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%
% ------------------------------------------------------------------------------
%
\title{Level by Level Inequivalence, Strong Compactness,
and GCH %Easton Functions
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
%level by level equivalence between strong
%compactness and supercompactness,
level by level inequivalence between strong
compactness and supercompactness, non-reflecting
stationary set of ordinals, Easton function,
Magidor iteration of Prikry forcing.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY 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/aapter\\
awapter@alum.mit.edu}
%\date{January 24, 2012}
\date{February 1, 2012\\
(revised July 5, 2012)}
\begin{document}
\maketitle
\begin{abstract}
We construct three models
containing exactly one supercompact cardinal in which
%$\gk$ in which
level by level inequivalence between
strong compactness and supercompactness holds.
In the first two models, below the supercompact cardinal $\gk$,
there is a non-supercompact strongly compact cardinal.
In the last model, any suitably defined ground model
Easton function is realized.
%(so $\gk$ is the second strongly compact cardinal).
%In the first of these models, the
%least strongly compact cardinal $\gl$ is the
%least measurable cardinal and has its strong
%compactness indestructible under $\gl$-directed
%closed forcing.
%In the second of these models, the least strongly
%compact cardinal is a limit of measurable cardinals.
%We also construct a model containing
%exactly one supercompact cardinal in which
%any suitably defined ground model Easton function
%is realized and in which level
%by level inequivalence between strong compactness
%and supercompactness holds.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Say that a model containing at least one supercompact cardinal
satisfies {\em level by level inequivalence between
strong compactness and supercompactness} if
for every non-supercompact measurable cardinal
$\gd$, there is some $\gg > \gd$ such that
$\gd$ is $\gg$ strongly compact yet $\gd$
is not $\gg$ supercompact.
Models containing exactly one supercompact
cardinal in which level by level inequivalence
between strong compactness and supercompactness
holds have been constructed in
\cite[Theorem 2]{A02}, \cite[Theorem 2]{A10},
and \cite[Theorem 1]{A11}.
%\footnote{Note
%that the dual notion of {\em level by level equivalence
%between strong compactness and supercompactness}
%was first studied by the author and Shelah in
%\cite{AS97a}, to which we refer readers for
%additional details.}
(See also \cite{AGH}.
Note
that the dual notion of {\em level by level equivalence
between strong compactness and supercompactness}
was first studied by the author and Shelah in
\cite{AS97a}, to which we refer readers for additional details.)
Key features of each of these models, however, are
rather restricted large cardinal structures and
fairly arbitrary GCH patterns.
In particular,
%there are no non-supercompact strongly compact cardinals
it is not possible to infer that there are any
strongly compact cardinals
below the supercompact cardinal
in any of these models
(although in the model of \cite[Theorem 1]{A11},
there are finitely many non-supercompact
strongly compact cardinals
above the supercompact cardinal).
In addition, GCH holds in the models of both
\cite[Theorem 2]{A02} and \cite[Theorem 2]{A10},
and the GCH pattern of the
model of \cite[Theorem 1]{A11} is controlled by
ground model indestructible supercompact cardinals.
This prompts us to ask the following two questions.
\bigskip\noindent {\bf Question 1:} Is it possible to construct models
containing at least one supercompact cardinal in which
level by level inequivalence between strong compactness
and supercompactness holds, and in which there is
a non-supercompact
strongly compact cardinal below some supercompact cardinal?
\bigskip\noindent {\bf Question 2:} Is it possible to construct a model
containing at least one supercompact cardinal in which
level by level inequivalence between strong compactness
and supercompactness holds, and in which the GCH pattern
on regular cardinals is precisely controlled?
\bigskip The purpose of this paper is to answer the
above questions in the affirmative.
Specifically, we will prove the following three theorems,
where we take as notation for this paper that if
$\ga$ is an ordinal, then $\gs_\ga$ is the
least inaccessible cardinal above $\ga$.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH + %There exist cardinals
$\gl < \gk_1 < \gk_2$ are such that $\gl$ and $\gk_1$ are
both supercompact and $\gk_2$ is inaccessible''.
%+ No cardinal $\gd > \gk_1$ is measurable''.
%satisfying the
%following properties:
%
%\begin{enumerate}
%
%\item $\gk_2$ is the least inaccessible cardinal above $\gk_1$.
%
%\item $\gl$ and $\gk_1$ are both $\gk_2$ supercompact''.
