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%
% ------------------------------------------------------------------------------
%
\title{\hbox{Some Structural Results Concerning
Supercompact Cardinals}}
%\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
%\thanks{Keywords: Supercompact cardinal, strongly compact
% cardinal, non-reflecting stationary set of ordinals}
% }
\author{Arthur W.~Apter\thanks{The author
wishes to thank the referee for her/his
useful comments.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{December 2, 1999\\
(revised November 26, 2000 and February 7, 2001)}
\begin{document}
\maketitle
\begin{abstract}
We show how the forcing of \cite{AS97b} can be
iterated so as to get a model containing
supercompact cardinals in which every
measurable cardinal $\gd$ is $\gd^+$
supercompact.
We then apply this iteration to prove
three additional theorems concerning
the structure of the class of
supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Since currently there is no inner model theory
pertaining to supercompactness, any results
concerning the structure of the class of
supercompact cardinals have been obtained, and
continue to be obtained, via forcing.
Examples include the theorems proven in
\hbox{\cite{Me}, \cite{Ma}, \cite{KM},
\cite{A97}, \cite{AS97a}, \cite{AS97b},
and \cite{A}}.
The purpose of this paper is to
continue in this same vein.
We show how to
iterate the forcing of \cite{AS97b}
to prove four new theorems concerning the
structure of the class of supercompact cardinals.
Specifically, we prove the following theorems.
\begin{theorem}\label{t1}
Suppose $V \models {\rm ZFC}$, and suppose
${\mathfrak K} \subseteq V$ is so that
${\mathfrak K} \neq \emptyset$ is the
(possibly proper) class of supercompact cardinals.
There is then a partial ordering
$\FP \subseteq V$ with $V^\FP \models
``$ZFC + The only supercompact cardinals are
the elements of ${\mathfrak K}$ + Every measurable
cardinal $\gd$ is $\gd^+$ supercompact''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models {\rm ZFC}$, and suppose
${\mathfrak K} \subseteq V$ is so that
${\mathfrak K} \neq \emptyset$ is the
(possibly proper) class of supercompact cardinals.
There is then a partial ordering
$\FP \subseteq V$ with $V^\FP \models
``$ZFC +
The only supercompact cardinals are the
elements of ${\mathfrak K}$ +
The supercompact and strongly compact
cardinals coincide precisely, except at
measurable limit points + Every measurable
cardinal $\gd$ is $\gd^+$ supercompact''.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH +
$\gk$ is both $\gk^{+2}$ supercompact
and a limit of supercompact cardinals''.
There is then a partial ordering
$\FP \in V$ so that
$V^\FP \models ``$ZFC + Every measurable
cardinal $\gd$ is $\gd^+$ supercompact +
The least
measurable limit of strongly compact
cardinals $\gk_0$ is $\gk^+_0$ supercompact +
The least measurable limit of supercompact
cardinals $\gk_1$ is $\gk^+_1$ supercompact''.
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + $\gk$ is
supercompact + No cardinal $\gl > \gk$
is measurable".
There is then a partial ordering
$\FP \in V$ so that
$V^\FP \models ``$ZFC + $\gk$ is
supercompact +
No cardinal $\gl > \gk$ is measurable +
For all regular cardinals $\gd < \gl < \gk$,
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact + Every measurable
cardinal $\gd$ is $\gd^+$ supercompact''.
\end{theorem}
We take this opportunity to make some
remarks concerning the above four theorems.
Theorems \ref{t2}-\ref{t4} are essentially
versions of the main
theorems of \cite{KM}, \cite{AS97b}, \cite{A},
and \cite{AS97a},
recast in the context that every measurable
cardinal $\gd$ is $\gd^+$ supercompact.
To a large extent, the proofs of these
theorems will rely on Hamkins' recent work of
\cite{H1}, \cite {H2}, and \cite{H3}, earlier
work of \cite{AS97a} and \cite{AS97b},
and a key new observation about the iterability
of the main forcing notion of \cite{AS97b}.
In particular, Theorem 1 of \cite{A}
and Theorem 2 of \cite{ANew} now
require no restrictions on the large
cardinal structure of the ground model
in their proofs.
%(See also the remarks after the proof
%of Theorem 1 of \cite{A}.)
And, in each of these theorems,
since every measurable cardinal $\gd$
is $\gd^+$ supercompact, it must be the
case that GCH fails at all measurable cardinals.
