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%
\title{On the Consistency Strength of Level by Level
Inequivalence
\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, nonreflecting
stationary set of ordinals, equiconsistency.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{This paper is dedicated to the memory of
Jim Baumgartner, a friend and inspiration
to all those who knew him.}
\thanks{The author wishes to thank Norman Perlmutter for
a helpful conversation on the subject matter of
this paper.}\\
% This paper is dedicated to the 60th
% birthdays of Peter Koepke and
% Philip Welch. It is truly a privilege
% to be able to contribute a paper to this
% Festschrift in their honor.}\\
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{October 22, 2013}
\date{January 17, 2014\\(revised August 2, 2014)}
\begin{document}
\maketitle
\begin{abstract}
We show that the theories ``ZFC + There is a supercompact
cardinal'' and ``ZFC + There is a supercompact cardinal +
Level by level inequivalence between strong compactness
and supercompactness holds'' are equiconsistent.
\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.
This can alternatively be stated by saying that
{\em level by level inequivalence between strong
compactness and supercompactness holds at every
measurable cardinal $\gd$}.
Models containing exactly one supercompact
cardinal in which level by level inequivalence
between strong compactness and supercompactness
holds may be found in %have been constructed in
\cite[Theorem 2]{A02}, \cite[Theorem 2]{A10},
\cite[Theorem 1]{A11}, \cite[Theorems 1--3]{A12},
\cite[Theorem 32(2)]{AGH}, and \cite[Theorem 1.1]{A13}.
%\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.}
(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.)
%For the purposes of this paper, this means that for
%any pair of regular cardinals $\gd < \gl$, $\gd$
%is $\gl$ strongly compact iff $\gd$ is $\gl$ supercompact.)
A key feature of all of these constructions, however,
is the use of hypotheses stronger in consistency strength
than ``ZFC + There is a supercompact cardinal''.
This prompts us to ask the following
\bigskip
\noindent Question: Are the theories ``ZFC + There is a supercompact
cardinal'' and ``ZFC + There is a supercompact cardinal +
Level by level inequivalence between strong compactness
and supercompactness holds'' equiconsistent?
\bigskip
The purpose of this paper is to answer the above Question
in the affirmative. Specifically, we prove the following.
\begin{theorem}\label{t1}
The theories
``ZFC + There is a supercompact
cardinal'' and ``ZFC + There is a supercompact cardinal +
Level by level inequivalence between strong compactness
and supercompactness holds'' are equiconsistent.
\end{theorem}
The proof of Theorem \ref{t1} raises the question of whether
there is something different about the models
for level by level inequivalence between strong
compactness and supercompactness constructed using
hypotheses stronger in consistency strength than
``ZFC + There is a supercompact cardinal''.
We will come back to this issue at the end of the paper.
Before beginning the proof of Theorem \ref{t1}, we very
briefly mention some preliminary information concerning
notation and terminology.
%We now very briefly give some
%preliminary information
%concerning notation and terminology.
%For anything left unexplained,
%readers are urged to consult \cite{A03},
%\cite{A01a},
%\cite{AS97a}, or \cite{AS97b}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
%and $p \decides \varphi$ means that
%{\em $p$ decides $\varphi$}.
%For $\gk$ a regular cardinal and $\gl$ an ordinal,
%$\add(\gk, \gl)$ is the standard partial ordering for adding
%$\gl$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
%$[\ga, \gb]$ and $(\ga, \gb]$ are as in standard interval notation.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
If $\FP$ is a reverse Easton iteration
such that at stage $\ga$, a nontrivial
forcing is done adding a subset
of $\gd$, then we will say that
{\em $\gd$ is in the field of $\FP$}.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
Suppose $\gk > \go$ is a regular cardinal.
The partial ordering
%$\FP$ is {\em $\gk$-directed closed} if
%every directed set of conditions
%of size less than $\gk$ has
%an upper bound.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga\mid \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even
(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\em ${\prec} \gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga\mid \ga < \gk\rangle$, where player I plays odd
stages and player II plays even
%(which of course includes limit)
stages (again choosing the trivial condition at stage 0),
player II has a strategy which ensures the game can always be continued.
$\FP$ is {\em ${<} \gk$-strategically closed} if
$\FP$ is $\gd$-strategically closed for every $\gd < \gk$.
Note that if $\FP$ is ${\prec} \gk$-strategically closed, then
$\FP$ is ${<} \gk$-strategically closed as well.
An example of a partial ordering
which is ${\prec} \gk$-strategically closed
and which will be used in the proof of Theorem \ref{t1} is
the partial ordering $\FP$ for adding a nonreflecting
stationary set of ordinals of cofinality $\go$ to $\gk$.
Specifically, $\FP = \{p \mid$ For some $\ga < \gk$,
$p : \ga \to \{0, 1\}$ 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, such that $\gb \in S_p$ implies $\gb > \go$ and
${\rm cof}(\gb) = \go\}$, 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 $S_p$.
For additional details, %and the exact definition,
readers are urged to consult \cite[second paragraph of
Section 1, page 106]{AS97a}.
