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\title{Level by Level Equivalence and
Strong Compactness
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, strong cardinal,
non-reflecting stationary set of ordinals,
level by level equivalence between
strong compactness and supercompactness.}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{June 5, 2003\\
September 24, 2003}
\begin{document}
\maketitle
\begin{abstract}
We force and construct
models in which there
are non-supercompact
strongly compact cardinals
which aren't measurable
limits of strongly compact
cardinals and in which level
by level equivalence between
strong compactness and
supercompactness holds
non-trivially
except at strongly compact cardinals.
In these models, every measurable
cardinal $\gk$ which isn't
either strongly compact or a witness to
a certain phenomenon first discovered
by Menas is such that for every
regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact.
%for every
%measurable cardinal which
%isn't strongly compact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
The notion of level by level equivalence
between strong compactness and
supercompactness was introduced
by Shelah and the author in
\cite{AS97a}, in which the following
theorem was proven.
\begin{theorem}\label{t0}
Let
$V \models ``$ZFC + $\K \neq \emptyset$
is the class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\K$ is the
class of supercompact cardinals +
For every pair of regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a limit of
cardinals which are $\gl$ supercompact''.
\end{theorem}
Models witnessing the last conclusion of
Theorem \ref{t0} are said to satisfy
level by level equivalence between
strong compactness and supercompactness.
%(or level by level equivalence between
%strong compactness and supercompactness
%for every measurable cardinal).
A measurable cardinal $\gk$ such that
for every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact is said
to witness
level by level equivalence between
strong compactness and supercompactness
at $\gk$.
A key feature of such models is that
the only time a measurable
cardinal $\gk$ can
be $\gl$ strongly compact and not
be $\gl$ supercompact when $\gl$
is regular is when
$\gk$ is a limit of
cardinals which are $\gl$
supercompact.\footnote{This is
not true if $\gl$ is singular, i.e.,
in a model for level by level
equivalence between strong
compactness and supercompactness
containing supercompact cardinals,
there will be cardinals
$\gk < \gl$ such that $\gk$ is
$\gl$ strongly compact, $\gk$
isn't $\gl$ supercompact,
$\gl$ is singular, and
$\gk$ isn't a limit of cardinals
which are $\gl$ supercompact.
Readers are urged to consult
Lemma 7 of \cite{AS97a} for
further details.}
This is since
by a result of Menas \cite{Me}, if
$\gk$ is the least measurable
cardinal which
is a limit of cardinals which are
either $\gk^+$ supercompact
or $\gk^+$ strongly compact,
%where $\gl$ is suitably definable,
$\gk$ is $\gk^+$ strongly compact but isn't
$2^\gk$ supercompact (which in the
case of GCH means that $\gk$
isn't $\gk^+$ supercompact).
Henceforth, any measurable
cardinal $\gk$
which is a limit of cardinals
which are $\gl$ supercompact
but which isn't $\gl$ supercompact
will be referred to as witnessing
the Menas exception at $\gl$.
%$2^{[\gl]^{< \gk}}$ supercompact.
%the least such $\gk$ must be $\gl$
%strongly compact but can't be $\gl$ supercompact.
%(Such a cardinal will henceforth
%be referred to as witnessing the
%Menas exception at $\gl$.)
Thus, in a model for level by
level equivalence between strong
compactness and supercompactness,
there can be non-supercompact
strongly compact cardinals, but
these must be rather ``large''
in the sense that all such
cardinals must be measurable
limits of supercompact cardinals.
This raises the following question:
Is it possible to have a model in
which there are non-supercompact
strongly compact cardinals
which aren't limits of strongly
compact cardinals
but which are limits of
measurable cardinals and in which
every measurable cardinal which
isn't strongly compact either
witnesses
level by level equivalence between
strong compactness and supercompactness
or some instance of the Menas exception?
%and in which level by level equivalence
%between strong compactness and
%supercompactness holds non-trivially
%for any measurable cardinal which
%isn't strongly compact?
Note that in Magidor's models
in which, for $n \in \go$,
there are exactly $n$
measurable cardinals and each
measurable cardinal is strongly
compact (see \cite{Ma} for
the case $n = 1$ and
\cite{AC1} for the case $n > 1$),
level by level equivalence between
strong compactness and supercompactness
holds vacuously for any measurable
cardinal which isn't strongly
compact, since no such cardinals exist.
Therefore, to avoid these sorts of
trivialities, we have asked the preceding
question so as to require all strongly
compact cardinals to be limits of
measurable cardinals.
Other trivial answers to the above
question if strongly compact
cardinals aren't required to be
limits of measurable cardinals
will be discussed below.
The purpose of this paper is to give
affirmative answers to the
aforementioned question.
Specifically, we prove the
following three theorems, where in
Theorem \ref{t3}, we take $\gk_0 = \go$.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + $\K \neq \emptyset$
is the class of supercompact
cardinals + $\gk$ is the
least supercompact cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is
the least strongly compact cardinal +
%$\gk$ isn't $2^\gk = \gk^+$ supercompact +
$\gk$ has trivial Mitchell rank but
is a limit of measurable cardinals +
$\K - \{\gk\}$ is the class of
supercompact cardinals +
Every measurable cardinal except for $\gk$
witnesses either level by level
equivalence between strong
compactness and supercompactness
or the Menas exception for some
$\gl > \gk$.
\end{theorem}
\begin{theorem}\label{t2}
Let
$V \models ``$ZFC + GCH +
$\K \neq \emptyset$
is the class of supercompact
cardinals +
Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal +
$\gk$ is the
least supercompact cardinal''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is
the least strongly compact cardinal +
$\gk$ is a limit of measurable cardinals
but isn't supercompact +
$\K - \{\gk\}$ is the class of
supercompact cardinals +
%No cardinal $\gd$ is both measurable
%and a limit of cardinals which
%are $\gd^+$ supercompact +
Level by level
equivalence between strong
compactness and supercompactness
%holds non-trivially both above and below $\gk$
holds for every measurable
cardinal except for $\gk$ +
$\gk$'s strong compactness is
indestructible under $\gk$-directed
closed forcing''.
\end{theorem}
\begin{theorem}\label{t3}
Let
$V \models ``$ZFC + For
$n \in \go$,
$\gk_1, \ldots, \gk_n$ are
supercompact +
%and are the only strongly compact cardinals +
No cardinal
$\gd > \gk_{n - 1}$ is $\gl$
supercompact for $\gl$
the least inaccessible
cardinal above $\gd$''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC +
$\gk_1, \ldots, \gk_n$ are
the only strongly compact
cardinals + No $\gk_i$ for
$i = 1, \ldots, n$ is supercompact +
%No cardinal $\gl > \gk_n$ is measurable +
For every $1 \le i \le n$ and every
$\gg \in (\gk_{i - 1}, \gk_i)$, there
is a cardinal $\gd \in [\gg, \gk_i)$
which is $\gd^{+ \gg}$ supercompact +
Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
except for $\gk_1, \ldots, \gk_n$''.
%non-strongly compact measurable cardinals''.
\end{theorem}
We take this opportunity to make
some remarks concerning
Theorems \ref{t1} - \ref{t3}.
In Theorem \ref{t1}, the structure
of the class of supercompact
cardinals in $V^\FP$
above the least strongly
compact cardinal $\gk$ can be
arbitrary and has no restrictions
placed upon it.
However, the only non-supercompact
strongly compact cardinal which
isn't a measurable limit of
supercompact cardinals is $\gk$.
This contrasts sharply with the
situation under Theorem \ref{t3},
where in $V^\FP$,
no cardinal is supercompact,
and since there are only
finitely many strongly compact
cardinals, no strongly compact
cardinal is a limit of
strongly compact cardinals.
It also contrasts sharply with the
situation under Theorem \ref{t2},
where in $V^\FP$,
there can be many supercompact
cardinals, but there is no measurable
limit of strongly compact cardinals
(meaning that the only non-supercompact
strongly compact cardinal is $\gk$).
It is also possible to prove a
trivial version of Theorem \ref{t1}
by forcing with Magidor's partial ordering
$\FP_M$ of \cite{Ma} which makes the
least strongly compact cardinal the
least measurable cardinal.
If this is done over a ground model $V$
for level by level equivalence between
strong compactness and supercompactness
in which $\gk$ is the least supercompact
cardinal and $\FP_M$ is defined
with respect to $\gk$, since
$|\FP_M| = \gk$, the L\'evy-Solovay results
\cite{LS} show that in
$V^{\FP_M}$, every measurable cardinal in
$\K - \{\gk\}$
witnesses either level by level equivalence
between strong compactness and supercompactness
or the Menas exception for some $\gl > \gk$.
Also, the only non-supercompact strongly
compact cardinal which isn't a limit of
supercompact cardinals is $\gk$.
In addition, by the work of \cite{Ma},
$\gk$ has become in
$V^{\FP_M}$ both the least strongly
compact and least measurable cardinal.
However, level by level equivalence
between strong compactness and
supercompactness holds only
trivially below $\gk$ in $V^{\FP_M}$,
since $\gk$ is the least measurable
cardinal in this model.
Theorem \ref{t2} shows that indestructibility
for a non-supercompact strongly compact
cardinal $\gk$ is consistent non-trivially
both above and below $\gk$
with level by level equivalence
between strong compactness and
supercompactness.
This extends Theorem $1$ of \cite{AG},
which shows that a supercompact
cardinal $\gk$ can be forced to be
both the least strongly compact
and least measurable cardinal
and have its strong compactness
indestructible under arbitrary
$\gk$-directed closed forcing.
Since this can be accomplished
by forcing with a partial
ordering having cardinality
$\gk$, the results of \cite{LS}
once again show that
if the ground model satisfies
level by level equivalence
between strong compactness
and supercompactness, the
generic extension will also,
except at $\gk$.
However, in this situation,
since there are no measurable
cardinals below $\gk$ in $V^\FP$,
level by level equivalence between
strong compactness and supercompactness
once more holds only trivially in $V^\FP$
below $\gk$.
Theorem \ref{t2} should also
be contrasted with Theorem $5$
of \cite{AH3}, which says that
if $\gk$ is an indestructible
supercompact cardinal and
level by level equivalence
between strong compactness and
supercompactness holds below $\gk$,
then no cardinal $\gl > \gk$ is
$2^\gl$ supercompact.