%
%\end{enumerate}
There is then a partial ordering $\FP \in V$, a
submodel $\ov V \subseteq V^\FP$, and $\gk \in (\gl, \gk_1)$
such that $\ov V \models ``$ZFC + GCH + $\gk$ is supercompact +
Level by level inequivalence between strong compactness
and supercompactness holds + No cardinal
$\gd > \gk$ is inaccessible''.
In $\ov V$,
$\gl$ is both the least strongly compact and least measurable
cardinal.
%, and $\gl$'s strong compactness is
%indestructible under $\gl$-directed closed forcing.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models ``$ZFC + GCH + %There exist cardinals
$\gl < \gk_1 < \gk_2$ are such that $\gl$ and $\gk_1$ are
both supercompact and $\gk_2$ is inaccessible''.
%+ No cardinal $\gd > \gk_1$ is measurable''.
%satisfying the
%following properties:
%
%\begin{enumerate}
%
%\item $\gk_2$ is the least inaccessible cardinal above $\gk_1$.
%
%\item $\gl$ and $\gk_1$ are both $\gk_2$ supercompact''.
%
%\end{enumerate}
There is then a partial ordering $\FP \in V$, a
submodel $\ov V \subseteq V^\FP$, and $\gk \in (\gl, \gk_1)$
such that $\ov V \models ``$ZFC + GCH + $\gk$ is supercompact +
Level by level inequivalence between strong compactness
and supercompactness holds + No cardinal
$\gd > \gk$ is inaccessible''.
In $\ov V$,
$\gl$ is the least strongly compact cardinal,
$\gl$ is not supercompact (in fact,
$\gl$ is not $2^\gl$ supercompact),
and $\gl$ is a limit of measurable cardinals.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH + $\gk_1 < \gk_2$
are such that $\gk_1$ is supercompact and $\gk_2$
is inaccessible + No cardinal $\gd > \gk_1$ is measurable''.
% + $F$ is an Easton
%function satisfying the properties of
%\cite[Theorem 18]{Me76}''.
Let $F$ be a class function defined on
the regular cardinals with range a subset
of the cardinals satisfying the following properties.
\begin{enumerate}
\item\label{i1} If
$\gd_1 < \gd_2$, then $F(\gd_1) \le F(\gd_2)$.
\item\label{i2} $F(\gd) \in (\gd, \gs_\gd)$
(or if $\gs_\gd$ does not exist, then $F(\gd) > \gd$).
\item\label{i3} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i4} $F$ is definable by a $\Delta_2$ function.
\end{enumerate}
\noindent There is then a partial ordering $\FP \subseteq V$, a
submodel $\ov V \subseteq V^\FP$, and $\gk < \gk_1$
such that $\ov V \models ``$ZFC + $\gk$ is supercompact +
Level by level inequivalence between strong compactness
and supercompactness holds + No cardinal
$\gd > \gk$ is inaccessible''.
In $\ov V$, $\gk$ is the least strongly compact cardinal, and
for every regular cardinal $\gd$,
$2^\gd = F(\gd)$.
\end{theorem}
We take this opportunity to make several remarks
concerning Theorem \ref{t3}.
We begin by observing that restriction (\ref{i4})
above on the Easton function $F$ is as a result of
Menas' proof of \cite{Me76}.
Restriction (\ref{i2}) is made so that inaccessible
cardinals are preserved between $V$ and $\ov V$,
thereby simplifying the proof of Theorem \ref{t3}.
(Restrictions (\ref{i1}) and (\ref{i3}) are of
course standard to any Easton function.)
These constraints are fairly innocuous, however,
and still allow us to contruct models $\ov V$
witnessing level by level inequivalence between
strong compactness and supercompactness in which
$2^\gd = \gd^{++}$ for every regular cardinal $\gd$,
$2^\gd = \gd^{+17}$ for every regular cardinal $\gd$, etc.
This is in sharp contrast to models in which level by
level equivalence between strong compactness
and supercompactness holds
and GCH fails significantly (such as, e.g.,
the models constructed in \cite{A06b}), where currently
available techniques seem to allow far less
flexibility in what can be forced to occur.
%(Readers are urged to consult
%(See ??? for a more extensive discussion of this phenomenon.)
%make several comments
%concerning Theorems \ref{t1} and \ref{t2}.
Before beginning the proof of Theorems \ref{t1} -- \ref{t3},
we elaborate briefly on our notation and terminology.
For $\ga < \gb$ ordinals, $(\ga, \gb)$,
$[\ga, \gb)$, and $(\ga, \gb]$
are as in standard interval notation.
