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\title{Universal Partial Indestructibility
and Strong Compactness
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, universal indestructibility,
Prikry forcing, Gitik iteration}}
\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.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{February 7, 2005\\
June 12, 2005}
\begin{document}
\maketitle
\begin{abstract}
For any ordinal $\gd$,
let $\gl_\gd$ be the
least inaccessible cardinal
above $\gd$.
We force and construct a
model in which the least
supercompact cardinal $\gk$
is indestructible under $\gk$-directed
closed forcing and in which every
measurable cardinal $\gd < \gk$
is ${<}\gl_\gd$ strongly compact
and has its ${<} \gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing
of rank less than $\gl_\gd$.
In this model, $\gk$ is also the
least strongly compact cardinal.
We also establish versions of this
result in which $\gk$ is the least
strongly compact cardinal but
isn't supercompact.
\end{abstract}
\baselineskip=24pt
%\newpage
\section{Introduction and Preliminaries}\label{s1}
In \cite{AH}, a model containing
one supercompact cardinal $\gk$
with no measurable cardinals
above $\gk$ was
constructed in which
{\it universal indestructibility for
supercompactness}
holds, i.e., in which every supercompact and
partially supercompact (including measurable)
cardinal $\gd$
has its degree of supercompactness
fully Laver indestructible
\cite{L} under $\gd$-directed
closed forcing.
(To be more explicit,
this just means that $\gk$ is
{\it indestructibly supercompact},
i.e., $\gk$ is supercompact and remains
supercompact after arbitrary
$\gk$-directed closed forcing,
and if $\gd$ is $\gl$ supercompact,
$\gd$ remains $\gl$ supercompact
after arbitrary $\gd$-directed
closed forcing.)
A version of this result was also
established for strong compactness,
in the sense that a model containing
one strongly compact cardinal $\gk$
with no measurable cardinals above $\gk$
was constructed in which every strongly
compact and partially strongly compact
(including measurable) cardinal $\gd$
has its degree of strong compactness
fully indestructible
under $\gd$-directed closed forcing.
In addition, a partial version of the
first of the above results
(called {\it universal partial
indestructibility for supercompactness})
was also established,
in the sense that a model containing
one supercompact cardinal $\gk$
with no measurable cardinals above $\gk$
was constructed in which
$\gk$ is indestructibly supercompact and
any cardinal $\gd < \gk$
which is $\gl$ supercompact
has its $\gl$ supercompactness indestructible
under $\gd$-directed closed forcing
having size at most $\gl$.
For the case $\gd = \gl$, i.e.,
when $\gd$ is measurable,
this just means that $\gd$'s
measurability is indestructible
under $\gd$-directed closed
partial orderings having size $\gd$,
i.e., only under the forcing for
adding a Cohen subset of $\gd$.
In this paper, we will focus
on obtaining generalized versions
of the last of the results mentioned
in the preceding paragraph.
%which incorporate strong compactness into the mixture.
We establish examples of
universal partial indestructibility
in which strong compactness is
incorporated into the mixture,
by demonstrating that there
can be non-trivial instances of indestructibility
for non-trivial degrees of
strong compactness. In particular,
we show that it is possible to construct
a model in which the least strongly
compact cardinal $\gk$ is also the
least supercompact cardinal and in which
not only is $\gk$'s supercompactness
fully indestructible, but every
measurable cardinal $\gd < \gk$ exhibits
a significant degree of strong compactness
which is indestructible under partial
orderings of rank less than $\gl_\gd$.
We also give versions of this result
when the least strongly compact
cardinal isn't supercompact.
Specifically, we prove
three theorems.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + GCH + $\gk$ is
$\gl = \gl_\gk$ supercompact''.
There is then a partial ordering
$\FP \in V$ such that for
$V^*$ defined as $V^\FP$ truncated at $\gl$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$,
$\gk$ is both the
least supercompact and least strongly
compact cardinal, and
$\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing. In addition,
in $V^*$, for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
\end{theorem}
\begin{theorem}\label{t2}
Let
$V \models ``$ZFC + GCH +
There is a cardinal $\gk$ which is
${<} \gl = \gl_\gk$ supercompact and
is a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact''.
There is then a partial ordering
$\FP \in V$ such that for
$V^*$ defined as $V^\FP$ truncated at $\gl$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$, $\gk$ is
the least strongly compact cardinal, and
$\gk$ isn't $2^\gk$ supercompact but is
a limit of measurable cardinals.
In addition, in $V^*$,
$\gk$'s strong compactness
is indestructible under $\gk$-directed
closed forcing, and for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
\end{theorem}
It is possible to obtain stronger
versions of Theorem \ref{t2} in which
$\gk$ is (at least) $\gk^+$ supercompact.
%exhibits a non-trivial degree of supercompactness.
A prototypical result
that can be established is as follows.
\begin{theorem}\label{t3}
Let
$V \models ``$ZFC + GCH + $\gk$ is
$\gl = \gl_\gk$ supercompact''.