This is certainly not necessarily
true in the
model witnessing the conclusions
of Theorem \ref{t2}, in which
there can be a proper class of
supercompact cardinals above
the indestructible non-supercompact
strongly compact cardinal $\gk$.
We note also that
the next to last
%property mentioned in the statement
%of Theorem \ref{t1} that
%$V^\FP$ satisfies
conclusion of Theorem \ref{t3}
guarantees that in $V^\FP$,
between any two strongly compact
cardinals, level by level
equivalence between strong compactness
and supercompactness must hold
non-trivially.
%This is since between any two
%strongly compact cardinals in
%$V^\FP$, there must be unboundedly
%many cardinals $\gd < \gl$ such that
%$\gd$ is $\gl$ supercompact and
%$\gl$ is a Ramsey cardinal.
In fact, as our proof will indicate,
depending upon how $\FP$ has
been defined, in $V^\FP$,
there may be unboundedly many
cardinals between
any two strongly compact cardinals
satisfying even larger degrees of
supercompactness.
As an example, using the appropriate
definition of $\FP$, for every
$1 \le i \le n - 1$ and every
$\gg \in (\gk_{i - 1}, \gk_i)$,
there will be in $V^\FP$
cardinals $\gd < \gl$,
$\gd, \gl \in
(\gg, \gk_i)$ such that
$\gd$ is $\gl$ supercompact
and $\gl$ is a Ramsey cardinal
(and perhaps even more).
We will comment on this further
at the end of the paper.
Before giving the proofs of
Theorems \ref{t1} - \ref{t3}, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in standard interval notation.
When forcing, $q \ge p$ will
mean that $q$ is stronger than $p$.
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 $\varphi$ is a sentence in the
forcing language with respect to $\FP$,
$p \decides \varphi$ will mean that
$p$ decides $\varphi$.
If we also have that $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an Easton support iteration of length $\gk$
so that at some stage $\ga$,
a non-trivial forcing is done
based on an ordinal $\gg$, then we will say that
$\gg$ is in the field of $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time,
confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a cardinal and $\FP$ is
a partial ordering,
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
every two elements
$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
$\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is
$\gk^+$-directed closed, then $\FP$ is
$\gk$-strategically closed.
Also, if $\FP$ is
$\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$\FP$ is ${\prec}\gk$-strategically closed if in the
%two person game in which the players construct an increasing
%sequence
%$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
%stages and player
%II plays even and limit stages (again choosing the
%trivial condition at stage 0),
%then player II has a strategy which
%ensures the game can always be continued.
Suppose $\gk < \gl$ are regular cardinals.
A partial ordering $\FP_{\gk, \gl}$ that will be used
in this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP_{\gk, \gl} =
\{s : s$ is a bounded subset of
$\gl$ consisting of ordinals of cofinality
$\gk$ such that for every $\ga < \gl$,
$s \cap \ga$ is non-stationary in $\ga\}$,
ordered by end-extension.
%\{ p$ : For some
%$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
%function of $S_p$, a subset of $\ga$ not stationary at its
%supremum nor having any initial segment which is stationary
%at its supremum, so that $\gb \in S_p$ implies
%$\gb > \gk$ and cof$(\gb) = \gk \}$,
%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 $S_p$.
Two things which can be shown
(see \cite{Bu} or \cite{A01})
are that
$\FP_{\gk, \gl}$ is $\gd$-strategically
closed for every $\gd < \gl$, and
if $G$ is $V$-generic over $\FP_{\gk, \gl}$,
in $V[G]$,
%if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced.
%the bounded subsets of $\gl$ are the same as those in $V$,
%and cardinals, cofinalities, and GCH
%have been preserved.
It is also virtually immediate that $\FP_{\gk, \gl}$
is $\gk$-directed closed.
In addition, by Theorem 4.8 of \cite{SRK}
and the succeeding remarks,
after forcing with $\FP_{\gk, \gl}$,
no cardinal
$\gd \in (\gk, \gl]$ is $\gl$
strongly compact, so in particular,
no cardinal
$\gd \in (\gk, \gl]$
is strongly compact.
We mention that we are assuming familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{K}
or \cite{SRK}
for further details.
We do wish to point out, however, that
unlike \cite{K}, we will say that
the cardinal $\gk$ is $\gl$ strong
for $\gl > \gk$ if there is
$j : V \to M$ an elementary embedding having
critical point $\gk$ such that
$j(\gk) > |V_\gl|$ and $V_\gl \subseteq M$.
As always, $\gk$ is strong if $\gk$ is $\gl$
strong for every $\gl > \gk$.
Also, a measurable cardinal $\gk$ has
trivial Mitchell rank if
there is no elementary embedding
$j : V \to M$ witnessing the
measurability of $\gk$ for which
$M \models ``\gk$ is measurable''.
An ultrafilter ${\cal U}$ generating
this sort of embedding will be said
to have trivial Mitchell rank as well.
%doesn't have a normal measure
%concentrating on measurable cardinals.
A cardinal $\gk$ having trivial
Mitchell rank is neither
$2^\gk$ supercompact nor
$\gk + 2$ strong.
\section{The Proof of Theorem \ref{t1}}\label{s1}
We turn now to the proof of Theorem \ref{t1}
\begin{proof}
Let $V$, $\gk$, and $\K$
be as in the hypotheses of Theorem \ref{t1}.
Without loss of generality, by first
forcing over $V$ using the
partial ordering of \cite{AS97a} if necessary,
we also assume that in $V$,
both GCH and level by level equivalence
between strong compactness and
supercompactness hold.
We are now in a position to
define the partial ordering $\FP$
used in the proof of Theorem \ref{t1}.
Let $\eta < \gk$ be a fixed but
arbitrary regular cardinal
below the least inaccessible
cardinal in $V$.
$\FP$ is then defined as the
Easton support iteration of
length $\gk$ which
begins by adding a Cohen subset
of $\go$ and then adds,
to each cardinal $\gd < \gk$ which
is in $V$ a measurable limit of
strong cardinals, a non-reflecting
stationary set of ordinals of
cofinality $\eta$.
By an induction similar to the one
given in Lemma 8 of \cite{AS97a},
it is easily seen that forcing
with $\FP$ preserves cardinals,
cofinalities, and GCH.
Further, since $\FP$
is essentially the partial
ordering defined in
Proposition 2.1 of
\cite{AH4}, by Lemmas 2.3 - 2.5 of
\cite{AH4}, in $V^\FP$,
$\gk$ has become the least
strongly compact cardinal
and has trivial Mitchell rank.
In addition, by the
results of \cite{LS},
since $\FP$ may be defined so
as to have cardinality $\gk$,
in $V^\FP$, $\K - \{\gk\}$ is
the class of supercompact cardinals,
and every measurable cardinal
above $\gk$ witnesses either
level by level equivalence
between strong compactness and
supercompactness or the Menas
exception for some $\gl > \gk$.
%Thus, the proof of Theorem \ref{t1}
%is completed by the proof of
%the following lemma.
\begin{lemma}\label{l1}
$V^\FP \models ``$Every measurable
cardinal $\gd < \gk$
witnesses either
level by level equivalence between
strong compactness and
supercompactness or the
Menas exception for some $\gl > \gd$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gk$ is measurable''.
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
In Hamkins' terminology of
\cite{H1}, \cite{H2}, and
\cite{H3}, $\FP$ ``admits a gap at
$\ha_1$'', so by the Gap Forcing Theorem
of \cite{H2} and \cite{H3}, $\gd$
must be measurable in $V$.
Let
$A = \{\gg < \gd : \gg$ is in $V$
a measurable limit of strong cardinals$\}$.
Rewrite $\FP = \FP_A \ast \dot \FQ$, where
$\FP_A$ is the portion of $\FP$ whose
field is composed of ordinals at or below
$\gd$, and $\dot \FQ$ is a term for
the rest of $\FP$.
It must be the case that
$\gd \not\in {\rm field}(\FP_A)$, since if
$\gd \in {\rm field}(\FP_A)$, by the
definition of $\FP$,
$V^{\FP_A} \models ``\gd$ contains a
non-reflecting stationary set of
ordinals of cofinality $\eta$ and hence
isn't weakly compact''.
As the definition of $\FP$ implies that
$\forces_{\FP_A} ``\dot \FQ$ is
$\gd$-strategically closed (and much
more)'',
$V^{\FP_A \ast \dot \FQ} = V^\FP \models
``\gd$ contains a non-reflecting
stationary set of ordinals of cofinality
$\eta$ and hence isn't weakly compact''.
This contradiction means
$\gd \not\in {\rm field}(\FP_A)$,
i.e., $\gd$ isn't in $V$ a limit
of strong cardinals.
Hence,
$|\FP_A| < \gd$. Therefore,
by the results of \cite{LS},
$V^{\FP_A} \models ``\gd$ witnesses
either level by level
equivalence between strong compactness
and supercompactness or the Menas
exception for some $\gl > \gd$''.
We consider now the following two cases.
\bigskip
\setlength{\parindent}{0pt}
Case 1: For every regular $\gl > \gd$,
in both $V$ and $V^{\FP_A}$,
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact.
Note that any such $\gl$ must be
below the least $V$-strong cardinal
$\gd^*$ above $\gd$.
This is since otherwise,
$V \models ``\gd$ is $\gr$
supercompact for every $\gr < \gd^*$ and
$\gd^*$ is strong'', so by
Lemma 1.1 of \cite{A02},
$V \models ``\gd$ is supercompact'',
contradicting that $\gk$ is the
least supercompact cardinal in $V$.
Thus, as
$\forces_{\FP_A} ``\dot \FQ$ is
$\gd^*$-strategically closed and
$\gd^*$ is inaccessible'',
$V^{\FP_A \ast \dot \FQ} = V^\FP \models
``$For every regular $\gl > \gd$,
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact'', i.e., in
$V^\FP$, level by level equivalence between
strong compactness and supercompactness
holds at $\gd$.
\bigskip
Case 2: There is some $\gl > \gd$ for which
$\gd$ is $\gl$ strongly compact, $\gd$
isn't $\gl$ supercompact, and
$\gd$ is a limit of cardinals which are
$\gl$ supercompact in both
$V$ and $V^{\FP_A}$. By the
same argument as given in Case 1,
$\gl < \gd^*$, so again, as
$\forces_{\FP_A} ``\dot \FQ$ is
$\gd^*$-strategically closed and
$\gd^*$ is inaccessible'',
$V^{\FP_A \ast \dot \FQ} = V^\FP \models
``$There is some $\gl > \gd$ for which
$\gd$ is $\gl$ strongly compact, $\gd$
isn't $\gl$ supercompact, and
$\gd$ is a limit of cardinals which are
$\gl$ supercompact'', i.e., in
$V^\FP$, $\gd$ is a witness to
the Menas exception at $\gl$.