%$\FP$ is {\em $(\gk, \infty)$-distributive} if
%given a sequence $\la D_\ga \mid \ga < \gk \ra$
%of dense open subsets of $\FP$,
%$\bigcap_{\ga < \gk} D_\ga$ is dense open as well.
%Note that if $\FP$ is $\gk$-strategically closed,
%then $\FP$ is $(\gk, \infty)$-distributive.
%Further, if $\FP$ is $(\gk, \infty)$-distributive and
%$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$ \FP$ is ${<}\gk$-strategically closed
%if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
We mention that
we are assuming complete familiarity with the notions of
measurability, strong compactness, and supercompactness.
Interested readers may consult \cite{K} %or \cite {SRK}
for further details.
We note only that all elementary embeddings witnessing the $\lambda$
supercompactness of $\k$ are presumed to come from some
fine, $\k$-complete, normal
ultrafilter ${\cal U}$ over $P_\k (\l) = \{ x \subseteq \l
\mid | x| < \k \}$, and all elementary embeddings witnessing the $\l$
strong compactness of $\k$ are presumed to come from
some fine, $\k$-complete ultrafilter ${\cal U}$ over
$P_\k(\l)$. An equivalent definition for $\k$
being $\l$ strongly compact is that there is an
elementary embedding $j : V \to M$ having critical
point $\k$ such that for any $x \subseteq M$ with
$|x| \le \l$, there is some $y \in M$ such that
$x \subseteq y$ and $M \models ``|y| < j(\k)$''.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorem \ref{t1}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\it admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2, H3},
$\FP$ is {\it mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size less than $\gk$ has
a ``nice'' name $\tau$
in $V$ of size less than $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{t2}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + $\gk$ is supercompact''.
Without loss of generality, by doing a preliminary
forcing and truncating the universe if necessary,
we assume in addition that
$V \models ``$GCH + There are no cardinals
$\gd < \gl$ such that $\gd$ is $\gg$ supercompact
for every $\gg < \gl$ and $\gl$ is measurable''.
%No cardinal $\gd$
%is supercompact up to a measurable cardinal''.
Note that this implies
$V \models ``$No cardinal $\eta > \gk$ is measurable''.
%For any cardinal $\gd$, let $\gl_\gd$ be the least
%singular strong limit cardinal above $\gd$ having cofinality $\gd$.
%Let $A = \{\gd < \gk \mid \gd$ is $\gg$ supercompact for every
%$\gg < \gl_\gd$ yet $\gd$ is not $\gl_\gd$ supercompact$\}$.
Let $B = \{\gd < \gk \mid \gd$ is measurable
and level by level inequivalence between
strong compactness and supercompactness holds at $\gd\}$.
It follows by a theorem of Magidor
(unpublished by him, but given as \cite[Lemma 7]{AS97a})
that $B$ is unbounded in $\gk$.
In fact, the proof of \cite[Lemma 7]{AS97a} actually shows
we may assume that $B \supseteq A$, where
$A = \{\gd < \gk \mid$ There exists $\gl > \gd$, $\gl < \gk$
such that $\gl$ has cofinality $\gd$,
$\gd$ is $\gg$ supercompact for every $\gg < \gl$,
$\gd$ is not $\gl$ supercompact, yet $\gd$ is $\gl$ strongly
compact$\}$.\footnote{The proof %A sketch of %this proof
that $A$ is unbounded in $\gk$
is as follows. Let $\gl > \gk$ be a singular
strong limit cardinal of cofinality $\gk$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of
$\gk$ such that $M \models ``\gk$ is not
$\gl$ supercompact''. Because $M^\gl \subseteq M$,
$M \models ``\gk$ is $\gg$ supercompact for every
$\gg < \gl$''.
Let now $\la \gg_\ga \mid \ga < \gk \ra$ be a sequence of
regular cardinals cofinal in $\gl$,
$\la \mu_\ga \mid \ga < \gk \ra$ be a sequence of
$\gk$-additive, fine ultrafilters over $P_\gk(\gg_\ga)$, and
$\mu$ be a $\gk$-additive measure over $\gk$.
If for any $A \subseteq P_\gk(\gl)$ and any $\ga < \gk$
we define
$A \rest \gg_\ga = A \cap P_\gk(\gg_\ga)$, then in analogy to
\cite[Lemma 7]{AS97a}) (see also the argument given in
\cite[(1) and (2) of Lemma 3]{DH}), the collection $\mu^*$
of subsets of $P_\gk(\gl)$ given by $A \in \mu^*$ iff
$\{\ga < \gd \mid A \rest \gg_\ga \in \mu_\ga\} \in \mu$
defines a $\gk$-additive, fine ultrafilter over $P_\gk(\gl)$.
Thus, $M \models ``\gk$ is $\gl$ strongly compact''.
By reflection, because
$M \models ``\gk$ is not $\gl$ supercompact'',
the set $A$ is unbounded in $\gk$.}
We are now in a position to define
the partial ordering $\FP$ with which we will force to
construct our model witnessing the conclusions of Theorem \ref{t1}.