There is then a partial ordering
$\FP \in V$ such that for
$V^*$ defined as $V^\FP$ truncated at $\gl$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$, $\gk$ is
the least strongly compact cardinal, and
$\gk$ is $\gk^{+}$ supercompact but isn't
$2^{\gk^{+}} = \gk^{++}$ supercompact.
In addition, in $V^*$,
both $\gk$'s strong compactness and
$\gk^{+}$ supercompactness
are indestructible under $\gk$-directed
closed forcing, and for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
\end{theorem}
Magidor's work of \cite{Ma} shows that
the least strongly compact cardinal
can be the least measurable cardinal,
and the result of \cite{A97}
(see also \cite{AC1}) indicates that
the least strongly compact cardinal
$\gk$ can be both $\gk^+$ supercompact
and the least measurable cardinal.
Therefore, to avoid trivialities,
we will need in Theorems \ref{t2} and
\ref{t3} that our strongly compact
cardinal $\gk$ is a limit of
measurable cardinals.
For both of these theorems, our
forcing construction will explicitly
guarantee this.
In particular,
for each of Theorems \ref{t1} - \ref{t3}, our
partial orderings will be defined
so as to have cardinality $\gk$,
which means since every ground
model satisfies GCH that GCH
will hold in the generic
extension for all cardinals
greater than or equal to $\gk$.
Consequently, by reflection, the
model constructed witnessing the
conclusions of Theorem \ref{t3}
will be such that $\gk$ is
a limit of measurable cardinals.
We would like to take this
opportunity to point out that
there is a sharp contrast
between the amount of
indestructibility obtained for
an arbitrary measurable cardinal
in each of the above three
theorems as compared with
the amount of indestructibility
for an arbitrary measurable cardinal
found in the universal partial
indestructibility result
(Theorem 16) of \cite{AH}.
As we indicated above, in \cite{AH},
if $\gd$ is an arbitrary measurable
cardinal, then we only know that
$\gd$'s measurability is indestructible
under Cohen forcing.
Here, in each of the above three theorems,
we know that $\gd$'s measurability
(along with its ${<} \gl_\gd$ strong
compactness) is indestructible under
{\it any} $\gd$-directed closed
partial ordering having rank
less than $\gl_\gd$.
We digress briefly to provide
some preliminary information.
Essentially, our notation and terminology
are standard, although exceptions
to this will be noted.
For $\ga < \gb$ ordinals,
$[\ga, \gb]$, $[\ga, \gb)$,
$(\ga, \gb]$, and $(\ga, \gb)$
are as in the usual interval notation.
When forcing, $q \ge p$ means that
$q$ {\it is stronger than} $p$.
For $\gk$ a regular cardinal,
the partial ordering $\FP$ is
{\it $\gk$-directed closed} if
every compatible set of conditions
of size less than $\gk$ has a
common extension.
For $\gl \ge \gk$, we write
$\add(\gk, \gl)$ for the usual
partial ordering for adding
$\gl$ Cohen subsets of $\gk$.
We abuse notation somewhat and
use both $V^\FP$ and $V[G]$
to denote the generic extension
by the partial ordering $\FP$.
Throughout the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik style iterations of
Prikry-like forcings.}
By this we will mean an Easton
support iteration
as first given by Gitik in \cite{G}
(and elaborated upon further in
\cite{G2}), where the ordering,
roughly speaking, is the
usual one associated with
Easton support iterations, except that
when extending Prikry conditions,
we take larger stems only finitely often.
For a more precise definition, we urge
readers to consult either \cite{G},
\cite{G2}, or \cite{AG}.
By Lemmas 1.2 and 1.3 of
\cite{G} and \cite{G2}
respectively and Lemma 1.4 of \cite{G},
if $\gd_0$ is the first stage in the
definition of $\FP$ at which a non-trivial
forcing is done, then forcing with
$\FP$ adds no bounded subsets to $\gd_0$,
assuming the forcing done at
any non-trivial stage $\gd$
is $\gd$-directed closed forcing
followed by either Prikry forcing over $\gd$
or trivial forcing.
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''.}
Finally, we mention that we are assuming
familiarity with the large cardinal
notions of measurability,
strong compactness, and supercompactness.
Interested readers may consult \cite{K}
%or \cite{SRK}
for further details.
We note only that $\gk$ is
{\it ${<} \gl$ strongly compact} (or
{\it ${<} \gl$ supercompact})
if $\gk$ is $\gd$ strongly compact
(or $\gd$ supercompact) for every
cardinal $\gd < \gl$.
\section{The Proofs of Theorems
\ref{t1} - \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + GCH + $\gk$ is
$\gl = \gl_\gk$ supercompact''.
%+ There is some supercompact ultrafilter
%${\cal U}$ over $P_\gk(\gl)$ such that
%for $j : V \to M$ the associated
%elementary embedding,
%$M \models ``\gk$ is $\gl$ supercompact''.
