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%
% ------------------------------------------------------------------------------
%
\title{Supercompactness and Measurable Limits of
Strong Cardinals II: Applications to
Level by Level Equivalence
% How Large can the Power Set of the Least
% Supercompact be Assuming Level by
% Level Equivalence?
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
level by level equivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grant
66489-00-35 and CUNY
Collaborative Incentive Grants.}
\thanks{The author wishes to thank
the referee for helpful comments
and suggestions which have considerably
improved the presentation of the
material contained herein.}\\
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{April 20, 2006\\
(revised July 18, 2006)}
\begin{document}
\maketitle
\begin{abstract}
We construct models
for the level by
level equivalence
between strong compactness
and supercompactness in
which for $\gk$ the least
supercompact cardinal and
$\gd \le \gk$ any cardinal
which is either a strong
cardinal or a measurable limit
of strong cardinals, $2^\gd > \gd^+$
and $\gd$ is ${<} 2^\gd$ supercompact.
In these models, the structure
of the class of supercompact
cardinals can be arbitrary, and
the size of the
power set of $\gk$
can essentially be made
as large as desired.
This extends and generalizes
Theorem 2 of \cite{A01a} and Theorem 4
of \cite{A01}.
We also sketch how our techniques
can be used to establish a
weak indestructibility result.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Suppose $V$ is a model of ZFC
in which for all regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
Such a universe will be said to
witness level by level
equivalence between strong
compactness and supercompactness.
%Any model witnessing level by
%level equivalence between strong
%compactness and supercompactness
%also witnesses the Kimchi-Magidor
%property \cite{KM} that the classes
%of strongly compact and supercompact
%cardinals coincide precisely,
%except at measurable limit points.
The exception
is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
%Models in which level by level
%equivalence between strong compactness
%and supercompactness holds nontrivially
%were first constructed in \cite{AS97a}.
As Woodin has pointed out,
%level by level equivalence between
%strong compactness and supercompactness
this is the sort of
phenomenon one might expect in an
inner model for a supercompact cardinal.
Nonetheless,
it has been shown (see \cite{A05}, \cite{A03},
\cite{A01}, \cite{A01a}, and \cite{AH4})
that this property is
consistent with exotic and unusual
occurrences that
one would not expect in an inner model,
such as significant failures of GCH.
It is thus reasonable to ask how
far these kinds of situations can be
extended.
The purpose of this paper is to
continue the studies referenced in
the preceding paragraph.
Specifically, we prove
the following theorem.
\begin{theorem} \label{t1}
Suppose
$V \models ``$ZFC + GCH + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal +
Level by level equivalence between
strong compactness and supercompactness holds''.
Let $h : \gk + 1 \to {\hbox{\rm Ord}}$
satisfy the following four conditions.
\begin{enumerate}
\item $h(\gd) = 0$ if $\gd$ isn't
either a strong cardinal or an inaccessible
limit of strong cardinals.
\item For $\gd$ either a strong cardinal
or an inaccessible limit of strong
cardinals, $h(\gd)$
has the properties that
$h(\gd) > \gd^+$, $h(\gd)$ is either inaccessible or is the
successor of a cardinal of cofinality
greater than $\gd$, and $h(\gd)$
is below the least strong cardinal above $\gd$.
\item\label{i3} If $\gd < \gk$ is
${<} h(\gd)$ supercompact, there is
$j_\gd : V \to M$ witnessing the
${<} h(\gd)$ supercompactness
of $\gd$ with
$j_\gd(h)(\gd) = h(\gd)$.
\item\label{i4} If $\gd \le \gk$ is $\gg$
supercompact and $\gg \ge h(\gd)$,
there is
$j_{\gd, \gg} : V \to M$ witnessing the $\gg$
supercompactness of $\gd$ with
$j_{\gd, \gg}(h)(\gd) = h(\gd)$.
\end{enumerate}
\noindent There is then
a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal''.
In $V^\FP$, level by level equivalence between
strong compactness and supercompactness holds,
and for every $\gd \le \gk$ which is
either a strong cardinal or an inaccessible
limit of strong cardinals,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$, and $\gd$ is
${<} h(\gd)$ supercompact
if $\gd$ is a measurable cardinal''.
\end{theorem}
Our techniques will also yield
a weak indestructibility result
for the least supercompact cardinal
$\gk$ in the context of a universe
in which the class of supercompact
cardinals can be arbitrary and level by
level equivalence between strong
compactness and supercompactness holds.
