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\title{Failures of GCH and the Level by Level Equivalence
between Strong Compactness and Supercompactness
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal, measurable 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{November 18, 2002\\
June 4, 2003}
\begin{document}
\maketitle
\begin{abstract}
We force and obtain three models
in which level by level equivalence
between strong compactness and
supercompactness holds and in which,
below the least supercompact cardinal,
GCH fails unboundedly often.
In two of these models, GCH fails
on a set having measure 1
with respect to certain
canonical measures.
There are no restrictions in all of
our models
%ground models or our generic extensions
on the structure of the
class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
Because of the lack of technology for
constructing any sort of inner model for
a supercompact cardinal, it is necessary
to study the structure of universes in
which supercompact cardinals exist via
forcing.
There are many examples of papers in
which this has been done, e.g.,
\cite{A97}, \cite{AS97a}, \cite{AS97b},
\cite{KM}, and \cite{Ma}.
Of special interest are theorems
in this vein which, in terminology
used by Woodin,
%to describe the main theorem of \cite{AS97a},
can be classified as
``inner model theorems proven via forcing''.
Such theorems concern models
which satisfy pleasant properties
one usually associates with
inner models, such as GCH.
In particular, the main theorem of
\cite{AS97a}
%in which the study of level by level equivalence between
%strong compactness and supercompactness was initiated by
%Shelah and the author,
is a theorem
which falls into this category.
More specifically, this theorem states
the following.
\begin{theorem}\label{t0}
Let
$V \models ``$ZFC + $\K$ 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 measurable limit
of cardinals $\gd$ which are $\gl$
supercompact''.
\end{theorem}
%As was mentioned in \cite{A02},
In any model witnessing the conclusions of
Theorem \ref{t0}, we will say that
level by level equivalence between
strong compactness and supercompactness holds.
We will also say that $\gk$ is a witness
to level by level equivalence between
strong compactness and supercompactness
iff for every regular $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact.
Note that the exception in Theorem \ref{t0}
is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
When this situation occurs, the
terminology we will henceforth
use is that $\gk$ is a witness
to the Menas exception at $\gl$.
Observe also that Theorem \ref{t0} is a
strengthening of the result of Kimchi and
Magidor \cite{KM}, who showed it is
consistent for the classes of strongly
compact and supercompact cardinals to
coincide precisely, except at measurable
limit points.
As is well known (see, e.g., the
discussion given at the beginning
of \cite{FMS}), if an inner model
for a supercompact cardinal were to
exist, it would have to have
properties drastically different
from those associated with
inner models for measurable,
strong, and Woodin cardinals.
Thus, one might expect that
a model for
Theorem \ref{t0} could be
permuted somewhat to give
a model in which the essential
property of level by level
equivalence between
strong compactness and
supercompactness holds, yet
GCH fails.
In fact, this turns out
to be the case.
In Theorem 4 of \cite{A99c},
a model containing
a supercompact cardinal $\gk$
%with no measurables above $\gk$ and
in which no cardinal
$\gd$ is $\gl$ supercompact for $\gl > \gd$ measurable
is produced in which there is precise level by level
equivalence between strong compactness
and supercompactness, yet for every measurable
cardinal $\gd$, $\gd$ is $\gd^+$ supercompact and
$2^\gd = \gd^{++}$.
Note, however, that here, there is
only one supercompact cardinal in
the universe, and no cardinal
$\gl > \gk$ is measurable.
There is also Theorem 3
of \cite{A02}, in which
there can be an arbitrary number
of supercompact cardinals in
the universe,
$2^\gd = \gd^+$ whenever $\gd$
isn't inaccessible,
$2^\gd = \gd^{++}$ whenever
$\gd$ is inaccessible, and
$\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact,
except possibly if $\gl$ is inaccessible,
%and $\gd$ is not a measurable limit
%of cardinals $\gg$ which are $\gl$ supercompact,
or for $\gl > \gd$ a regular cardinal,
$\gd$ is a witness to the Menas exception at $\gl$.
In this model, though, there is a
question whether there is
precise level by level equivalence
between strong compactness and supercompactness.
Finally, there is Theorem 8 of \cite{AH4},
where a model containing a supercompact
cardinal $\gk$ with no $\gl > \gk$ which
is $2^\gl$ supercompact is constructed
in which $\gk$ satisfies a weak form of
indestructibility and level by level
equivalence between strong compactness
and supercompactness holds.
The properties of this model ensure
that GCH fails on an unbounded set
of cardinals below $\gk$,
and with a further forcing,
GCH can fail on a set having measure 1
with respect to certain normal
measures over $\gk$.
Notice that in the preceding theorems,
although GCH fails significantly,
there is either only one supercompact
cardinal in the universe along with
level by level equivalence between
strong compactness and supercompactness,
or there are many supercompact
cardinals in the universe, but there
is an ambiguity as to whether level
by level equivalence between strong
compactness and supercompactness
always holds.