\setlength{\parindent}{1.5em}
\bigskip
Cases 1 and 2 complete the
proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l1a}
$V^\FP \models ``\gk$ is a
limit of measurable cardinals''.
\end{lemma}
\begin{proof}
We observe that
any $V$-measurable cardinal $\gd < \gk$
which isn't a limit of $V$-strong cardinals
must have the set $A$ as defined
in Lemma \ref{l1} bounded below $\gd$.
The factorization of $\FP$ as
$\FP_A \ast \dot \FQ$ given
in the proof of Lemma \ref{l1}
along with the proof of Lemma \ref{l1}
thus yield that $\gd$ is
measurable in $V^\FP$
(and in fact exhibits
in $V^\FP$ either the same degree of
level by level equivalence between
strong compactness and supercompactness
as it did in $V$ or the Menas exception
for some $\gl > \gd$).
Since there are unboundedly in $\gk$ many
such $\gd < \gk$,
this completes the proof of
Lemma \ref{l1a}.
\end{proof}
We note that by Theorem 4.8 of
\cite{SRK} and the succeeding
remarks, since
$V^\FP \models ``$Unboundedly
many cardinals
$\gd \in (\eta, \gk)$ contain
non-reflecting stationary sets
of ordinals of cofinality $\eta$'',
$V^\FP \models ``$Every cardinal
$\gd \in (\eta, \gk)$ has its
degree of strong compactness
bounded below $\gk$''. This means that
$V^\FP \models ``\gk$ isn't a witness
to the Menas exception for any
$\gl > \gk$''.
This observation, along with
Lemmas \ref{l1} and \ref{l1a},
complete the proof of Theorem \ref{t1}.
\end{proof}
We conclude Section \ref{s1}
by remarking that the large
cardinal structure below
$\gk$ in $V^\FP$ can be
rather complicated, and depends
to a great extent on the
large cardinal structure of
our ground model $V$.
%In fact, whatever degrees of
%strong compactness or
%supercompactness a cardinal
%at or above $\gk$ exhibits
%will be reflected unboundedly
%often below $\gk$ in both
%$V$ and $V^\FP$.
To see this,
suppose $\gd < \gk$ is a
$V$-strong cardinal which
isn't a limit of $V$-strong cardinals.
%Note that
As in the first sentence
of the proof of Lemma \ref{l1a},
any $V$-measurable cardinal
$\gg$ between $\gd$ and the least
$V$-strong cardinal $\gd^*$
above $\gd$
will have the set $A$
of Lemma \ref{l1}
bounded below it. Therefore,
since $\gd^*$ will reflect
unboundedly often below $\gd^*$
in $V$ the degrees of
supercompactness any
measurable cardinal $\eta$
above $\gd^*$
manifests in $V$ or any instance
of the Menas exception
occurring above $\gd^*$ in $V$,
and since by Lemma 2.1 of \cite{AC2},
there are unboundedly in $\gk$ many
$V$-strong cardinals below $\gk$,
by the proof of Lemma \ref{l1a},
there will be unboundedly in $\gk$ many
measurable cardinals $\gg$
in both $V$ and $V^\FP$
witnessing either level by level
equivalence between strong compactness
and supercompactness or the Menas
exception for some $\gl > \gg$.
%which are a $\gd$ as above.
%These cardinals will exhibit
%level by level
%equivalence between strong
%compactness and supercompactness
%non-trivially in $V^\FP$, and
%in fact, will exhibit the
%same amount of level by level
%equivalence between strong
%compactness and supercompactness
%in $V^\FP$ as they did in $V$.
This means there may be in $V^\FP$, e.g.,
unboundedly many below $\gk$ cardinals
$\gg$ which are measurable limits
of cardinals which are, say,
$\gg^{+ 17}$ supercompact.
On the other hand, there
will definitely be unboundedly
many below $\gk$ cardinals
$\gd$ which are in $V^\FP$, e.g.,
$\gd^{+ \gd + 1}$ supercompact.
To see this, note that by
the discussion given in Case 1
of Lemma \ref{l1}, since
$\gk$ is the least supercompact
cardinal in $V$, no cardinal
$\gg < \gk$ can be $\ga$
supercompact in $V$
for every $\ga < \gk$.
%since such a cardinal would itself
%have to be supercompact.
Thus, if $j : V \to M$ is an
elementary embedding witnessing the
$\gk^{+ \gk + 2}$ supercompactness
of $\gk$ such that
$M \models ``\gk$ is $\gk^{+ \gk + 1}$
supercompact but isn't
$\gk^{+ \gk + 2}$ supercompact'',
by closure,
it is also the case that
$M \models ``$No cardinal
$\gg < \gk$ is $\ga$ supercompact
for every $\ga < \gk$ and
$\gk$ isn't a witness to the
Menas exception for any $\gl > \gk$''.
Further, by elementarity,
$M \models ``$Level by level
equivalence between strong compactness
and supercompactness holds at $\gk$''.
This will reflect
%unboundedly often below $\gk$
to produce in both
$V$ and $V^\FP$ unboundedly many
below $\gk$ cardinals $\gd$ which
witness level by level equivalence
between strong compactness and
supercompactness
%are not witnesses to the Menas exception
%at $\gd$ for any cardinal $\gl > \gd$,
and which
are $\gd^{+ \gd + 1}$ supercompact
but aren't $\gd^{+ \gd + 2}$
supercompact. This results in a rich
exhibition of level by level
equivalence between strong
compactness and supercompactness
below $\gk$ in $V^\FP$.
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + GCH +
$\K \neq \emptyset$ is the
class of supercompact cardinals +
Level by level equivalence
between strong compactness and
supercompactness holds for
every measurable cardinal +
$\gk$ is the least supercompact cardinal''.
We therefore know that, in particular,
there is no measurable cardinal $\gd$ in $V$
which is a witness to the Menas
exception at $\gl$ for some $\gl > \gd$.
We are now in a position to define
the partial ordering $\FP$
that will be used in the
proof of Theorem \ref{t2}.
$\FP$ will be a modification
of the partial orderings
used in the proofs of
Theorems 1 and 2 of \cite{AG}.
We begin by fixing $f$ as a
Laver function \cite{L} for
$\gk$, i.e., as a function
$f : \gk \to V_\gk$ such that
for every $x \in V$ and any
$\gl \ge |{\rm TC}(x)|$, there
is some supercompact ultrafilter
and associated elementary embedding
$j : V \to M$ with $j(f)(\gk) = x$.
We also assume without loss of generality
that $f$ is non-trivial only on
cardinals $\gd$ which are
$\gd^*$ supercompact in $V$, where
$\gd^*$ is the least
beth fixed point above $\gd$.
We can now use $f$ to define an
Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$ of length $\gk$.
%in a manner roughly analogous
%to the definition given in \cite{AG}.
As in both \cite{AG} and \cite{L},
at each stage $\ga$, we choose an
ordinal $\gr_\ga$.
We begin with $\FP_0 = \{\emptyset\}$ and
$\gr_0 = 0$, and we take
$\gr_\ga = \bigcup_{\gb < \ga} \gr_\gb$ if
$\ga$ is a limit ordinal.
We then set
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for
$\{\emptyset\} \ast \{\dot \emptyset\}$, and
$\gr_{\ga + 1} = \gr_\ga$, except if we
are in one of the following two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: For all $\gb < \ga$,
$\gr_\gb < \ga$,
$f(\ga) = \la \dot \FQ, \gs \ra$
for $\dot \FQ$ such that
$\forces_{\FP_\ga} ``\dot \FQ$ is
$\ga$-directed closed'' and
$\gs \ge \max(|{\rm TC}(\dot \FQ)|,
\ga)$ is an ordinal, and
$\forces_{\FP_\ga} ``\ga$ is inaccessible''.
In this case,
$\FP_{\ga + 1} = \FP_\ga \ast
\dot \FQ \ast \dot \FQ'$ and
$\gr_{\ga + 1} = \gs$, where
$\dot \FQ'$ is
a term for $\{\emptyset\}$ if
$\forces_{\FP_\ga \ast \dot \FQ}
``$There are no measurable cardinals
in the interval $[\ga, \gs]$'',
but if
$\forces_{\FP_\ga \ast \dot \FQ}
``$There are measurable cardinals
in the interval $[\ga, \gs]$'', then
$\dot \FQ'$ is a term such that
$\forces_{\FP_\ga \ast \dot \FQ}
``\dot \FQ'$ is Magidor's iteration
\cite{Ma} of Prikry forcing which
destroys all measurable cardinals
$\gd \in [\ga, \gs]$ which are either
$\gd^*$ strongly compact or for which
level by level equivalence between
strong compactness and supercompactness
fails''.
More explicitly,
$\dot \FQ'$ is a term for the
following partial ordering. Let
$\la \gd_\gb : \gb < \eta \le \gs \ra$
enumerate all measurable cardinals in
$V^{\FP_\ga \ast \dot \FQ}$ which are
in the interval $[\ga, \gs]$. Let
$\FR_\gb$ be the Magidor iteration
defined through stage $\gb$, with
$\FR_0 = \{\emptyset\}$.
$\FR_{\gb + 1} = \FR_\gb \ast \dot \FS$,
where $\dot \FS$ is a term for the
trivial partial ordering
$\{\emptyset\}$, except if
$\forces_{\FR_\gb} ``\gd_\gb$ is either
$\gd_\gb^*$ strongly compact or
level by level equivalence between
strong compactness and supercompactness
fails for $\gd_\gb$''. If this
is the case, then
$\FR_{\gb + 1} = \FR_\gb \ast \dot \FS$,
where $\dot \FS$ is a term for Prikry
forcing over $\gd_\gb$ defined
with respect to some normal measure
over $\gd_\gb$.
\bigskip
Case 2: $\forces_{\FP_\ga} ``\ga$ is
$\ga^*$ strongly compact'', yet Case 1
doesn't hold.
