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\title{A Note on Indestructibility and
Strong Compactness
% Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact, strongly
compact,
indestructibility,
non-reflecting stationary set of ordinals.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grants
and CUNY
Collaborative Incentive Grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{February 20, 2008}
\begin{document}
\maketitle
\begin{abstract}
If $\gk < \gl$
are such that $\gk$ is
both supercompact and
%suitably indestructible
indestructible under $\gk$-directed
closed forcing which is also
$(\gk^+, \infty)$-distributive
and $\gl$ is $2^\gl$
supercompact, then by
\cite[Theorem 5]{AH4},
$\{\gd < \gk \mid \gd$
is $\gd^+$ strongly compact
yet $\gd$ isn't $\gd^+$
supercompact$\}$
must be unbounded in $\gk$.
We show that the large cardinal
hypothesis on $\gl$ is necessary by
constructing a model containing a
supercompact cardinal $\gk$ in which
no cardinal $\gd > \gk$ is $2^\gd = \gd^+$
supercompact, $\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing which is also
$(\gk^+, \infty)$-distributive, and
for every measurable cardinal $\gd$,
$\gd$ is $\gd^+$ strongly compact iff
$\gd$ is $\gd^+$ supercompact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{AH4}, it was shown
(see Theorem 5) that
if $\gk < \gl$ are such that
$\gk$ is indestructibly supercompact and
$\gl$ is $2^\gl$ supercompact, then
$\{\gd < \gk \mid \gd$
is $\gd^+$ strongly compact
yet $\gd$ isn't $\gd^+$
supercompact$\}$
must be unbounded in $\gk$.
The only use of indestructibility
in this proof is that
$\gk$ remains supercompact
after forcing with the partial
ordering which first (if necessary) makes
$2^\gl = \gl^+$ and $2^{\gl^+} = \gl^{++}$
and then does
a reverse Easton iteration
of length $\gl$ which adds a
non-reflecting stationary set
of ordinals of cofinality $\gk$
to each measurable cardinal in a final segment of
the open interval $(\gk, \gl)$.
Thus, we actually have the following result.
\begin{theorem}\label{t1}
Suppose $\gk^+ \le \gg < \gl$ are such that
$\gk$ is supercompact, $\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing which is also $(\gg, \infty)$-distributive,
and $\gl$ is $2^\gl$ supercompact.
Then
$A = \{\gd < \gk \mid \gd$
is $\gd^+$ strongly compact
yet $\gd$ isn't $\gd^+$ supercompact$\}$
is unbounded in $\gk$.
\end{theorem}
The purpose of this note is to show
that the large cardinal hypothesis
on $\gl$ in Theorem \ref{t1} is necessary.
Specifically, we prove the following theorem.
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is
$2^\gd = \gd^+$ supercompact + For every cardinal
$\gd$, $\gd$ is $\gd^+$ strongly compact iff
$\gd$ is $\gd^+$ supercompact''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$
is $2^\gd = \gd^+$ supercompact''. In
$V^\FP$, $\gk$'s supercompactness is
indestructible under $\gk$-directed closed
forcing which is also $(\gk^+, \infty)$-distributive.
Further, in $V^\FP$, $\gd$ is $\gd^+$ strongly
compact iff $\gd$ is $\gd^+$ supercompact.
\end{theorem}
The existence of models $V$
satisfying the hypotheses of Theorem \ref{t2}
(and much more) was first shown in \cite{AS97a}.
By a result of Menas \cite{Me},
$V \models ``$No cardinal $\gd < \gk$ is both
measurable and a limit of cardinals $\gg$
which are either $\gd^+$ strongly compact
or $\gd^+$ supercompact'', since if $\gd$
is the least such cardinal, then
$V \models ``\gd$ is $\gd^+$ strongly
compact but not $\gd^+$ supercompact''.
Hence, there must of necessity be some
restrictions on the large cardinal
structure of $V$ below $\gk$.
We conclude Section \ref{s1}
with a very brief discussion of
some preliminary material.
We presume a basic knowledge
of large cardinals and forcing.
A good reference in this
regard is \cite{J}.
We also mention that the partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D$ of
conditions of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
$\FP$ is {\em $(\gk, \infty)$-distributive}
if the intersection of $\gk$ many
dense-open subsets of $\FP$ is dense open.