We will briefly return to this towards
the end of the paper.
We take this opportunity to make
several remarks concerning
the above theorem.
First, we note that although
the conditions on $h$ appear
to be rather technical in
nature, they are actually
satisfied by many naturally
occurring functions.
For instance, if $\gd$ is
either a strong cardinal or
an inaccessible limit of strong
cardinals and $h(\gd)$ is
defined as $\gd^{+ 17}$, or
the least Mahlo cardinal above
$\gd$, or the successor of the least Ramsey
cardinal above $\gd$, etc.,
then $h$ satisfies the conditions
given above.
In addition, Theorem \ref{t1}
provides a generalization of
Theorem 2 of \cite{A01a} and
Theorem 4 of \cite{A01}, but
in the context of a universe
in which the structure of
the class of supercompact cardinals
can be {\em arbitrary}, and not limited
as in either of these two aforementioned
theorems.
Further, Theorem \ref{t1} also extends
Theorem 2 of \cite{A01a} to a
model in which level by level equivalence
between strong compactness and
supercompactness holds.
%which is not necessarily true in \cite{A01a}.
Finally, and perhaps
most importantly, in Theorem \ref{t1}, it
can be the case that $2^\gk > \gk^{++}$.
This contrasts sharply with the
results found in
\cite{A03} and \cite{A05},
where the structure of the class
of supercompact cardinals can
be arbitrary,
level by level equivalence between
strong compactness and supercompactness holds,
yet if GCH fails at the least supercompact
cardinal $\gk$,
$2^\gk = \gk^{++}$.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{A03},
\cite{A01a}, \cite{AS97a}, or
\cite{AS97b}.
When forcing, $q \ge p$ means that
$q$ is stronger than $p$.
%and $p \decides \varphi$ means that $p$ decides $\varphi$.
For $\gk$ a regular cardinal and $\gl$
an ordinal,
$\add(\gk, \gl)$ is the
standard partial ordering for adding
$\gl$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
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 $\FP$ is a reverse Easton iteration
such that at stage $\ga$, a nontrivial
forcing is done adding a subset
of $\gd$, then we will say that
$\gd$ is in the field of $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
The partial ordering
$\FP$ is $\gk$-directed closed if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is $\gk$-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),
player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$ \FP$ is ${<}\gk$-strategically closed
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
We take this opportunity to
elaborate a bit more on the notion of
level by level equivalence between strong
compactness and supercompactness.
Any model of ZFC with this property
also witnesses the Kimchi-Magidor
property \cite{KM} that the classes
of strongly compact and supercompact
cardinals coincide precisely,
except at measurable limit points.
In addition,
models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
We assume familiarity with the
large cardinal notions of
measurability, strongness, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} for further details.
%We do mention, however, that
%the cardinal $\gk$ is ${<} \gl$
%strongly compact or ${<} \gl$ supercompact if
%$\gk$ is $\gg$ strongly compact or $\gg$
%supercompact for every $\gg < \gl$.
We do note, however, that
we will say $\gk$ is supercompact
up to the strong cardinal $\gl$ if
$\gk$ is $\gg$ supercompact for every
$\gg < \gl$.
Finally, we mention that
although readers need not be familiar
with all of the details of \cite{AS97a},
\cite{AS97b}, and \cite{A01a},
we strongly recommend that copies
of these papers be kept
close at hand to improve
comprehensibility when reading this paper.
\section{Forcing Notions from \cite{AS97a}
and \cite{AS97b}}\label{s2}
In order to present in a meaningful way the iteration
to be used in the proof of Theorem \ref{t1},
we first recall the definitions and properties of the
fundamental building blocks
of this partial ordering.
In particular, we describe now
a specific form of
the partial orderings
$\FP^0_{\gd, \gl}$,
$\FP^1_{\gd, \gl}[S]$,
and
$\FP^2_{\gd, \gl}[S]$
of Section 4 of \cite{AS97b}.
So that readers are not overly burdened, we
abbreviate our definitions and descriptions somewhat.
Full details may be found by consulting
\cite{AS97b}, along with
the relevant portions of \cite{AS97a}.
Fix $\gd < \gl$, $\gl > \gd^+$ regular cardinals in our
ground model $V$, with $\gd$ inaccessible and $\gl$ either
inaccessible or the successor of a cardinal of cofinality
greater than $\gd$.
We assume GCH holds for all
cardinals $\eta \ge \gd$.
