\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{amssymb}
%\usepackage{diagram}
\newcommand{\ga}{\alpha}
\newcommand{\gb}{\beta}
\renewcommand{\gg}{\gamma}
\newcommand{\gd}{\delta}
\newcommand{\gep}{\epsilon}
\newcommand{\gz}{\zeta}
\newcommand{\gee}{\eta}
\newcommand{\gth}{\theta}
\newcommand{\gi}{\iota}
\newcommand{\gk}{\kappa}
\newcommand{\gl}{\lambda}
\newcommand{\gm}{\mu}
\newcommand{\gn}{\nu}
\newcommand{\gx}{\xi}
\newcommand{\gom}{\omicron}
\newcommand{\gp}{\pi}
\newcommand{\gr}{\rho}
\newcommand{\gs}{\sigma}
\newcommand{\gt}{\tau}
\newcommand{\gu}{\upsilon}
\newcommand{\gph}{\phi}
\newcommand{\gch}{\chi}
\newcommand{\gps}{\psi}
\newcommand{\go}{\omega}
\newcommand{\gA}{A}
\newcommand{\gB}{B}
\newcommand{\gG}{\Gamma}
\newcommand{\gD}{\Delta}
\newcommand{\gEp}{E}
\newcommand{\gZ}{Z}
\newcommand{\gEe}{H}
\newcommand{\gTh}{\Theta}
\newcommand{\gI}{I}
\newcommand{\gK}{K}
\newcommand{\gL}{\Lambda}
\newcommand{\gM}{M}
\newcommand{\gN}{N}
\newcommand{\gX}{\Xi}
\newcommand{\gOm}{O}
\newcommand{\gP}{\Pi}
\newcommand{\gR}{P}
\newcommand{\gS}{\Sigma}
\newcommand{\gT}{T}
\newcommand{\gU}{\Upsilon}
\newcommand{\gPh}{\Phi}
\newcommand{\gCh}{X}
\newcommand{\gPs}{\Psi}
\newcommand{\gO}{\Omega}
\newcommand{\bA}{{\bf A}}
\newcommand{\bB}{{\bf B}}
\newcommand{\bG}{\boldGamma}
\newcommand{\bD}{\boldDelta}
\newcommand{\bEp}{{\bf E}}
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bEe}{{\bf H}}
\newcommand{\bTh}{\boldTheta}
\newcommand{\bI}{{\bf I}}
\newcommand{\bK}{{\bf K}}
\newcommand{\bL}{{\bf L}}
\newcommand{\bM}{{\bf M}}
\newcommand{\bN}{{\bf N}}
\newcommand{\bX}{\boldXi}
\newcommand{\bOm}{{\bf O}}
\newcommand{\bP}{\boldPi}
\newcommand{\bR}{{\bf P}}
\newcommand{\bS}{\boldSigma}
\newcommand{\bT}{{\bf T}}
\newcommand{\bU}{\boldUpsilon}
\newcommand{\bPh}{\boldPhi}
\newcommand{\bCh}{{\bf X}}
\newcommand{\bPs}{\boldPsi}
\newcommand{\bO}{\boldOmega}
\newcommand{\rest}{\restriction}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\ov}{\overline}
\newcommand{\add}{{\rm Add}}
\newcommand{\K}{{\mathfrak K}}
%
% Hebrew letters
%
\newcommand{\ha}{\aleph}
\newcommand{\hb}{\beth}
\newcommand{\hg}{\gimel}
\newcommand{\hd}{\daleth}
%
% basic set theory constructions
%
\newcommand{\setof}[2]{{\{\; #1 \; \vert \; #2 \; \} } }
\newcommand{\seq}[1]{{\langle #1 \rangle} }
\newcommand{\card}[1]{{\vert #1 \vert} }
\newcommand{\ot}[1]{\hbox{o.t.($#1$)}}
\newcommand{\forces}{\Vdash}
\newcommand{\decides}{\parallel}
\newcommand{\ndecides}{\nparallel}
\renewcommand{\models}{\vDash}
\newcommand{\powerset}{{\cal P}}
\newcommand{\bool}{{\bf b} }%
%
% stuff for use inside math formulae
%
\newcommand{\dom}{{\rm dom}}
\newcommand{\rge}{{\rm rge}}
\newcommand{\crit}{{\rm crit}}
\renewcommand{\top}{{\rm top}}
\newcommand{\supp}{{\rm supp}}
\newcommand{\support}{{\rm support}}
\newcommand{\cf}{{\rm cf}}
\newcommand{\lh}{{\rm lh}}
\newcommand{\lp}{{\rm lp}}
\newcommand{\up}{{\rm up}}
\newcommand{\FP}{{\mathbb P}}
\newcommand{\FQ}{{\mathbb Q}}
\newcommand{\FR}{{\mathbb R}}
\newcommand{\FS}{{\mathbb S}}
\newcommand{\FT}{{\mathbb T}}