It therefore follows that forcing with
any partial ordering $\FP$
which is both
$\gk$-directed closed and $(\gk^+, \infty)$-distributive
preserves either the $\gk^+$ strong compactness or
$\gk^+$ supercompactness of $\gk$,
since forcing with $\FP$ preserves $P_\gk(\gk^+)$.
We abuse notation slightly and
take $V^\FP$ as being the generic
extension of $V$ by $\FP$.
An {\em indestructibly supercompact
cardinal} is one as first given
by Laver in \cite{L}, i.e.,
$\gk$ is indestructibly supercompact
if $\gk$'s supercompactness is
preserved in any generic extension
via a $\gk$-directed closed
partial ordering.
For $\gd$ any ordinal, $\gd'$ is
the least cardinal $\gg > \gd$ such that
$V \models ``\gg$ is $\gg^+$ supercompact''.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorem \ref{t2}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ is {\em mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$
that admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is
$2^\gd = \gd^+$ supercompact + For every cardinal
$\gd$, $\gd$ is $\gd^+$ strongly compact iff
$\gd$ is $\gd^+$ supercompact''.
Let $f$ be a Laver function
\cite{L} for $\gk$, i.e.,
$f : \gk \to V_\gk$ is such that
for every $x \in V$ and every
$\gl \ge \card{{\rm TC}(x)}$, there is
an elementary embedding
$j : V \to M$ generated by a
supercompact ultrafilter over
$P_\gk(\gl)$ such that
$j(f)(\gk) = x$.
The partial ordering $\FP$
which is used to establish
Theorem \ref{t2} is the
reverse Easton iteration of
length $\gk$ which begins by
adding a Cohen subset of
$\go$ and then (possibly) does nontrivial
forcing only at those
cardinals $\gd < \gk$ which
are at least $\gd^+$ supercompact in $V$.
At such a stage $\gd$, if
$f(\gd) = \dot \FQ$ and
$\forces_{\FP_\gd} ``\dot \FQ$
is a $\gd$-directed
closed, $(\gd^+, \infty)$-distributive partial
ordering having rank below $\gd'$'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$.
If this is not the case, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$,
where $\dot \FQ$ is a term for trivial forcing.
%we perform trivial forcing.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing which is also
$(\gk^+, \infty)$-distributive''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A06}.
Let $\FQ \in V^\FP$ be such that
$V^\FP \models ``\FQ$ is
$\gk$-directed closed and
$(\gk^+, \infty)$-distributive''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{\FP} ``\dot \FQ$ is
$\gk$-directed closed and
$(\gk^+, \infty)$-distributive''.
Suppose $\gl \ge
\max(\gk^{++}, \card{{\rm TC}(\dot \FQ)})$
is an arbitrary cardinal, and let
$\gg = 2^{\card{[\gl]^{< \gk}}}$. Take
$j : V \to M$ as an elementary
embedding witnessing the $\gg$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gg)$ such that
$j(f)(\gk) = \dot \FQ$. Since
$V \models ``$No cardinal $\gd$ above
$\gk$ is $2^\gd = \gd^+$ supercompact'', $\gg \ge
2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
$M \models ``\gk$ is $2^\gk = \gk^+$ supercompact and
no cardinal $\gd$ in the half-open interval
$(\gk, \gg]$ is $2^\gd = \gd^+$ supercompact''. Hence,
the definition of $\FP$ implies that
$j(\FP \ast \dot \FQ) = \FP \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first stage at which
$\dot \FR$ is forced to do nontrivial forcing
is well above $\gg$.
%$2^{[\gg]^{< \gk}}$.
Laver's original argument from \cite{L} now applies
and shows
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be $V$-generic over
$\FP \ast \dot \FQ \ast \dot \FR$,
lift $j$ in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
$V[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ)$ containing $p$, lift $j$
again in $V[G_0][G_1][G_2][G_3]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gg^+$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gl))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0][G_1]$.)
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of
Lemma \ref{l1}.
\end{proof}
Since trivial forcing is both
$\gk$-directed closed and $(\gk^+, \infty)$-distributive,
Lemma \ref{l1} implies that
$V^\FP \models ``\gk$ is supercompact''.
Also, because $\FP$ may be defined so that
$\card{\FP} = \gk$, standard arguments in
tandem with the results of \cite{LS} show that
$V^\FP \models ``$No cardinal $\gd > \gk$ is either
$2^\gd = \gd^+$ strongly compact or supercompact''.
\begin{lemma}\label{l2}
If $V \models ``\gd$ is $\gd^+$ supercompact'', then
$V^\FP \models ``\gd$ is $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose
$V \models ``\gd$ is $\gd^+$ supercompact''.