Suppose $\gk < \gl$ are cardinals.
The partial ordering $\FP$ is {\em $\gk$-directed closed}
if every directed subset of $\FP$ of cardinality
less than $\gk$ has a common extension.
We will abuse notation slightly and use $V^\FP$
to denote the generic extension of $V$ by $\FP$.
$\gk$ is {\em ${<} \gl$ supercompact
(${<} \gl$ strongly compact)} if
$\gk$ is $\gd$ supercompact
($\gd$ strongly compact) for every $\gd < \gl$.
\section{The Proofs of Theorems \ref{t1} -- \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + GCH + There exist cardinals
$\gl < \gk_1 < \gk_2$ such that $\gl$ and $\gk_1$ are
both supercompact and $\gk_2$ is inaccessible''.
%+ No cardinal $\gd > \gk_1$ is measurable''.
Without loss of generality, by \cite[Theorem 2]{A01} and
the remarks at the end of \cite{A01},
we may assume
in addition that $V \models ``$Every measurable cardinal
$\gd$ is $\gs_\gd$ strongly compact''.
%This additional property of $V$
%is achieved by first iteratively destroying any measurable
%cardinal $\gd$ which is not $\gs_\gd$ strongly
%compact by adding non-reflecting
%stationary sets of ordinals.
%, where $\gs_\gd$ is the least inaccessible cardinal above $\gd$''.
Let $\FP$ be Magidor's
partial ordering of \cite[Theorem 3.5]{Ma}
which iteratively changes the cofinality of
every measurable cardinal $\gd < \gl$ to
$\go$ via Prikry forcing.
%defined with respect to $\gl$.
By the work of \cite{Ma},
$V^\FP \models ``$GCH + $\gl$ is both the least strongly compact
and least measurable cardinal''.
%and has its strong compactness
%indestructible under $\gl$-directed closed forcing''.
Since $\FP$ may be defined so that $\card{\FP} = \gl$,
by the L\'evy-Solovay results \cite{LS},
$V^\FP \models ``$Every measurable cardinal $\gd > \gl$
is $\gs_\gd$ strongly compact + $\gk_1$ is $\gk_2$
supercompact and $\gk_2$ is inaccessible''.
By reflection, we may therefore let $\gk \in (\gl, \gk_1)$
be the least cardinal such that
$V^\FP \models ``\gk$ is ${<} \gs_\gk$ supercompact''.
It is then the case that for $\ov V = (V_{\gs_\gk})^{V^\FP}$,
$\ov V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
Further, in $\ov V$, $\gl$ is both the least
strongly compact and least measurable cardinal.
As any cardinal $\gd$ which is $2^\gd$ supercompact
must be a limit of measurable cardinals, this
means that
$\ov V \models ``\gl$ is not $2^\gl = \gl^+$ supercompact''.
Consequently, because $\ov V \models ``$Every measurable
cardinal $\gd \in (\gl, \gk)$ is $\gs_\gd$ strongly
compact'',
$\ov V \models ``$Level
by level inequivalence between strong compactness and
supercompactness holds''.
%and $\gl$'s strong compactness is indestructible under
%$\gl$-directed closed forcing.
This completes the proof of Theorem \ref{t1}.
\end{proof}
We observe that by replacing the partial ordering
$\FP$ used in the proof of Theorem \ref{t1}
with the partial ordering of \cite[Theorem 1]{AG},
it is possible to assume that in addition,
$\gl$ has its strong compactness indestructible
under $\gl$-directed closed forcing.
This is the exact analogue of Laver's result of
\cite{L} for strongly compact, rather than
supercompact, cardinals.
If this has been done, GCH will no longer hold below
$\gl$ in the final model. GCH will, however, continue to be true at
and above $\gl$, since the partial ordering of
\cite[Theorem 1]{AG} can be defined so as to have
cardinality $\gl$.
In particular,
%since the partial ordering used in the proof of
%Theorem \ref{t1} can also be defined so as to have
%cardinality $\gl$,
it will also be the case
%this implies that in both Theorem \ref{t1} and
%this modification of Theorem \ref{t1},
(as it was in Theorem \ref{t1})
that $\gl$ is not $2^\gl = \gl^+$ supercompact.
\begin{pf}
Turning now to the proof of Theorem \ref{t2},
once again,
let $V \models ``$ZFC + GCH + There exist cardinals
$\gl < \gk_1 < \gk_2$ such that $\gl$ and $\gk_1$ are
both supercompact and $\gk_2$ is inaccessible''.