\newcommand{\implies}{\Longrightarrow}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}[definition]{Remark}
\newtheorem{fact}[definition]{Fact}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{claim}[definition]{Claim}
\newtheorem{conjecture}{Conjecture}
\newenvironment{proof}{\noindent{\bf
Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
\par\bigskip}
\newenvironment{sketch}{\noindent{\bf
Sketch of Proof:}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newenvironment{pf}{\indent{${}$}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newcommand{\lra}{\longrightarrow}
\newcommand{\commtriangle}[6]
{
\medskip
\[
\setlength{\dgARROWLENGTH}{6.0em}
\begin{diagram}
\node{#1} \arrow[2]{e,t}{#6} \arrow{se,b}{#4} \node[2]{#3} \\
\node[2]{#2} \arrow{ne,r}{#5}
\end{diagram}
\]
\medskip
}
%
% This picture tells you what order to put the arguments in
%
%
%
%
% #6
% #1 --------- #3
% \ /
% \#4 / #5
% \ /
% #2/
%
\setlength{\topmargin}{-0.62in}
\setlength{\textheight}{9.10in}
\setlength{\oddsidemargin}{-0.15in}
\setlength{\textwidth}{6.95in}
\setlength{\parindent}{1.5em}
\setcounter{section}{+1}
% IndWC.tex
% The following macros are a selection from Joel's general math
% macros used in the document below
%
\def\tlt{\triangleleft}
\def\k{\kappa}
\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\s{\sigma}
\def\t{\tau}
\def\l{\lambda}
\def\lted{{{\leq}\d}}
\def\ltk{{{<}\k}}
%
% Arthur, in the next two macro definitions, use blackboard bold for
% \bm. Latex uses a different name I think.
%
\def\P{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Qdot{\dot\Q}
\def\Pforces{\forces_{\P}}
\def\of{{\subseteq}}
%\def\card#1{\left|#1\right|}
\def\boolval#1{\mathopen{\lbrack\!\lbrack}\,#1\,\mathclose{\rbrack\!
\rbrack}}
\def\restrict{\mathbin{\mathchoice{\hbox{\am\char'26}}{\hbox{\am\char'
26}}{\hbox{\eightam\char'26}}{\hbox{\sixam\char'26}}}}
\def\st{\mid}
\def\set#1{\{\,{#1}\,\}}
\def\th{{\hbox{\fiverm th}}}
\def\muchgt{>>}
\def\cof{\mathop{\rm cof}\nolimits}
\def\iff{\mathrel{\leftrightarrow}}
\def\intersect{\cap}
\def\minus{\setminus}
\def\Union{\bigcup}
\def\union{\cup}
\def\and{\mathrel{\kern1pt\&\kern1pt}}
\def\image{\mathbin{\hbox{\tt\char'42}}}
\def\elesub{\prec}
\def\iso{\cong}
\def\<#1>{\langle\,#1\,\rangle}
\def\ot{\mathop{\rm ot}\nolimits}
%
% ------------------------------------------------------------------------------
%
\title{Universal Indestructibility is Consistent with
Two Strongly Compact Cardinals
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, indestructibility, universal indestructibility.}}
\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{June 3, 2005\\
(revised June 18, 2005)}
\begin{document}
\maketitle
\begin{abstract}
We show that universal indestructibility
for both strong compactness and supercompactness
is consistent
with the existence of two strongly
compact cardinals.
This is in contrast to the fact that if $\gk$
is supercompact and universal indestructibility
for either strong compactness or
supercompactness holds, then no cardinal
$\gl > \gk$ is measurable.