In addition, there is nothing special about
each measurable cardinal $\gd$ being $\gd^+$
supercompact. Roughly speaking, our proof
techniques will allow us to construct models
in which any measurable cardinal $\gd$
exhibits greater degrees of supercompactness,
so long as the power set of $\gd$ is made
large enough. Interested readers should be
able to provide their own generalizations
after finishing our proofs.
Finally, Theorem 2 of \cite{AS97b} provides an
instance of a model in which every measurable
cardinal $\gd$ is $\gd^+$ supercompact.
In this model, however, there are no supercompact
cardinals, and this paper provides the first
examples of models containing supercompact cardinals
in which every measurable cardinal $\gd$ is
$\gd^+$ supercompact.
We mention now some
preliminary material.
If $\ga < \gb$ are ordinals, then
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
When forcing, $q \ge p$ will mean that
$q$ is stronger than $p$.
If $\FP$ is our partial ordering,
$V^\FP$ and $V[G]$ will be used
interchangeably to denote the generic extension
when forcing with $\FP$.
The partial ordering
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$. $\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-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 $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
%Note that trivially, if $\FP$ is ${<}\gk$-closed, then $\FP$ is
%${<}\gk$-strategically
%closed and ${\prec}\gk $-strategically closed. The converse of
%both of these facts is false.
Finally, we mention that although readers need not be familiar
with all of the details of \cite{AS97a} and \cite{AS97b},
we strongly recommend that copies of these papers be kept
close at hand to improve comprehensibility when reading this
paper.
\section{The Forcing Conditions of \cite{AS97b}}\label{s2}
In order to define in a meaningful way the iterations
to be used in the proofs of Theorems \ref{t1}-\ref{t4},
we first recall the definitions and properties of the
fundamental building blocks of these iterations.
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
$> \gd$. We recall now the partial orderings
$\FP^0_{\gd, \gl}$, $\FP^1_{\gd, \gl}[S]$, and
$\FP^2_{\gd, \gl}[S]$
of \cite{AS97b}.
We assume GCH holds for all
cardinals $\gk \ge \gd$.
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 of cofinality
$\gd$ to $\gl$.
Specifically, $\FP^0_{\gd,\gl} = \{ p$ : For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\gb \in S_p$ implies
$\gb > \gd$ and cof$(\gb) = \gd\}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP^0_{\gd, \gl}$ (see
\cite{Bu} or \cite{KM}), in $V[G]$,
since GCH holds in $V$ for all cardinals
$\gk \ge \gd$, a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G \} \subseteq \gl$
of ordinals of cofinality $\gd$ has been introduced, the
bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH at cardinals
$\gk \ge \gd$ have been preserved.
It is also virtually immediate that $\FP^0_{\gd, \gl}$
is $\gd$-directed closed, and it can be shown
(see \cite{Bu} or \cite{KM}) that
$\FP^0_{\gd, \gl}$
is ${\prec}\gl$-strategically closed.
Work now in $V_1 = V^{\FP^0_{\gd, \gl}}$, letting $\dot S$
be a term always forced to denote the above set $S$.
$\FP^2_{\gd, \gl}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ non-stationary).
Specifically, $\FP^2_{\gd, \gl} [S] = \{ p$ : For
some successor ordinal $\ga < \gl$,
$p : \ga \to \{0,1\}$ is a characteristic function of
$C_p$, a club subset of $\ga$, so that
$C_p \cap S = \emptyset \}$,
ordered by $ q \ge p $ iff $C_q$ is an end extension of $C_p$.
It is again well-known (see \cite{MekS}) that for $H$
$V_1$-generic over $\FP^2_{\gd, \gl}[S]$, a club set
$C = C[H] = \bigcup \{C_p : p \in H \}
\subseteq \gl$ which is disjoint to $S$ has been introduced,
the bounded subsets of $\gl$
are the same as those in $V_1$,
and cardinals, cofinalities, and GCH
for cardinals $\gk \ge \gd$ have been preserved.
The following lemma is proven in
\cite{AS97a} and \cite{AS97b}.
\begin{lemma}\label{lem} (Lemma 1 of
\cite{AS97a} and \cite{AS97b})
$\forces_{\FP^0_{\gd, \gl}} ``
\clubsuit({\dot S})$'', i.e., $V_1 \models ``$There is a
sequence $\langle x_\alpha : \alpha \in S \rangle$ so that
for each $\alpha \in S$, $ x_\alpha \subseteq \alpha$ is
cofinal in
$\alpha$, and for any $A \in
{[\gl]}^{\gl}$, $\{\alpha \in S : x_\alpha
\subseteq A \}$ is stationary''.
\end{lemma}
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\ga : \ga \in S \ra$.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gd, \gl}
[S] $.