As $V \models ``$No cardinal $\gd > \gk$
is $2^\gd = \gd^+$ supercompact'' and
$V^\FP \models ``\gk$ is supercompact'',
we may assume that $\gd < \gk$.
Write $\FP = \FP_\gd \ast \dot \FP^\gd$. Since by
the definition of $\FP$,
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is both
$\gd$-directed closed
and $(\gd^+, \infty)$-distributive'', to show
$V^\FP = V^{\FP_\gd \ast \dot \FP^\gd} \models
``\gd$ is $\gd^+$ supercompact'', it suffices
to show that $V^{\FP_\gd} \models ``\gd$ is
$\gd^+$ supercompact''. To do this,
we consider the following two cases.
\medskip\noindent
Case 1: $\card{\FP_\gd} < \gd$. If this
occurs, then by the results of \cite{LS},
$V^{\FP_\gd} \models ``\gd$ is $\gd^+$
supercompact''.
\hfill$\square$
\medskip\noindent
Case 2: $\card{\FP_\gd} \ge \gd$.
In this situation, by the definition of $\FP$,
$\card{\FP_\gg} < \gd$ for every $\gg < \gd$,
and $\gd$ is a limit of cardinals $\gg$
which are $\gg^+$ supercompact. Hence,
$\card{\FP_\gd} = \gd$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gd^+$ supercompactness of
$\gd$ generated by a supercompact ultrafilter
over $P_\gd(\gd^+)$ such that
$M \models ``\gd$ isn't $\gd^+$ supercompact''.
%Consequently,
%by the definition of $\FP$,
We may now infer that
only trivial forcing is done at stage
$\gd$ in $M$ in the definition of $j(\FP_\gd)$.
%This means that
It then follows that
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$,
where the first stage at which $\dot \FQ$
is forced to do nontrivial forcing is
well above $\gd^+$.
A standard diagonalization argument (see, e.g., the proof of
\cite[Lemma 8.1]{AH4})
now shows that $V^{\FP_\gd} \models ``\gd$
is $\gd^+$ supercompact''.
\hfill$\square$
\medskip Cases 1 and 2 complete the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``\gd$ is $\gd^+$ strongly compact
iff $\gd$ is $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd$ is $\gd^+$ strongly compact''.
By Lemma \ref{l2} and our remarks above,
we may assume without loss
of generality that $\gd < \gk$ and
$V \models ``\gd$ isn't $\gd^+$ supercompact''.
Let $\gg = \sup(\{\ga < \gd \mid \ga$ is
$\ga^+$ supercompact$\})$, and write
$\FP = \FP_\gg \ast \dot \FQ$. By the
definition of $\FP$,
$\forces_{\FP_\gg} ``\dot \FQ$ is both
$\gd'$-directed closed and
$({(\gd')}^+, \infty)$-distributive''
(from which it follows that
$\forces_{\FP_\gg} ``\dot \FQ$ is both
$\gd$-directed closed and $(\gd^+, \infty)$-distributive'').
Consequently,
$V^{\FP_\gg} \models ``\gd$ is $\gd^+$ strongly compact''.
Further, by its definition, $\FP_\gg$
admits a gap at $\ha_1$.
If $\card{\FP_\gg} < \gd$, then by the results
of \cite{LS}, $V \models ``\gd$ is $\gd^+$
strongly compact''. Hence, by our hypotheses on $V$,
$V \models ``\gd$ is $\gd^+$ supercompact'',
which is contradictory to our assumptions.
If $\card{\FP_\gg} \ge \gd$, then we first assume that
$\FP_\gg$ is mild with respect to $\gd$.
Under these circumstances,
by Theorem \ref{t3}, $V \models ``\gd$
is $\gd^+$ strongly compact'', which means we
reach the same contradiction as when
$\card{\FP_\gg} < \gd$.
Thus, we may assume
without loss of generality that
$\FP_\gg$ isn't mild with respect to $\gd$.
%Under these circumstances,
%by the definition of $\FP$,
%there must be a largest nontrivial stage of
%forcing $\gs < \gd$, and
%$\FP_\gg = \FP_{\gs} \ast \dot \FQ_{\gs}$.