The first notion of forcing $\FP^0_{\gd, \gl}$ is just
the standard notion of forcing
for adding a non-reflecting stationary
set of ordinals $S$ of cofinality
$\go $ to $\gl$.
Next, work in
$V_1 = V^{\FP^0_{\gd, \gl}}$, letting $\dot S$
be a term always forced to denote $S$.
$\FP^2_{\gd, \gl}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ non-stationary).
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\ga : \ga \in S \ra$,
the existence of which is given by
Lemma 1 of \cite{AS97a} and \cite{AS97b}.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gd, \gl}
[S] $.
First, since each element of
$S$ has cofinality $\go$, the proof of Lemma 1
of \cite{AS97a} and \cite{AS97b}
shows each $x \in X$ can be assumed to be
such that order type$(x) = \go$. Then,
$\FP^1_{\gd, \gl}[ S]$ is defined as the set of all
4-tuples $\la w, \ga, \bar r, Z \ra$ satisfying the
following properties.
\begin{enumerate}
\item $w \in {[\gl]}^{< \gd}$.
\item $\ga < \gd$.
\item $ \bar r = \la r_i : i \in w \ra$ is a
sequence of functions from $\ga$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\ga$.
\item $Z \subseteq \{x_\gb : \gb \in S\}$
is a set such that if $z \in Z$, then for some
$y \in {[w]}^\go$, $y \subseteq z$ and $z - y$
is bounded in the $\gb$ such that $z = x_\gb$.
\end{enumerate}
%\noindent As in Section 4 of \cite{AS97b}
%and Section 1 of \cite{AS97a}, the
%definition of $Z$ implies
%$|Z| < \gd$.
\noindent
The ordering on $\FP^1_{\gd, \gl}[S]$ is given by
$\la w^1, \ga^1, \bar r^1, Z^1 \ra \le
\la w^2, \ga^2, \bar r^2, Z^2 \ra$ iff the following hold.
\begin{enumerate}
\item $w^1 \subseteq w^2$.
\item $\ga^1 \le \ga^2$.
\item If $i \in w^1$, then $r^1_i
\subseteq r^2_i$.
\item $Z^1 \subseteq Z^2$.
\item If $z \in Z^1 \cap {[w^1]}^\go$ and
$\ga^1 \le \ga < \ga^2$, then $|\{i \in z :
r^2_i(\ga) = 0\}| = |\{i \in z : r^2_i(\ga) = 1\}| = \go$.
\end{enumerate}
The proof of Lemma 4 of \cite{AS97a} shows that
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$ is equivalent to
${\hbox{\rm Add}}(\gl, 1) \ast \dot {\hbox{\rm Add}}(\gd, \gl)$.
The proofs of Lemmas 3 and 5 of \cite{AS97a}
and Lemma 6 of \cite{AS97b} show that
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ preserves cardinals
and cofinalities, is $\gl^+$-c.c.,
is ${<}\gd$-strategically closed, and is such that
$V^{\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]} \models
``2^\eta = \gl$ for every cardinal $\eta \in [\gd, \gl)$
and $\gd$ is non-measurable''.
\section{The Proof of Theorem \ref{t1}}\label{s3}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V$, $h$, and $\gk$ are as in
the hypotheses for Theorem \ref{t1}.
The partial ordering $\FP$ used in
the proof of Theorem \ref{t1}
is the reverse Easton iteration having
length $\gk + 1$ which begins by
adding a Cohen real
and then does trivial forcing,
except at stages $\gd \le \gk$
which are in $V$ either strong
cardinals or inaccessible limits
of strong cardinals. If such a $\gd$
is not ${<} h(\gd)$ supercompact, then
the forcing done at stage $\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}]$,
where $\dot S_{h(\gd)}$ is a term
for the non-reflecting stationary
set of ordinals of cofinality $\go$
introduced by $\FP^0_{\gd, h(\gd)}$.
If such a $\gd$ is ${<} h(\gd)$ supercompact,
then the forcing done at stage $\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
(\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}] \times
\FP^2_{\gd, h(\gd)}[\dot S_{h(\gd)}])$,
where $\dot S_{h(\gd)}$ is as in the
previous sentence.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is
the least supercompact cardinal''.
\end{lemma}
\begin{proof}
Suppose $\gl > \gk$ is any
cardinal, and $j : V \to M$
is any elementary embedding witnessing
the $\gl$ supercompactness of $\gk$.
Note that
$M \models ``$No cardinal $\gd
\in (\gk, \gl]$ is strong''.