First, since each element of
$S$ has cofinality $\gd$, the proof of Lemma 1 of \cite{AS97b}
shows each $x \in X$ can be assumed to be
so that order type$(x) = \gd$.
$\FP^1_{\gd, \gl} [S]$ is then the set of all $5$-tuples
$\la w, \ga, \bar r, Z, \Gamma \ra $ satisfying the following properties.
\begin{enumerate}
\item $w \subseteq \gl$ is so that $|w| = \gd$.
\item $\ga < \gd$.
\item $\bar r = \la r_i : i \in w \rangle $ is a sequence of
functions from $\ga $ to $\{ 0, 1 \}$, i.e., a sequence
of subsets of $\ga$.
\item $Z$ is a function so that:
\begin{enumerate}
\item dom$(Z) \subseteq \{x_\beta : \beta \in
S \}$ and range$(Z) \subseteq \{0, 1\}$.
\item If $z \in$ dom$(Z)$, then for some
$y \in {[w]}^\gd$, $y \subseteq z$ and $z - y$ is
bounded in the $\gb$ so that $z = x_\gb$.
\end{enumerate}
\item $\Gamma$ is a function so that:
\begin{enumerate}
\item dom$(\Gamma) =$ dom$(Z)$.
\item If $z \in$ dom$(\Gamma)$, then $\Gamma(z)$ is a
closed, bounded subset of $\ga$ such that if $\gg$ is
inaccessible, $\gg \in \Gamma(z)$, and $\gb$ is the
$\gg^{\hbox{\rm th}}$ element of $z$, then $\gb \in w$, and for some
$\gb' \in \gb \cap w \cap z$, $r_{\gb'}(\gg) = Z(z)$.
\end{enumerate}
\end{enumerate}
\noindent Note that the definitions of $Z$ and $\Gamma$ imply
$|$dom$(Z)| = |$dom$(\Gamma)| \le \gd$.
The ordering on $\FP^1_{\gd, \gl} [S]$ is given by $\la w^1, \ga^1, \bar
r^1,
Z^1, \Gamma^1 \ra\le \la w^2, \ga^2, \bar r^2, Z^2,\Gamma^2\ra$ 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 $r^1_i \subseteq r^2_i$
and $|\{i \in w^1 : r^2_i \rest (\ga_2 - \ga_1)
$ is not constantly $0 \}| < \gd$.
\item $Z^1 \subseteq Z^2.$
\item dom$(\Gamma^1) \subseteq$ dom$(\Gamma^2)$.
\item If $z \in$ dom$(\Gamma^1)$, then $\Gamma^1(z)$ is
an initial segment of $\Gamma^2(z)$ and
$|\{z \in {\hbox{\rm dom}}(\Gamma^1) :
\Gamma^1(z) \neq \Gamma^2(z) \}| < \gd$.
\end{enumerate}
The intuition behind the above definition of
$\FP^1_{\gd, \gl}[S]$
is given in \cite{AS97b} and \cite{A}.
For the convenience of readers,
we repeat and expand upon it here as well.
If $\gd$ is measurable, then $\gd$ must carry a normal measure.
The partial ordering $\FP^1_{\gd, \gl}[S]$ has specifically been designed
to destroy this fact. (See Lemma 3 of \cite{AS97b} for a proof.)
It has been designed, however, to
destroy the measurability of $\gd$ ``as lightly as possible'',
making little damage, assuming $\gd$ is ${<}\gl$
supercompact. Specifically, if $\gd$ is ${<}\gl$
supercompact, then the non-reflecting stationary set $S$,
having been added to $\gl$, does not kill the ${<}\gl$
supercompactness of $\gd$ by itself. The additional partial ordering
$\FP^1_{\gd, \gl}[S]$ is necessary to do the job and has been
designed so as not only to destroy the ${<}\gl$
supercompactness of $\gd$ but to destroy the measurability of
$\gd$ as well. The partial ordering $\FP^1_{\gd, \gl}[S]$, however, has
been designed so that
%$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$
%is ${<}\gd$-strategically closed and
%$\FP^0_{\gd, \gl} \ast
%(\FP^1_{\gd, \gl}[\dot S] \times
%\FP^2_{\gd, \gl}[\dot S])$
%has a dense subset which is $\gd$-directed closed and so that
if necessary, we can resurrect the
${<}\gl$ supercompactness of $\gd$ by forcing further with
$\FP^2_{\gd, \gl}[S]$, assuming the appropriate
iteration has been done.