$\FP$ is the reverse Easton iteration having length $\gk$ which
begins by adding a Cohen subset of $\go$ and then
%forcing with $\add(\go, 1)$ and then
does nontrivial forcing only when
$\gd < \gk$ is measurable and $\gd \not\in A$.
At such a stage, we force with the partial ordering
adding a nonreflecting stationary set of ordinals
of cofinality $\go$ to $\gd$.
Since by its definition, forcing with $\FP$ preserves
all cofinalities (and hence all cardinals as well), the expression
``$\cof(\gl) = \gd$'' may be written without fear of ambiguity.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is supercompact''.
\end{lemma}
\begin{proof}
Let $\gl > \gk$ be an arbitrary singular strong
limit cardinal having cofinality $\gk$.
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 $M \models ``\gk$ is not $\gl$ supercompact''.
Because $M^\gl \subseteq M$ and $\gl$ is a
strong limit cardinal in $M$, it follows that
$M \models ``\gk$ is $\gg$ supercompact for
every $\gg < \gl$''. By the proof %of \cite[Lemma 7]{AS97a},
found in the first footnote,
as $M \models ``{\rm cof}(\gl) = \gk$'',
$M \models ``\gk$ is $\gl$ strongly compact''.
Thus, $M \models ``$Level by level inequivalence between
strong compactness and supercompactness holds at $\gk$
and is witnessed by the singular strong limit
cardinal $\gl$ of cofinality $\gk$'', i.e.,
$M \models ``\gk \in j(A)$''.
Further, since $V \models ``$No cardinal $\eta > \gk$ is
measurable'' and $M^\gl \subseteq M$,
$M \models ``$No cardinal in the half-open interval
$(\gk, \gl]$ is measurable''.
This means that $j(\FP) = \FP \ast \dot \FQ$, where
$\gk$ is a trivial stage of forcing, and the
first nontrivial stage of forcing $\gd > \gk$
is such that $\gd > \gl$.
If we now let $G$ be $V$-generic over $\FP$ and
$H$ be $V[G]$-generic over $\FQ$, since $j '' G \subseteq G \ast H$,
standard arguments show that $j$ lifts in $V[G][H]$ to
$j : V[G] \to M[G][H]$. Further, because $\FP$ is a reverse
Easton iteration having length $\gk$, $\FP$ is $\gk$-c.c. Thus,
since $M^\gl \subseteq M$ in $V$,
$M[G]^\gl \subseteq M[G]$ in $V[G]$ as well.
As $\gd > \gl$, the definition of $\FP$
implies that $\FQ$ is ${<} \gl^+$-strategically closed in
$M[G]$. The fact $M[G]^\gl \subseteq M[G]$ in $V[G]$ implies that
$\FQ$ is also ${<} \gl^+$-strategically closed in
$V[G]$.
%Let $\gg \in (\gk, \gl)$ be an arbitrary cardinal.
Let $\gg > \gk$, $\gg < \gl$ be an arbitrary cardinal.
Since $\gl$ is a strong limit cardinal in
both $V[G]$ and $M[G]$ and $\FQ$ is
${<} \gl^+$-strategically closed in each of these models,
the supercompact ultrafilter $\U_\gg \in V[G][H]$ over
$(P_\gk(\gg))^{V[G]}$ defined by $x \in \U_\gg$ iff
$\la j(\ga) \mid \ga < \gg \ra \in j(x)$ is such that
$\U_\gg \in V[G]$. Hence, $V[G] \models ``\gk$ is $\gg$
supercompact for every $\gg < \gl$''. As $\gl > \gk$ was arbitrary,
$V[G] \models ``\gk$ is supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
Suppose $V^\FP \models ``\gd < \gk$ is measurable''.
Then for some $\gl > \gd$, $\gl < \gk$ such that
$\cof(\gl) = \gd$, $V^\FP \models ``\gd$ is $\gl$
strongly compact''.
\end{lemma}
\begin{proof}
Assume $\gd < \gk$ is such that
$V^\FP \models ``\gd$ is measurable''.
Write $\FP = \FP^0 \ast \dot \FP^1$, where
$\card{\FP^0} = \go$, $\FP^0$ is nontrivial,
and $\forces_{\FP^0} ``\dot \FP^1$ is $\ha_1$-strategically closed''.
By this factorization and its definition,
$\FP$ admits a gap at $\ha_1$.
Hence, by Theorem \ref{t2}, it must also be the case that
$V \models ``\gd$ is measurable''. Further, it must be true
that $\gd \in A$. This is since otherwise,
by the definition of $\FP$, $V^\FP \models ``\gd$ contains a
nonreflecting stationary set of ordinals of cofinality $\go$
and hence is nonmeasurable''. Consequently, there must be some
$\gl > \gd$, $\gl < \gk$ such that $\cof(\gl) = \gd$ and
$V \models ``\gd$ is $\xi$ supercompact for every $\xi < \gl$,
$\gd$ is not $\gl$ supercompact, yet $\gd$ is $\gl$ strongly compact''.
%We show now that
Let $\gg < \gl$ be such that $\gg = \gz^{++}$
for some cardinal $\gz \ge \gd$. Since $\gg < \gl$,
$V \models ``\gd$ is $\gg$ supercompact''.
