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%
% ------------------------------------------------------------------------------
%
\title{Failures of SCH and Level by Level Equivalence
% and the Level by Level
% Equivalence between Strong Compactness
% and Supercompactness
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
Gitik iteration, Prikry forcing,
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
James Cummings for a helpful
discussion which led to the concluding
remarks of this paper.
The author also wishes to thank
the referee for helpful suggestions
which have considerably
improved the presentation of the
material contained herein. In
particular, a number of the referee's
remarks have been incorporated almost
verbatim into the body of the paper.}\\
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{March 12, 2006\\
(revised April 24, 2006)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model
for the level by
level equivalence
between strong compactness
and supercompactness in
which below the least
supercompact cardinal $\gk$,
there is a stationary set
of cardinals on which SCH fails.
In this model, the structure
of the class of supercompact
cardinals can be arbitrary.
%This result is best possible.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
The study of possible failures
of GCH that are consistent with
the level by level equivalence between
strong compactness and supercompactness
is one which has been addressed,
in part, in \cite{A03}, \cite{A05},
\cite{A01}, and \cite{AH4}.
In each of these articles, however,
any failure of GCH occurring at a
singular cardinal has been induced
by a failure of GCH at a regular
cardinal, and hence has been at a
singular cardinal which isn't a strong limit.
%non-strong limit singular cardinal.
%In each of these articles, however,
%all failures of GCH have occurred at
%regular cardinals.
To this point, there had been
no results concerning possible failures
of GCH at singular strong limit cardinals that
are consistent with the level by level
equivalence between strong compactness
and supercompactness.
The purpose of this paper is to
rectify this situation by
proving the following theorem.
\begin{theorem}\label{t1}
%Suppose we start with a model
Assume the existence of a model
for ZFC + GCH containing at least
one supercompact cardinal in which level
by level equivalence between
strong compactness and supercompactness
holds.
There is then a forcing extension,
containing exactly the same
supercompact cardinals and preserving
level by level equivalence between
strong compactness and supercompactness,
in which GCH fails on a stationary
subset of the least supercompact cardinal
composed of singular strong limit cardinals
of cofinality $\go$.
In particular, this forcing extension
has a stationary set of violations of SCH
below the least supercompact cardinal.
\end{theorem}
We note that by Solovay's celebrated theorem
of \cite{S}, GCH must hold at any
singular strong limit cardinal above
a strongly compact cardinal.
Thus, any failures of GCH that occur
on singular strong limit cardinals
must of necessity take place below
the least strongly compact cardinal.
In addition, any set having measure
1 with respect to a normal measure
over a measurable cardinal must of
course concentrate on regular cardinals.
Therefore, one cannot improve
Theorem \ref{t1} by having
violations of
SCH above the least supercompact cardinal,
or by changing ``stationary''
to normal measure one.
%It is for these two reasons that
%Theorem \ref{t1} gives a result
%that is best possible in terms of
%possible failures of SCH that are
%consistent with the level by level
%equivalence between strong compactness
%and supercompactness.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{A03}.
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,
$\add(\gk, \gk^{++})$ is the
standard partial ordering for adding
$\gk^{++}$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
$(\ga, \gb]$ and $[\ga, \gb]$
are as in standard interval notation.
%is the usual half-open
%interval of ordinals which doesn't
%include $\ga$ but includes $\gb$.
For $\gk$ a cardinal, the
partial ordering $\FP$ is
$\gk$-directed closed if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
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 $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
Suppose $V$ is a model of ZFC
%containing supercompact cardinals
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 model will be said to
witness level by level
equivalence between strong
compactness and supercompactness.
We will also say that $\gk$ is a witness
to level by level equivalence between
strong compactness and supercompactness
iff for every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact.
Note that 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.
When this situation occurs, the
terminology we will henceforth
use is that $\gk$ is a witness
to the Menas exception at $\gl$.
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} and \cite{K} for further details.
We just mention that a cardinal $\gk$
will be said to be either supercompact or
strong up to a strong cardinal $\gl$ if
$\gk$ is either $\gd$ supercompact or
$\gd$ strong for every $\gd < \gl$.
Also, a cardinal $\gk$
will be said to have trivial Mitchell
rank with respect to $\gk^+$ supercompactness
if there is no embedding $j : V \to M$ witnessing
the $\gk^+$ supercompactness of
$\gk$ for which
$M \models ``\gk$ is $\gk^+$ supercompact''.