(See Lemma 9 of \cite{AS97b} for a proof.)
Note, however, that by Lemma 6 of \cite{AS97b}, the partial orderings
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ and
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$ both collapse $\gd^+$
while preserving every other cardinal and cofinality.
Also, by their definitions, both of these
partial orderings have cardinality $\gl$,
meaning by the L\'evy-Solovay results \cite{LS}
that forcing with either of these
partial orderings preserves the ground model
large cardinal structure above $\gl$.
\section{The Proofs of Theorems \ref{t1}-\ref{t3}}\label{s3}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models {\rm ZFC}$, and let
${\mathfrak K} \subseteq V$ be the
class of supercompact cardinals.
By doing a preliminary forcing if necessary,
we may assume without loss of generality that
$V \models {\rm GCH}$.
Let
$\la \gd_\ga : \ga < \Omega \ra$, where
$\Omega$ is the appropriate ordinal if
${\mathfrak K}$ is a set but is the
class of all ordinals otherwise, enumerate in $V$
all $V$-measurable cardinals.
%$\{\gd : \gd$ is a measurable cardinal$\}$.
We define an Easton support iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \Omega \ra$ as follows.
\begin{enumerate}
\item $\FP_0$ is the
partial ordering which adds a Cohen subset to
$\omega$.
\item $\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
the partial ordering
$\FP^0_{\gd_\ga, \gd^{+3}_\ga} \ast
\FP^1_{\gd_\ga, \gd^{+3}_\ga}[\dot S]$ if
$\gd_\ga$ isn't $\gd^{+2}_\ga$ supercompact.
\item $\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
$\FP^0_{\gd_\ga, \gd^{+3}_\ga} \ast
(\FP^1_{\gd_\ga, \gd^{+3}_\ga}[\dot S] \times
\FP^2_{\gd_\ga, \gd^{+3}_\ga}[\dot S])$ if
$\gd_\ga$ is $\gd^{+2}_\ga$ supercompact.
\end{enumerate}
The intuition behind the above definition of
$\FP$ is quite simple.
We wish to destroy all measurable cardinals
$\gd$ which aren't candidates for eventually
being $\gd^+$ supercompact while preserving
both the measurability and $\gd^+$
supercompactness of those measurable cardinals
which will remain in our generic extension.
In addition, we wish to preserve the
supercompactness of all $V$-supercompact cardinals,
while creating no new supercompact cardinals.
$\FP$ has been designed to carry out these tasks.
By Lemma 6 of \cite{AS97b},
for any $\ga$,
$\FP^0_{\gd_\ga, \gd^{+3}_\ga} \ast
\FP^1_{\gd_\ga, \gd^{+3}_\ga}[\dot S]$ is
${<}\gd_\ga$-strategically closed, and by
Lemma 4 of \cite{AS97b},
$\FP^0_{\gd_\ga, \gd^{+3}_\ga} \ast
(\FP^1_{\gd_\ga, \gd^{+3}_\ga}[\dot S] \times
\FP^2_{\gd_\ga, \gd^{+3}_\ga}[\dot S])$ has a
dense subset which is
$\gd_\ga$-directed closed.
Thus, the standard Easton arguments yield that
$V^\FP \models {\rm ZFC}$.
\begin{lemma}\label{l1}
$V^\FP \models ``\gd$ is measurable iff
$\gd$ is $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Write $\FP = \FP_0 \ast \dot \FQ$.
By the definition of $\FP$,
$|\FP_0| = \omega$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
Thus, by Hamkins' work of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ ``admits a gap at
$\ha_1$'', so by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
any measurable cardinal in $V^\FP$ had to
have been measurable in $V$, and any
supercompact cardinal in $V^\FP$ had to
have been supercompact in $V$.
Further, since the definition of $\FP$
ensures that for any $V$-cardinal $\gk$,
any set in $V^\FP$ of size below $\gk$
has a term denoting it also of size
below $\gk$, the results of
\cite{H2} and \cite{H3} yield that forcing
with $\FP$ can't increase the degree of
strong compactness of any $V$-cardinal.
In particular, any strongly compact cardinal in
$V^\FP$ had to have been strongly compact in $V$.
Fixing now $\gd = \gd_\ga$ as a $V$-measurable cardinal,
this allows us to consider the following two cases.
\setlength{\parindent}{0pt}
Case 1: $\gd$ isn't $\gd^{+2}$ supercompact.