In this case, $\gr_{\ga + 1} = \gr_\ga$,
and for some term $\dot \mu$ such that
$\forces_{\FP_\ga} ``\dot \mu$ is a
normal measure over $\ga$'',
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ
\ast \dot \FQ'$, where $\dot \FQ$ is
a term for $\{\emptyset\}$ and
$\forces_{\FP_\ga \ast \dot \FQ}
``\dot \FQ'$ is Prikry forcing defined
with respect to $\dot \mu$''.
\bigskip
\setlength{\parindent}{1.5em}
$\FP$ is ordered via the
usual Easton support iteration
ordering, with the exception
that, roughly speaking,
``the stems of Prikry conditions
are extended non-trivially only
finitely often.''
(This is as in \cite{AG},
using the ordering first given in
\cite{G86}; the more
precise definition of the
ordering used for $\FP$
%can be found
is given
in both of these papers.)
By the arguments found in \cite{AG},
which remain applicable
virtually unchanged here,
in $V^\FP$,
%no cardinal $\gd < \gk$ is strongly compact,
$\gk$ is the least strongly
compact cardinal, $\gk$ isn't supercompact
(and in fact, no cardinal $\gd < \gk$ is
either $\gd^*$ supercompact
or $\gd^*$ strongly compact),
and $\gk$'s strong compactness is
indestructible under $\gk$-directed
closed forcing.
Also, since $\FP$ may be defined
so as to have cardinality $\gk$,
as in the proof of Theorem \ref{t1},
the arguments of \cite{LS} tell us
that in $V^\FP$,
$\K - \{\gk\}$ is the class of
supercompact cardinals,
and level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
above $\gk$.
Thus, the proof of Theorem \ref{t2}
is completed by the proof of
the following two lemmas.
\begin{lemma}\label{l2}
$V^\FP \models ``$Level by level
equivalence between strong
compactness and supercompactness
holds for every measurable
cardinal below $\gk$''.
\end{lemma}
\begin{proof}
Suppose $\gd < \gk$ is
measurable in $V^\FP$. We consider
the following two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: There are $\ga$, $\gs$ with
$\ga \le \gd \le \gs$ such that
$\ga$ is a stage at which
Case 1 in the definition of $\FP$
occurs, where $\gs$ is the ordinal
associated with $\ga$ and
$\dot \FQ$, $\dot \FQ'$ are terms
for the associated partial
orderings. If this is the situation,
then by the definition of $\FP$,
$V^{\FP_\ga \ast \dot \FQ \ast \dot \FQ'} \models
``$Level by level equivalence between
strong compactness and supercompactness
holds at $\gd$ and no cardinal
$\gg \in [\ga, \gs]$
(including $\gd$) is $\gg^*$
strongly compact''. By writing
$\FP = \FP_\ga \ast \dot \FQ
\ast \dot \FQ' \ast \dot \FR$
and once again using the definition of
$\FP$, which ensures that the first
non-trivial stage of forcing after
$\ga$ is above $\gs$,
it must be the case that
$\forces_{\FP_\ga \ast \dot \FQ \ast \dot \FQ'}
``$Forcing with $\dot \FR$ adds no bounded
subsets of the least Mahlo cardinal above $\gd$''.
We are then able to infer immediately that
$V^{\FP_\ga \ast \dot \FQ \ast \dot \FQ'
\ast \dot \FR} = V^\FP \models ``$Level by
level equivalence between strong compactness
and supercompactness holds at $\gd$''.
\bigskip
Case 2: For any $\ga$, $\gs$ as in
Case 1 of the definition of $\FP$, where
$\gs$ is the ordinal associated with
$\ga$, either
$\ga \le \gs < \gd$ or
$\gd < \ga \le \gs$.
If this is the situation, let
$\eta > \gd$ be such that
$V^\FP \models ``\gd$ is $\eta$
strongly compact and $\eta$
is regular''.
We are not in Case 2 of the
definition of $\FP$ at stage
$\gd + 1$ of the definition of $\FP$,
since if we were, in both
$V^{\FP_{\gd + 1}}$ and $V^\FP$,
$\gd$ would have cofinality $\go$,
which contradicts that $\gd$ is
measurable in $V^\FP$.
As we are not in Case 1 of the
%definition of $\FP$,
proof of this lemma, we may hence
write $\FP = \FP_\gd \ast \dot \FP^\gd$,
where
%$\FP_\gd$ is $\FP$ defined up
%through stage $\gd$, and
$\gd \not\in {\rm field}(\dot \FP^\gd)$.
Since
$V^\FP \models ``\gd$ isn't $\gd^*$
strongly compact'' and
$\forces_{\FP_\gd} ``\dot \FP^\gd$ adds
no bounded subsets to the least Mahlo
cardinal above $\gd$ which must
also be above $\gd^*$'', it must
actually be true that
$V^{\FP_\gd} \models ``\gd$ is $\eta$
strongly compact'',
and $\eta < \gd^*$ in both
$V^\FP$ and $V^{\FP_\gd}$.
Therefore, if
$|\FP_\gd| < \gd$, by the results of
\cite{LS},
$V \models ``\gd$ is $\eta$ strongly
compact''.
By level by level equivalence between
strong compactness and supercompactness
at all measurable cardinals in $V$,
$V \models ``\gd$ is $\eta$ supercompact''.
The results of \cite{LS} and the
fact that forcing with $\FP^\gd$
over $V^{\FP_\gd}$ adds no
bounded subsets to the least
Mahlo cardinal above $\gd$ then allow us
to infer that in both
$V^{\FP_\gd}$ and
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP$,
$\gd$ is $\eta$ supercompact.
\setlength{\parindent}{1.5em}
Assume now $|\FP_\gd| \ge \gd$.
If this is so, then by
%the fact we are in Case 2 of
the fact we are not in Case 1
of the proof of this lemma and
the definition of $\FP$,
$|\FP_\gd| = \gd$, and
$\FP_\gd$ is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
This means that
since $\FP_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FP_\gd}$
(this follows since $\gd$ is
measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$
is a subordering of the
direct limit of
$\la \FP_\ga : \ga < \gd \ra$
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
$\gg \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gg$ in $V$.
Therefore, since the $\eta$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FP_\gd}$-regular cardinal
$\gg \in [\gd, \eta]$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FP_\gd}$,
and since the fact $\FP_\gd$
is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$
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
$\gg \in [\gd, \eta]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
%once again
then implies that
$\gd$ is $\eta$ strongly
compact in $V$.
As in the preceding paragraph,
$V \models ``\gd$ is $\eta$
supercompact''.
Let
$j : V \to M$ be an elementary
embedding witnessing the
$\eta$ supercompactness of
$\gd$ such that
$M \models ``\gd$ isn't
$\eta$ supercompact''.
As we observed in the
first paragraph of the
current case (Case 2),
$\eta$ is below
the least beth fixed point
above $\gd$ in both $V^\FP$ and
$V^{\FP_\gd}$. By the
fact $\gd$ is measurable
and hence Mahlo in $V$
and $\FP_\gd$
is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$,
$\eta$ is below
the least beth fixed point
above $\gd$ in $V$, and
therefore, by the closure
properties of $M$, in
$M$ as well.
The definitions of $\FP$
and $\FP_\gd$ now tell
us that the forcing done at
stage $\gd + 1$ in $M$ in the
definition of $j(\FP_\gd)$ is
trivial.
To see this, observe that since
$M \models ``\eta < \gd^*$'',
Case 1 in the definition of
$j(\FP)$ clearly doesn't occur
at stage $\gd + 1$ in $M$.
If Case 2 in the definition of
$j(\FP)$ occurred at stage $\gd + 1$
in $M$, it would have to be
the case that in $M$, since
$\FP_\gd = {(j(\FP))}_\gd$,
$\forces_{\FP_\gd} ``\gd$ is
$\gd^*$ strongly compact''.
We may then use the argument
given in the preceding paragraph
to infer that $\gd$ is
$\gd^*$ supercompact in $M$,
which contradicts that
$M \models ``\gd$ isn't
$\eta$ supercompact and
$\eta < \gd^*$''.
The fact we are not in Case 1
of the proof of this lemma
tells us this exhausts all
possibilities for what happens
at stage $\gd + 1$ in the
definition of $j(\FP)$ in $M$. Hence,
since GCH holds in $V$, we may
therefore now apply the argument
given in the proof of Lemma 1.5 of
\cite{G86} to infer that in both
$V^{\FP_\gd}$ and
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP$,
$\gd$ is $\eta$ supercompact.\footnote{An
outline of the argument that $\gd$ is
$\eta$ supercompact in
$V^{\FP_\gd}$ is as follows.
Let $G$ be $V$-generic over
$\FP_\gd$.
By GCH and the regularity of
$\eta$, we can let
$\la \gt_\ga : \ga < \eta^+ \ra$
be an enumeration in $V$ of all
canonical $\FP_\gd$-names for
subsets of
${(P_\gd(\eta))}^{V[G]} =
{(P_\gd(\eta))}^{M[G]}$.
Since the forcing done at stage
$\gd + 1$ in $M$ is trivial,
we can use Lemmas 1.4 and 1.2 of
\cite{G86} to define an increasing
sequence $\la p_\ga : \ga <
\eta^+ \ra$ of elements of
$j(\FP_\gd)/G$ such that for every
$\ga < \eta^+$,
$p_{\ga + 1} \decides
``\la j(\gb) : \gb < \eta \ra \in
j(\gt_\ga)$''.
The ultrafilter ${\cal U}$
defined over ${(P_\gd(\eta))}^{V[G]}$
by $x \in {\cal U}$ iff for some
$\ga < \eta^+$ and some
$\FP_\gd$-name $\gt_\ga$ for $x$,
$p_{\ga + 1} \forces_{j(\FP_\gd)/G}
``\la j(\gb) : \gb < \eta \ra \in
j(\gt_\ga)$'' witnesses the
$\eta$ supercompactness of $\gd$
in $V[G]$.}
As $\eta$ was arbitrary,
$V^\FP \models ``$Level by level
equivalence between strong
compactness and supercompactness
holds at $\gd$''.
\bigskip
Cases 1 and 2 complete the
proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l2a}
In $V^\FP$, $\gk$
is a limit of measurable cardinals.
\end{lemma}
\begin{proof}
Let $\ga$ be a non-trivial
stage in the definition of
$\FP$ at which Case 1
occurs, with $\gs$
the ordinal associated
with $\ga$.
Fix the least
$\gd > \gs$ such that in $V$,
$\gd$ is $\gd^{+ \gb}$ supercompact
for some suitably definable
$\gb$ below the least
beth fixed point above $\gd$
(such as, e.g., $\gb =
\ha_1 + 1$).