The partial ordering $\FP$
used in the proof of Theorem \ref{t1}
is a Gitik style iteration of
Prikry-like forcings of length $\gk$,
$\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$,
which does a non-trivial forcing
only at those ordinals $\gd < \gk$ which are
$V$-measurable cardinals.
At such a stage $\gd$,
%if $V \models ``\gd$ isn't
%${<} \gl_\gd$ supercompact'',
we first force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank less than
$\gl_\gd$ which are
$\gd$-directed closed
in $V^{\FP_\gd}$.
If $V \models ``\gd$ isn't ${<} \gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If the preceding conjunction of
conditions doesn't hold,
i.e., if either $V \models ``\gd$
is ${<} \gl_\gd$ supercompact''
or $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ isn't measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is a
${<} \gl$ supercompact
cardinal whose ${<} \gl$
supercompactness is
indestructible under forcing with
partial orderings having rank less than $\gl$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\FQ$ is a $\gk$-directed
closed partial ordering having rank
less than $\gl$''.
We show that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is ${<} \gl$ supercompact''.
To do this, we combine arguments
found in the proofs of Lemmas 2
and 3 of \cite{AG}.
For any missing details, readers
are urged to consult
%the aforementioned lemmas of
\cite{AG}.
Let $\gd > \max(\gk, \card{{\rm TC}(\dot \FQ)})$,
$\gd < \gl$
be an arbitrary $V$-cardinal large
enough so that
${(2^{[\gd]^{< \gk}})}^V = \rho =
{(2^{[\gd]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$.
%and $\rho$ is regular in both
%$V$ and $V^{\FP \ast \dot \FQ}$.
%and let $\gs = \rho^+$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness
of $\gk$.
%such that $M \models ``\gk$ isn't $\gs$ supercompact''.
By the definition of $\FP$,
$\FQ$ is an allowable choice in the
stage $\gk$ lottery held in
$M^{\FP_\gk} = M^\FP$
in the definition of $j(\FP)$.
In addition, $M \models ``\gk$ is
${<} \gl$ supercompact''.
Consequently, $j(\FP \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS$ is a term for trivial forcing.
Further, since
$M \models ``$There are no measurable
cardinals in the interval
$(\gk, \gl]$'', the next non-trivial
stage in the definition of
$j(\FP)$ after $\gk$ takes place
well above $\gl$.
Hence, 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 $\dot \FQ$.
Let $K$ be $V[G][H]$-generic over
$\FS \ast \dot \FR \ast j(\dot \FQ)$.
Define an embedding
$j^* : V[G][H] \to M[G][H][K]$ by
$j^*(i_{G \ast H}(\ov \tau)) =
i_{G \ast H \ast K}(j(\ov \tau))$ for any term
$\ov \tau$ denoting a set in $V[G][H]$.
Since the closure properties of $M$ imply any term for a
condition in $j(\dot \FQ)$ can be assumed to extend the
``master condition'' $\tau$ above,
as in Lemma 3 of \cite{AG}, $j^*$ is a
well-defined elementary embedding lifting $j$ which can
be used to define a supercompact ultrafilter
${\cal U} \in V[G][H][K]$ over
${(P_\gk(\gd))}^{V[G][H]}$ by
$X \in {\cal U}$ iff $\la j(\ga) : \ga < \gd \ra
\in j^*(X)$.
Since $\FP$ is $\gk$-c.c$.$,
the usual arguments show that
$M[G][H]$ remains $\gr$ closed
with respect to $V[G][H]$. Hence,
since forcing with
$\FS \ast \dot \FR \ast j(\dot \FQ)$
over either $M[G][H]$ or $V[G][H]$
adds no subsets of $\gr$,
${\cal U} \in V[G][H]$, i.e.,
$V[G][H] \models ``\gk$ is $\gd$ supercompact''.
Since $\gd < \gl$ was an arbitrarily
large enough $V$-cardinal,
and since trivial forcing is
$\gk$-directed closed,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ was ${<} \gl_\gd$ supercompact in $V$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gk$ is measurable''.
Write
$\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$.
Since by the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ adds
no subsets of $\gl_\gd$'', it must be the case that
$\forces_{\FP_{\gd + 1}} ``\gd$ is measurable''.
Note now that
$V \models ``\gd$ is measurable''.
For, if this were not the case, then
again by the definition of $\FP$,
since $\dot \FQ_\gd$ is a term
for trivial forcing, it must
be true that
$\forces_{\FP_\gd} ``\gd$ is measurable''.
As $\FP$ is a Gitik style iteration
of Prikry-like forcings, $\FP_\gd$
is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
This means that since $\FP_\gd$
satisfies $\gd$-c.c$.$ in
$V^{\FP_\gd}$ (this follows because
$\gd$ is measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$ is
%isomorphic to
a subordering of the Easton support product of
$\la \FQ_\ga : \ga < \gd \ra$ as
calculated in $V^{\FP_\gd}$),
the proof of Lemma 3 of \cite{AC1}
tells us that every $\gd$-additive
ultrafilter over $\gd$ present in
$V^{\FP_\gd}$ must in fact be
present in $V$. This contradiction
to our supposition that
$V \models ``\gd$ isn't measurable''
consequently yields that
$V \models ``\gd$ is measurable''.