%To begin,
Write $\FP = \FP_\gd \ast \dot \FP^\gd$.
By the definition of $\FP$, it is the case that
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is ${<} \eta$-strategically closed
for $\eta$ the least measurable cardinal above $\gd$''.
Since again by the definition of $\FP$,
$\card{\FP_\gd} \le \gd$, by the L\'evy-Solovay results \cite{LS},
$\eta$ is the least measurable cardinal above $\gd$ in
both $V$ and $V^{\FP_\gd}$.
Consequently, by the fact that
$V \models ``$There are no cardinals $\gs < \gr$ such that
$\gs$ is $\xi$ supercompact for every $\xi < \gr$
and $\gr$ is measurable'',
%and $V \models ``\gl$ is singular'',
$\gg, \gl < \eta$.
Thus, to show that either
$V^\FP \models ``\gd$ is $\gg$ strongly compact'' or
$V^\FP \models ``\gd$ is $\gl$ strongly compact'',
it suffices to prove that either
$V^{\FP_\gd} \models ``\gd$ is $\gg$ strongly compact''
or
$V^{\FP_\gd} \models ``\gd$ is $\gl$ strongly compact''.
We first demonstrate that
$V^{\FP_\gd} \models ``\gd$ is $\gg$ strongly compact'' by
considering the following two cases.
\bigskip
\noindent Case 1: $\card{\FP_\gd} < \gd$.
%In this situation,
By the results of \cite{LS},
$V^{\FP_\gd} \models ``\gd$ is $\gg$ supercompact''
(and so of course, $V^{\FP_\gd} \models ``\gd$
is $\gg$ strongly compact'').
\bigskip
\noindent Case 2: $\card{\FP_\gd} = \gd$.
We use an argument analogous to the one found
in the proof of \cite[Lemma 2.3]{AH03}, from which
we feel free to quote liberally when necessary. Specifically,
let $k_1 : V \to M$ be an elementary
embedding witnessing the $\gg$ supercompactness
of $\gd$
%generated by a supercompact ultrafilter over $P_\gd(\gg)$
such that $M \models ``\gd$ is not $\gg$ supercompact''.
%By the choice of $\gg$, the
%cardinal $\gd$ is ${<}\gg$ supercompact in $M$.
Since $\gg > \gd^+ = 2^\gd$, we know that
$\gd$ is measurable in $M$.
Therefore, there is a normal measure
%is a normal measure of trivial Mitchell rank
over $\gd$ in $M$ yielding an embedding $k_2 : M
\to N$ with critical point $\gd$ such that
$N \models ``\gd$ is not measurable''.
It is easy to verify using the
embedding definition of $\gg$
strong compactness given in Section \ref{s1}
that the composed embedding
$j = k_2 \circ k_1:V\to N$ witnesses
%has the $\gg$-cover property, and therefore witnesses
the $\gg$ strong compactness of $\gd$. We will
show that $j$ lifts to $j : V^{\FP_\gd} \to
N^{j(\FP_\gd)}$. This lifted embedding will
witness the $\gg$ strong compactness of $\gd$ in
$V^{\FP_\gd}$.
To do this, factor $j(\FP_\gd)$ as $\FP_\gd \ast \dot \FQ
\ast \dot \FR$, where $\dot \FQ$ is a term for
the portion of $j(\FP_\gd)$ from stage $\gd$ up to
and including stage $k_2(\gd)$, and $\dot \FR$ is
a term for the rest of $j(\FP_\gd)$, from stage
$k_2(\gd)+1$ up to $j(\gd)$. Since $N
\models ``\gd$ is not measurable'', we know that
$\gd \not\in {\rm field}(\dot \FQ)$.
Further, by GCH, $M \models ``\gd$ is $\gz^+$
supercompact''. Consequently,
because $M \models ``\gd$ is not $\gg = \gz^{++}$ supercompact'', there
cannot be some $M$-singular cardinal $\gr > \gd$ such that
$M \models ``\gd$ is $\xi$ supercompact for every
$\xi < \gr$, $\gd$ is not $\gr$ supercompact,
yet $\gd$ is $\gr$ strongly compact''.
This means that $\gd \not\in k_1(A)$ and
$k_2(\gd) \not\in k_2(k_1(A))$.
Thus,
the field of $\dot \FQ$ is composed of a subset of the
$N$-measurable cardinals in the interval $(\gd, k_2(\gd)]$
(and in particular, $k_2(\gd)$ is
in the field of $\dot \FQ$), and the field of $\dot
\FR$ is composed of a subset of the $N$-measurable cardinals
in the interval $\bigl(k_2(\gd), k_2(k_1(\gd))\bigr)$.
Let $G_0$ be $V$-generic over $\FP_\gd$. We will
construct in $V[G_0]$ an $N[G_0]$-generic object
$G_1$ over $\FQ$ and an $N[G_0][G_1]$-generic
object $G_2$ over $\FR$. Since $\FP_\gd$ is an Easton
support iteration of small forcing, with a direct
limit at stage $\gd$ and no forcing right at
stage $\gd$, the construction of $G_1$ and $G_2$
ensures that $j '' G_0 \subseteq G_0 \ast G_1
\ast G_2$. It follows that $j : V \to N$ lifts to
$j : V[G_0] \to N[G_0][G_1][G_2]$ in $V[G_0]$.