This sort of $\gd$ must have
$|\FP_\gd| < \gd$,
so by the proof given
in the first paragraph of Case 2
of Lemma \ref{l2},
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds at $\gd$ and
$\gd$ is $\gd^{+ \gb}$ supercompact''.
Since there are unboundedly in $\gk$ many
such $\gd < \gk$,
this completes the proof of Lemma \ref{l2a}.
\end{proof}
Lemmas \ref{l2} and \ref{l2a}
complete the proof of
Theorem \ref{t2}.
\end{proof}
We conclude Section \ref{s2}
by making two remarks.
First, the proof of
Lemma \ref{l2a} shows that
not only is $\gk$ a limit
of measurable cardinals
in $V^\FP$, but that there are
unboundedly many in $\gk$ cardinals
$\gd < \gk$ such that
$\gd$ is $\gd^+$ supercompact,
$\gd$ is $\gd^{++}$ supercompact,
$\gd$ is $\gd^{+ \ha_1 + 6}$ supercompact,
etc. This once again results in a
rich exhibition of level by level
equivalence between strong
compactness and supercompactness
below $\gk$ in $V^\FP$.
Finally, the definition
of $\gd^*$ as given in the proof
of Theorem \ref{t2} is totally
arbitrary.
With suitable modifications to
the proof of Theorem \ref{t2},
$\gd^*$ could be redefined as
$\gd^{+ 17}$, the second beth
fixed point above $\gd$, the
least Mahlo cardinal above $\gd$
(assuming there is a Mahlo cardinal
above $\gk$ in $V$), etc.
Readers are urged to look at the
possible choices for $\varphi$
in the proof of Theorem 2 of
\cite{AG} for further details.
\section{The Proof of Theorem \ref{t3}}\label{s3}
We turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Let
$V \models ``$ZFC + For $n \in \go$,
$\gk_1, \ldots, \gk_n$ are
supercompact +
%and are the only strongly compact cardinals +
No cardinal $\gd > \gk_{n - 1}$ is
$\gl$ supercompact for $\gl$
the least inaccessible cardinal above $\gd$''.
Without loss of generality,
by the results of \cite{A98},
we may also assume that in $V$,
each $\gk_i$ for $i = 1, \ldots, n$
satisfies GCH and
has its supercompactness indestructible
under $\gk_i$-directed closed forcing
and that $\gk_1, \ldots, \gk_n$ are
the first $n$ strongly compact cardinals.
We are now in a position to define the
partial ordering $\FP$ to be used
in the proof of Theorem \ref{t3}.
We will do this by first specifying
component partial orderings
$\FP_1, \ldots, \FP_n$ and then
letting $\FP$ be
$\FP_1 \times \cdots \times \FP_n$.
$\FP_n$ is taken as the partial ordering
$\FQ^* \ast \dot \FQ_0 \ast \dot \FQ_1$,
where $\FQ^*$ adds a Cohen
subset of $\gk_{n - 1}$,
$\dot \FQ_0$ is a term for the
partial ordering
forcing GCH and level by level
equivalence between strong
compactness and supercompactness
for all cardinals at or above
the least inaccessible cardinal
above $\gk_{n - 1}$
employing the partial ordering $\FQ$
%for the general case
of \cite{AS97a}
defined using components
guaranteeing that $\FQ$ is
$\gk_{n -1 }$-directed closed, and
$\dot \FQ_1$ is a term for
the Easton support iteration
which begins by adding a Cohen
subset of $\gk^{++}_{n - 1}$ and then
adds, to every $\gd \in (\gk_{n - 1}, \gk_n)$
which is in $V^{\FQ^* \ast
\dot \FQ_0}$ a measurable limit of
strong cardinals, a non-reflecting
stationary set of ordinals of cofinality
$\gk_{n - 1}$. Note that
$\FQ_1$ is essentially
the partial ordering
used in the proof of Theorem \ref{t1}
defined over $V^{\FQ^* \ast \dot \FQ_0}$.
%Taking $\gk_0 = \go$,
$\FP_i$ for $i = 1, \ldots, n - 1$ is
taken as
the Easton support iteration which
begins by adding a Cohen subset of
$\gk_{i - 1}$ and then adds a non-reflecting
stationary set of ordinals of cofinality
$\gk_{i - 1}$ to each $V$-measurable cardinal
$\gd \in (\gk_{i - 1}, \gk_i)$ for which
either GCH fails in $V$, or for which
there is no final segment of
cardinals less than or equal to $\gd$
on which level by level equivalence between
strong compactness and supercompactness
holds in $V$, or which is $\gl$ supercompact for
$\gl$ the least inaccessible cardinal
above $\gd$ in $V$.
Note that by Theorem 5 of \cite{AH3},
since $\gk_n > \gk_i$ is
supercompact and $\gk_i$
has its supercompactness indestructible
under $\gk_i$-directed closed
forcing, there are
unboundedly many in $\gk_i$
measurable cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$ in $V$
for which level by level equivalence
between strong compactness and
supercompactness fails, i.e., there are
unboundedly many in $\gk_i$
measurable cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$ in $V$
for which there is no final segment
of cardinals less than or equal to $\gd$
on which level by level equivalence
between strong compactness and
supercompactness holds.
Further, again using the fact
$\gk_n > \gk_i$, by reflection,
in $V$, there are unboundedly
many in $\gk_i$ cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
which are $\gl$ supercompact for
$\gl$ the least inaccessible
cardinal above $\gd$.
\begin{lemma}\label{l3}
$V^\FP \models ``$Level
by level equivalence
between strong compactness
and supercompactness holds
for every measurable cardinal
except possibly for
$\gk_1, \ldots, \gk_n$''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3} is
via a ``downwards induction'' on $n$.
By the definition of $\FP_n$ and
the proof of Theorem \ref{t1},
$V^{\FP_n} \models ``$Level by level
equivalence between strong compactness
and supercompactness holds for every
measurable cardinal in the interval
$(\gk_{n - 1}, \gk_n)$''. Since
%and every measurable cardinal above $\gk_n$''. Since
$|\FP_1 \times \cdots \times \FP_{n - 1}|
= \gk_{n - 1}$, by the results of \cite{LS},
$V^{\FP_n \times \FP_{n - 1} \times
\cdots \times \FP_1} = V^\FP
\models ``$Level by level
equivalence between strong compactness
and supercompactness holds for every
measurable cardinal in the interval
$(\gk_{n - 1}, \gk_n)$''.
%and every measurable cardinal above $\gk_n$''.
Assume now by induction that
$1 \le i < n$ and
$V^{\FP_n \times \FP_{n - 1}
\times \cdots \times \FP_{i + 1}} \models
``$Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
in the intervals $(\gk_{n - 1}, \gk_n),
\ldots, (\gk_i, \gk_{i + 1})$''.
%and every measurable cardinal above $\gk_n$''.
We show that
$V^{\FP_n \times \FP_{n - 1}
\times \cdots \times \FP_{i + 1}
\times \FP_i} \models
``$Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
in the intervals $(\gk_{n - 1}, \gk_n),
\ldots, (\gk_i, \gk_{i + 1}),
(\gk_{i - 1}, \gk_i)$''.
%and every measurable cardinal above $\gk_n$''.
Since it will be the case that
$|\FP_1 \times \cdots \times \FP_{i - 1}|
= \gk_{i - 1}$, the results of \cite{LS} will
once again yield that
$V^{\FP_n \times \cdots \times \FP_1} =
V^\FP \models
``$Level by level equivalence between
strong compactness and supercompactness
holds for every measurable cardinal
in the intervals $(\gk_{n - 1}, \gk_n),
\ldots, (\gk_i, \gk_{i + 1}),
(\gk_{i - 1}, \gk_i)$''.
%and every measurable cardinal above $\gk_n$''.
To do this, we first observe that
by the definition of $\FP_\ell$ for
$\ell = i + 1, \ldots, n$,
$\FP_n \times \cdots \times \FP_{i + 1}$
is $\gk_i$-directed closed.
This has as an immediate consequence that
the measurable cardinals in the
interval $(\gk_{i - 1}, \gk_i)$ exhibit
the same degrees of strong compactness
and supercompactness in both
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
and $V$, since otherwise, there must
be a cardinal $\gd \in (\gk_{i - 1}, \gk_i)$
which is in $V$
either $\gg$ strongly compact for every
$\gg < \gk_i$ or $\gg$ supercompact for
every $\gg < \gk_i$, so by Lemma 1.1 of
\cite{A02}, $\gd$ is either
strongly compact or supercompact in $V$,
contradicting the fact that
$\gk_1, \ldots, \gk_n$ are in $V$
the first $n$ strongly compact and
supercompact cardinals. Thus,
%the definition of $\FP_i$ remains the same
$\FP_i$ has the same properties
in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
meaning that in
$V^{\FP_n \times \cdots \times \FP_{i}}$,
there are no cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
which are measurable
in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
on which
either GCH fails, or which are
$\gl$ supercompact in $V$
and $V^{\FP_n \times \cdots \times \FP_{i + 1}}$
for $\gl$
the least inaccessible cardinal above
$\gd$, or for which there is a
final segment of cardinals less
than or equal to $\gd$ on
which level by level
equivalence between strong compactness
and supercompactness fails in $V$
and $V^{\FP_n \times \cdots \times \FP_{i + 1}}$.
Let now
$\gd \in (\gk_{i - 1}, \gk_i)$ be
a measurable cardinal in
$V^{\FP_n \times \cdots \times \FP_{i}}$.
Note that by its definition,
$\FP_i$ is in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
``mild'' with respect to $\gd$,
where as in \cite{H2} and \cite{H3},
a partial ordering $\FR$
in a model $\ov V \models {\rm ZFC}$
is mild
with respect to a cardinal $\gg$
in $\ov V$
iff every set of ordinals $x$ in
${\ov V}^\FR$ of size below $\gg$ has a
``nice'' name $\gt$ in $\ov V$
of size below $\gg$, i.e., in
$\ov V$,
there is a set $y$, $|y| < \gg$,
such that any ordinal forced by a condition
in $\FR$ to be in $\gt$ is an element of $y$.
Further,
in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
we can write
$\FP_i = \FR_0 \ast \dot \FR_1$, where
$|\FR_0| = \gk_{i - 1}$ and
$\forces_{\FR_0} ``\dot \FR_1$ is
$\gk^+_{i - 1}$ strategically closed''.