This is since otherwise, it would be
the case that
$M \models ``\gk$ is supercompact
up to a strong cardinal'', so by Lemma
1.1 of \cite{A02},
$M \models ``\gk$ is supercompact''.
By reflection, it must then be true that
$\{\gd < \gk : \gd$ is supercompact$\}$
is unbounded in $\gk$ in $V$, which
contradicts that $\gk$ is in $V$
the least supercompact cardinal.
From this, it immediately follows that
$j(\FP) = \FP \ast \dot \FQ$,
where the first ordinal in the
field of $\dot \FQ$ is well above $\gl$.
The preceding sentence now immediately
yields that the argument of Lemma 4.1 of
\cite{A01a} goes through
unchanged to show that
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$For every
$\gd \le \gk$ which is either
a strong cardinal or an inaccessible
limit of strong cardinals,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$,
and $\gd$ is
${<} h(\gd)$ supercompact
if $\gd$ is a measurable cardinal''.
\end{lemma}
\begin{proof}
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
By Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}, this
factorization of $\FP$ indicates
that any $\gd \le \gk$ which
is either in $V^\FP$ a strong
cardinal or an inaccessible limit of
strong cardinals had to have been in $V$
either a strong cardinal or an inaccessible
limit of strong cardinals.
Further, suppose $\gd < \gk$ is such that
$V \models ``\gd$ is a strong cardinal
which isn't a limit of strong cardinals''.
Since $V \models {\rm GCH}$, Lemma 3.1 of
\cite{A01a} tells us that
$V \models ``\gd$ isn't $2^\gd = \gd^+$
supercompact''. Consequently, by the
definition of $\FP$,
$V^\FP \models ``\gd$ isn't a
measurable cardinal''. We may therefore
infer that any cardinal $\gd < \gk$
which is a strong cardinal in $V^\FP$
had to have been in $V$ a strong cardinal
which is a limit of strong cardinals.
This last sentence now tells us that the
argument of Lemma 4.2 of \cite{A01a}
goes through unchanged to show that
$V^\FP \models ``$For every
$\gd \le \gk$ which is either
a strong cardinal or an inaccessible
limit of strong cardinals,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$,
and $\gd$ is
${<} h(\gd)$ supercompact
if $\gd$ is a measurable cardinal''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gl$ are
regular cardinals such that
$\gd$ is $\gl$ strongly compact and
$\gd$ isn't a measurable limit of
cardinals $\gg$ which are
$\gl$ supercompact''.
Assume first $\gd > \gk$.
Since $V^\FP \models ``2^\gk = h(\gk)$'',
if $V^\FP \models ``\gd$ is measurable'',
it must be the case that $\gd > h(\gk)$.
Hence, by the results of \cite{LS}, since $\FP$ can
be defined so that $|\FP| = h(\gk)$,
Lemma \ref{l3} is true if $\gd > \gk$.
By Lemma \ref{l1}, Lemma \ref{l3}
is true if $\gd = \gk$.
It therefore
suffices to prove Lemma \ref{l3}
when $\gd < \gk$, which we
assume for the duration of the
proof of this lemma.
Let $A = \{\gg \le \gd : \gg$
is a strong cardinal or
an inaccessible limit of
strong cardinals$\}$. Write
$\FP = \FP_A \ast \dot \FQ$, where
$\FP_A$ is the portion of $\FP$
whose field is composed of
ordinals at most $\gd$, and
$\dot \FQ$ is a term for the
rest of $\FP$, i.e., the portion
of $\FP$ whose field contains
ordinals above $\gd$. We claim that
since $\gd < \gk$, it follows
that $\gl$ is below the
least $V$-strong cardinal $\gz$ above
$\gd$. This is because otherwise,
$V \models ``\gd$ is $\gg$
strongly compact for all
$\gg < \gz$ and $\gz$ is
a strong cardinal'',
so by Lemma 1.1 of \cite{A02},
$V \models ``\gd$ is strongly compact'',
a contradiction to the fact that $\gd < \gk$ and
$\gk$ is both the least $V$-supercompact
and least $V$-strongly compact cardinal.
As $\forces_{\FP_A} ``\dot \FQ$ is
$\gz$-strategically closed and
$\gz$ is inaccessible'',
$V^{\FP_A} \models ``\gd$
is $\gl$ strongly compact and $\gd$ isn't
a measurable limit of cardinals
$\gg$ which are $\gl$ supercompact''.
Further, to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact'',
it hence suffices to show that
$V^{\FP_A} \models ``\gd$ is $\gl$
supercompact''.