To build $G_1$, note that since $k_2$ is
generated by an ultrafilter ${\cal U}$ over $\gd$
and GCH holds in both $V$ and $M$, we
know %$|k_2(2^{[\gg]^{< \gd}})| =
$|k_2(2^\gd)| = |k_2(\gd^+)| = |\{ f \mid f :
\gd \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gd| = \gd^+$. Thus, as $N[G_0]
\models ``|\wp(\FQ)| = k_2(2^\gd)$'', we can let
$\la D_\ga \mid \ga < \gd^+ \ra$ be an enumeration
in $V[G_0]$ of the dense open subsets of $\FQ$
present in $N[G_0]$. Since the $\gd$ closure of
$N$ with respect to either $M$ or $V$ implies
that the least element of the field of $\FQ$ is
above $\gd^+$, the definition of $\FQ$ as the
Easton support iteration which adds a
nonreflecting stationary set of ordinals of
cofinality $\go$ to the appropriate subset
of the $N$-measurable cardinals in the interval
$(\gd, k_2(\gd)]$ implies that $N[G_0]
\models ``\FQ$ is
${\prec} \gd^+$-strategically closed''. Since the
standard arguments show that forcing with the
$\gd$-c.c$.$ partial ordering $\FP_\gd$ preserves
that $N[G_0]$ remains $\gd$-closed with respect
to either $M[G_0]$ or $V[G_0]$, we know that
$\FQ$ is ${\prec} \gd^+$-strategically closed in
both $M[G_0]$ and $V[G_0]$. We now construct
$G_1$ in either $M[G_0]$ or $V[G_0]$ as follows.
Fix a winning strategy ${\cal S}$
for player II in the game
of length $\gd^+$ for the partial ordering
$\FQ$ and use it to construct a play $\$ of the game.
%(Note: our convention
%is that player II plays all even moves, including move $q_0$, which is
%required to be the trivial condition of $\FQ$.)
Since player II's moves are determined by
${\cal S}$, we need only specify the moves of the
first player. Specifically, if player II has just played the
condition $q_{2\alpha}$ at the (even) stage
$2\alpha$, player I selects and
then plays a condition $q_{2\alpha+1}$ above
$q_{2\alpha}$ from the dense set $D_\alpha$ using ${\cal S}$.
Since ${\cal S}$ is used at limit stages, this
completes the construction of the play
$\$. Let $G_1 = \{p
\in \FQ \mid \exists \ga < \gd^+\, (q_\ga \ge p)\}$
be the filter generated by this increasing
sequence of conditions. By construction, this
filter meets all the dense sets $D_\alpha$, and
so it is $N[G_0]$-generic over $\FQ$.
It remains to construct in $V[G_0]$ the desired
$N[G_0][G_1]$-generic object $G_2$ over $\FR$. To
do this, we first observe that as $M \models
``\gd \not\in k_1(A)$'',
we can factor $k_1(\FP_\gd)$ as $\FP_\gd
\ast \dot \FS \ast \dot \FT$, where
$\dot \FS$ is a term for the partial ordering
adding a nonreflecting stationary set of ordinals
of cofinality $\go$ to $\gd$,
%$\forces_{\FP_\gd} ``\dot \FS = \dot \FP_{\eta,\gd}$'',
%adds a nonreflecting stationary set of
%ordinals of cofinality $\eta$ to $\gd$'',
and $\dot \FT$ is a term for the rest of
$k_1(\FP_\gd)$.
Since $M \models ``$There are no measurable
cardinals in the half-open interval
$(\gd, \gg]$'' (which follows because
$\gg < \eta$), the field of $\dot \FT$ is
composed of a subset of the
$M$-measurable cardinals in the interval $(\gg, k_1(\gd))$.
This implies that in $M$, $\forces_{\FP_\gd \ast
\dot \FS} ``\dot \FT$ is ${\prec}
\gg^+$-strategically closed''. Further, since
$\gg$ is regular, GCH implies that
$|{[\gg]}^{< \gd}| =
\gg$. By GCH, we know that
$2^\gg = \gg^+$. Therefore, as $k_1$ is
generated by an ultrafilter over
$P_\gd(\gg)$, we may calculate $|2^{k_1(\gg)}|^M
= |k_1(2^\gg)| = |k_1(\gg^+)| = |\{ f \mid f :
P_\gd(\gg) \to \gg^+$ is a function$\}| =
|{[\gg^+]}^\gg| = \gg^+$.