Hence,
by the results of \cite{H2} and \cite{H3},
if $\gg \ge \gd$ is a cardinal such that
$V^{\FP_n \times \cdots \times \FP_{i}} \models
``\gd$ is $\gg$ strongly compact'',
$V^{\FP_n \times \cdots \times \FP_{i + 1}} \models
``\gd$ is $\gg$ strongly compact'' as well.
Therefore, if $\gg$ is in addition regular,
by the last sentence of the preceding paragraph,
$V^{\FP_n \times \cdots \times \FP_{i + 1}}
\models ``\gd$ is $\gg$ supercompact'',
and $\gd$ isn't $\gl$ supercompact
for $\gl$ the least inaccessible cardinal
above $\gd$ in $V^{\FP_n \times \cdots \times \FP_{i + 1}}$.
In addition, in $V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
there must be a final
segment of cardinals less than or equal
to $\gd$ on which level by level
equivalence between strong compactness
and supercompactness holds, and GCH
at $\gd$ must be true as well.
If we therefore let
$j : V^{\FP_n \times \cdots \times \FP_{i + 1}}
\to M$ be an elementary
embedding witnessing the $\gg$
supercompactness of $\gd$ such that
$M \models ``\gd$ isn't $\gg$
supercompact'', then since $\gg$
is below the least inaccessible
cardinal above $\gd$ in
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
by the closure properties of $M$,
$M \models ``\gd$ isn't $\gz$
supercompact for $\gz$ the least
inaccessible cardinal above $\gd$''.
Further, by elementarity, if we let
$\eta < \gd$ be such that
$V^{\FP_n \times \cdots \times \FP_{i + 1}}
\models ``$For every measurable cardinal
$\gr \in (\eta, \gd]$, level by level
equivalence between strong compactness
and supercompactness holds'',
$M \models ``$For every measurable cardinal
$\gr \in (\eta, j(\gd)]$, level by level
equivalence between strong compactness
and supercompactness holds''.
In particular, in $M$, GCH and level by level
equivalence between strong compactness
and supercompactness hold at $\gd$, and
$\gd$ has a final segment of measurable
cardinals below it on which level by
level equivalence between strong compactness
and supercompactness holds.
Working in
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
if we let $\FP^*_i$ be ${(\FP_i)}_\gd$,
this means we can write
$j(\FP_i^*) = \FP_i^* \ast \dot \FR$, where
$\gd \not\in {\rm field}(\dot \FR)$ and
$\forces_{\FP_i^*} ``\dot \FR$ is
$\gg$-strategically closed''.
Therefore, since GCH holds at $\gd$ in
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
if $G_n, \ldots, G_{i + 1}$ is a sequence
of $V$-generic objects over
$\FP_n, \ldots, \FP_{i + 1}$ and
$G^*_i$ is $V$-generic over $\FP^*_i$,
the Product Lemma allows us in
$V[G_n \times \cdots \times G_{i + 1} \times G^*_i]$
to use the usual diagonalization
techniques (see, e.g., the proof
of Lemma 8.1 of \cite{AH3}) to construct an
$M[G_n \times \cdots \times G_{i + 1} \times G^*_i]$-generic
object over $\FR$ and lift $j$ to an
embedding witnessing the $\gg$ supercompactness
of $\gd$ in
$V[G_n \times \cdots \times G_{i + 1} \times G^*_i]$.
Since $\FP_i = \FP^*_i \ast \dot \FS$ where
$\forces_{\FP^*_i} ``\dot \FS$ is
$2^{[\gg]^{< \gd}}$-strategically closed'',
$V^{\FP_n \times \cdots \times \FP_i} \models
``\gd$ is $\gg$ supercompact and
$2^\gd = \gd^+$''. As
$|\FP_{i - 1} \times \cdots \times
\FP_1| = \gk_{i - 1} < \gd$, the results of
\cite{LS} tell us that
$V^{\FP_n \times \cdots \times
\FP_i \times \FP_{i - 1} \times \cdots
\times \FP_1} = V^\FP \models
``\gd$ is $\gg$ supercompact''.
Finally, since
$V \models ``$No cardinal
$\eta > \gk_{n - 1}$ is
$\gl$ supercompact for $\gl$
the least inaccessible cardinal
above $\eta$ and $\gk_n > \gk_{n - 1}$
is supercompact'',
$V \models ``$There are no inaccessible
cardinals above $\gk_n$''.
This means
$V^\FP \models ``$There are no inaccessible
cardinals above $\gk_n$'', so in addition,
$V^\FP \models ``$There are no measurable
cardinals above $\gk_n$''.
We thus have vacuously that
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds for every
measurable cardinal above $\gk_n$''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l3a}
$V^\FP \models ``$No cardinal
$\gd$ which is either above
$\gk_{n - 1}$ or is an
element of the interval
$(\gk_{i - 1}, \gk_i)$ for
$i = 1, \ldots, n - 1$
is $\gl$ supercompact for
$\gl$ the least inaccessible
cardinal above $\gd$''.
\end{lemma}
\begin{proof}
As in the proof of
Lemma \ref{l3}, we proceed
via a ``downwards induction''
on $n$.
By hypothesis,
$V \models ``$No cardinal
$\gd > \gk_{n -1}$ is $\gl$
supercompact for $\gl$
the least inaccessible
cardinal above $\gd$''.
Since we can write
$\FP_n = \FR_0 \ast \dot \FR_1$,
where $|\FR_0| = \gk_{n - 1}$ and
$\forces_{\FR_0} ``\dot \FR_1$ is
$\gk_{n - 1}^+$-strategically closed'',
the results of \cite{H1},
\cite{H2}, and \cite{H3} imply that
any cardinal $\gd > \gk_{n - 1}$
which is $\gg$ supercompact in
$V^{\FP_n}$ for any cardinal
$\gg$ had to have been $\gg$
supercompact in $V$ as well.
This immediately yields that
$V^{\FP_n} \models ``$No cardinal
$\gd > \gk_{n -1}$ is $\gl$
supercompact for $\gl$
the least inaccessible
cardinal above $\gd$''.
Since $|\FP_1 \times \cdots
\times \FP_{n - 1}| = \gk_{n - 1}$,
the results of \cite{LS} imply that
$V^{\FP_n \times \FP_{n - 1} \times
\cdots \times \FP_1} = V^\FP \models
``$No cardinal
$\gd > \gk_{n -1}$ is $\gl$
supercompact for $\gl$
the least inaccessible
cardinal above $\gd$''.
Assume now by induction that
$1 \le i < n$ and
$V^{\FP_n \times \FP_{n - 1}
\times \cdots \times \FP_{i + 1}} \models
``$No cardinal $\gd$ which is
%either above $\gk_{n - 1}$ or is
an element of one of the intervals
$(\gk_{n - 1}, \gk_n), \ldots,
(\gk_i, \gk_{i + 1})$ is
$\gl$ supercompact for $\gl$ the
least inaccessible cardinal above $\gd$''.
Since in either $V$ or
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
we can write
$\FP_i = \FR_0 \ast \dot \FR_1$,
where $|\FR_0| = \gk_{i - 1}$ and
$\forces_{\FR_0} ``\dot \FR_1$ is
$\gk_{i - 1}^+$-strategically closed'',
the results of \cite{H1},
\cite{H2}, and \cite{H3} again imply that
any cardinal $\gd > \gk_{i - 1}$
which is $\gg$ supercompact in
$V^{\FP_n \times \cdots \times \FP_{i}}$
for any cardinal
$\gg$ had to have been $\gg$
supercompact in $V^{\FP_n \times
\cdots \times \FP_{i + 1}}$ as well.
Since forcing with $\FP_i$ over
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
by the definition of $\FP_i$ destroys
all cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
which are $\gl$ supercompact in either
$V$ or $V^{\FP_n \times \cdots \times
\FP_{i + 1}}$ for $\gl$ the least
inaccessible cardinal above $\gd$,
this fact and the results of \cite{LS}
immediately yield that
$V^{\FP_n \times \cdots \times \FP_{i + 1}
\times \FP_i} \models
``$No cardinal $\gd$ which is
%either above $\gk_{n - 1}$ or is
an element of one of the intervals
$(\gk_{n - 1}, \gk_n), \ldots,
(\gk_i, \gk_{i + 1}),
(\gk_{i - 1}, \gk_i)$ is
$\gl$ supercompact for $\gl$ the
least inaccessible cardinal above $\gd$''.
Since $|\FP_1 \times \cdots
\times \FP_{i - 1}| = \gk_{i - 1}$,
the results of \cite{LS} imply that
$V^{\FP_n \times \FP_{n - 1} \times
\cdots \times \FP_1} = V^\FP \models
``$No cardinal $\gd$ which is
%either above $\gk_{n - 1}$ or is
an element of one of the intervals
$(\gk_{n - 1}, \gk_n), \ldots,
(\gk_i, \gk_{i + 1}),
(\gk_{i - 1}, \gk_i)$ is
$\gl$ supercompact for $\gl$ the
least inaccessible cardinal above $\gd$''.
This completes the proof of Lemma \ref{l3a}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$For every
$1 \le i \le n$ and every
$\gg \in (\gk_{i - 1}, \gk_i)$,
there is a cardinal
$\gd \in [\gg, \gk_i)$ which is
$\gd^{+ \gg}$ supercompact''.
\end{lemma}
\begin{proof}
By the remarks found at the
end of the proof of
Theorem \ref{t1} and the
definition of $\FP_n$,
in $V^{\FP_n}$, for every
$\gg \in (\gk_{n - 1}, \gk_n)$,
there is a cardinal
$\gd \in [\gg, \gk_n)$ which is
$\gd^{+ \gg}$ supercompact.
Since
$|\FP_{n - 1} \times \cdots
\times \FP_1| = \gk_{n - 1} < \gd$, the results
of \cite{LS} tell us that in
$V^{\FP_n \times \FP_{n - 1} \times
\cdots \times \FP_1} = V^\FP$, for every
$\gg \in (\gk_{n - 1}, \gk_n)$,
there is a cardinal
$\gd \in [\gg, \gk_n)$ which is
$\gd^{+ \gg}$ supercompact.
Suppose now $1 \le i < n$.