However, if it isn't the case that
$\gd$ is ${<} \gl_\gd$
supercompact in $V$, then by the
definition of $\FP$,
$V^{\FP_{\gd + 1}} \models ``\gd$
%contains a Prikry sequence and hence
isn't measurable''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$''.
\end{lemma}
\begin{proof}
By Lemma \ref{l2}, we know that any
measurable cardinal $\gd < \gl_\gd$
had to have been ${<} \gl_\gd$
supercompact in $V$.
As in Lemma \ref{l2},
by the definition of $\FP$,
write
$\FP = \FP_{\gd + 1} \ast
\dot \FP^{\gd + 1}$, where
$\forces_{\FP_{\gd + 1}} ``$Forcing
with $\dot \FP^{\gd + 1}$
doesn't add any subsets of $\gl_\gd$''.
Thus, to prove Lemma \ref{l3}, it
suffices to show that
its conclusions hold in $V^{\FP_{\gd + 1}}$.
Towards this end, let
$\FQ \in V^{\FP_{\gd + 1}}$ be such that
$V^{\FP_{\gd + 1}} \models
``\FQ$ is $\gd$-directed closed and
has rank less than $\gl_\gd$''.
By the definition of $\FP$, since
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact'',
$\FP_{\gd + 1} \ast \dot \FQ$ is forcing
equivalent to $\FP_\gd \ast \dot \FQ'
\ast \dot \FS_\gd \ast \dot \FQ$, where $\dot \FQ'$ is
a term for the partial ordering of rank
less than $\gl_\gd$ selected in the
stage $\gd$ lottery held in the
definition of $\FP$, and
$\dot \FS_\gd$ is a term for trivial forcing.
Since $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$
can be taken to be a term in the forcing language
with respect to $\FP_\gd$ for a
$\gd$-directed closed partial
ordering having rank less than $\gl_\gd$,
we abuse notation in what follows and
assume without loss of generality that
what we will write as $\dot \FQ$ is
actually $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$.
We proceed now in analogy to the
argument given in the second
paragraph of the proof of Lemma \ref{l1}.
Specifically,
let $\gg > \max(\gd, \card{{\rm TC}(\dot \FQ)})$,
$\gg < \gl_\gd$
be an arbitrary $V$-regular cardinal large
enough so that
${(2^{[\gg]^{< \gd}})}^V = \rho =
{(2^{[\gg]^{< \gd}})}^{V^{\FP_\gd \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP_\gd \ast \dot \FQ}$,
and let $\gs = \rho^+$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gd$.
%such that $M \models ``\gd$ isn't $\gs$ supercompact''.
By the definition of $\FP_\gd$,
$\FQ$ is an allowable choice in the
stage $\gd$ lottery held in $M^{\FP_\gd}$
in the definition of $j(\FP_\gd)$.
Consequently, $j(\FP_\gd \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd ' \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS_\gd '$ is either
a term for trivial forcing or for
Prikry forcing over $\gd$ defined
using the appropriate normal measure.
Further, since
$M \models ``$There are no inaccessible
cardinals in the interval
$(\gd, \gs]$'', as before, the next non-trivial
stage in the definition of
$j(\FP_\gd)$ after $\gd$ takes place
well above $\gs$.
Hence, as in Lemma \ref{l1} and
Lemma 2 of \cite{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP_\gd)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP_\gd \ast \dot \FQ$,
$\forces_{j(\FP_\gd)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is
once again a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga : \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP_\gd \ast \dot \FQ$ names of subsets of
${(P_\gd(\gg))}^{V[G \ast H]}$,
we can as
%we did earlier in the proof of Lemma \ref{l1}
is done in the proof of Lemma 2 of \cite{AG}
define in $M$ a sequence of
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) : \gb < \gg \ra \in
j(\dot A_\ga)$'$\ $'', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb : \gb < \eta \ra$''.
If we then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gg]^{< \gd}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gd(\gg)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FS_\gd '$ either
the trivial condition (if $\FS_\gd '$
is trivial forcing) or of the form
$\la \emptyset, B \ra$ (if $\FS_\gd '$ is
Prikry forcing),
in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) : \gb < \gg \ra \in
j(\dot X)$'' for some name $\dot X$ of $X$,
then as in Lemma 2 of \cite{AG},
${\cal U}$ is a $\gd$-additive, fine
ultrafilter over
${(P_\gd(\gg))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gd$ is
$\gg$ strongly compact''.
Since $\gg$ was arbitrary,
and since trivial forcing is
${<} \gl_\gd$-directed closed
and can be defined so as to
have rank less than $\gl_\gd$, this
completes the proof of Lemma \ref{l3}.