We consider now the following two cases.
Our argument is analogous to the one
given in the proof of \cite[Lemma 2.3]{A03}.
\medskip\noindent
Case 1: ${(\gd^+)}^V < {(\gd^+)}^{V^{\FP_\gg}}$.
If this is the situation, then as
$\gd$ is measurable and hence a cardinal in $V^{\FP_\gg}$,
$V^{\FP_\gg} \models ``|{(\gd^+)}^V| = \gd$''.
Therefore,
since for any ordinal $\rho$
having cardinality $\gd$,
$\gd$ is measurable iff $\gd$ is $\rho$
strongly compact iff $\gd$ is $\rho$ supercompact,
$V^{\FP_\gg} \models ``\gd$ is ${(\gd^+)}^V$
supercompact''.
By Theorem \ref{t3},
$V \models ``\gd$ is ${(\gd^+)}^V = \gd^+$
supercompact'', an immediate contradiction.
\hfill$\square$
\medskip\noindent
Case 2: ${(\gd^+)}^V = {(\gd^+)}^{V^{\FP_\gg}}$.
To handle when this occurs, we use an
idea due to Hamkins, which has also appeared in
\cite{H03} in a more general context
(as well as in this context in \cite[Lemma 2.3]{A03}).
%and which shows that
%$V \models ``\gd$ is $\gd^+$ strongly compact''.
Hamkins' argument is as follows. Let
$G$ be $V$-generic over ${\FP_\gg}$, and let
$j : V[G] \to M[j(G)]$ be
an elementary embedding witnessing
the $\gd^+$ strong compactness of
$\gd$ generated by a $\gd$-additive,
fine ultrafilter over $P_\gd(\gd^+)$
present in $V[G]$.
As %${\FP_\gg}$ admits a gap at $\ha_1$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$,
by Theorem \ref{t3}, the embedding
$j^* = j \rest V : V \to M$ is definable in $V$.
Note that $j$ and $j^*$ agree on the ordinals.
Since $j$ is a $\gd^+$ strong compactness
embedding in $V[G]$, there is some
$X \subseteq j(\gd^+)$, $X \in M[j(G)]$ with
$j '' \gd^+ \subseteq X$ and
$M[j(G)] \models ``|X| < j(\gd^+)$''.
Therefore, since $\gd^+$ is regular in
$V[G]$, $j(\gd^+)$ is regular in $M[j(G)]$,
%and $\gd^+$ is a regular cardinal,
so we can find an
$\ga < j(\gd^+)$ with $\ga > \sup(X)
\ge \sup(j''\gd^+)$.
This means that if $x \subseteq \gd^+$
is such that
$x \subseteq \gb < \gd^+$,
$j(\ga) \not\in j(x) \subseteq j(\gb)$.
But then,
${\cal U} = \{x \subseteq \gd^+ \mid
\ga \in j^*(x)\}$ defines in $V$
a $\gd$-additive,
uniform ultrafilter over $\gd^+$
which gives measure 1 to sets having
size $\gd^+$.
By a theorem of Ketonen \cite{Ke},
$\gd$ is $\gd^+$ strongly compact in $V$.
Again by our hypotheses on $V$,
$V \models ``\gd$ is $\gd^+$ supercompact'',
a contradiction.
\hfill$\square$
\medskip
Thus, assuming $V^\FP \models ``\gd$ is
$\gd^+$ strongly compact'' leads to the
conclusion that $V \models ``\gd$ is
$\gd^+$ supercompact''. Since this
contradicts our initial assumptions,
the proof of Lemma \ref{l3} is now complete.
\end{proof}
Lemmas \ref{l1} -- \ref{l3} and the intervening
remarks complete the proof of Theorem \ref{t2}.
\end{proof}
We take this opportunity to observe that
our preceding work actually shows that if
$V^\FP \models ``\FQ$ is both
$\gk$-directed closed and
$(\gk^+, \infty)$-distributive'', then
$V^{\FP \ast \dot \FQ} \models ``\gd$ is
$\gd^+$ strongly compact iff
$\gd$ is $\gd^+$ supercompact''. This easily follows
for $\gd \le \gk$, since any forcing which
is both $\gk$-directed closed and
$(\gk^+, \infty)$-distributive will
preserve the conclusions of Lemma \ref{l3}.
%for such a $\gd$.