As in the proof of Theorem \ref{t1}, we also assume that
$V \models ``$Every measurable cardinal
$\gd$ is $\gs_\gd$ strongly compact''.
Let $A = \{\gd < \gl \mid \gd$ is a measurable cardinal
which is a limit of measurable cardinals$\}$.
Let $\FP$ be Magidor's partial ordering of \cite{Ma}
which iteratively changes the cofinality of every
$\gd \in A$ to $\go$ via Prikry forcing.
By the work of \cite{Ma}
(in particular, by \cite[Theorem 3.4]{Ma}),
$V^\FP \models ``$GCH + $\gl$ is a strongly compact cardinal''.
\begin{lemma}\label{l1}
$V^\FP \models ``\gl$ is not $2^\gl = \gl^+$ supercompact''.
\end{lemma}
\begin{proof}
%Suppose $\gd \in \gl - A$ is a measurable cardinal,
%i.e., suppose that
%$V \models ``\gd$ is a measurable cardinal which
%is not a limit of measurable cardinals''.
By \cite[Theorem 3.1]{Ma}, any cardinal which
is measurable in $V^\FP$ had to have been measurable in $V$.
Thus, if $V^\FP \models ``\gd < \gl$ is a measurable
cardinal which is a limit of measurable cardinals'', then
$V \models ``\gd$ is a measurable
cardinal which is a limit of measurable cardinals''.
However, by the definition of $\FP$,
$V^\FP \models ``{\rm cof}(\gd) = \go$''.
This means that $V^\FP \models ``$Below $\gl$,
there are no measurable %cardinals which are
limits of measurable cardinals''.
Consequently, $V^\FP \models ``\gl$ is not
$2^\gl = \gl^+$ supercompact'', since if it were, then
$V^\FP \models ``\gl$ is a limit of measurable
limits of measurable cardinals''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal $\gd < \gl$
is strongly compact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 3.1]{A06}.
As was mentioned in the proof of Lemma \ref{l1},
since $V \models ``\gl$ is $2^\gl$ supercompact'',
$V \models ``\gl$ is a limit of measurable
limits of measurable cardinals''.
Hence, by the definition of $\FP$,
in $V^\FP$, unboundedly in $\gl$
many $\gd < \gl$ contain Prikry sequences.
However, by \cite[Theorem 11.1]{CFM},
the presence of a Prikry sequence
implies the presence of a
non-reflecting stationary set
of ordinals of cofinality $\go$.
Therefore, since \cite[Theorem 4.8]{SRK} and the
succeeding remarks imply such a set
cannot exist above a
strongly compact cardinal,
we may
now immediately infer that no
cardinal $\gd < \gl$ is strongly
compact.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$If $\gd < \gl$, $\gd\not\in A$ is
measurable in $V$, then $\gd$ is $\gs_\gd$ strongly compact
and is not $2^\gd = \gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose $\gd$ is as in the hypotheses of Lemma \ref{l3}.
Write $\FP = \FP_\gd \ast \dot \FP^\gd$, where
$\FP_\gd$ is the portion of $\FP$ defined on
measurable cardinals below $\gd$, and $\dot \FP^\gd$
is a term for the rest of $\FP$.
Since
$V \models ``\gd$ is a measurable cardinal which is
not a limit of measurable cardinals'',
it follows that $\card{\FP_\gd} < \gd$ and
the first ordinal at which $\dot \FP^\gd$ is
forced to do nontrivial forcing is a $V$-measurable
cardinal above $\gd$.
The results of \cite{LS} therefore imply that
$V^{\FP_\gd} \models ``\gd$ is $\gs_\gd$ strongly compact
and is not a limit of measurable cardinals''.
Then, because \cite[Lemma 2.4]{Ma} implies that
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$
adds no new subsets of $\gs_\gd$'',
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models ``\gd$
is $\gs_\gd$ strongly compact and is not a limit
of measurable cardinals''.
Since as we have already observed in the
proof of Theorem \ref{t1},
any $\gd$ which is $2^\gd$ supercompact
must be a limit of measurable cardinals,
it immediately follows that
$V^\FP \models ``\gd$ is not $2^\gd = \gd^+$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
We now argue as in the proof of Theorem \ref{t1}
to complete the proof of Theorem \ref{t2}.