\end{abstract}
\baselineskip=24pt
%\newpage
%\section{Introduction and Preliminaries}\label{s1}
In \cite{AH},
the concepts of {\it universal indestructibility}
for both strong compactness and
supercompactness were introduced.
Specifically, we say that
{\it universal indestructibility for
supercompactness}
holds in a model $V$ for ZFC if
every $V$-supercompact and
partially supercompact (including measurable)
cardinal $\gd$
has its degree of supercompactness
fully Laver indestructible
\cite{L} under $\gd$-directed
closed forcing. Analogously,
{\it universal indestructibility for
strong compactness}
holds in a model $V$ for ZFC if
every $V$-strongly compact and
partially strongly compact (including measurable)
cardinal $\gd$
has its degree of strong compactness
fully indestructible under $\gd$-directed
closed forcing.
%The notion of universal indestructibility
%for strongness was also introduced in \cite{AH}.
%Analogously, universal indestructibility for
%strongness holds in a model $V$ for ZFC if
%every $V$-strong and
%partially strong (including measurable) cardinal $\gd$
%has its degree of strong compactness
%fully indestructible under $\gd$-directed
Readers are urged to consult \cite{AH}
for further details.
One of the key results of \cite{AH}
is Theorem 10, which states that
if there are two supercompact cardinals,
then universal indestructibility fails
for partial supercompactness
(and, as can be inferred from its
proof, for partial strong
compactness as well).
In particular, if $\gk$ is $\gl^+$
supercompact where $\gl > \gk$
is measurable, then universal
indestructibility fails for
both partial supercompactness
and partial strong compactness.
%Further, Theorem 11 of \cite{AH}
%states that if $\gk$ is
This is because under these
circumstances, it must be the case that
there are unboundedly many in $\gk$
cardinals $\gd < \gk$ whose
measurability can be destroyed
by adding a Cohen subset of $\gd$.
The proof, however, is heavily dependent
on the fact that
supercompactness embeddings are
into highly closed inner models.
Since strongly compact cardinals
don't necessarily possess
such embeddings,
this leads to the following
\bigskip\setlength{\parindent}{0pt}
Question: Is universal indestructibility
for either strong compactness or
supercompactness consistent with the
existence of more than one
strongly compact cardinal?
\bigskip\setlength{\parindent}{1.5em}
The purpose of this note is to show
that the answer to the above Question
is yes. Specifically, we prove the
following theorem.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + GCH + There is
a high-jump cardinal''.
There is then a
%transitive
model of ZFC containing
two strongly compact cardinals
in which universal indestructibility
for supercompactness holds.
In addition, there is a
model of ZFC containing two
strongly compact cardinals
in which universal
indestructibility for strong
compactness holds.
\end{theorem}
We note that the assumption
of GCH is made for convenience
and ease of presentation.
At the end of this note,
we will indicate how to
prove Theorem \ref{t1}
under arbitrary circumstances.
We recall from \cite{AH}
that the cardinal $\gk$ is a
{\it high-jump cardinal} if
there is an elementary embedding
$j : V \to M$ having critical
point $\gk$ such that for
some $\theta$ we have that
$M^\theta \subseteq M$ and
$j(f)(\gk) < \theta$ for every
function $f : \gk \to \gk$.
As Lemma 2 of \cite{AH} indicates,
if $\gk$ is almost huge, then
$\gk$ is the $\gk^{\rm th}$
high-jump cardinal.
Further, Lemma 3 of \cite{AH}
tells us that if $\gk$ is a
high-jump cardinal, then
$V_\gk \models ``$ZFC + There is a
proper class of supercompact cardinals''.
Thus, in terms of consistency strength,
the property of being a high-jump
cardinal lies strictly in between
supercompactness and almost hugeness.
\begin{pf}
To prove Theorem \ref{t1},
we combine the methods of
\cite{AH} with the techniques
of \cite{AG}.
We begin, however, with the
following lemma, which shows that
the L\'evy-Solovay results \cite{LS}
of preservation of large cardinal
properties under small forcing
are true for high-jump cardinals.
\begin{lemma}\label{l1}
Suppose
$V \models ``$ZFC + $\gk$ is
a high-jump cardinal +
$\FP$ is a partial ordering
such that $\card{\FP} < \gk$''.