In this case, write
$\FP = \FP_{\ga + 1} \ast \dot \FQ^{*}$, where
$\forces_{\FP_{\ga + 1}} ``\dot \FQ^{*}$ is
$\gg$-strategically closed for $\gg$ the least
cardinal above $\gd$ which is inaccessible in either
$V$ or $V^{\FP_{\ga + 1}}$''.
By the definition of $\FP$,
%$|\FP_{\ga + 1}| < \gg$ and
$V^{\FP_{\ga + 1}} \models ``\gd$
isn't measurable''. Thus,
$V^{\FP_{\ga + 1} \ast \dot \FQ^{*}} = V^\FP \models
``\gd$ isn't measurable''.
Case 2: $\gd$ is $\gd^{+2}$ supercompact.
In this case, write
$\FP = \FP_{\ga + 1} \ast \dot \FQ^{**}$, where
$\forces_{\FP_{\ga + 1}} ``\dot \FQ^{**}$ is
$\gg$-strategically closed for $\gg$ the least
cardinal above $\gd$ which is inaccessible in either
$V$ or $V^{\FP_{\ga + 1}}$''.
By Lemma 9 of \cite{AS97b},
$V^{\FP_{\ga + 1}} \models ``\gd$ is $\gd^+$
supercompact'', so
$V^{\FP_{\ga + 1} \ast \dot \FQ^{**}} = V^\FP \models
``\gd$ is $\gd^+$ supercompact''.
\setlength{\parindent}{1.5em}
Cases 1 and 2 complete the proof of Lemma \ref{l1}.
\end{proof}
We remark that the proof of Lemma 9 of
\cite{AS97b} shows that in $V^\FP$,
for every measurable cardinal $\gd$,
there is an elementary embedding
$j : V^\FP \to M$ witnessing the
$\gd^+$ supercompactness of $\gd$
so that
$M \models ``\gd$ isn't measurable''.
\begin{lemma}\label{l2}
$V^\FP \models ``$The only supercompact cardinals
are the elements of ${\mathfrak K}$''.
\end{lemma}
\begin{proof}
By the third sentence of the first paragraph of
the proof of Lemma \ref{l1}, the proof of
Lemma \ref{l2} will be complete once we have shown
$V^\FP \models ``$If $\gk \in {\mathfrak K}$, then
$\gk$ is supercompact''.
Fix $\gk \in {\mathfrak K}$ an arbitrary
$V$-supercompact cardinal, and let
$\gl > \gk$ be a regular cardinal so that
$\FP$ factors as
$\FP_\gk \ast \dot \FR \ast \dot \FS$, where
$|\FP_\gk \ast \dot \FR| < \gl$ and
$\gl$ is below the least ordinal in the
domain of $\dot \FS$.
%$\forces_{\FP_\gk \ast \dot \FR} ``|\dot \FS| < \gl$''.
The remarks on page 2014 of \cite{AS97b}
after the proof of Lemma 3 show that
although any partial ordering
$\FT = \la \FP^0_{\gd, \gd^{+3}} \ast
\FP^1_{\gd, \gd^{+3}}[\dot S], \le \ra$
is not $\gd$-directed closed
under the order relation $\le$ defined in Section \ref{s2}
(recall that Lemma 3 of \cite{AS97b} shows that
forcing with $\FT$ destroys $\gd$'s measurability),
for any cardinal $\gg < \gd$,
$\le$
can be redefined to an order relation $\le^\gg$
so that
$\FT^\gg = \la \FP^0_{\gd, \gd^{+3}} \ast
\FP^1_{\gd, \gd^{+3}}[\dot S], \le^\gg \ra$ is
$\gg^+$-directed closed, and
$G$ is (appropriately) generic with respect to
$\FT$ iff $G$ is (appropriately) generic
with respect to $\FT^\gg$.
Combining this with the remarks made in the
paragraph immediately preceding the proof of
Lemma \ref{l1} therefore yields that if
$G_0$ is $V$-generic over $\FP_\gk$, then any
$G_1$ which is $V[G_0]$-generic over $\FR$
must be $V[G_0]$-generic over an equivalent
partial ordering
$\FR^*$ which is $\gk$-directed closed.
Hence, if we fix
$j : V \to M$ as an elementary embedding
witnessing the
$\sigma = 2^{[\gl]^{< \gk}}$
supercompactness of $\gk$, then $M$
witnesses that
$j(\FP_\gk \ast \dot \FR) =
\FP_\gk \ast \dot \FR \ast \dot \FR^{**}$, and
in both $V$ and $M$,
$\forces_{\FP_\gk \ast \dot \FR} ``\dot \FR^{**}$ is
equivalent to a partial ordering that is
$\sigma$-directed closed''.