Work until otherwise specified in $M$. Consider
the ``term forcing'' partial ordering $\FT^*$
(see \cite{F} for the first published account of
term forcing or \cite[Section 1.2.5, page 8]{C};
the notion is originally due to Laver) associated
with $\dot \FT$, i.e., $\tau \in \FT^*$
essentially iff $\tau$ is a term in the forcing
language with respect to $\FP_\gd \ast \dot \FS$ and
$\forces_{\FP_\gd \ast \dot \FS} ``\tau \in \dot
\FT$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP_\gd \ast \dot \FS} ``\tau \ge
\sigma$''. Since this definition, taken
literally, would produce a proper class, we
restrict the terms appearing in it to a
sufficiently large set-sized collection (such that
any term $\tau$ forced by the trivial condition
to be in $\dot\FT$ will be forced by the trivial
condition to be equal to an element of $\FT^*$)
of size $k_1(\gg)$ in $M$.\footnote{In the
official definition of $\FT^*$, the basic idea is
to include only the canonical terms. Since $\dot
\FT$ is forced to have cardinality $k_1(\gg)$,
there is a set $\{\tau_\alpha \mid
\alpha \gd$.
By Cases 1 and 2 above, for each $\gl^{++}_\ga$,
we may choose a $\gd$-additive, fine ultrafilter
$\mu_\ga \in V^{\FP_\gd}$ over $P_\gd(\gl^{++}_\ga)$.
Because $V^{\FP_\gd} \models ``{\rm cof}(\gl) = \gd$'',
%Let $\mu \in V^{\FP_\gd}$
%be a $\gd$-additive measure over $\gd$.
%For any $A \subseteq P_\gd(\gl)$, define
%$A \rest \gl^{++}_\ga = A \cap P_\gd(\gl^{++}_\ga)$.
%\{p \cap P_\gd(\gl_\ga) \mid p \in A\}$.
by the argument given in the first footnote,
%In analogy to \cite[Lemma 7]{AS97a}
%(see also the argument given in
%\cite[(1) and (2) of Lemma 3]{DH}), the collection $\mu^*$
%of subsets of $P_\gd(\gl)$ defined in $V^{\FP_\gd}$ by
%$A \in \mu^*$ iff
%$\{\ga < \gd \mid A \rest \gl^{++}_\ga \in \mu_\ga\} \in \mu$
%defines a $\gd$-additive, fine ultrafilter over $P_\gd(\gl)$, i.e.,
$V^{\FP_\gd} \models ``\gd$ is $\gl$ strongly compact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level inequivalence between
strong compactness and supercompactness holds''.
\end{lemma}
\begin{proof}
Suppose $\gd$ is such that $V^\FP \models
``\gd$ is measurable''. Since $\FP$ may be defined such that
$\card{\FP} = \gk$ and $V \models ``$No cardinal $\eta > \gk$
is measurable'', %by the L\'evy-Solovay results \cite{LS},
by the results of \cite{LS},
$\gd \le \gk$. Since by Lemma \ref{l1},
$V^\FP \models ``\gk$ is supercompact'', we assume without
loss of generality that $\gd < \gk$. Further, by the proof of
Lemma \ref{l2}, we know that
there must be some
$\gl > \gd$, $\gl < \gk$ such that $\cof(\gl) = \gd$ and
$V \models ``\gd$ is $\gg$ supercompact for every $\gg < \gl$,
$\gd$ is not $\gl$ supercompact, yet $\gd$ is $\gl$ strongly compact''.
By Lemma \ref{l2}, $V^\FP \models ``\gd$ is $\gl$ strongly compact''.
In addition, by the factorization of $\FP$ given in Lemma \ref{l2} and
Theorem \ref{t2}, if $\gg$ is any cardinal such that
$V^\FP \models ``\gd$ is $\gg$ supercompact'', then
$V \models ``\gd$ is $\gg$ supercompact'' as well.
Hence, since $V \models ``\gd$ is not
$\gl$ supercompact'', $\gg < \gl$,
so every degree of supercompactness witnessed by $\gd$ in $V^\FP$ must
be below $\gl$. This means that $V^\FP \models ``$Level by level
inequivalence between strong compactness and supercompactness
holds at $\gd$'', so since $\gd$ was arbitrary,
$V^\FP \models ``$Level by level inequivalence between strong compactness
and supercompactness holds''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Since clearly Con(ZFC + There is a supercompact cardinal +
Level by level inequivalence between strong compactness
and supercompactness holds) $\implies$ Con(ZFC + There is
a supercompact cardinal),
Lemmas \ref{l1} -- \ref{l3} complete the proof of Theorem \ref{t1}.
\end{proof}
In conclusion to this paper, we return to the question raised before
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
All of the previous models have the property that when
a non-supercompact measurable cardinal $\gd$ witnesses
level by level inequivalence in $V^\FP$
(or in some intermediate generic extension
used to construct the final model $V^\FP$),
\hfill\break$\ $\break\noindent
(*) there is always some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that
for every $\gg \in [\gr, \gl)$, $\gd$ is $\gg$ strongly compact
yet $\gd$ is not $\gg$ supercompact.
\hfill\break$\ $\break\noindent
The construction given here does not necessarily do this,
and relies on singular instances of
level by level inequivalence between strong
compactness and supercompactness.
To see that there may be measurable cardinals
for which property (*) does not hold,
we may suppose by the results of
\cite{AS97a} and by truncating the universe if necessary
that our ground model $V$ is such that
$V \models ``$ZFC + GCH + $\gk$ is supercompact + There are
no cardinals $\gd < \gl$ such that $\gd$ is $\gg$ supercompact
for every $\gg < \gl$
and $\gl$ is inaccessible + For every pair of regular cardinals
$\gd < \gl$, $\gd$ is $\gl$ strongly compact iff $\gd$
is $\gl$ supercompact''.