Then $V^\FP \models ``\gk$ is
a high-jump cardinal''.
\end{lemma}
\begin{proof}
Let $j : V \to M$ and $\theta$
%be such that $j$
witness that in $V$,
$\gk$ is a high-jump cardinal.
By standard arguments
(see, e.g., the proof of
the Main Theorem of \cite{HW}),
since $\card{\FP} < \gk$,
in $V^\FP$, $j$ lifts to
$j^* : V^\FP \to M^{j(\FP)}$.
Also, $M^{j(\FP)}$ remains
$\theta$ closed with respect
to $V^\FP$.
Thus, the proof of Lemma \ref{l1}
will be complete once we have
shown that $j^*$ and $\theta$ continue to
witness that $\gk$ is a high-jump
cardinal in $V^\FP$.
To do this,
let $p \in \FP$ and $\dot f$
be such that
$p \forces ``\dot f : \gk \to \gk$
is a function''.
Define in $V$ a function
$g : \gk \to \gk$ by
$g(\ga) = \sup(\{\gb < \gk \mid$
For some $q$ extending $p$,
$q \forces ``\dot f(\ga) = \gb$''$\})$.
Since $\card{\FP} < \gk$ and
$\gk$ is regular,
$g$ is well-defined.
It is then the case that
$p \forces ``$For every $\ga < \gk$,
$\dot f(\ga) \le g(\ga)$'', from which
it can be immediately inferred that
$M^{j(\FP)} \models ``j^*(f)(\gk)
\le j^*(g)(\gk) = j(g)(\gk) < \theta$''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
Given Lemma \ref{l1}, it now becomes
possible to prove Theorem \ref{t1}. Suppose
$V \models ``$ZFC + GCH + $\gk$ is a
high-jump cardinal''.
By our earlier remarks, let
$\gl$ be the least cardinal such that
$V_\gk \models ``$ZFC + $\gl$ is supercompact''.
Working in $V_\gk$, let $\FP$ be
the partial ordering of Theorem 1
of \cite{AG}, defined with respect
to $\gl$. Since $V_\gk \models {\rm GCH}$,
the arguments of \cite{AG} show that
$V^\FP_\gk \models ``$ZFC + $\gl$ is both the
least strongly compact and least
measurable cardinal + $\gl$'s strong
compactness is indestructible under
$\gl$-directed closed forcing''.
Since $\FP$ may be defined so that
$\card{\FP} = \gl < \gk$,
standard arguments show that
GCH holds at $\gl$ after
forcing with $\FP$. Further,
by Lemma \ref{l1},
$V^\FP \models ``\gk$ is a high-jump cardinal''.
Working now in $V^\FP$, let
$\FQ$ be the partial ordering
of either Theorem 5 or Theorem 6 of \cite{AH},
%This partial ordering forces universal
%indestructibility for supercompactness.
with the
first non-trivial stage of forcing
taking place at or above the least
$V^\FP$-weakly compact cardinal $\gs$ above $\gl$.
Let $\gg < \gk$ be the stage at which
the construction of $\FQ$ terminates,
and let $\gd < \gk$ be the least weakly compact
cardinal above $\gg$ in
$V^{\FP \ast \dot \FQ}$.
By the arguments of \cite{AH},
$V^{\FP \ast \dot \FQ}_\gd \models ``$ZFC +
Universal indestructibility for supercompactness
holds for every measurable cardinal
above $\gl$ + $\gg$ is the
least supercompact cardinal''.
Since the definition of $\FQ$
from \cite{AH} ensures that
$\FQ$ is $\gs$-directed closed in both
$V^\FP$ and $V^\FP_\gd$, we may
infer using $\gl < \gs < \gg < \gd < \gk$ that
$V^{\FP \ast \dot \FQ}_\gd \models ``$GCH
holds at $\gl$ + $\gl$ is both the
least strongly compact and least
measurable cardinal + $\gl$'s strong
compactness is indestructible under
$\gl$-directed closed forcing''.
However, since
$V^{\FP \ast \dot \FQ}_\gd \models
``2^\gl = \gl^+$ + $\gl$ is
the least measurable cardinal'',
we may immediately infer that
$V^{\FP \ast \dot \FQ}_\gd \models ``\gl$
isn't $2^\gl = \gl^+$ supercompact''.