An ultrafilter ${\cal U}$ generating
this sort of embedding will be said to
have trivial Mitchell rank with
respect to $\gk^+$ supercompactness as well.
If $\gk$ is $\gk^+$ supercompact,
there will always be a $\gk^+$ supercompact
ultrafilter over $P_\gk(\gk^+)$ having
trivial Mitchell rank with respect to
$\gk^+$ supercompactness.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose
$\ov V \models ``$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''.
Without loss of generality, we assume
that the forcing of Theorem 1 of \cite{A03}
has been done to generically extend
$\ov V$ to a model $V$ such that
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals + $\gk$ is the
least supercompact cardinal + Level by
level equivalence between strong compactness
and supercompactness holds +
$2^\gd = \gd^{++}$ if $\gd$ is in
$\ov V$ an inaccessible limit of
strong cardinals + $2^\gd = \gd^+$ otherwise''.
Note that the partial ordering used to
construct $V$ is the reverse Easton
iteration $\FP^*$ of length $\gk + 1$ which begins
by adding a Cohen subset of $\go$
and then, at each cardinal $\gd \le \gk$
which is in $\ov V$ an inaccessible
limit of strong cardinals, forces with
$\add(\gd, \gd^{++})$.
At all other stages, the forcing is trivial.
Consequently, by Hamkins' Gap Forcing
Theorem of \cite{H2} and \cite{H3},
any cardinal which in $V$ is
a measurable limit of strong cardinals
had to have been in $\ov V$ a
measurable limit of strong cardinals as well.
We may therefore infer that
$V \models ``$For any $\gd \le \gk$ which
is a measurable limit of strong cardinals,
$2^\gd = 2^{\gd^+} = \gd^{++}$''.
In addition, if $\gd < \gk$ and $\gl$ are
such that
$V \models ``\gd$ is $\gl$ supercompact'',
then again by the Gap Forcing Theorem,
$\ov V \models ``\gd$ is $\gl$ supercompact''
as well. It is consequently the
case that $\gl$ is smaller than the
least $\ov V$-strong cardinal
above $\gd$, for if not, then
$\gd$ is supercompact up to a cardinal
which is strong in $\ov V$.
By Lemma 2.4 of \cite{AC2},
$\gd$ is supercompact in $\ov V$,
which contradicts that
$\ov V \models ``\gk$ is the
least supercompact cardinal''.
By the construction of $V$, it
must therefore be the case that
$V \models ``2^\gl = \gl^+$''.
Take $V$ as our ground model. Let
$A = \{ \gd < \gk \mid \gd$ is
$\gd^+$ supercompact,
$\gd$ has trivial Mitchell rank
with respect to $\gd^+$ supercompactness, and
$\gd$ is a limit of strong cardinals$\}$.
Our partial ordering $\FP$ is then
the Gitik iteration of length $\gk$\footnote{This is an
Easton support iteration of
length $\gk$ with the usual ordering,
except,
roughly speaking, the stems of
Prikry conditions are extended
nontrivially only finitely often ---
for a more precise definition
and explanation of associated
terminology, including what it
means for one condition to be
an Easton extension of another,
see \cite{G}.} which, for
$\gd \in A$, does Prikry forcing
with respect to the appropriately chosen normal
measure over $\gd$. At all other stages,
the forcing is trivial.
If $V \models {\rm GCH}$, then by
Lemma 1.5 of \cite{G}, the definition of
$\FP$ just given is valid.
The situation here, however, is far
less clear. This is since without GCH,
we don't know if it is always possible,
for $\gd \in A$, to define
$\FP_{\gd + 1}$ from $\FP_\gd$.
In other words, we don't know
if it is always possible,
for $\gd \in A$, to show that
$\forces_{\FP_\gd} ``\gd$ is measurable''.
We consequently begin with the following lemma.
\begin{lemma}\label{l1}
$\FP$ is well-defined.
In particular, for every
$\gd \in A$, $\forces_{\FP_\gd} ``\gd$
is $\gd^+$ supercompact
(and hence is measurable)''.
\end{lemma}
\begin{proof}
Assume that for $\gd \in A$,
$\FP_\gd$ has been defined.
We show that it is possible
to define $\FP_{\gd + 1}$.