Since $\FP$ may be defined so that $\card{\FP} = \gl$,
%as before,
%$V^\FP \models ``$Every measurable cardinal $\gd > \gl$
%is $\gs_\gd$ strongly compact + $\gk_1$ is $\gk_2$
%supercompact and $\gk_2$ is inaccessible''.
%By reflection, we may therefore
%once again let $\gk \in (\gl, \gk_1)$
%be the least cardinal such that
%$V^\FP \models ``\gk$ is ${<} \gs_\gk$ supercompact''.
%It is then the case that for $\ov V = (V_{\gs_\gk})^{V^\FP}$,
we may define $\ov V$ as in the proof of Theorem \ref{t1}
and infer that
$\ov V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is
inaccessible''.
By Lemmas \ref{l1} -- \ref{l3},
in $\ov V$, $\gl$ is the least
strongly compact cardinal,
$\gl$ is not $2^\gl$ supercompact,
$\gl$ is a limit of measurable cardinals,
and every measurable cardinal $\gd < \gl$
is $\gs_\gd$ strongly compact yet is
not $\gd^+$ supercompact.
Consequently, as was the case previously,
%because $\ov V \models ``$Every measurable cardinal
%$\gd \in (\gl, \gk)$ is $\gs_\gd$ strongly compact'',
$\ov V \models ``$Level by level inequivalence between
strong compactness and supercompactness holds''.
This completes the proof of Theorem \ref{t2}.
\end{pf}
We remark that in each of the models $\ov V$
constructed above, there are no supercompact
cardinals below $\gk$. This is since if there were,
then there would be some $\gd < \gk$ which
is ${<} \gs_\gd$ supercompact in both $\ov V$
and $V^\FP$. This, of course, contradicts the
choice of $\gk$. On the other hand,
the question of whether there can be more
than one non-supercompact strongly
compact cardinal below $\gk$ is quite
intriguing.
If we begin our constructions by forcing
over a model $V$ such that
$V \models ``$ZFC + GCH + $\gl < \gk_1 < \gk_2$
are such that $\gl$ and $\gk_1$ are both
supercompact and $\gk_2$ is inaccessible +
$\gl$ and $\gk_1$ are the only strongly
compact cardinals'' (such as a model in \cite{AS97a}),
and then force as in \cite{A01} to obtain the
additional property that each measurable cardinal
$\gd$ is $\gs_\gd$ strongly compact, then
the answer is no.
This follows from Hamkins' gap forcing results of
\cite{H2, H3}, since the forcing of
\cite{A01} is both ``mild''
and can be formulated to ``admit a low
enough gap''
(both in the sense of \cite{H2, H3})
so that it cannot create any
new strongly compact cardinals.
However, when the ground model $V$
does not satisfy this additional property,
we do not currently know an answer to this question.
\begin{pf}
Turning now to the proof of Theorem \ref{t3},
let $V \models ``$ZFC + GCH + $\gk_1 < \gk_2$
are such that $\gk_1$ is supercompact and $\gk_2$
is inaccessible + No cardinal $\gd > \gk_1$ is measurable''.
%+ $F$ is an Easton function satisfying the properties of
%\cite[Theorem 18]{Me76}''.
Suppose $F$ is a function satisfying properties
(\ref{i1}) -- (\ref{i4}) of the hypotheses of Theorem \ref{t3}.
%Without loss of generality, we assume
%in addition that
%$V \models ``$No cardinal $\gd > \gk_1$ is measurable''.
By \cite[Theorem 18]{Me76}, we further assume that
$V$ has been extended to a model $V_*$
via class forcing $\FQ$ such that
$V_* \models ``$ZFC + $\gk_1 < \gk_2$
are such that $\gk_1$ is supercompact +
No cardinal $\gd > \gk_1$ is
measurable + For every regular cardinal
$\gd$, $2^\gd = F(\gd)$''.
Property (\ref{i2}) of $F$ implies that
$V_* \models ``\gk_2$ is inaccessible'' as well.
%Since property (\ref{i2}) of $F$ ensures that
%$F(\gk_1) < \gk_2$, we may also assume that
%$V_* \models ``\gk_2$ is inaccessible'' as well.
Let $A = \{\gd < \gk_1 \mid \gd$ is a measurable cardinal
which is not ${<} \gs_\gd$ supercompact$\}$.
Let $\FR$ be Magidor's partial ordering of \cite{Ma}
which iteratively changes the cofinality of every
$\gd \in A$ to $\go$ via Prikry forcing.