Let $\FP$ be as before.
It will still be true that $V^\FP$ is a model for
level by level inequivalence between strong compactness and supercompactness.
However, in $V^\FP$, no measurable cardinal
will satisfy property (*).
This follows since if $\gd$ is measurable and
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that for every $\gg \in [\gr, \gl)$,
$\gd$ is $\gg$ strongly compact yet $\gd$ is not
$\gg$ supercompact, then
by the factorization of $\FP$ given in the proof of Lemma \ref{l2}
and the fact $\FP$ is mild with respect to $\gd$,
it must also be the case that for every $\gg \in [\gr, \gl)$,
$V \models ``\gd$ is $\gg$ strongly compact''.
By our assumptions on $V$, we have in addition that
$V \models ``\gd$ is $\gg$ supercompact''.
This means that in $V$,
$\gl > \gd$ is inaccessible and
such that $\gd$ is $\gg$ supercompact
for every $\gg \in [\gr, \gl)$,
which contradicts our assumptions on $V$
that there are no cardinals $\gd < \gl$ such that
$\gd$ is $\gg$ supercompact for every $\gg < \gl$
and $\gl$ is inaccessible.
We therefore %ask
end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every non-supercompact
measurable cardinal $\gd$,
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that for every $\gg \in [\gr, \gl)$,
$\gd$ is $\gg$ strongly compact
yet $\gd$ is not $\gg$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
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\end{document}
The same argument as found in the proofs of
\cite[Lemma 2.4]{AC2} and \cite[Lemma 2.3]{AH03}
now shows that $V^{\FP_\gd} \models ``\gd$ is
$\gl$ strongly compact''. An outline
of this argument is as follows. Let
$k : M \to N$ be an elementary embedding
generated by a normal measure ${\cal U} \in M$
having trivial Mitchell rank. The elementary
embedding $i = k \circ j$ witnesses the
$\gl$ strong compactness of $\gd$ in $V$.
As in the proof of Lemma \ref{l4}, this
embedding lifts in $V^{\FP_\gd}$
to an elementary embedding
$i : V^{\FP_\gd} \to N^{i(\FP_\gd)}$
witnessing the $\gl$ strong compactness of $\gd$.
This is shown by writing $i(\FP_\gd) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$,
%$j(\FP_\gd) = \FP_\gd \ast \dot \FR^1 \ast \dot \FR^2$,
taking $G_0$ to be
$V$-generic over $\FP_\gd$, and building in
$V[G_0]$ generic objects $G_1$ and $G_2$
for $\FQ^1$ and $\FQ^2$ respectively.
The construction of $G_1$ is the same as
that found for the generic object $G_1$
in the proof of Lemma \ref{l4}.
The construction of $G_2$ first requires
building an $M$-generic object $G^{**}_2$
for the term forcing partial ordering
$\FT$ associated with $\dot \FR^2$ and defined
in $M$ with respect to $\FP_\gd \ast
\dot \FR^1$.
%Unlike the argument given in the
%proof of Lemma \ref{l4} for $G^{**}_2$, however,
The current $G^{**}_2$ is built using the fact that since
$M^\gl \subseteq M$, $\FT$ is ${\prec}\gl$-strategically
closed in both $M$ and $V$, which means
that the diagonalization argument employed
in the construction of $G_1$ may be applied
in this situation as well.
$k '' G^{**}_2$ now generates an $N$-generic
object $G^{***}_2$ for $k(\FT)$ and an
$N[G_0][G_1]$-generic object $G_2$ for $\FQ^2$
%in the same way that $k '' G^{**}_2$ does
as in the proof of Lemma \ref{l4}. This means that
$i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
In all of the earlier models constructed, when
a measurable cardinal $\gd$ witnesses level by level
inequivalence, there is always some maximal {\em regular}
cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact. In our situation, this is
never going to be true.
%not necessarily going to be true
%(e.g., when in the ground model $V$,
%To see this,
This follows since by the proof of Lemma \ref{l3},
in the model $V^\FP$ witnessing the conclusions of Theorem \ref{t1},
for any measurable cardinal $\gd < \gk$,
there is always a singular limit cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
(So in particular, for any regular $\gg < \gl$ such that
$V^\FP \models ``\gd$ is $\gg$ strongly compact yet
$\gd$ is not $\gg$ supercompact'', $\gg^+ < \gl$ is also such that
$V^\FP \models ``\gd$ is $\gg^+$ strongly compact yet
$\gd$ is not $\gg^+$ supercompact.)
We therefore end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every measurable cardinal $\gd$, there is
some maximal regular cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
In all of the earlier models constructed, when
a measurable cardinal $\gd$ witnesses level by level
inequivalence, there is always some {\em regular}
cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact. In our situation, this may
not be true.
