.
Thus, by the definitions of
$\FP$ and $\FR$,
since every ordinal in the
realization of $\dot \FR$
is above $\gg$,
$\FR$ is $\gg$-strategically closed in
$M[{G}][G_1]$ and hence by closure in
$V[{G}][G_1]$ as well.
Also, since we are dealing with
the usual sort of reverse
Easton iterations throughout,
if $G_2$ is $V[{G}][G_1]$-generic over $\FR$,
$M[{G}][G_1][G_2]$ remains $\gg$ closed with respect to
$V[{G}][G_1][G_2]$, and
$j$ lifts in $V[{G}][G_1][G_2]$ to
$j : V[{G}] \to M[{G}][G_1][G_2]$.
This means we can now by closure
and the definition of $j(\FQ)$ find in
$V[{G}][G_1][G_2]$ a master condition $q$ for
$j''G_1$ and let $G_3$ be a
$V[{G}][G_1][G_2]$-generic object over $j(\FQ)$
containing $q$.
Again since we are dealing with
reverse Easton iterations,
$j$ lifts in $V[{G}][G_1][G_2][G_3]$ to
$j : V[{G}][G_1] \to M[{G}][G_1][G_2][G_3]$, so
$V[{G}][G_1][G_2][G_3] \models ``\gk_2$ is
$\gl$ supercompact''.
As $G_2 \ast G_3$ is
$V[{G}][G_1]$-generic over $\FR \ast j(\dot \FQ)$,
a partial ordering which is $\gg$-strategically
closed in $V[{G}][G_1]$,
$V[{G}][G_1] \models ``\gk_2$ is $\gl$ supercompact''.
Since $\gl$ was arbitrary,
$V^{\FP \ast \dot \FQ} \models ``\gk_2$ is
supercompact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Theorem \ref{t1} now follows from
Lemmas \ref{l1} - \ref{l4}.
By Lemma \ref{l1}, there are no strongly
compact cardinals in $V^\FP$ in
the interval $(\gk_1, \gk_2)$
Therefore, by Lemma \ref{l2} and the
fact that $\FP$ is defined so as to be
$\gk_1$-directed closed, in $V^\FP$,
$\gk_1$ is both the least strongly
compact and least measurable cardinal
and retains the indestructibility of
its strong compactness under
$\gk_1$-directed closed forcing, and
$\gk_2$ is the second strongly
compact cardinal.
By Lemma \ref{l3}, $\gk_2$ isn't supercompact.
By Lemma \ref{l4}, $\gk_2$ remains strongly
compact after forcing with a
$\gk_2$-directed closed partial
ordering $\FQ$ that can be written in
the form
${\rm Add}(\gk_2, 1) \ast \dot \FQ'$,
and $\gk_2$ is clearly strongly
compact after doing a trivial forcing
over $V^\FP$.
This completes the proof
of Theorem \ref{t1}.
\end{proof}
We remark that in the proof of
Theorem \ref{t1} just given,
it is not necessary for
$\gk_1$ to be both the least
strongly compact and least
measurable cardinal.
By Theorem 2 of \cite{AG}, we
may assume, e.g., that our model $V$
has been constructed to be so that
$V \models ``\gk_1$ is both the
least strongly compact cardinal
and least cardinal $\gd$
which is $\gd^+$
supercompact + $\gk_1$'s strong
compactness and
$\gk^+_1$ supercompactness are
indestructible under
$\gk_1$-directed closed forcing +
GCH holds for all cardinals greater
than or equal to $\gk_1$ + $\gk_2$
is supercompact''.
We may then define the partial ordering
$\FP$ used in the proof of Theorem \ref{t1}
as before.
Since $\FP$ is defined so as to be
$\gk_1$-directed closed, in
$V^\FP$, $\gk_1$ remains as both
the least strongly compact and
least cardinal $\gd$ which is
$\gd^+$ supercompact and
retains the indestructibility of its
strong compactness and its
$\gk^+_1$ supercompactness under $\gk_1$-directed
closed forcing.
The remainder of Theorem \ref{t1}
then follows as earlier.
\section{Concluding Remarks}\label{s3}
In some sense, Theorem \ref{t1}
raises more questions than it answers.
We conclude this paper
by listing some of them.
In particular:
\begin{enumerate}
\item Is it possible to prove
Theorem \ref{t1} without the
requirement that indestructibility
for $\gk_2$ for non-trivial forcing
be under partial orderings which
begin by adding a Cohen subset of $\gk_2$?
(With the current proof of Theorem \ref{t1},
if we drop this requirement in the
definition of $\FP$, then
$\gk_2$ remains a supercompact
cardinal after forcing with $\FP$.