%By the work of \cite{Ma},
%(in particular, by \cite[Theorem 3.4]{Ma}),
%$V^\FR \models ``$GCH + $\gl$ is a strongly compact cardinal''.
\begin{lemma}\label{l4}
$V^\FR_* \models ``\gk_1$ is supercompact''.
\end{lemma}
\begin{proof}
Suppose $\gl \ge \gk_2$ is arbitrary.
Let $j : {V_*} \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk_1$
generated by a supercompact ultrafilter over $P_{\gk_1}(\gl)$.
%such that $M \models ``\gk_1$ is not $\gl$ supercompact''.
Since $M^\gl \subseteq M$ and
$V_* \models ``$No cardinal $\gd > \gk_1$ is
measurable'', $M \models ``\gk_1$ is
${<} \gk_2$ supercompact + $\gk_2$ is inaccessible +
No cardinal $\gd \in (\gk_1, \gl]$ is measurable''.
Thus, since $j$ is generated by a supercompact
ultrafilter over $P_{\gk_1}(\gl)$,
by the definition of $\FR$, $j(\FR) =
\FR \ast \dot \FR'$, where the first ordinal $\gg$
at which $\dot \FR'$ is forced to do nontrivial
forcing is well above $\gl$.
We follow now the proof of \cite[Lemma 1.2]{A11}
(which itself follows the proof of
the Lemma of \cite{A00}) to show that
$V^\FR_* \models ``\gk_1$ is $\gl$ supercompact''.
Let $|\ \ \ |$ be the distance function of \cite{Ma}.
Define a term $\dot {\cal U}$ in ${{V_*}}$ by
$p \forces ``\dot B \in \dot {\cal U}$''
iff $p \forces ``\dot B \subseteq
{(P_{\gk_1}(\gl))}^{V^\FR_*}$''
and there is $q \in j(\FR)$ such that
$q \ge j(p)$ ($q$ extends $j(p)$),
$|j(p) - q| = 0$,
$j(p) \rest {\gg} = q \rest {\gg} =
j(p) \rest {\gk_1} = q \rest \gk_1 = p$, and
$q \forces ``\la j(\b) \mid \b < \gl \ra \in j(\dot B)$''.
By \cite[Theorem 3.4]{Ma}, $\dot{\cal U}$ is a
well-defined term for a strongly compact measure over $
{(P_{\gk_1}(\gl))}^{{V^\FR_*}}
$ in ${V^\FR_*}$.
To see that $\forces_\FR ``\dot {\cal U}$ is normal'', let
$p \forces ``\dot f :
{(P_{\gk_1}(\gl))}^{{V^\FR_*}}
\to \gl$ is a function such that $\dot f(s) \in s$
for all $s \in \dot B$ where $\dot B \in \dot {\cal U}$''.
%Let $\la \varphi_\a \mid \a < \gl \ra$ be the sequence of
%statements in the forcing language with respect to
%$j(\FR)$, where $\varphi_\a$ is the statement
%$``j(\dot f)(\la j(\b) \mid \b < \gl \ra) = j(\a)$''.
Let $\varphi_\ga$ for $\ga < \gl$ be the statement
$``j(\dot f)(\la j(\b) \mid \b < \gl \ra) = j(\a)$''
in the forcing language with respect to $j(\FR)$, and
consider the sequence $\la \varphi_\a \mid \a < \gl \ra$.
Since $M^\gl \subseteq M$,
$\la \varphi_\a \mid \a < \gl \ra \in M$. Thus, since
$\gg$ is the least $M$ cardinal
in the half-open interval $[{\gk_1}, j({\gk_1}))$ at which
%$j(\FR)$
$\dot \FR'$ is forced to do nontrivial forcing and
$\gg > \gl$, we can apply \cite[Lemma 2.4]{Ma} in
$M$ to $\la \varphi_\a \mid \a < \gl \ra$ and obtain a condition
$q \ge j(p)$, $q \in j(\FR)$ such that
$|j(p) - q| = 0$,
$j(p) \rest {\gg} = q \rest {\gg} =
j(p) \rest {\gk_1} = q \rest \gk_1 = p$, and
%$j(p) \rest \gk_1 = q \rest \gk_1 =
%j(p) \rest {\gl} = q \rest {\gl} = p$,
if $q' \ge q$, $q'$ decides $\varphi_\a$ for some
$\a < \gl$, then
$q' \rest {\gk_1} \cup (q - p)$ decides $\varphi_\a$
in the same way. Hence, since
$p \forces ``\dot B \in \dot {\cal U}$'' implies we can assume
(by extending $q$ if necessary) that
$q \forces ``\la j(\b) \mid \b < \gl \ra \in j(\dot B)$'',
there must be some $\a < \gl$ such that for some
$q' \ge q$, $q' \forces \varphi_\a$, i.e., such that
$q' \forces ``j(\dot f)(\la j(\b) \mid \b < \gl \ra) = j(\a)$''.