For $\gd > \gk$, the arguments of Lemma \ref{l3}
with $\FP \ast \dot \FQ$ replacing
$\FP_\gg$ show that if
$V^{\FP \ast \dot \FQ} \models ``\gd$ is
$\gd^+$ strongly compact'', then
$V \models ``\gd$ is $\gd^+$ supercompact''.
This, of course, contradicts our initial
hypotheses on $V$.
Thus, we may in fact infer that
$V^{\FP \ast \dot \FQ} \models ``$No
cardinal $\gd > \gk$ is $\gd^+$ strongly compact''.
The methods we have used still leave open
some interesting questions, with which we conclude
this note.
Specifically, is it possible to prove an analogue
of Theorem \ref{t2} in which $\gk$ is
(fully) indestructibly supercompact?
Is it possible to prove an analogue of Theorem \ref{t2}
in which, e.g., for every cardinal $\gd$,
$\gd$ is $\gd^{++}$ strongly compact iff
$\gd$ is $\gd^{++}$ supercompact?
Hamkins' idea of \cite{H03}
%, which is key to the proof of Lemma \ref{l3},
used in the proof of Lemma \ref{l3}
does not yet
seem to generalize to the situation where
$\gd$ is $\gg$ strongly compact but $\gg \ge \gd^{++}$.
Finally, in a question first posed in \cite{AH4},
is it possible to construct a model containing
an indestructibly supercompact cardinal $\gk$ in which
for every pair of regular cardinals $\gd < \gg$,
$\gd$ is $\gg$ strongly compact iff
$\gd$ is $\gg$ supercompact?
As Theorem \ref{t1} indicates, an answer to
this final question would take place
in a model with some restrictions on
its large cardinal structure.
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%\bibitem{A97} A.~Apter, ``Patterns of
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\end{document}
We note that the related
question of whether,
in a universe containing sufficiently
few large cardinals, it is
possible to have both an indestructibly
supercompact cardinal and full
level by level equivalence between
strong compactness and supercompactness
is one which has been addressed, in part,
in \cite{A06} and \cite{AH4}.
The question, unfortunately, remains open.
The methods we develop here, however,
will allow us to delve into what
we believe are some interesting variants.
As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra : \FP \in {\cal A}$
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
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. The terminology ``lottery sum''
is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
Suppose $V \models ``$ZFC + GCH +
$\gk$ is supercompact +
No cardinal is supercompact up
to a cardinal $\gd$ which is
$2^\gd = \gd^+$ supercompact''.
If necessary, by first doing
the appropriate form of the forcing
found in \cite{AS97a},
we may assume
in addition that
$V \models ``$For every cardinal
$\gd$, $\gd$ is $\gd^+$ strongly compact
iff $\gd$ is $\gd^+$ supercompact''.
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal
is supercompact up to a
measurable cardinal''.
\end{lemma}
\begin{proof}
Write
$\FP = \FR_0 \ast \dot \FR_1$,
where $\card{\FR_0} = \go$,
$\FR_0$ is nontrivial, and
$\forces_{\FR_0} ``\dot \FR_1$ is
$\ha_2$-directed closed''.
Since $\FP$ admits the factorization
just given,
%by Hamkins' Gap Forcing Theorem of \cite{H2} and \cite{H3},
by Theorem \ref{t3},
for any pair of cardinals
$\gd \le \gl$, if
$V^\FP \models ``\gd$ is $\gl$
supercompact'', then
$V \models ``\gd$ is $\gl$ supercompact''
as well. Consequently, since
$V \models ``$No cardinal is
supercompact up to a measurable cardinal'',
it is also the case that
$V^\FP \models ``$No cardinal is
supercompact up to a measurable cardinal''.
%This completes the proof of Lemma \ref{l2}.
\end{proof}
The existence of models $V$
satisfying the hypotheses of Theorem \ref{t2}
(and much more) was first shown in \cite{AS97a}.
By a result of Menas \cite{Me}, there must
be some restrictions on the large cardinal
structure of $V$ beyond those of the
hypotheses, since if $\gk$ is the least
cardinal which is both measurable and a limit
of cardinals $\gd$ which are $\gk^+$ strongly
compact, then $\gk$ is $\gk^+$ strongly compact
but not $\gk^+$ supercompact.
On the other hand, the large cardinal structure
of $V$ can still be fairly rich, as the
existence of, e.g., inaccessible limits of
supercompact cardinals below $\gk$ is not
precluded by our assumptions.