Consider now the following two cases. \bigskip
\noindent Case 1: $\sup(A) = \gs < \gd$.
If this is true, it must be the
case that $\card{\FP_A} < \gd$. This is
since if $\card{\FP_A} \ge \gd$,
by the definition of $\FP$,
the order type of $A$ is a successor ordinal,
and $\gs$ must be the largest member of $A$.
We may then infer that
$V^{\FP_A} \models ``2^\gs \ge \gd$'', which
contradicts the measurability of $\gd$ in
$V^{\FP_A}$. Thus, since
$\card{\FP_A} < \gd$, by the results of \cite{LS},
$V \models ``\gd$ is $\gl$ strongly compact
and $\gd$ isn't a measurable limit of cardinals
$\gg$ which are $\gl$ supercompact''.
Hence, by the level by level equivalence
between strong compactness and supercompactness
in $V$, $V \models ``\gd$ is $\gl$ supercompact'',
so again by the results of \cite{LS},
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''
as well. \bigskip
\noindent Case 2: $\sup(A) = \gd$.
As in the proof of Lemma \ref{l2},
we can write
$\FP_A = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
Further, it is easily seen that
any subset of $\gd$ in
$V^\FP$ of size below $\gd$
has a name of size
below $\gd$ in $V$.
Therefore, by the results of
\cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
In addition, as in Case 1 above, it is
the case that
$V \models ``\gd$ isn't a measurable
limit of cardinals $\gg$ which are
$\gl$ supercompact''.
This is since otherwise,
as $V \models ``\gd$ is a
a limit of strong cardinals'',
some cardinal $\gg < \gd < \gk$
must be supercompact up to
a strong cardinal in $V$.
By Lemma 1.1 of \cite{A02}, $\gg$
is then supercompact in $V$, which
contradicts that $V \models ``\gk$ is
the least supercompact cardinal''.
Thus, by the level by level
equivalence between strong compactness
and supercompactness in $V$,
$V \models ``\gd$ is $\gl$ supercompact''.
It must be the case that
$V \models ``\gd$ is
${<} h(\gd)$ supercompact'', because
otherwise, by the definition of $\FP$,
$V^{\FP_A} \models ``\gd$ isn't
measurable''.
Since by Lemma \ref{l2},
$V^\FP \models ``\gd$ is
${<} h(\gd)$ supercompact'',
which means by the definition of
$\FP$ that
$V^{\FP_A} \models ``\gd$ is
${<} h(\gd)$ supercompact'' as well,
we may assume without loss of
generality that $\gl \ge h(\gd)$.
Consequently, let $j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gd$
satisfying condition (\ref{i4})
of Theorem \ref{t1} (so
$j$ is such that
$j(h)(\gd) = h(\gd)$).
Since $\gl \ge h(\gd)$,
$\FP_A$ is forcing equivalent to
$\FP_\gd \ast \dot \FQ^*$, where
$\forces_{\FP_\gd} ``|\dot \FQ^*|
= h(\gd) \le \gl$ and
$\dot \FQ^*$ is $\gd$-directed closed''.
($\FQ^*$ is forcing equivalent to
$\add(h(\gd), 1) \ast
\dot \add(\gd, h(\gd))$.)
In addition, the same reasoning as
found in the proof of Lemma \ref{l1}
shows that
$M \models ``$No cardinal in the
half-open interval $(\gd, \gl]$ is strong''.
Thus, $j(\FP_\gd \ast \dot \FQ^*)$ is
forcing equivalent to
$\FP_\gd \ast \dot \FQ^* \ast \dot \FR
\ast j(\dot \FQ^*)$, where the first
ordinal in the field of $\dot \FR$
is well above $\gl$.
Standard lifting arguments (as given,
e.g., in the proof of
Theorem 4 of \cite{A01}) are then
applicable and show that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.\bigskip
Cases 1 and 2 now complete
the proof of Lemma \ref{l3}.
\end{proof}
Since $h(\gk)$ lies below the least
$V$-strong cardinal above $\gk$,
by the results of \cite{LS},
$V^\FP \models ``\K - \{\gk\}$
is the class of supercompact cardinals
above $\gk$''. This observation,
together with Lemmas \ref{l1} -- \ref{l3},
complete the proof of Theorem \ref{t1}.
\end{proof}
By Theorem 5 of \cite{AH4},
if $\gk$ is an indestructibly
supercompact cardinal (in the sense of
\cite{L}) and level by level equivalence
between strong compactness and
supercompactness holds below $\gk$, then
no cardinal $\gl > \gk$ is $2^\gl$ supercompact.