Thus, the standard reverse Easton arguments
(see, e.g., the proof of Lemma 2 of \cite{A98})
show that if $G_2$ is
$V[G_0][G_1]$-generic over $\FR^{**}$ and
contains a master condition for $G_1$,
then $j$ extends in
$V[G_0][G_1][G_2]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2]$.
This embedding can then be used to generate
a supercompact ultrafilter over
$P_\gk(\gl)$ which is present in
$V[G_0][G_1]$. Since
$\forces_{\FP_\gk \ast \dot \FR} ``\dot \FS$
is $\zeta$-strategically closed for $\zeta$
the least inaccessible above $\sigma$'',
this shows that
$V^{\FP_\gk \ast \dot \FR \ast \dot \FS} =
V^\FP \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ can be chosen arbitrarily large,
this completes the proof of Lemma \ref{l2}.
\end{proof}
Lemmas \ref{l1} and \ref{l2} complete the
proof of Theorem \ref{t1}.
\end{proof}
We note that in the proof just given,
$\dot \FS$ can be a term for the
trivial partial ordering, depending upon
the large cardinal structure of $V$
above $\gk$. If this is the case, however,
this causes no real change in the proof
of Lemma \ref{l2}.
Also, by Lemmas 4-6 of \cite{AS97b}, in $V^\FP$,
every measurable cardinal $\gd$ will be so that
$2^\gd = \gd^{+2}$.
In addition, the nature of $\FP$ ensures that
GCH will hold at a proper class of regular
cardinals, e.g., at any cardinal of the form
$\gd^{+5}$, or, if the universe is rich enough,
at any inaccessible cardinal not measurable in $V$.
\begin{pf}
The proofs of Theorems \ref{t2} and \ref{t3}
now follow as easy corollaries to the proof
of Theorem \ref{t1}.
To see this, consider first the proof of
Theorem \ref{t2}. Let
$V$ be as in Theorems \ref{t1} and \ref{t2}.
Without loss of generality, by assuming
that $V$ is a model constructed as in one of
the proofs given in \cite{A98}, \cite{AS97a}, or
\cite{KM}, we may further assume that
$V \models ``$ZFC + GCH + The supercompact and strongly compact
cardinals coincide precisely, except at measurable
limit points''.
If we then force over $V$ with the partial ordering
$\FP$ used in the proof of Theorem \ref{t1}, then
by the remarks contained in the first paragraph of
the proof of Lemma \ref{l1},
since no new strongly compact or supercompact cardinals
are created,
the proof of Theorem \ref{t1} yields that
$V^\FP \models ``$ZFC + The only supercompact
cardinals are the elements of ${\mathfrak K}$ +
The strongly compact and
supercompact cardinals coincide precisely, except
at measurable limit points + Every measurable cardinal
$\gd$ is $\gd^+$ supercompact''.
Consider now the proof of Theorem \ref{t3}. Let
$V \models ``$ZFC + GCH + $\gk$ is both $\gk^{+2}$
supercompact and a limit of supercompact cardinals''.
If we once again force over $V$ with the partial
ordering $\FP$ used in the proof of Theorem \ref{t1},
then the proof of Theorem \ref{t1} tells us that
$V^\FP \models ``$ZFC + $\gk$ is both $\gk^+$
supercompact and a limit of supercompact cardinals +
Every measurable cardinal $\gd$ is $\gd^+$
supercompact''. Thus, in $V^\FP$, there must be
measurable limits of strongly compact cardinals, so
$V^\FP \models ``$ZFC + The least measurable limit
$\gk_0$ of strongly compact cardinals is
$\gk^+_0$ strongly compact + The least measurable
limit $\gk_1$ of supercompact cardinals is
$\gk^+_1$ supercompact''.
This completes the proofs of Theorems \ref{t2} and
\ref{t3}.
\end{pf}
We conclude Section \ref{s3} by observing that if
$\gk$ in Theorem \ref{t3} is taken as the least
cardinal which is both $\gk^{+2}$ supercompact
and a limit of supercompact cardinals, then
$\gk = \gk_1$.
This is since any measurable limit of supercompact
cardinals $\gd$ below $\gk$ can't be
$\gd^{+2}$ supercompact by assumption,
so since forcing with $\FP$ destroys both
$\gd$'s measurability and creates neither any
new supercompact cardinals nor increases the
degree of supercompactness of any $V$-cardinal
by the work of \cite{H1}, \cite{H2}, and
\cite{H3}, it must then be the case that
$\gk = \gk_1$.