Thus, since $\gl$ is an indestructible
strongly compact cardinal in
$V^{\FP \ast \dot \FQ}_\gd$,
$V^{\FP \ast \dot \FQ}_\gd \models ``\gl$'s
degree of supercompactness (namely
measurability) is indestructible''.
Hence, $V^{\FP \ast \dot \FQ}_\gd$ is
a model of ZFC in which universal indestructibility
for supercompactness holds and there are
two strongly compact cardinals
(namely $\gl$ and $\gg$).
Finally, if we change the definition
of $\FQ$ to be the partial ordering
of Theorem 7 of \cite{AH} but keep
the meanings of $\gl$, $\gs$, $\gg$, and
$\gd$ as before, then by the
arguments of \cite{AH},
$V^{\FP \ast \dot \FQ}_\gd \models ``$ZFC +
Universal indestructibility for strong compactness
holds for every measurable cardinal
above $\gl$ + $\gg$ is the
least strongly compact cardinal
above $\gl$''.
Since once again the definition of
$\FQ$ ensures that $\FQ$ is
$\gs$-directed closed in both
$V^\FP$ and $V^\FP_\gd$,
$V^{\FP \ast \dot \FQ}_\gd \models
``\gl$ is an indestructible
strongly compact cardinal''.
Consequently, $V^{\FP \ast \dot \FQ}_\gd$ is
a model of ZFC in which universal indestructibility
for strong compactness holds and there are
two strongly compact cardinals
(namely $\gl$ and $\gg$).
This completes the proof of Theorem \ref{t1}.
\end{pf}
As was mentioned earlier,
it is possible to prove
Theorem \ref{t1} without
the additional assumption
of GCH. To see how this is done,
note that the forcing $\FP$
given above is a Gitik style
iteration of Prikry-like forcings
as described in \cite{G1} and \cite{G2}.
Such iterations are possible
regardless of any GCH assumptions
in the ground model.
We therefore must show that
the arguments given in Theorem 1
of \cite{AG}, which were presented
using GCH, are possible when GCH
doesn't necessarily hold.
That this can be accomplished is found by
a close examination of the reasoning
done in \cite{AG}.
Theorem 1 of \cite{AG} is proven
via Lemmas 1 and 2 of that paper,
and the proof of Lemma 1 does not
require GCH.
As can be verified by examining its proof,
any use of GCH in Lemma 2 of \cite{AG}
may be replaced by choosing initially
a large enough singular strong limit cardinal
of sufficiently high cofinality satisfying
GCH,
which is possible by Solovay's theorem of
\cite{So}.
Thus, if we force with the partial ordering
$\FP$ of Theorem \ref{t1} over a model
in which GCH isn't necessarily true,
we are still able to verify that the
cardinal $\gl$ of Theorem \ref{t1}
is indestructibly strongly compact
and the least measurable
cardinal in $V^\FP_\gk$.
If we then force GCH at $\gl$ (if
it doesn't already hold) by adding a Cohen
subset of $\gl^+$, $\gl$ remains
indestructibly strongly compact
and the least measurable cardinal.
The remainder of the proof of Theorem
\ref{t1} then goes through as before.
Let us note that the partial
ordering $\FP$ for Theorem \ref{t1}
does not force universal indestructibility
for Ramseyness and weak
compactness\footnote{{\it Universal
indestructibility for Ramseyness}
holds in a model $V$ for ZFC if
every $V$-Ramsey cardinal $\gd$ has its
Ramseyness fully indestructible under
$\gd$-directed closed forcing.
Similarly, {\it universal
indestructibility for weak compactness}
holds in a model $V$ for ZFC if
every $V$-weakly compact cardinal $\gd$ has its
weak compactness fully indestructible under
$\gd$-directed closed forcing.},
as do the partial orderings of
Theorems 6 and 7 of \cite{AH}.
Thus, we can ask if universal indestructibility
for Ramseyness and weak compactness,
together with universal indestructibility
for either supercompactness or strong
compactness, is consistent with the
existence of two or more
strongly compact cardinals.