This is since the proof of Lemma \ref{l3}
becomes invalid without this requirement,
and the proof of Lemma \ref{l4}
becomes valid for trivial forcing.)
Note, however, that in practice, when
indestructibility is used in the
context of supercompactness,
beginning a construction by adding a
Cohen subset to a supercompact cardinal
is essentially harmless.
\item Is it possible to prove
Theorem \ref{t1} such that after
doing a non-trivial
$\gk_2$-directed closed
forcing, $\gk_2$'s supercompactness
isn't resurrected, i.e., such that after
doing a non-trivial
$\gk_2$-directed closed
forcing, $\gk_2$ remains a
non-supercompact
strongly compact cardinal?
In particular, is it possible
to prove a version of
Theorem \ref{t1} in which
$\gk_2$ is both the second
strongly compact and second
measurable cardinal?
\item In general, for $\ga > 2$,
is it consistent,
relative to anything, for the
first $\ga$ strongly compact
cardinals $\gk_\ga$
all to be non-supercompact and
to have their strong compactness
indestructible under (some version
of) $\gk_\ga$-directed closed forcing?
\end{enumerate}
Note that
as the proof of Theorem \ref{t1}
currently stands, the forcing $\FP$
can't be iterated, in the sense that
if we define a similar type of
partial ordering above $\gk_2$,
forcing with it resurrects the
supercompactness of $\gk_2$.
Thus, the techniques used in proving
Theorem \ref{t1} are inadequate for
answering the last of the above
questions.
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\end{thebibliography}
\end{document}
\begin{graveyard}
Although the introductory section of
\cte{A02} provides a good discussion
of much of the recent history of
this endeavor, for the convenience
of readers, we summarize now
the highlights of some of the work done
in this regard.
Suppose $V \models ``$ZFC +
$\gk_1$ and $\gk_2$ are supercompact''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk_1$ and
$\gk_2$ are the first two strongly
compact cardinals + $\gk_1$ and
$\gk_2$ aren't supercompact
yet satisfy significant indestructibility
properties for their strong
compactness''.
Specifically, in $V^\FP$,
$\gk_1$'s strong compactness
is indestructible under
arbitrary $\gk_1$-directed
closed forcing, and $\gk_2$'s
strong compactness is
indestructible under either
trivial forcing or $\gk_2$-directed
closed forcing that can be written
in the form
$\add(\gk_2, 1) \ast \dot \FQ'$.
Let $\gl > 2^{\gk_2}$
be any regular cardinal.
%of cofinality
%greater than $\gk_2$ at which GCH holds
%(such as an appropriate
%singular strong limit
%cardinal or a regular cardinal).
Let $j : V \to M$ be an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk_2$
such that $M \models ``\gk_2$
isn't $\gl$ supercompact''.
In $M$, $\gk_2$ is a
measurable limit of
strong cardinals, meaning
by the definition of $\FP$
it is possible to opt for
%the forcing adding a Cohen subset of $\gk_2$ in
${\rm Add}(\gk_2, 1)$ in
the stage $\gk_2$ lottery
held in $M$ in the
definition of $j(\FP)$.
Further, since
$M \models ``$No cardinal
$\gd \in (\gk_2, \gl]$ is
strong''
(otherwise, $\gk_2$
is in $M$ supercompact
up to a strong cardinal
and hence fully
supercompact), the next non-trivial
forcing in the
definition of $j(\FP)$ takes place
well above $\gl$.
Since $\gl$ has been chosen
large enough, we can let
$\mu \in M$ be a normal measure
over $\gk_2$ minimal in the
Mitchell ordering and
$k : M \to N$ be the
elementary embedding generated
by the ultrapower via $\mu$.
$i = k \circ j$ is therefore an
elementary embedding with
${\rm cp}(i) = \gk_2$ such that
$i : V \to N$ witnesses the
$\gl$ strong compactness of
$\gk_2$ and
$N \models ``\gk_2$ isn't a
measurable cardinal''.
Thus, only trivial forcing
takes place at stage $\gk_2$
in $N$ in the definition of
$i(\FP)$.
This means it is now possible to use
the standard reverse Easton arguments
to lift $j$ in $V^{\FP \ast \dot \FQ}$
to an elementary embedding
$j : V^{\FP \ast \dot \FQ} \to
M^{j(\FP \ast \dot \FQ)}$ witnessing
the $\gl$ supercompactness of $\gk_2$.
Since $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l4}.
At such a stage $\gd$, the
forcing done is the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$
which begin by
adding a Cohen subset to $\gd$,
followed by the partial ordering
which adds a non-reflecting
stationary set of ordinals of
cofinality $\gk_1$ to $\gd'$.
%the least $V$-strong cardinal above $\gd$.
\end{graveyard}