\end{proof}
We explicitly note that Lemma \ref{l3}
actually tells us that in $V^\FP$, any cardinal
$\gd < \gk$ which was ${<}\gl_\gd$ strongly
compact in $V$ is ${<}\gl_\gd$
strongly compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$ (so in particular
is measurable in $V^\FP$).
This observation will be used in the
proof of Theorem \ref{t2}.
\begin{lemma}\label{l4}
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
We argue in analogy to the proof
of Lemma 4 of \cite{AG}.
Let $\gd > \gk^{++}$ be any
sufficiently large cardinal, e.g.,
the least strong limit cardinal
above $\gk$.
Take $j : V \to M$ as an elementary
embedding witnessing the $\gd$
supercompactness of $\gk$ such that
$M \models ``\gk$ isn't $\gd$ supercompact'',
which means since $\gd < \gl$ that
$M \models ``\gk$ isn't
${<} \gl_\gk$ supercompact''.
By the choice of $\gd$, it is possible to
opt for $\add(\gk, \gk^{++})$ at stage
$\gk$ in $M^\FP =
M^{\FP_\gk}$ in the definition of $j(\FP)$.
Further, by Lemma \ref{l1} and the fact
$M^\gd \subseteq M$,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}
\models ``\gk$ is measurable''.
By the definition of $\FP$, this
therefore means that
above the appropriate condition,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR$,
where $\dot \FS_\gk$ is a term for
Prikry forcing over $\gk$
defined with respect to the
appropriate normal measure, and
$\forces_{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk} ``$Forcing with $\dot \FR$ does not
add any subsets to $\gk$''.
Consequently,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR} =
M^{j(\FP)} \models ``\gk$ is a singular
strong limit cardinal violating GCH''.
By reflection, this just means that
$V^\FP \models ``$There are unboundedly
in $\gk$ many singular strong limit
cardinals below $\gk$ violating GCH''.
By a theorem of Solovay \cite{So},
we may infer that
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Let now $V^*$ be
the model $V^\FP$ truncated at $\gl$.
By Lemmas \ref{l1} - \ref{l4},
their proofs, and the fact that
$\FP$ may be defined so as
to have cardinality $\gk$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$,
$\gk$ is both the
least supercompact and least strongly
compact cardinal, and
$\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing. In addition,
in $V^*$, for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
This completes the proof of
Theorem \ref{t1}.
\end{proof}
\begin{pf}
Theorem \ref{t2}
follows as a fairly
easy corollary to the
proof of Theorem \ref{t1}.
To prove Theorem \ref{t2},
let
$V \models ``$ZFC + GCH + $\gk$ is
${<} \gl = \gl_\gk$ supercompact and
is a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact''.
Without loss of generality, we
assume that $\gk$ is the least
such cardinal.
We then force with the same partial
ordering $\FP$ used in the proof
of Theorem \ref{t1}, defined
through stage $\gk$.
Since again $\FP$ may be defined so
as to have cardinality $\gk$,
as above, we
take $V^*$ to be the model $V^\FP$
truncated at $\gl$.
Note that
$V \models ``$There are unboundedly
many in $\gk$ cardinals $\gg$
which are ${<} \gl_\gg$ supercompact''.
Therefore,
the proof of Lemma \ref{l3}
(which goes through exactly as before),
together with the remark found
after the proof of Lemma \ref{l3},
show that
$V^\FP \models ``$There are unboundedly
many in $\gk$ measurable cardinals''.
The proofs of Lemma \ref{l2} and
Lemma \ref{l4} also go through in exactly
the same way as earlier, which, in
tandem with (the appropriate analogue
of) Lemma \ref{l3},
allow us to infer that
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$ +
No cardinal $\gd < \gk$ is
strongly compact''.
Also, if $\FQ \in V^\FP$ is
$\gk$-directed closed and
has rank less than $\gl$, then
we may use the same argument
as given in the third paragraph
of the proof of Lemma \ref{l3}
to infer that
$V^\FP \models ``\gk$
is ${<} \gl_\gk$ strongly
compact and has its ${<} \gl_\gk$ strong
compactness indestructible under
$\gk$-directed closed forcing having
rank less than $\gl_\gk$''.
Finally, to see that
$V^\FP \models ``\gk$ isn't $2^\gk$
supercompact'', we observe that
$V^\FP \models ``$No measurable
cardinal $\gd < \gk$ is a limit
of measurable cardinals''.
This is since
$V \models ``\gk$ is the least
cardinal which is ${<} \gl_\gk$
supercompact and is the
limit of cardinals $\gg$ which
are ${<} \gl_\gg$ supercompact'',
so (the appropriate analogues of)
Lemmas \ref{l2} and \ref{l3}
immediately yield that
$V^\FP \models ``\gk$ is the
least measurable limit of
measurable cardinals''.
This completes the proof of Theorem \ref{t2}.