%This raises the following question:
Thus, the the following question is raised:
Is it possible to obtain a model
in which the structure of the class
of supercompact cardinals is arbitrary,
there is level by level equivalence
between strong compactness and
supercompactness, and in which
there is a significant class of
cardinals at which GCH fails?
The purpose of this paper is to
provide an affirmative answer to
the above question.
Specifically, we prove the
following two theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + $\K$
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 + $\K$
is the class of supercompact
cardinals + $\gk$ is the
least supercompact cardinal +
Level by level equivalence
between strong compactness
and supercompactness holds +
$2^\gd = \gd^{++}$ if
$\gd \le \gk$ is in $V$ an
inaccessible limit of strong
cardinals + $2^\gd = \gd^+$
otherwise''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\K$
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 + $\K$
is the class of supercompact
cardinals + $\gk$ is the
least supercompact cardinal +
Level by level equivalence
between strong compactness
and supercompactness holds +
%$\gk$'s supercompactness
%is indestructible under forcing
%with $\add(\gk, \gk^{++})$
%(and more) +
%that preserves level by level equivalence between strong
%compactness and supercompactness +
GCH holds for all cardinals
at and above $\gk$''.
In addition, after forcing over $V^\FP$ with
$\add(\gk, \gk^{++})$ (and some additional
partial orderings as well),
$\K$ is the class of supercompact cardinals
(so $\gk$ remains the least supercompact
cardinal), and level by level equivalence
between strong compactness and
supercompactness continues to hold.
\end{theorem}
We take this opportunity
to make some remarks on
Theorems \ref{t1} and \ref{t2}.
In Theorem \ref{t1},
the definition of $\FP$
in tandem with
Hamkins' gap forcing results of
\cite{H1}, \cite{H2}, and \cite{H3}
will yield that in $V^\FP$,
any $\gd < \gk$ which is an
inaccessible limit of strong
cardinals was also an inaccessible
limit of strong cardinals in $V$.
By Lemma 2.1 of \cite{AC2}
and the succeeding remark, if
$j : V^\FP \to M$ is an elementary
embedding witnessing (at least)
the measurability of
$\gk$ and $\mu$ is the normal
measure over $\gk$ associated with $j$,
$M \models ``\gk$ is an
inaccessible limit of
strong cardinals at which
GCH fails'', and
$\{\gd < \gk : \gd$ is an
inaccessible limit of strong
cardinals at which GCH
fails$\} \in \mu$ by reflection.
This means that in $V^\FP$,
not only does level by level
equivalence between strong
compactness and supercompactness hold,
but since $\gk$ is supercompact,
GCH fails on a set which has
measure 1 for $2^{2^\gk}$ many
normal measures over $\gk$.
Theorem \ref{t2} is a generalization
of the aforementioned Theorem 8 of
\cite{AH4}, proven in the context
of a universe in which the
class of supercompact cardinals
has no restrictions placed on it.
As with Theorem 8 of \cite{AH4},
the fact $\gk$'s supercompactness
becomes indestructible when forcing
with $\add(\gk, \gk^{++})$ and
other partial orderings
%the amount of indestructibility obtained
does not immediately
indicate the exact nature of
%it is certainly not immediately
%apparent why the amount of
%indestructibility obtained implies
the desired failures of GCH.
However, the fact that in Theorem \ref{t2},
$V^{\FP \ast \dot \add(\gk, \gk^{++})}
\models ``\gk$ is supercompact''
indicates that in both
$V^\FP$ and
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
there is an unbounded set of cardinals
below $\gk$ on which GCH fails.
This will be discussed in greater detail
after the proof of Theorem \ref{t2} has
been given.
We would also like to point out
that for
Theorems \ref{t1} and \ref{t2},
the structure of the set of
measurable cardinals below
$\gk$ can be quite complicated
in each of $V$ and $V^\FP$.
As an example, if there is
a measurable limit of supercompact
cardinals above $\gk$ in $V$,
this will reflect unboundedly
often below $\gk$ in both $V$ and
$V^\FP$ to produce many
different instances of the
Menas exception, such as cardinals
$\gd$ which are $\gd^+$ strongly
compact, aren't $\gd^+$ supercompact,
and are measurable limits of cardinals
$\gg$ which are $\gd^+$ supercompact.
Thus, even though in
the universes witnessing the
conclusions of
Theorems \ref{t1} and \ref{t2},
there are no failures of GCH
at any cardinal above $\gk$,
the failures of GCH that exist
below $\gk$ are of a much more
complicated nature than any
failures of GCH that exist in
a universe in which there is
level by level equivalence
between strong compactness and
supercompactness and a restricted
number of large cardinals.