By the L\'evy-Solovay results \cite{LS},
if $\card{\FP_\gd} < \gd$,
$\forces_{\FP_\gd} ``\gd$ is
$\gd^+$ supercompact (and hence
is measurable)''.
Hence, it is possible to do Prikry
forcing over $\gd$ in $V^{\FP_\gd}$
with respect to
any appropriately chosen normal measure,
which means that
$\FP_{\gd + 1}$ can be defined.
We therefore assume that
$\card{\FP_\gd} = \gd$.
If this is the case, then in
spite of the fact
$V \models ``2^\gd = \gd^{++}$'',
the argument of Lemma 1.5 of
\cite{G} is still applicable. Specifically,
let $G$ be $V$-generic over $\FP_\gd$.
%By the definition of $\FP_\gd$,
%$j '' G = G$.
Since the choice of $V$ tells us that
$V \models ``2^{\gd^+} = \gd^{++}$'', we may let
$\la \dot x_\ga \mid \ga < \gd^{++} \ra$ be an
enumeration in $V$ of all of the
canonical $\FP_\gd$-names of subsets of
$P_\gd(\gd^+)$. Then,
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''.
This means that
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gd^+$. Consequently,
by Lemmas 1.4 and 1.2 of \cite{G} and the
fact $M^{\gd^+} \subseteq M$, we may
define an increasing sequence
$\la p_\ga \mid \ga < \gd^{++} \ra$
of elements of $j(\FP_\gd)/G$
such that if $\ga < \gb < \gd^{++}$,
$p_\gb$ is an
Easton extension of $p_\ga$,
every initial segment of
the sequence is in $M$, and for every
$\ga < \gd^{++}$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gd^+ \ra \in j(\dot x_\ga)$''.
The remainder of the argument of
Lemma 1.5 of \cite{G} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_\gd(\gd^+))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_\gd(\gd^+))}^{V[G]}$ and
for some $\ga < \gd^{++}$ and some
$\FP_\gd$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j(\FP_\gd)/G}
``\la j(\gb) \mid
\gb < \gd^+ \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gd(\gd^+)$.
Thus, $\forces_{\FP_\gd} ``\gd$ is
$\gd^+$ supercompact (and hence is measurable)''.
It is therefore possible to do Prikry forcing
over $\gd$ in $V^{\FP_\gd}$ with respect to any
appropriately chosen normal measure,
%over $\gd$ in $V^{\FP_\gd}$,
which means that
$\FP_{\gd + 1}$ can be defined.
This completes the proof of Lemma \ref{l1}.
\end{proof}
We note that the argument given in Lemma \ref{l1}
for the preservation of
(a certain amount of) supercompactness
will be used in the proofs of both
Lemmas \ref{l3} and \ref{l4}.
\begin{lemma}\label{l2}
In both $V$ and $V^\FP$,
$A$ is a stationary subset of $\gk$.
\end{lemma}
\begin{proof}
%In addition, $A$ is in $V$ a
%stationary subset of $\gk$.
%To see this,
Since $\gk$ is
supercompact, let
${\cal U}_0 \lhd {\cal U}_1$
be such that ${\cal U}_0$ has
trivial Mitchell rank with
respect to $\gk^+$ supercompactness and
${\cal U}_1$ is an immediate
successor of ${\cal U}_0$ in the Mitchell ordering of
normal ultrafilters over $P_\gk(\gk^+)$.
Take $j : V \to M$ as the elementary
embedding witnessing the $\gk^+$
supercompactness of $\gk$ generated
by ${\cal U}_1$.
It is the case that
$M \models ``\gk$ is $\gk^+$
supercompact and has trivial
Mitchell rank with respect to
$\gk^+$ supercompactness'', and
by Lemma 2.1 of \cite{AC2}
and the succeeding remarks,
$M \models ``\gk$ is a limit
of strong cardinals''.
Thus, $A \in \mu$, where
$\mu$ is the normal measure
over $\gk$ generated by $j$.
This of course immediately
yields that $A$ is in $V$
a stationary subset of $\gk$.
Since $\FP$ satisfies $\gk$-c.c.,
by Exercise H2, page 247 of
\cite{Ku}, $A$ remains stationary
in $V^\FP$.
This completes the proof of
Lemma \ref{l2}.