\end{pf}
Having finished the proof of
Theorem \ref{t2},
we turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Let
$V \models ``$ZFC + GCH + $\gk$ is
$\gl = \gl_\gk$ supercompact''.
The partial ordering $\FP$
used in the proof of Theorem \ref{t3}
is a modification of the one used
in the proof of Theorem \ref{t1}.
Once again, $\FP$
is a Gitik style iteration of
Prikry-like forcings of length $\gk$,
$\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$,
which does a non-trivial forcing
only at those ordinals $\gd < \gk$ which are
$V$-measurable cardinals.
At such a stage $\gd$,
we first force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank less than
$\gl_\gd$ which are
$\gd$-directed closed
in $V^{\FP_\gd}$.
If $V \models ``\gd$ isn't ${<} \gl_\gd$ supercompact''
and $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If $V \models ``\gd$ is ${<} \gl_\gd$ supercompact''
and $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is $\gd^{+}$ supercompact'', then again,
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If the preceding two cases don't hold,
i.e., if either
$V \models ``\gd$
is ${<} \gl_\gd$ supercompact''
and $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ isn't $\gd^{+}$ supercompact''
or $V \models ``\gd$ isn't
${<}\gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ isn't measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is a
${<} \gl$ strongly compact
cardinal whose ${<} \gl$
strong compactness is
indestructible under forcing with
partial orderings having rank less than $\gl$''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\FQ$
is $\gk$-directed closed and
has rank less than $\gl$''.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$.
As in the proof of Lemma \ref{l1},
$\FQ$ is an allowable choice in
the stage $\gk$ lottery held in
$M^\FP$ in the definition of
$j(\FP)$.
We may therefore now apply the
argument given in the third
paragraph of the proof of
Lemma \ref{l3} to infer that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is ${<} \gl$ strongly compact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l5a}
$V^\FP \models ``\gk$ is a
$\gk^{+}$ supercompact
cardinal whose $\gk^{+}$
supercompactness is
indestructible under forcing with
partial orderings having rank less than $\gl$''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\FQ$
is $\gk$-directed closed and
has rank less than $\gl$''.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$.
As above,
$\FQ$ is an allowable choice in
the stage $\gk$ lottery held in
$M^\FP$ in the definition of
$j(\FP)$.
If $V^{\FP \ast \dot \FQ} \models ``\gk$ is
$\gk^{+}$ supercompact'',
then we're done, so we assume
that this is not the case.
By the closure properties of $M$,
it is then true that
$M^{\FP \ast \dot \FQ} \models ``\gk$ is not
$\gk^{+}$ supercompact''.
Consequently, by the definition of
$\FP$, $j(\FP)$ is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS
\ast \dot \FR \ast j(\dot \FQ)$,
where $\dot \FS$ is a term for
trivial forcing.
We may therefore apply the argument
given in the second and third paragraphs of
the proof of Lemma \ref{l1}
to show that
$V^{\FP \ast \dot \FQ} \models ``\gk$ is
$\gk^{+}$ supercompact''.
This contradiction completes
the proof of Lemma \ref{l5a}.
\end{proof}
We note that since $\FP$
may be defined so as to
have cardinality $\gk$,
$V^\FP \models ``$GCH holds
for all cardinals greater than
or equal to $\gk$''.
Thus, Lemma \ref{l5a} and
reflection together imply that
$\{\gd < \gk : \gd$ is measurable$\}$
%and $2^\gd = \gd^{+}\}$ is unbounded in $\gk$.
%This further implies that in $V^\FP$,
%$\{\gd < \gk : \gd$ is measurable
%but $\gd$ isn't $\gd^+$ supercompact$\}$
is unbounded in $\gk$ in $V^\FP$.
\begin{lemma}\label{l6}
If $V \models ``\gd < \gk$ is
${<} \gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd}
\models ``\gd$ isn't
$\gd^+$ supercompact'', then
$V^\FP \models ``\gd$
is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness
%and degree of supercompactness
indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$''.
\end{lemma}
\begin{proof}
As in the proofs of Lemmas
\ref{l2} and \ref{l3},
we may write
$\FP = \FP_{\gd + 1} \ast
\dot \FP^{\gd + 1}$, where
$\forces_{\FP_{\gd + 1}} ``$Forcing
with $\dot \FP^{\gd + 1}$
doesn't add any subsets of $\gl_\gd$''.
Thus, to prove Lemma \ref{l6}, it
suffices to show that its conclusions
hold in $V^{\FP_{\gd + 1}}$.
Towards this end, let
$\FQ \in V^{\FP_{\gd + 1}}$ be such that
$V^{\FP_{\gd + 1}} \models
``\FQ$ is $\gd$-directed closed and
has rank less than $\gl_\gd$''.