Since by our initial assumptions,
$\gk_i$ has its supercompactness
indestructible under $\gk_i$-directed
closed forcing and $\FQ^* \ast \dot
\FQ_0$ as given
in the definition of $\FP$ is
$\gk_i$-directed closed, $\gk_i$
is supercompact in
$V^{\FQ^* \ast \dot \FQ_0}$.
By the definition of
$\FQ^* \ast \dot \FQ_0$,
in $V^{\FQ^* \ast \dot \FQ_0}$, there is
a cardinal $\gr \in (\gk_{n -1}, \gk_n)$
which is $\gr^{+ \gr + 1}$ supercompact,
isn't $\gr^{+ \gr + 2}$ supercompact,
and has a final segment of cardinals
less than or equal to $\gr$ on which
both GCH and level by level equivalence
between strong compactness and
supercompactness hold.
This will reflect to produce in
$V^{\FQ^* \ast \dot \FQ_0}$
unboundedly many in $\gk_i$ cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
which are $\gd^{+ \gd + 1}$ supercompact,
aren't $\gd^{+ \gd + 2}$ supercompact,
and have a final segment of cardinals
less than or equal to $\gd$ on which
both GCH and level by level equivalence
between strong compactness and
supercompactness hold.
Since $\FQ^* \ast \dot \FQ_0$
and $\FP_n \times \cdots
\times \FP_{i + 1}$ are both $\gk_i$-directed
closed, there will be unboundedly many
in $\gk_i$ such cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$ in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$.
By the proof of Lemma \ref{l3}, there
must be in $V^\FP$
%$V^{\FP_n \times \cdots \times \FP_i}$
unboundedly many
$\gd \in (\gk_{i - 1}, \gk_i)$ which
are $\gd^{+ \gd + 1}$ supercompact.
%and for which level by level
%equivalence between strong compactness
%and supercompactness holds.
Thus, in $V^\FP$,
%$V^{\FP_n \times \cdots \times \FP_i}$,
for every $\gg \in (\gk_{i - 1}, \gk_i)$,
there must be a cardinal
$\gd \in [\gg, \gk_i)$ which is
$\gd^{+ \gg}$ supercompact.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$Each $\gk_i$ for
$i = 1, \ldots, n$ is a
non-supercompact strongly compact cardinal''.
\end{lemma}
\begin{proof}
As in the proof of Theorem \ref{t1}, in
$V^{\FP_n}$, $\gk_n$ is a non-supercompact
strongly compact cardinal, so since
%by the definition of $\FP_\ell$ for
%$i = 1, \ldots, n - 1$,
%$1 \le \ell \le n - 1$,
$|\FP_{n - 1} \times \cdots \times \FP_1|
= \gk_{n - 1} < \gk_n$,
the results of \cite{LS} yield that in
$V^{\FP_n \times \FP_{n - 1} \times
\cdots \times \FP_1} = V^\FP$,
$\gk_n$ is a
non-supercompact strongly compact cardinal.
In addition, by Lemma \ref{l4},
in $V^\FP$, every
$\gk_i$ for $1 \le i < n$ has a
cardinal above it which is inaccessible.
Therefore, no such
$\gk_i$ is supercompact in $V^\FP$, since if it
were, there would have to be unboundedly many
in $\gk_i$ cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$ which are
$\gl$ supercompact for $\gl$ the least
inaccessible cardinal above $\gd$,
which by Lemma \ref{l3a} is impossible.
Hence, the proof of Lemma \ref{l5} will
be complete once we have shown that
every $\gk_i$ for $1 \le i < n$ is
strongly compact.
We once again proceed via a
``downwards induction'' on $n$.
Let $1 \le i < n$, and assume
by induction that
$V^{\FP_n \times \cdots
\times \FP_{i + 1}} \models
``\gk_{i + 1}$ is strongly
compact''. Since
$\FP_n \times \cdots \times \FP_{i + 1}$
is $\gk_i$-directed closed, by indestructibility,
let $j : V^{\FP_n \times \cdots \times \FP_{i + 1}}
\to M$ be an elementary embedding witnessing the
$\gk_{i + 1}$ supercompactness of $\gk_i$ such that
$M \models ``\gk_i$ isn't $\gk_{i + 1}$
supercompact''.
Note that
$M \models ``\gk_{i + 1}$ isn't measurable'',
since otherwise,
$M \models ``\gk_i$ is $\gd$ supercompact
for every $\gd < \gk_{i + 1}$ and
$\gk_{i + 1}$ is measurable'', which
immediately implies that
$M \models ``\gk_i$ is
$\gk_{i + 1}$ supercompact''.
We therefore have that
$j(\FP_i) = \FP_i \ast \dot \FS
\ast \dot \FT$, where
$\dot \FS$ is a term for
$\FP_{\gk_{i - 1}, \gk_i}$
(this is since by our remarks
immediately preceding the
statement of Lemma \ref{l3}
there are unboundedly many
measurable cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
which do not satisfy level by level
equivalence between strong
compactness and supercompactness
in either $V$,
$V^{\FP_n \times \cdots \times
\FP_{i + 1}}$ or $M$),
and the field
of $\dot \FT$ is composed of
ordinals in the interval
$(\gk_{i + 1}, j(\gk_i))$
(this is since by the definition of
$\FP_{i + 1}$, the proofs of
Lemmas \ref{l3} and \ref{l3a},
and the closure properties of $M$,
there are no measurable cardinals
$\gd \in (\gk_i, \gk_{i + 1})$ for
which either GCH or level by level
equivalence between strong compactness
and supercompactness fails, or which
are $\gl$ supercompact for $\gl$
the least inaccessible cardinal
above $\gd$).
As the definition of
$\FP_n \times \cdots \times \FP_{i + 1}$
ensures that GCH holds at
both $\gk_i$ and $\gk_{i + 1}$ in
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
the argument given in the proof of
Lemma 4 of \cite{AC1}
(see also the arguments found
in the proofs of Lemma 3.2 of
\cite{A01}, Lemma 2.4 of \cite{AC2},
Lemma 2.3 of \cite{AH4}, and
Lemma 5.1 of \cite{AH3})
goes through
unchanged to show that
$V^{\FP_n \times \cdots \times
\FP_{i + 1} \times \FP_i} \models
``\gk_i$ is $\gk_{i + 1}$
strongly compact''.\footnote{An
outline of the argument is
as follows. Let
$\ov V = V^{\FP_n \times
\cdots \times \FP_{i + 1}}$.
Fix $k : M \to N$ an elementary
embedding generated by a normal
ultrafilter over $\gk_i$
in $N$ having trivial
Mitchell rank, and let
$h : \ov V \to N$ be defined by
$h = k \circ j$. Working in
$\ov V$, $h(\FP_i) =
\FP_i \ast \dot \FQ \ast \dot \FR$,
where the field of $\dot \FQ$
is composed of ordinals in the interval
$(\gk_i, k(\gk_i)]$, and the
field of $\dot \FR$ is composed
of ordinals in the interval
$(k(\gk_i), h(\gk_i))$.
If $G_0$ is $\ov V$-generic over
$\FP_i$, since
$N \models ``\gk_i$ isn't measurable''
and GCH holds at $\gk_i$ in $\ov V$, it is
possible to use the standard diagonalization
techniques to construct in $\ov V[G_0]$ an
$N[G_0]$-generic object $G_1$ over $\FQ$.
Since GCH holds at $\gk_{i + 1}$
in $\ov V$ and no cardinal
$\gd \in [\gk_i, \gk_{i + 1}]$
is in the field of $\dot \FT$,
we can again use the standard
diagonalization techniques to
construct in $\ov V[G_0]$ an $M$-generic
object for the term forcing
partial ordering associated with
$\dot \FT$ defined with respect to
$\FP_i \ast \dot \FS$, transfer it
using $k$, and realize the transferred
generic using $G_0 \ast G_1$ to
obtain an $N[G_0][G_1]$-generic
object $G_2$ for $\FR$.
$h$ then lifts to
$h : \ov V[G_0] \to N[G_0][G_1][G_2]$,
which witnesses the $\gk_{i + 1}$
strong compactness of $\gk_i$ in
$\ov V[G_0]$, meaning that
$\gk_i$ is $\gk_{i + 1}$ strongly
compact in
$V^{\FP_n \times \cdots \times \FP_i}$.}
As $|\FP_{i}| = \gk_i < \gk_{i + 1}$,
the results of \cite{LS} tell us that
$V^{\FP_n \times \cdots \times
\FP_{i + 1} \times \FP_i} \models
``\gk_{i + 1}$ is strongly compact''.
By a theorem of DiPrisco \cite{DH}, because
$V^{\FP_n \times \cdots \times
\FP_{i + 1} \times \FP_i} \models
``\gk_i$ is $\gk_{i + 1}$
strongly compact and $\gk_{i + 1}$
is strongly compact'',
$V^{\FP_n \times \cdots \times
\FP_{i + 1} \times \FP_i} \models
``\gk_i$ is strongly compact''.
By the results of \cite{LS}, since
$|\FP_{i - 1} \times \cdots
\times \FP_1| = \gk_{i - 1} < \gk_i$,
$V^{\FP_n \times \cdots \times
\FP_i \times \FP_{i - 1} \times
\cdots \times \FP_1} = V^\FP \models
``\gk_i$ is strongly compact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``\gk_1, \ldots, \gk_n$ are
the only strongly compact cardinals''.
\end{lemma}
\begin{proof}
As we remarked at the
end of the proof of Lemma \ref{l3},
$V^\FP \models ``$No cardinal
$\gd > \gk_n$ is either
measurable or inaccessible''.
In addition, if
$i = 1, \ldots, n$ and
$\gd \in (\gk_{i - 1}, \gk_i)$
is measurable in $V^\FP$,
Lemma \ref{l3} tells us that
level by level equivalence between
strong compactness and supercompactness
holds for $\gd$ in $V^\FP$, and Lemma \ref{l3a}
tells us that $\gd$ isn't $\gl$ supercompact
in $V^\FP$ for $\gl$ the least inaccessible
cardinal above $\gd$. Thus,
%since $\gk_i > \gl > \gd$,
$\gd$ isn't strongly compact in
$V^\FP$, since if it were, it would have
to be $\gl$ strongly compact in $V^\FP$
and consequently, by level by level
equivalence between strong compactness
and supercompactness, $\gl$ supercompact
in $V^\FP$.