\end{proof}
By Lemmas 1.4 and 1.2 of \cite{G}
and the fact $\FP$ is a Gitik
iteration of Prikry forcing,
$V^\FP \models ``$Every $\gd \in A$
is a singular strong limit cardinal
of cofinality $\go$ violating GCH'', i.e.,
$V^\FP \models ``$Every $\gd \in A$
is a singular strong limit cardinal
of cofinality $\go$ violating SCH''.
Because $\FP$ may be defined so that
$\card{\FP} = \gk$, by the results of
\cite{LS}, $V^\FP \models ``\K - \{\gk\}$
is the class of supercompact cardinals
above $\gk$''. Thus, the proof of
Theorem \ref{t1} is completed
by the following two lemmas.
\begin{lemma}\label{l3}
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
\end{lemma}
\begin{proof}
Since $V^\FP \models ``$GCH fails on
a stationary (and hence unbounded)
set of singular strong
limit cardinals below $\gk$'', by
Solovay's theorem of \cite{S},
$V^\FP \models ``$There are no
strongly compact cardinals below $\gk$''.
Thus, the proof of Lemma \ref{l3}
will be complete once we have shown that
$V^\FP \models ``\gk$ is supercompact''.
To do this, note that
by the definition of $\FP^*$,
$V \models ``$GCH holds for all
cardinals greater than or equal
to $\gk^{+}$''. Consequently, let
$\gl > \gk^+$ be an arbitrary
regular cardinal at which GCH holds, and let
$j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of
$\gk$ generated by a supercompact ultrafilter over
$P_\gk(\gl)$ such that
$M \models ``\gk$ isn't $\gl$ supercompact''.
It is the case that
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''.
This is since otherwise, $\gk$
is supercompact up to a strong cardinal in $M$,
and thus, as was mentioned earlier,
%as in Lemma 2.4 of \cite{AC2},
$M \models ``\gk$ is supercompact'', a contradiction.
Further, by the choice of $\gl$,
$M \models ``\gk$ is (at least) $\gk^+$
supercompact and has nontrivial Mitchell rank
with respect to $\gk^+$ supercompactness''.
This means that
$j(\FP) = \FP \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well above $\gl$.
%The remainder of the proof of Lemma \ref{l3} is now
%as in the proof of Lemma \ref{l1}.
We may now show that
$V^\FP \models ``\gk$ is $\gl$ supercompact''
using the same argument as given in
the proof of Lemma \ref{l1}. Since $\gl$
was arbitrary,
this completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds''.
\end{lemma}
\begin{proof}
Since $\FP$ may be defined so that
$\card{\FP} = \gk$, and since
$V \models ``$Level by level equivalence
between strong compactness and
supercompactness holds'', by the results
of \cite{LS},
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds above $\gk$''.
By Lemma \ref{l3},
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds at $\gk$''.
Thus, the proof of Lemma \ref{l4}
will be complete once we have shown that
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds below $\gk$''.
To do this, let $\gd < \gk$ and
$\gl > \gd$ be such that
$V^\FP \models ``\gd$ is $\gl$ strongly
compact and $\gl$ is regular''. Let
$\gg = \sup(\{\ga < \gd \mid \ga$ is
a nontrivial stage of forcing$\})$, and write
%be the supremum of the nontrivial
%stages of forcing below $\gd$, and write
$\FP = \FP_\gg \ast \dot \FP^\gg$.
By the definition of $\FP$, let
$\eta > \gd$, $\eta < \gk$ be the
least cardinal
at which $\dot \FP^\gg$ is forced to
do a nontrivial forcing.
It must then be the case that
$V^\FP \models ``\eta$ contains a Prikry sequence''.
%in the domain of $\dot \FP^\gg$.
By Theorem 11.1 of \cite{CFM},
$V^\FP \models ``\eta^+$ contains a
non-reflecting stationary set of ordinals
of cofinality $\go$'', so by Theorem
4.8 of \cite{SRK} and the succeeding remarks,
$V^\FP \models ``\gd$ isn't $\eta^+$
strongly compact''. Since
$V^\FP \models ``{\rm cof}(\eta) = \go$'',
it must consequently be the case that
$\gl < \eta$.
Therefore, as Lemmas 1.4 and 1.2 of \cite{G}
imply that
$\forces_{\FP_\gg} ``$Forcing with
$\dot \FP^\gg$ adds no bounded subsets to $\eta$
and turns $\eta$ into a singular strong
limit cardinal of cofinality $\go$
violating GCH'', we may hence infer that
$\forces_{\FP_\gg} ``\gd$ is $\gl$
strongly compact'' iff
$\forces_{\FP} ``\gd$ is $\gl$
strongly compact'', i.e.,
$V^{\FP_\gg} \models ``\gd$ is
$\gl$ strongly compact''.