We remark that since the
forcing of Theorem 1 of \cite{AG}
adds bona fide Prikry sequences,
by Theorem 11.1(1) of \cite{CFM},
this forcing adds non-reflecting
stationary sets of ordinals
of cofinality $\go$. By
Theorem 4.8 of \cite{SRK} and
the succeeding remarks,
such a set of ordinals can't
exist above a strongly compact cardinal.
Thus, the forcing $\FP$ of
Theorem \ref{t1} of this paper
can't be iterated in order to
obtain a version of Theorem \ref{t1}
in which there are more than
two strongly compact cardinals.
Since the methods of \cite{AH}
by themselves do not allow for
the construction of a model for
universal indestructibility for
either strong compactness or
supercompactness containing
more than one strongly compact cardinal,
we conclude by asking
if it is possible to have models for
universal indestructibility for
either strong compactness or
supercompactness containing
more than two strongly compact cardinals.
\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{CFM} J.~Cummings, M.~Foreman,
M.~Magidor, ``Squares, Scales,
and Stationary Reflection'',
{\it Journal of Mathematical Logic 1},
2001, 35--98.
\bibitem{G1} 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{HW} J.~D.~Hamkins, W.~H.~Woodin,
``Small Forcing Creates neither Strong nor
Woodin Cardinals'', {\it Proceedings
of the American Mathematical Society 128},
2000, 3025--3029.
%\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{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
``Strong Axioms of Infinity and Elementary Embeddings'',
{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\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.
\end{thebibliography}
\end{document}
\begin{bibgr}
\bibitem{Alot} A.~Apter, ``Some Remarks
on Indestructiblity and Hamkins'
Lottery Preparation'',
{\it Archive for Mathematical Logic 42},
2003, 717--735.
\bibitem{A01} A.~Apter, ``Strong Compactness,
Measurability, and the Class of Supercompact
Cardinals'', {\it Fundamenta Mathematicae 167},
2001, 65--78.
\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness II: Strong Cardinals'',
{\it Archive for Mathematical Logic 40}, 2001, 25--38.
\bibitem{AH2} A.~Apter, J.~D.~Hamkins,
``Indestructible Weakly Compact Cardinals
and the Necessity of Supercompactness
for Certain Proof Schemata'',
{\it Mathematical Logic Quarterly 47}, 2001, 563--571.
\bibitem{Bu} J.~Burgess, ``Forcing'', in:
J.~Barwise, editor, {\it Handbook of Mathematical
Logic}, North-Holland, Amsterdam, 1977, 403--452.
\bibitem{A02} A.~Apter,
``Aspects of Strong Compactness,
Measurability, and Indestructibility'',
{\it Archive for Mathematical Logic 41},
2002, 705--719.
\bibitem{A98} A.~Apter,
``Laver Indestructibility and the
Class of Compact Cardinals''
{\it Journal of Symbolic Logic 63},
1998, 149--157.
\bibitem{AH3} A.~Apter, J.~D.~Hamkins,
``Exactly Controlling the Non-Supercompact
Strongly Compact Cardinals'',
{\it Journal of Symbolic Logic 68}, 2003, 669--688.
\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{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{H5} J.~D.~Hamkins, ``A Generalization
of the Gap Forcing Theorem'',
submitted for publication to
{\it Fundamenta Mathematicae}.
%\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{HW} J.~D.~Hamkins, W.~H.~Woodin,
``Small Forcing Creates neither Strong nor
Woodin Cardinals'',
{\it Proceedings of the American Mathematical
Society 128}, 2000, 3025--3029.
%\bibitem{MS} D.A.~Martin, J.~Steel, ``A Proof of Projective
%Determinacy'', {\it Journal of the American Mathematical
%Society 2}, 1989, 71--125.
%\bibitem{MekS} A.~Mekler, S.~Shelah,
%``Does $\gk$-Free Imply Strongly $\gk$-Free?'', in:
%{\it Proceedings of the Third Conference on Abelian
%Group Theory}, Gordon and Breach, Salzburg, 1987, 137--148.
\end{bibgr}
\begin{graveyard}
The proof is heavily dependent
on the fact that a supercompact
cardinal reflects properties
that hold above it.
Since strongly compact cardinals
don't necessarily manifest
reflection, however,
this leads to the following
\end{graveyard}