Since any cardinal $\gd$ which is
measurable in $V^\FP$ must be
either a $\gk_i$ for
$i = 1, \ldots, n$ or an
element of $(\gk_{i - 1}, \gk_i)$ for
$i = 1, \ldots, n$,
this completes the proof of
Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l3} - \ref{l6} complete
the proof of Theorem \ref{t3}.
\end{proof}
We conclude Section \ref{s3}
and this paper with two remarks.
First, if the definition of $\FP_i$ for
$i = 1, \ldots, n - 1$ is changed
so that instead of adding
non-reflecting stationary sets of
ordinals of cofinality $\gk_{i - 1}$
to cardinals $\gd$ which
are $\gl$ supercompact for $\gl$
the least inaccessible cardinal
above $\gd$, non-reflecting
stationary sets of ordinals of
cofinality $\gk_{i - 1}$ are added
to cardinals $\gd$ which are
$\gl$ supercompact for
$\gl$ the least Ramsey limit of
Ramsey cardinals above $\gd$, then
our methods of proof tell us that in
$V^\FP$, for every
$i = 1, \ldots, n - 1$ and
$\gg \in (\gk_{i - 1}, \gk_i)$,
there will be cardinals $\gd < \gl$,
$\gd, \gl \in (\gg, \gk_i)$ such that
$\gd$ is $\gl$ supercompact and
$\gl$ is a Ramsey cardinal.
In fact, there will be unboundedly
many cardinals
$\gd \in (\gk_{i - 1}, \gk_i)$
with even greater degrees of
supercompactness.
This allows the non-strongly
compact measurable cardinals in
$V^\FP$ to manifest even more
degrees of level by level equivalence
between strong compactness and
supercompactness than
mentioned in the statement of
Theorem \ref{t3}.
Finally, the question remains
as to whether it is possible
to prove some sort of version of
Theorems \ref{t1} - \ref{t3} for
infinitely many strongly
compact cardinals, e.g.,
whether it is possible to
obtain a model in which
$\la \gk_n : n < \go \ra$ are
the first $\go$ strongly compact
cardinals, each
$\gk \in \la \gk_n : n < \go \ra$
isn't supercompact,
and every measurable cardinal
$\gd \not\in \la \gk_n : n < \go \ra$
witnesses either
level by level
equivalence between strong compactness
and supercompactness or the Menas
exception for some $\gl > \gd$,
regardless if
$\gk \in \la \gk_n : n < \go \ra$
is a limit of measurable cardinals.
We note that not only do the methods
of this paper fail to provide a
means of constructing such a model,
%to provide an
%answer to this intriguing question,
but that either a positive or
negative answer to this intriguing
question seems to be beyond the
scope of current set-theoretic
techniques.
%\section{Concluding Remarks}\label{s4}
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\end{document}
\begin{graveyard}
\begin{abstract}
Starting with a model in which
there are $n \in \go$
supercompact cardinals,
we force and construct a model in which
there are $n$
strongly compact cardinals and
no strongly compact cardinal
is supercompact.
Further, in this model,
level by level
equivalence between strong
compactness and supercompactness
holds non-trivially unboundedly often
between any two strongly compact cardinals.
%for any measurable cardinal which isn't strongly compact.
\end{abstract}
The structure of this paper
is as follows.
Sections \ref{s0} contains
our introductory comments and
preliminary remarks.
Section \ref{s1} contains
the proof of Theorem \ref{t1}
for the case $n = 1$.
Section \ref{s2} contains the
proof of Theorem \ref{t1}
for the case $n > 1$.
Section \ref{s3} contains our
concluding remarks.
\begin{lemma}\label{l1}
If $V \models ``\gd < \gl < \gk$
are such that $\gd$ is $\gl$
supercompact and $\gl$ is regular, then
$V^\FP \models ``\gd$ is $\gl$
supercompact''.
\end{lemma}
\begin{proof}
Fix $\gd$ and $\gl$ as in
the hypotheses of Lemma \ref{l1}.
Let
$A = \{\gg < \gd : \gg$ is in $V$
a measurable limit of strong cardinals$\}$.
Write $\FP = \FP_A \ast \dot \FQ$, where
$\FP_A$ is the portion of $\FP$ whose
field is composed of ordinals below
$\gd$, and $\dot \FQ$ is a term for
the rest of $\FP$.
\end{proof}
Assume now $|\FP_\gd| = \gd$.
If this is the case, then by
the definition of $\FP$,
there must either be a final
segment on which $\FP_\gd$
is a Magidor iteration of Prikry
forcing, or if not, $\FP_\gd$
is a full Gitik style iteration
of Prikry like forcings in the sense
of \cite{G86}.
In the former case,
(the proof of)
Theorem 3.1 of \cite{Ma}
tells us that every $\gd$-additive
uniform ultrafilter over a cardinal
$\gg \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gg$ in $V$.
In the latter case,
since $\FP_\gd$ satisfies
$\gd$-c.c$.$ 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})
tells us that every $\gd$-additive
uniform ultrafilter over a cardinal
$\gg \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gg$ in $V$.
Therefore, since the $\eta$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies by
Ketonen's theorem of \cite{Ke}
that every regular cardinal
$\gg \in [\gd, \eta]$ carries
a $\gd$-additive uniform ultrafilter,
the preceding two sentences imply
that this must be true in
$V$ as well.
Ketonen's theorem of \cite{Ke}
once again implies that
$\gd$ is $\eta$ strongly
compact in $V$.
As in the preceding paragraph,
$V \models ``\gd$ is $\eta$
supercompact''.
Because by Lemma \ref{l4}
there are measurable and hence
inaccessible cardinals above $\gk_i$ in
$V^{\FP_n \times \cdots \times \FP_{i}}$,
and because if $\gk_i$ were
supercompact in
$V^{\FP_n \times \cdots \times \FP_{i}}$,
this would be reflected to unboundedly
in $\gk_i$ many cardinals in
$(\gk_{i - 1}, \gk_i)$,
We now know that any measurable
cardinal below $\gk$ in $V^\FP$
must satisfy level by level
equivalence between strong
compactness and supercompactness.
The proof of Lemma \ref{l1} will
therefore be complete once we have
shown that there are indeed measurable
cardinals below $\gk$ in $V^\FP$.
To see that this is indeed so,
By Cases 1 and 2 just given,
we know that level by level
equivalence between strong compactness and
supercompactness holds below $\gk$
in $V^\FP$. The proof of Lemma \ref{l2}
will now be complete if we can show that
this equivalence is non-trivial.
$\gd$ isn't $\gd^{+ \ga + 1}$
supercompact, and
$\gd$ is not a limit of
non-trivial stages of forcing
(such as the least $\gd$ as just
specified above a non-trivial
stage of forcing).
Since
$|\FP_{i - 1} \times \cdots \FP_1| = \gk_i$,
the results of \cite{LS} tell us this
same fact must be true in
$V^{\FP_n \times \cdots \times \FP_i
\times \FP_{i - 1} \times \cdots \FP_1} =
V^\FP$ as well.
Since
$V \models ``$No cardinal $\gd > \gk_{n - 1}$
is $\gl$ supercompact for $\gl$ the
least inaccessible cardinal above $\gd$'',
$\gk_n > \gk_{n - 1}$, and $\gk_n$ is
supercompact,
$V \models ``$No cardinal $\gd > \gk_n$
is inaccessible''. Hence,
$V^\FP \models ``$No cardinal $\gd > \gk_n$
is inaccessible'', and consequently,
$V^\FP \models ``$No cardinal $\gd > \gk_n$
is strongly compact''.
there can be more than
one non-supercompact strongly
compact cardinal, but there are
only finitely many strongly compact
cardinals, and no cardinal is supercompact.
No cardinal
$\gd$ is both measurable and
a limit of cardinals which are
$\gd^+$ supercompact
$V$, $\gk$, and $\K$
be as in the hypotheses of Theorem \ref{t2}.
As in the proof of Theorem \ref{t1},
without loss of generality, by first
forcing over $V$ using the
partial ordering of \cite{AS97a} if necessary,
we also assume that in $V$,
both GCH and level by level equivalence
between strong compactness and
supercompactness hold.
Further, as no cardinal $\gd < \gk$ is
$\gd^*$ supercompact, no cardinal
$\gd < \gk$ is both measurable and
a limit of cardinals
which are $\gd^+$ supercompact, since
a cardinal $\gg < \gd$ which is
$\gd^+$ supercompact where
$\gd$ is measurable would
automatically have to be
$\gg^*$ supercompact.
no cardinal $\gd > \gk$ is both
measurable and a limit of cardinals
which are $\gd^+$ supercompact,
and the fact that in $V$,
no cardinal $\gg$ is both measurable
and a limit of cardinals which are
$\gg^+$ supercompact,
$\gd$ is not a witness to the Menas
exception at $\eta$ in $V$, and
suppose now $\gd < \gk$ is a
$V$-strong cardinal which
isn't a limit of $V$-strong cardinals.
Note that any $V$-measurable cardinal
$\gg$ between $\gd$ and the least
$V$-strong cardinal $\gd^*$
above $\gd$
will have the set $A$
of Lemma \ref{l1a}
bounded below
it as well. Therefore,
since $\gd$ will reflect
unboundedly often below $\gd$
in $V$ the degrees of
supercompactness any cardinal $\gg$
above $\gd$
manifests in $V$ or any instance
of the Menas exception
occurring above $\gd$ in $V$,
and since by Lemma 2.1 of \cite{AC2},
there are unboundedly in $\gk$ many
$V$-strong cardinals below $\gk$,
there will be unboundedly many in $\gk$
measurable cardinals $\gg$
in both $V$ and $V^\FP$
witnessing either level by level
equivalence between strong compactness
and supercompactness or the Menas
exception for some $\gl > \gg$.
%which are a $\gd$ as above.
%These cardinals will exhibit
%level by level
%equivalence between strong
%compactness and supercompactness
%non-trivially in $V^\FP$, and
%in fact, will exhibit the
%same amount of level by level
%equivalence between strong
%compactness and supercompactness
%in $V^\FP$ as they did in $V$.
Finally, although the proof
just given requires that
level by level equivalence
between strong compactness
and supercompactness holds
for every measurable cardinal
in $V$, we conjecture
that this restriction can
be removed, i.e., so that
%so that the
%class $\K$ of supercompact
%cardinals in $V$ can be
%arbitrary, i.e., so that
there may exist in $V$ a
cardinal $\gk$ which is both
measurable and a limit of
cardinals which are $\gk^+$
supercompact.
\end{graveyard}