%Whether it is consistent, however, for
%$\gk_0$ to be strictly below $\gk_1$
%remains an intriguing open question.
\section{The Proof of Theorem \ref{t4}}\label{s4}
We turn now to the proof of Theorem \ref{t4}.
\begin{proof}
Let $V$ be as in the hypotheses of Theorem \ref{t4}.
By first doing the forcing of \cite{AS97a}, we may
assume that
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gl > \gk$ is measurable + For all
regular cardinals $\gd < \gl < \gk$,
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact''.
We may further assume, by truncating the universe
if necessary, that
$V \models ``$No cardinal $\gd < \gk$ is
$\gg$ supercompact for $\gg$ the least
measurable cardinal above $\gd$''.
Let $\FP = \la \la \FP_\ga,
\dot \FQ_\ga \ra : \ga \le \gk \ra$
be the partial ordering used
in the proof of Theorem \ref{t1}.
By the properties of $\FP$ established earlier,
we know that
$V^\FP \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gl > \gk$ is measurable + Every
measurable cardinal $\gd$ is $\gd^+$ supercompact''.
It thus suffices to show that
$V^\FP \models ``$For all regular cardinals
$\gd < \gl < \gk$,
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact''.
To show this, let $\gd < \gl < \gk$, $\gd$ and $\gl$
both regular cardinals, be so that
$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
By the work of \cite{H2} and \cite{H3} and the
remarks in the first paragraph of the proof of
Lemma \ref{l1},
$V \models ``\gd$ is $\gl$ strongly compact''.
Therefore, by our assumptions on $V$,
$V \models ``\gd$ is $\gl$ supercompact''.
In analogy to the proof of Theorem \ref{t1}, let
$\la \gd_\ga : \ga \le \gk \ra$ enumerate in
$V$ all of the measurable cardinals. Write
$\gd = \gd_\ga$ and
$\FP = \FP_{\ga + 1} \ast \dot \FQ$, where
$\ga < \gk$ and
$\forces_{\FP_{\ga + 1}} ``\dot \FQ$ is
${<}\gg$-strategically closed for $\gg$ the
least cardinal above $\gd$ which is measurable
in either $V$ or $V^{\FP_{\ga + 1}}$''.
By our earlier remarks, we know that
$\gl < \gg$.
This means that
$V^\FP \models ``\gd$ is $\gl$ supercompact'' iff
$V^{\FP_{\ga + 1}} \models ``\gd$ is $\gl$
supercompact'', so it suffices to show that
$V^{\FP_{\ga + 1}} \models ``\gd$ is $\gl$ supercompact''.
To see this last fact, we first note that the proof
of Lemma \ref{l1} tells us that
$\gl \ge {(\gd^{+2})}^V$.
If $\gl = {(\gd^{+2})}^V$, then Lemma 9 of
\cite{AS97b} yields that
$V^{\FP_{\ga + 1}} \models ``\gd$ is $\gl = \gd^+$
supercompact''. If
$\gl > {(\gd^{+2})}^V$, then we first fix
$j : V \to M$ witnessing the $\gl$
supercompactness of $\gd$. Write
$j(\FP_\ga \ast \dot \FQ_\ga) =
\FP_\ga \ast \dot \FQ_\ga \ast
\dot \FQ' \ast \dot \FQ''$.
Since
$|\FP_{\ga + 1}| \le \gl$, by the
definition of $\FP$, if we let
$G = G_0 \ast G_1$ be so that
$G_0$ is $V$-generic over $\FP_\ga$ and
$G_1$ is $V[G_0]$-generic over $\FQ_\ga$,
standard arguments then show that
$M[G_0][G_1]$ remains $\gl$-closed
with respect to $V[G_0][G_1]$.
Thus, as $j$ can be assumed to be
generated by a normal ultrafilter
${\cal U}$ over $P_\gd(\gl)$, it follows
by GCH in $V$ that any $M$ cardinal
$\sigma \in [j(\gd), j(\gl^+)]$ has
$V$-cardinality $\gl^+$. Hence,
by the definition of $\FP$, in
$M[G]$, $\FQ'$ and $\wp(\FQ')$ both have
cardinality ${(\gl^+)}^V$.