Since $V \models ``\gd < \gk$ is
${<} \gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd}
\models ``\gd$ isn't $\gd^+$
supercompact'',
by the definition of $\FP$,
$\FP_{\gd + 1} \ast \dot \FQ$
is forcing equivalent to $\FP_\gd \ast \dot \FQ'
\ast \dot \FS_\gd \ast \dot \FQ$, where $\dot \FQ'$ is
a term for the partial ordering of rank
less than $\gl_\gd$ selected in the
stage $\gd$ lottery held in the
definition of $\FP$, and
$\dot \FS_\gd$ is a term for trivial forcing.
The remainder of the argument given
in the proof of Lemma \ref{l3}
now applies to show that
$V^{\FP_{\gd + 1} \ast \dot \FQ} \models
``\gd$ is ${<} \gl_\gd$ strongly compact''.
%In addition, if
%$V^\FP \models ``\gd$ is $\gd^+$
%supercompact'',
%the argument given in the last two
%paragraphs of the proof of Lemma \ref{l1}
%may be used to show that
%$V^{\FP_{\gd + 1} \ast \dot \FQ} \models
%``\gd$ is $\gd^+$ supercompact''.
Since $\FQ$ was arbitrary,
this completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l6a}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is $\gd^{+}$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$V^\FP \models ``\gd < \gk$ is
$\gd^{+}$ supercompact''.
By the factorization of $\FP$
given in the first paragraph of
the proof of Lemma \ref{l6},
it must therefore also be
the case that
$V^{\FP_{\gd + 1}} \models ``\gd$
is $\gd^{+}$ supercompact''.
This means by the definition of
$\FP$ that $\FP_{\gd + 1}$ is
forcing equivalent to
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd$,
where $\dot \FQ$ is a term for
the partial ordering of rank less than
$\gd$ selected in the stage $\gd$
lottery held in the definition of
$\FP$, and $\dot \FS_\gd$ is a term
for trivial forcing.
This further implies that
$V^{\FP_\gd \ast \dot \FQ} \models
``\gd$ is $\gd^{+}$ supercompact'',
which means by the definition of $\FP$
that $\dot \FS_\gd$ must actually be a
term for Prikry forcing defined
with respect to the appropriate
normal measure over $\gd$.
This contradiction completes the
proof of Lemma \ref{l6a}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness
%and degree of supercompactness
indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$''.
\end{lemma}
\begin{proof}
By replacing the phrase
``${<} \gl_\gd$ supercompact in $V$'' with
``${<} \gl_\gd$ supercompact in $V$ and isn't
$\gd^+$ supercompact in
$V^{\FP_\gd \ast \dot \FQ^*_\gd}$'',
the same proof as given in Lemma \ref{l2}
then applies to show that
any measurable cardinal $\gd < \gk$
in $V^\FP$ had to have been in $V$
${<} \gl_\gd$ supercompact but not
$\gd^+$ supercompact in
$V^{\FP_\gd \ast \dot \FQ^*_\gd}$.
Lemma \ref{l7} then immediately
follows from Lemma \ref{l6}.
This completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
$V^\FP \models ``\gk$ isn't
$2^{\gk^{+}} = \gk^{++}$ supercompact''.
\end{lemma}
\begin{proof}
As we have already mentioned,
$V^\FP \models ``$GCH holds
for all cardinals greater than
or equal to $\gk$''.
Thus, if
$V^\FP \models ``\gk$ is
$2^{\gk^{+}} = \gk^{++}$ supercompact'',
then by reflection,
$\{\gd < \gk : \gd$ is $\gd^+$
supercompact$\}$ is unbounded in $\gk$.
This, however, contradicts Lemma \ref{l6a}.
This completes the proof of Lemma \ref{l8}.
\end{proof}
By Lemma \ref{l8},
$V^\FP \models ``\gk$
isn't $2^{\gk^{+}} =
\gk^{++}$ supercompact''.
%and by Lemmas \ref{l6} and \ref{l7},
%$V^\FP \models ``\gk$ is a limit of measurable cardinals''.
Further,
by replacing a reference to Lemma \ref{l1}
with a reference to Lemma \ref{l5},
the proof of Lemma \ref{l4}
goes through as before
to show that
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
Consequently, we may now infer
as we did earlier, using
Lemmas \ref{l5} - \ref{l8},
their proofs, and the
intervening remarks, that for
$V^*$ the model $V^\FP$
truncated at $\gl$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$,
$\gk$ is the least strongly compact cardinal, and
$\gk$ is $\gk^{+}$ supercompact.
%but isn't $2^{\gk^{+}} = \gk^{++}$ supercompact.
In addition, in $V^*$,
both $\gk$'s strong compactness and
$\gk^{+}$ supercompactness
are indestructible under $\gk$-directed
closed forcing, and for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
This completes the proof of
Theorem \ref{t3}.
\end{proof}
We note that it is possible
to establish versions of
Theorem \ref{t3} in which
$\gk$ is the least strongly compact cardinal,
every measurable cardinal
$\gd < \gk$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$,
$\gk$ is $\gg$ supercompact
for $\gg = \gk^{++}$,
$\gg = \gk^{+++}$, etc.,
$\gk$ isn't $2^\gg$ supercompact,
yet both $\gk$'s strong compactness
and degree of supercompactness
are indestructible under $\gk$-directed
closed forcing.