We consider now two cases. \bigskip
\noindent Case 1: $\gg < \gd$.
In this situation, by the
definition of $\FP$,
$\card{\FP_\gg} < \gd$.
Thus, by the results of \cite{LS},
$V^{\FP_\gg} \models ``\gd$ is $\gl$ strongly compact''
iff
$V \models ``\gd$ is $\gl$ strongly compact''.
Since $V \models ``$Level by level equivalence
between strong compactness and supercompactness holds'',
either $V \models ``\gd$ is $\gl$ supercompact'', or
$V \models ``\gd$ is a witness to the Menas
exception at $\gl$''. Again by the results of
\cite{LS},
either $V^{\FP_\gg} \models ``\gd$ is $\gl$ supercompact'', or
$V^{\FP_\gg} \models ``\gd$ is a witness to the Menas
exception at $\gl$''. Regardless of which
of these occurs, $\gd$ does not witness a
failure of the level by level equivalence
between strong compactness and supercompactness. \bigskip
\noindent Case 2: $\gg = \gd$.
If this occurs,
then by the definition of
$\FP$, it must be the case that
$\card{\FP_\gd} = \gd$.
Note that since $\gd$
is measurable in $V^{\FP_\gd}$,
$\gd$ must be Mahlo in $V^{\FP_\gd}$
and thus also Mahlo in $V$. Consequently,
$\FP_\gd$ is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$, and
$\FP_\gd$ satisfies $\gd$-c.c$.$ in $V$.
This means that
since $\FP_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FP_\gd}$ as well
(this follows because $\gd$ is Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$
is a subordering of the
direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$
as calculated in $V^{\FP_\gd}$),
(the proof of) Lemma 8 of
\cite{A97} (see in particular
the argument found starting in
the third paragraph on page 111
of \cite{A97}) or (the proof of)
Lemma 3 of \cite{AC1}
tells us that every $\gd$-additive
uniform ultrafilter over a cardinal
$\gb \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gb$ in $V$.
Therefore, since the $\gl$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FP_\gd}$-regular cardinal
$\gb \in [\gd, \gl]$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FP_\gd}$,
and since the fact $\FP_\gd$
is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$
tells us the regular cardinals
at or above $\gd$ in
$V^{\FP_\gd}$ are the same
as those in $V$,
the preceding sentence implies
that every $V$-regular cardinal
$\gb \in [\gd, \gl]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gl$ strongly
compact in $V$.
Observe now that $\gd$ cannot
witness in $V$ the Menas exception
at $\gl$. The reason is that
if this were the case, then
$\gd$ would have to be a limit of
cardinals which are $\gl$ supercompact in $V$.
However, by the definition of
$\FP$, any such cardinal $\gb$
would have to be in $V$ supercompact up
to a strong cardinal, which as we have
already observed,
%in the proof of Lemma \ref{l3},
implies that
$\gb$ is supercompact in $V$.
This is a contradiction, since
$\gb < \gk$, and $\gk$ is the
least supercompact cardinal in $V$.
Thus, by the level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gd$ is $\gl$ supercompact''.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$ such that
$M \models ``\gd$ isn't $\gl$ supercompact''.
Write
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$.
As in Lemma \ref{l3}, $M \models ``$No cardinal
$\gb \in (\gd, \gl]$ is strong''. Also,
$M \models ``\gd$ doesn't have trivial
Mitchell rank with respect to
$\gd^+$ supercompactness''.
If $\gl = \gd^+$, this follows from
the fact that $M \models ``\gd$ isn't
$\gd^+$ supercompact''. If $\gl > \gd^+$,
then since
$V \models ``2^{[\gd^+]^{< \gd}} =
2^{\gd^+} = \gd^{++} \le \gl$'',
%$V \models ``2^\gl = \gl^+$'',
$M \models ``\gd$ is $\gd^+$ supercompact
and has nontrivial Mitchell rank with
respect to $\gd^+$ supercompactness''.
Putting this together, we may now
infer that the first
nontrivial stage in $\dot \FQ$
%ordinal in the domain of $\dot \FQ$
is well above $\gl$.