We can thus let
$\la D_\gb : \gb < {(\gl^+)}^V \ra$ enumerate in
$V[G]$ the dense open subsets of $\FQ'$ present in
$M[G]$. Since the definition of $\FP$ and the remarks
made in the proof of Lemma \ref{l2} ensure that
$\FQ'$ is equivalent to a partial ordering which is
$\gl^+$-directed closed in both $M[G]$ and $V[G]$,
working in $V[G]$, we can define a sequence of
conditions
$\la p_\gb : \gb < \gl^+ \ra$ so that
$p_0 \in D_0$ and for
$\gb \in [1, \gl^+)$,
$p_\gb \in D_\gb$ extends
every element of the sequence
$\la p_\rho : \rho < \gb \ra$.
$G_2 = \{p \in \FQ' : p$ is extended by
some element of the sequence
$\la p_\gb : \gb < \gl^+ \ra\}$ can
easily be verified as being
$M[G]$-generic over $\FQ'$. Since
the definition of $\FP_\ga$ as an
Easton support iteration tells us that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$,
in $V[G]$, $j$ extends to
$j : V[G_0] \to M[G_0][G_1][G_2] = M[G][G_2]$.
Because
$M[G][G_2]$ remains $\gl$-closed with respect to
$V[G][G_2] = V[G]$ and $G_1$ has cardinality
$\gl$ in $V[G]$,
in $V[G]$,
$\{j''p : p \in G_1\}$ generates a master condition
$q$ for $\FQ''$. By the fact
$\gl > {(\gd^{+2})}^V$,
$\FQ''$ has cardinality in $M[G][G_2]$ at most
$j(\gl)$, so by GCH in $V$, the number of dense open subsets
of $\FQ''$ present in $M[G][G_2]$ is at most
$j(\gl^+)$. Since $\FQ''$ is equivalent to a
partial ordering which is $\gl^+$-directed
closed in both $V[G]$ and $M[G][G_2]$,
we can use the argument just given to construct an
$M[G][G_2]$-generic object $G_3$ containing $q$.
Thus, in $V[G]$, $j$ once again extends to
$j : V[G] \to M[G][G_2][G_3]$ so that
$j '' G \subseteq G \ast G_2 \ast G_3$, i.e.,
$\gd$ is $\gl$ supercompact in $V[G]$.
This completes the proof of Theorem \ref{t4}.
\end{proof}
We conclude Section \ref{s4} by observing
that Theorem \ref{t4} indicates how
unusual an inner model for a supercompact
cardinal might be.
Woodin, in a conversation with the author,
described the main theorem of \cite{AS97a}
as an ``inner model theorem proven by forcing.''
The ground model $V$ used in the proof
of Theorem \ref{t4} provides an
instance of the sort of model constructed in
\cite{AS97a}, i.e., a model in which GCH holds,
there is a supercompact cardinal, and there is
also level by level equivalence where possible
between the notions of $\gl$ strong compactness
and $\gl$ supercompactness.
The model witnessing the conclusions of
Theorem \ref{t4} also provides an instance of
perhaps the key property one might expect in
an inner model for a supercompact cardinal,
i.e., the level by level equivalence
just mentioned. Yet, this model in addition contains
a feature one doesn't find in the usual
sort of inner model, namely a very strong
failure of GCH, as witnessed by the fact
that every measurable cardinal $\gd$ is
$\gd^+$ supercompact.
\section{Concluding Remarks}\label{s5}
In conclusion to this paper, we note
that in the proof of Theorem \ref{t4}
just given, it is critical that
there be no measurable cardinals above
the supercompact cardinal $\gk$.
To see this, if there were a measurable
cardinal above $\gk$, then there would
be cardinals $\gd < \gl < \gk$ so that
$\gd$ is $\gl$ supercompact,
$\gl$ is the least measurable
above $\gd$, and $\gd$ isn't
$\gl^+$ supercompact.
Since $\gl$ is the least measurable
above $\gd$,
the forcing $\FP^0_{\gd, \gl} \ast
\FP^1_{\gd, \gl}[\dot S]$
would then have to be done
at $\gl$, and although the
techniques of this paper show that
$\gd$ remains ${<}\gl$ supercompact
after doing this forcing,
these same techniques do not show
if $\gd$ remains $\gl$ supercompact,
if $\gd$ becomes $\gl$ strongly compact
but not $\gl$ supercompact, or if the
$\gl$ strong compactness of $\gd$ is destroyed.
Thus, whether it is possible to have a model
containing more than one supercompact cardinal
in which every measurable cardinal $\gd$ is
$\gd^+$ supercompact and in which, for every
$\gd < \gl$ so that $\gd$ and $\gl$ are both
regular, $\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact, except possibly
if $\gd$ is a measurable limit of cardinals
$\sigma$ which are $\gl$ supercompact
(the situation in its full generality found
in \cite{AS97a}), remains open.
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\end{document}