By choice of $q$, $q' \rest {\gk_1} \cup (q - p) \forces \varphi_\a$,
i.e., $q' \rest {\gk_1} \ge p$ is such that for some
$r \in j(\FR)$ ($r$ can be taken as
$q' \rest {\gk_1} \cup (q - p))$,
$|j(q' \rest {\gk_1}) - r| = 0$,
$j(q' \rest {\gk_1}) \rest {\gk_1} = r \rest {\gk_1} =
q' \rest {\gk_1}$, and
$r \forces \varphi_\a$. Since $r \forces \varphi_\a$,
$r \forces
``\la j(\b) \mid \b < \gl \ra \in
j(\{ s \in \dot B \mid \dot f(s) = \a \})$'', so
$q' \rest {\gk_1} \ge p$ is such that $q' \rest {\gk_1} \forces
``\{ s \in \dot B \mid \dot f(s) = \a \} \in \dot {\cal U}$''.
Thus, $V^\FR_* \models ``\gk_1$ is $\gl$ supercompact''.
Since $\gl \ge \gk_2$ was arbitrary,
this completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FR_* \models ``$If $\gd < \gk_1$ is measurable, then
$\gd$ is ${<} \gs_\gd$ strongly compact''.
\end{lemma}
\begin{proof}
Suppose $V^\FR_* \models ``\gd < \gk_1$ is measurable''.
By \cite[Theorem 3.1]{Ma}, $V_* \models ``\gd$ is measurable''.
Further, it must be the
case that $V_* \models ``\gd$ is ${<} \gs_\gd$ supercompact''.
This is since otherwise, by the definition of $\FR$,
$V^\FR_* \models ``{\rm cof}(\gd) = \go$'', contradicting
the measurability of $\gd$ in $V^\FR_*$. Therefore,
again by the definition of $\FR$,
as in Lemma \ref{l3},
we may write
$\FR = \FR_\gd \ast \dot \FR^\gd$.
By the proof of \cite[Theorem 3.4]{Ma},
$V^{\FR_\gd}_* \models ``\gd$ is ${<} \gs_\gd$
strongly compact''.
Then, again as in Lemma \ref{l3}, because
\cite[Lemma 2.4]{Ma} implies that
$\forces_{\FR_\gd} ``$Forcing with $\dot \FR^\gd$
adds no new subsets of $\gs_\gd$'',
$V^{\FR_\gd \ast \dot \FR^\gd}_* =
V^\FR_* \models ``\gd$ is ${<} \gs_\gd$ strongly compact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
Since $\FR$ may be defined so that $\card{\FR} = \gk_1$,
$V^\FR_* \models ``\gk_2$ is inaccessible''.
Therefore, since
$V^\FR_* \models ``\gk_1$ is supercompact'',
by reflection, let $\gk < \gk_1$ be the least
cardinal which is ${<} \gs_\gk$ supercompact.
By \cite[Theorem 3.1]{Ma}, $V_* \models ``\gk$ is measurable''.
Further, because $V^\FR_* \models ``\gd < \gk_1$ has
cofinality $\go$ if $\gd$ is measurable but not
${<} \gs_\gd$ supercompact in $V_*$'', $V_* \models ``\gk$ is
${<} \gs_\gk$ supercompact''. Consequently,
$A \cap \gk$ is unbounded in $\gk$, and
as in the proof of Lemma \ref{l2},
%because $A$ is unbounded in $\gk_1$,
$V^\FR_* \models ``$No cardinal $\gd < \gk$ is
strongly compact''.
As in the proofs of Theorems \ref{t1} and \ref{t2},
if we now let
$\ov V = (V_{\gs_\gk})^{V^\FR_*}$,
$\ov V \models ``$ZFC + $\gk$ is supercompact +
$\gk$ is the least strongly compact cardinal +
Level by level inequivalence between strong compactness
and supercompactness holds + No cardinal
$\gd > \gk$ is inaccessible''.