To see this, suppose that our ground model $V$,
in addition to satisfying the properties stipulated
earlier, also is as in \cite{AS97a}, i.e., satisfies
level by level equivalence between strong compactness
and supercompactness. Let $\FP$ be as in the proof of Theorem \ref{t1}.
In $V^\FP$, let $\gd$ be the least member of $A$,
i.e., $\gd$ is the least cardinal below
our supercompact cardinal $\gk$ such that for some
$\gl > \gd$ having cofinality $\gd$, $\gd$ is $\gg$ supercompact for
every $\gg < \gl$, $\gd$ is not $\gl$ supercompact, yet $\gd$ is $\gl$
strongly compact. We claim that
$V^\FP \models ``$There is no regular cardinal $\xi$ such that
$\gd$ is $\xi$ strongly compact yet $\gd$ is not $\xi$ supercompact''.
This follows since if $\xi$ were such a cardinal, then
by the choice of $\gd$, $\xi \ge \gl^+$.
%Without loss of generality, we may assume that $\xi = \gl^+$.
By Theorem \ref{t2} and the fact that
$\xi$ must be regular in $V$,
%and the fact that forcing with $\FP$
%preserves all cardinals and cofinalities,
it would have to be the case that
$V \models ``\gd$ is $\xi$ strongly compact''.
%, and $\gd$ is not $\xi$ supercompact''.
Since level by level equivalence between strong compactness and
supercompactness holds in $V$,
$V \models ``\gd$ is $\xi$ supercompact''.
We therefore end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every measurable cardinal $\gd$, there is
some maximal regular cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier in which level by level inequivalence between
strong compactness and supercompactness holds.
In fact, there is a key difference.
In all of the earlier models,
for any measurable cardinal $\gd$
at which level by level inequivalence holds,
there is always some regular cardinal $\gl > \gd$
such that $\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
The construction given here does not necessarily do this,
and relies in many cases on singular instances of
level by level inequivalence between strong
compactness and supercompactness.
To see that there may be measurable cardinals
for which there is no regular witness to
level by level inequivalence, we may suppose by the results of
\cite{AS97a} and by truncating the universe if necessary
that our ground model $V$ is such that
$V \models ``$ZFC + GCH + $\gk$ is supercompact + There are
no cardinals $\gd < \gl$ such that $\gd$ is $\gl$ supercompact
and $\gl$ is inaccessible + For every pair of regular cardinals
$\gd < \gl$, $\gd$ is $\gl$ strongly compact iff $\gd$
is $\gl$ supercompact''.
Let $\FP$ be as before.
It will still be true that $V^\FP$ is a model for
level by level inequivalence between strong compactness and supercompactness.
However, in $V^\FP$, there will be unboundedly in
$\gk$ many measurable cardinals $\gd$ such that for no regular cardinal
$\gl > \gd$ is it the case that $\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
In fact, any $\gd \in A$ for $A$ as earlier
(defined in $V^\FP$) is such a measurable cardinal.
This follows since if $\gd \in A$, then there is some singular
$\eta > \gd$ of cofinality $\gd$
such that $\gd$ is $\gg$ supercompact for every $\gg < \eta$,
$\gd$ is not $\eta$ supercompact, yet $\gd$ is $\eta$ strongly compact.
If there were some regular $\gl > \eta$ such that
$\gd$ is $\gl$ strongly compact in $V^\FP$ (and of necessity, since
$\gl > \eta$, $\gd$ is not $\gl$ supercompact in $V^\FP$), then
by the factorization of $\FP$ given in the proof of Lemma \ref{l2}
and the fact $\FP$ is mild with respect to $\gd$,
it must also be the case that
$V \models ``\gd$ is $\gl$ strongly compact''.
By our assumptions on $V$, we have in addition that
$V \models ``\gd$ is $\gl$ supercompact''. However,
We therefore %ask
end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every non-supercompact
measurable cardinal $\gd$,
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
When all of the earlier models were constructed,
essential use was made of the property that when
a measurable cardinal $\gd$ witnesses
level by level inequivalence in $V^\FP$,
there is always some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gg$ supercompact.
The construction given here does not do this,
and relies on singular witnesses to
level by level inequivalence between strong
compactness and supercompactness.
We therefore %ask
end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every non-supercompact
measurable cardinal $\gd$,
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gg$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
compact$\}$.\footnote{A sketch of %this proof
the proof that $A$ is unbounded in $\gk$
is as follows. Let $\gl > \gk$ be a singular
strong limit cardinal of cofinality $\gk$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of
$\gk$ such that $M \models ``\gk$ is not
$\gl$ supercompact''. Because $M^\gl \subseteq M$,
$M \models ``\gk$ is $\gg$ supercompact for every
$\gg < \gl$''. By the argument from \cite[Lemma 7]{AS97a})
(which will be given in some detail in
the last paragraph of the proof of Lemma \ref{l2}),
since $M \models ``\gk$ is $\gg$ supercompact for
every $\gg < \gl$ and $\gl$ has cofinality greater
than or equal to $\gk$'',
$M \models ``\gk$ is $\gl$ strongly compact''.
By reflection, because
$M \models ``\gk$ is not $\gl$ supercompact'',
the set $A$ is unbounded in $\gk$.}