To do this, we change in the
definition of the partial ordering
$\FP$ for Theorem \ref{t3} the clause
``$\gd$ is $\gd^+$ supercompact'' to
``$\gd$ is $\gd^{++}$ supercompact'',
``$\gd$ is $\gd^{+++}$ supercompact'', etc.
We leave additional details of the construction
and proof to the readers of this paper.
Also, since we require in both
Theorems \ref{t1} and \ref{t3} that
$\gk$ exhibit indestructibility
for at least $\gk^+$ supercompactness
under arbitrary $\gk$-directed closed
partial orderings (such as the
L\'evy collapse of any cardinal
$\gg \in (\gk, \gl)$ to $\gk$),
it is natural that the proofs of
these theorems use stronger hypotheses
than the proof of Theorem \ref{t2}.
We observe that
because of our methods of proof,
there are limitations on the
conclusions obtained in each
of our theorems.
For instance, the
degrees of strong compactness
of all measurable cardinals
below our strongly compact or
supercompact cardinal $\gk$
are restricted.
Is this really necessary?
In particular, is it possible
to have, in Theorems \ref{t1} -
\ref{t3}, each measurable
cardinal $\gd < \gk$ be
fully strongly compact
and have its strong compactness
indestructible under
$\gd$-directed closed forcing
(of any rank)?
Is it possible to have some
version of Theorems \ref{t1} - \ref{t3}
in which there is more than one
strongly compact cardinal,
subject to the restriction given
by Theorem 10 of \cite{AH} that
if (full) universal indestructibility
holds and $\gk$ is supercompact, then
no cardinal $\gd > \gk$ is measurable?
As the proof of Lemma \ref{l4}
indicates, if the answer to any
of the preceding questions posed in
this paragraph is to be yes,
then we will need a
different method of proof.
Also, it is unclear if, in
Theorem \ref{t1},
the degrees of supercompactness
for every measurable cardinal
$\gd < \gk$ are indestructible
under $\gd$-directed closed
forcing having rank less
than $\gl_\gd$.\footnote{This is true in
Theorem \ref{t2}, since the forcing
used ensures that for each
measurable cardinal $\gd < \gk$,
$2^\gd = \gd^+$ in $V^\FP$. This means,
since $\gk$ is the least measurable
limit of measurable cardinals in
$V^\FP$, that no measurable cardinal
$\gd < \gk$ in $V^\FP$ is
$2^\gd = \gd^+$ supercompact.
It is also true in Theorem \ref{t3},
since by Lemma \ref{l6a},
$V^\FP \models ``$No cardinal
$\gd < \gk$ is $\gd^+$ supercompact''.}
We therefore conclude by asking if
in either Theorem \ref{t1} or
the generalized versions of
Theorem \ref{t3} mentioned above,
where there will be cardinals less
than $\gk$ exhibiting non-trivial
degrees of supercompactness,
this is indeed
possible, or if these degrees of
supercompactness can be indestructible
under partial orderings having larger rank.
\begin{thebibliography}{99}
\bibitem{A97} A.~Apter,
``More on the Least Strongly Compact Cardinal'',
{\it Mathematical Logic Quarterly 43}, 1997, 427--430.
\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{AH} A.~Apter, J.~D.~Hamkins,
``Universal Indestructibility'',
{\it Kobe Journal of Mathematics 16}, 1999, 119--130.
%\bibitem{C} J.~Cummings, handwritten notes.
\bibitem{G} M.~Gitik, ``Changing Cofinalities
and the Nonstationary Ideal'',
{\it Israel Journal of Mathematics 56},
1986, 280--314.
\bibitem{G2} M.~Gitik, ``On Closed Unbounded
Sets Consisting of Former Regulars'',
{\it Journal of Symbolic Logic 64}, 1999, 1--12.
%\bibitem{GS} M.~Gitik, S.~Shelah, ``On
%Certain Indestructibility of Strong
%Cardinals and a Question of Hajnal'',
%{\it Archive for Mathematical Logic 28},
%1989, 35--42.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
{\it Annals of Pure and Applied Logic 101},
2000, 103--146.
%\bibitem{J} T.~Jech, {\it Set Theory},
%Academic Press, New York and San
%Francisco, 1978.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite}, Springer-Verlag,
Berlin and New York, 1994.
%\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{Ma2} M.~Magidor, ``On the Role of Supercompact
%and Extendible Cardinals in Logic'', {\it Israel Journal
%of Mathematics 10}, 1971, 147--157.
\bibitem{So} R.~Solovay, ``Strongly Compact Cardinals
and the GCH'', in: {\it Proceedings of the Tarski
Symposium}, {\bf Proceedings of Symposia in Pure
Mathematics 25}, American Mathematical Society,
Providence, 1974, 365--372.
%\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}