In addition, since by the work of \cite{Ma},
the Magidor iteration of Prikry forcing preserves
both cardinals and the sizes of their power sets,
%$\FR$ preserves cofinalities, and is such that
%for any regular cardinal $\gd \le \gk_1$ and factorization
%$\FR = \FR_\gd \ast \dot \FR^\gd$,
%$\card{\FR_\gd} \le \gd$ and
%$\forces_{\FR_\gd} ``$Forcing with $\dot \FR^\gd$
%adds no new subsets of $\gs_\gd$''. Therefore,
in $V^\FR_*$ and $\ov V$, for every regular cardinal $\gd$,
$2^\gd = F(\gd)$.
This completes the proof of Theorem \ref{t3}
(with $\FP$ defined as $\FQ \ast \dot \FR$).
\end{pf}
\section{Concluding Remarks}\label{s3}
We observe that a key difference between the
proof of Theorem \ref{t3} and the proofs of
Theorems \ref{t1} and \ref{t2} is that in
Theorems \ref{t1} and \ref{t2}, we are able to
%take as an initial assumption that
assume initially, without loss of generality, that
every measurable cardinal $\gd$ is $\gs_\gd$ strongly compact.
This is accomplished by forcing over a model satisfying GCH.
Without GCH, the proofs from \cite{A01} do not go through
and allow us to assume that every measurable cardinal
satisfies this degree of strong compactness.
Since GCH will be false in $V_*$, a different approach
%becomes necessary
is used in order to prove Theorem \ref{t3}.
We note that it is possible to prove
Theorems \ref{t1} -- \ref{t3} using
slightly weaker hypotheses.
Theorems \ref{t1} and \ref{t2} may be established using
the existence of cardinals $\gl < \gk_1 < \gk_2$
such that
$\gl$ and $\gk_1$ are both $\gk_2$ supercompact
and $\gk_2$ is inaccessible.
Theorem \ref{t3} may be established using the existence of
cardinals $\gk_1 < \gk_2$ such that $\gk_1$ is
$\gk_2$ supercompact and $\gk_2$ is inaccessible.
To avoid excessive technicalities and simplify
our exposition, however,
we have established these theorems using
the hypotheses previously mentioned.
Finally, it is of course the case that each of
the models $\ov V$ constructed above has a
rather limited large cardinal structure.
By slightly modifying the proofs of Theorems \ref{t1} -- \ref{t3} and
truncating the universe not at $\gs_\gk$ but at the least weakly compact
cardinal above $\gk$, the least Ramsey cardinal above $\gk$, or in
general, at some suitable large cardinal which is provably below the
least measurable cardinal above $\gk$, it is possible to assume
that $\ov V$ has a nontrival, although still rather
restricted, large cardinal structure above $\gk$.
We consequently conclude by asking in general
what the possible large cardinal
structures are in a universe containing
supercompact cardinals which satisfies
level by level inequivalence between strong
compactness and supercompactness.
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\end{document}
Note that by Lemma \ref{l2} and the
L\'evy-Solovay results \cite{LS},
$V_{**} \models ``\gk_1$ is $\gl$ supercompact +
$\la \gk_i \mid i < \go \ra$ is a sequence of
measurable cardinals such that for each $i < \go$,
$\gk_{i + 1}$ is the least measurable cardinal
greater than $\gk_i$ +
$\gk_i$ is $\gk_{i + 1}$ strongly compact''.
Therefore, by reflection,
%for each $\gg < \gk_1$,
%there is a sequence $\la \gd_i \mid i < \go \ra$
%of measurable cardinals with $\gd_0 > \gg$
%having the properties that for each
%$i < \go$, $\gd_{i + 1}$ is the least measurable
%cardinal greater than $\gd_i$, $\gd_i$ is
%$\gd_{i + 1}$ strongly compact, and $\gd_0$ is
%$\gl'$ supercompact for $\gl' = \sup_{i < \go} \gd_i$.
%This means that
we can let $\gl_0 < \gl_1 < \cdots < \gl_n < \gk_1$
be a sequence of measurable cardinals such that
We finish with some additional questions.
First, we ask if it is possible to
prove Theorem \ref{t1} using somewhat
weaker hypotheses. Our current methods
of proof seem to require something along the
lines of the existence
of an $\go$ sequence of supercompact cardinals.
Second, we note that the large cardinal structure
of the model witnessing the conclusions of
Theorem \ref{t1} remains somewhat limited.
We conclude by asking if it is possible
to remove the restrictions inherent to our proof,
and obtain results
analogous to those of this paper in which
the large cardinal structure of the universe
can be arbitrary.