Hence, since
as we observed at the beginning
of the proof of Theorem \ref{t1},
%once again using the fact that
$V \models ``2^\gl = \gl^+$'',
we may apply the same argument as
given in the proof of Lemma \ref{l1}
to infer that
$V^{\FP_\gd} \models ``\gd$ is
$\gl$ supercompact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} -- \ref{l4} and the
intervening remarks complete
the proof of Theorem \ref{t1}.
\end{proof}
Since $\card{\FP} = \gk$,
$V^\FP \models ``2^\gk = \gk^{++}$''.
We note as our final observation
%In conclusion to this paper, we note
that it is possible
to obtain a model witnessing
the conclusions of Theorem \ref{t1}
in which GCH holds at the
least supercompact cardinal.
%cardinal $\gk$, $2^\gk = \gk^+$.
An outline of the argument
accomplishing this is as follows.
Let $\ov V$, $\K$,
and $\gk$ be as
before. Redefine $\FP^*$ as the reverse Easton
iteration of length $\gk$ which begins by
adding a Cohen subset of $\gk$ and then,
at each $\gd < \gk$ which is in $\ov V$
an inaccessible limit of strong cardinals,
forces with the lottery sum of trivial
forcing and
$\add(\gd, \gd^{++})$.\footnote{As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\rm lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra \mid \FP \in {\cal A}$
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.''}
At all other stages, the forcing is trivial.
By the same reasoning as found in
the proof of Theorem 2 of \cite{A03},
$\gk$ is the least
supercompact cardinal in either
$\ov V^{\FP^*} = V$ or
$\ov V^{\FP^* \ast \dot \add(\gk, \gk^{++})} = V_1$,
$\K$ is the class of supercompact cardinals in
$V$ and $V_1$,
and level by level equivalence between strong
compactness and supercompactness holds
in both of these models.
We observe that the set of strong
cardinals below $\gk$ is the
same in both $V$ and $V_1$.
To see this, let $\gd < \gk$
be a $V$-strong cardinal.
Since $\add(\gk, \gk^{++})$ is
$\gk$-directed closed, $\gd$
is strong up to $\gk$ in $V_1$.
Since $\gk$ is supercompact in
$V_1$ and hence also strong in
$V_1$, by Lemma 2.1 of \cite{AC2}
and the succeeding remarks,
$\gd$ is strong in $V_1$. And,
if $\gd < \gk$ is strong in $V_1$,
again by the fact $\add(\gk, \gk^{++})$
is $\gk$-directed closed, $\gd$ is
strong up to $\gk$ in $V$.
Since $\gk$ is supercompact in $V$,
as we have just noted, $\gd$ must
be strong in $V$ as well.
Consequently, since forcing with
$\add(\gk, \gk^{++})$ doesn't
change the cardinals $\gd < \gk$
which are $\gd^+$ supercompact,
have trivial Mitchell rank with
respect to $\gd^+$ supercompactness,
and are such that $2^\gd = \gd^{++}$,
the set $A$ of Theorem \ref{t1}
slightly redefined as
$\{ \gd < \gk \mid \gd$ is
$\gd^+$ supercompact,
$\gd$ has trivial Mitchell rank
with respect to $\gd^+$ supercompactness,
$2^\gd = \gd^{++}$, and
$\gd$ is a limit of strong cardinals$\}$
is composed of the same members in both
$V$ and $V_1$.
We therefore write without fear
of ambiguity $A$ for either
$A^V$ or $A^{V_1}$.
Further, since
$V_1 \models ``2^\gk = \gk^{++}$'', the proof of
Lemma \ref{l2} shows that $A$ is
a stationary subset of $\gk$ in $V_1$,
%so since $A$ is the same in both $V$ and $V_1$,
so $A$ is a stationary
subset of $\gk$ in $V$ as well.
Therefore, if the partial ordering
$\FP$ of Theorem \ref{t1} is redefined
using the current version of $A$
(which of course preserves that
$\card{\FP} = \gk$),
since GCH holds at $\gk$ in $V$, and
since our earlier arguments
%as given earlier in the proofs of
%Lemmas \ref{l2} -- \ref{l4}
remain valid with trivial modifications,
$V^\FP$ is a model witnessing the
conclusions of Theorem \ref{t1} in
which GCH holds at the least supercompact cardinal.
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\end{document}