Thus, in a universe in which
there is more than one supercompact cardinal
%the structure of the class of supercompact cardinals is arbitrary
and there is level by level
equivalence between strong compactness
and supercompactness, the least supercompact
cardinal $\gk$ cannot exhibit
indestructibility under arbitrary
$\gk$-directed closed partial orderings.
The proof of Theorem 5 of \cite{AH4},
however, does not rule out all forms
of indestructibility for the least
supercompact cardinal $\gk$ in a
universe in which
there is more than one supercompact cardinal,
%the structure of the class of supercompact cardinals can be arbitrary,
assuming level by
level equivalence between strong
compactness and supercompactness.
In fact, as Theorem 15 of \cite{AH4}
shows, under these circumstances,
%under the circumstances set forth in the preceding sentence,
it is possible for $\gk$'s
supercompactness to be indestructible under
forcing with the L\'evy collapse
${\rm Coll}(\gk, \gl)$, where $\gl \ge \gk$
is an arbitrary regular cardinal.
Left open by the work of \cite{AH4}, however,
is the question of whether,
regardless of the large cardinal
structure of the universe,
level by level equivalence between
strong compactness and supercompactness
can hold and the least supercompact cardinal
$\gk$ can have its supercompactness
indestructible under trivial forcing
and partial orderings of the form
$\add(\gk, \gl)$, where $\gl \ge \gk$
is some ordinal.
%Indeed, this question is unanswered in
%\cite{AH4}, regardless of the large
%cardinal structure of the universe.
Unfortunately, this question remains
open. However, we do have the following
result, which provides a very weak answer.
\begin{theorem} \label{t2}
Suppose
$V \models ``$ZFC + GCH + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal +
Level by level equivalence between
strong compactness and supercompactness holds''.
Let $h : \gk + 1 \to {\hbox{\rm Ord}}$
be as in Theorem \ref{t1}.
There is then
a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal''.
In $V^\FP$, $2^\gk = \gk^+$,
level by level equivalence between
strong compactness and supercompactness holds,
and $\gk$'s supercompactness is indestructible
under trivial forcing, $\add(\gk, \gk^{++})$,
${\rm Coll}(\gk, \gl)$ for $\gl \ge \gk$
any regular cardinal, and $\add(h(\gk), 1)
\ast \dot \add(\gk, h(\gk))$.
\end{theorem}
We will not prove this theorem,
since it falls far short of answering
the above question. We will, however,
give a brief idea of its methods of proof.
The partial ordering $\FP$ used in the proof
of Theorem \ref{t2} is a restricted version
of the partial ordering used in the proof of
Theorem 2 of \cite{A03}.
Specifically, for any inaccessible
cardinal $\gd < \gk$, first define $\FR_\gd$ as follows.
If $\gd$
is not ${<} h(\gd)$ supercompact, then
$\FR_\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}]$,
and if $\gd$ is ${<} h(\gd)$ supercompact,
$\FR_\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
(\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}] \times
\FP^2_{\gd, h(\gd)}[\dot S_{h(\gd)}])$,
where these partial orderings are as in
the proof of Theorem \ref{t1}.
$\FP$ is then a reverse Easton
iteration of length $\gk$ which begins
by adding a Cohen real.
At an inaccessible cardinal
$\gd < \gk$ which is in $V$ a limit
of strong cardinals, we force with
the lottery sum of trivial forcing,
$\add(\gd, \gd^{++})$, ${\rm Coll}(\gd, \gl)$
where $\gl \ge \gd$ is a regular cardinal below
the least $V$-strong cardinal above $\gd$, and
$\FR_\gd$.\footnote{As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\rm lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra : \FP \in {\cal A}$
and $p \in \FP\} \bigcup \{0\}$, 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.''}
At all other stages, the forcing
is trivial.
The methods of proof used in Theorem 2 of
\cite{A03}, in tandem with the proofs of
Theorem 15 of \cite{AH4} and
Theorem \ref{t1} of this paper, then
can be used to establish Theorem \ref{t2}.
However, unfortunately, the specific
question mentioned above, or the more
general question of whether, regardless
of the large cardinal structure of
the universe,
level by level equivalence between
strong compactness and supercompactness can
hold and the least supercompact cardinal
$\gk$ can have its supercompactness
indestructible under trivial forcing and
partial orderings which increase the
size of its power set arbitrarily high,
remains open.
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\end{document}