Before presenting the proofs
of our theorems, 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 somewhat and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
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 stage $\ga$, a non-trivial forcing is done
adding a subset of $\gd_\ga$, then we will say that
$\gd_\ga$ is in the field of $\FP$.
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 $V$.
Let $\gk$ be a regular cardinal.
The partial ordering
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
In the proof of Theorem \ref{t1},
we will employ
the standard Cohen partial ordering
for adding $\gl$ subsets of $\gk$,
$\add(\gk, \gl)$.
As opposed to the most common usage,
however, where $\gk < \gl$ are both
regular cardinals, we will allow
$\gl$ to be an arbitrary ordinal when appropriate.
Assuming GCH holds for cardinals at and above
$\gk$, this will not change the fact that
$\add(\gk, \gl)$ is
$\gk$-directed closed and $\gk^+$-c.c.
We recall for the benefit of readers Hamkins'
definition from Section 3 of \cite{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the lottery sum is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings'', ``side-by-side forcing'', and
``choosing which partial ordering to force
with generically''.}
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}
for further details.
We note only that the cardinal
$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
supercompact for every cardinal $\gd < \gl$.
\section{The Proof of Theorem \ref{t1}}\label{s1}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals + $\gk$ is the
least supercompact cardinal''.
Without loss of generality, by first doing
a preliminary forcing if necessary, we may
also assume
that $V$ is as in Theorem \ref{t0}, i.e.,
that GCH and level by level equivalence between
strong compactness and supercompactness
hold in $V$.
This allows us to define in $V$ our partial
ordering $\FP$ as the Easton support iteration
of length $\gk + 1$
which first adds a Cohen real and then,
at any inaccessible cardinal $\gd \le \gk$
which is a limit of strong cardinals, forces with
$\add(\gd, \gd^{++})$.
At all other stages, the forcing is trivial.
Since we can write
$\FP = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed'',
the results of \cite{H1}, \cite{H2},
and \cite{H3} tell us that in
$V^\FP$, any supercompact cardinal
had to have been supercompact in $V$,
and any inaccessible limit of strong
cardinals had to have been an
inaccessible limit of strong
cardinals in $V$.
Standard arguments in tandem with the
L\'evy-Solovay results \cite{LS} then show
%in tandem with the results of
%\cite{H1}, \cite{H2}, and \cite{H3} then show
that cardinals and cofinalities are preserved
when forcing with $\FP$ and
$V^\FP \models ``$ZFC + $\K - \{\gk\}$ is the class of
supercompact cardinals above $\gk$ + $2^\gd = \gd^{++}$ if
$\gd \le \gk$ is in $V$
an inaccessible limit of strong cardinals +
$2^\gd = \gd^+$ otherwise''.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
\end{lemma}
\begin{proof}
By our remarks in the preceding
paragraph, since forcing with $\FP$
creates no new supercompact cardinals,
it suffices to show that
$V^\FP \models ``\gk$ is supercompact''.
To do this,
let $\gl > \gk^+ = 2^\gk$ be a regular cardinal,
and let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$ such that
$M \models ``\gk$ isn't $\gl$ supercompact''.
As we remarked immediately following
the statements of Theorems \ref{t1}
and \ref{t2},
$M \models ``\gk$ is an inaccessible
limit of strong cardinals''.
Also, as in Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''.
This means we can write
$j(\FP) = \FP_\gk \ast \dot
\add(\gk, \gk^{++}) \ast \dot \FQ
\ast \dot \add(j(\gk), j(\gk^{++}))$,
where $\FP_\gk$ is the partial
ordering $\FP$ defined through
stage $\gk$, and the first
ordinal in the field of $\dot \FQ$
is above $\gl$.
Thus, since $M^\gl \subseteq M$,
it is the case that
in both $V$ and $M$,
$\forces_{\FP_\gk \ast \dot
\add(\gk, \gk^{++})} ``\dot \FQ$ is
$\gl^+$-directed closed'', i.e.,
$\forces_{\FP} ``\dot \FQ$ is
$\gl^+$-directed closed''.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be
$V[G]$-generic over
$\add(\gk, \gk^{++})$.
Standard arguments show that
$M[G][H]$ remains $\gl$ closed
with respect to $V[G][H]$.
Since
$M^\FP \models ``|\FQ| = j(\gk)$'' and
GCH holds in both $V$ and $M$,
there are $2^{j(\gk)} = j(\gk^+)$
dense open subsets of $\FQ$
present in $M^\FP$. However, since
$|j(\gk^+)| =
|\{f : f : P_\gk(\gl) \to \gk^+$
is a function$\}| = \gl^+$ by GCH,
we can use the standard diagonalization
arguments
(as given, e.g., in the construction
of the generic object $G_1$ in
Lemma 2.4 of \cite{AC2})
to construct in
$V[G][H]$ an $M[G][H]$-generic
object $H'$ over $\FQ$ and lift $j$ to
$j : V[G] \to M[G][H][H']$ in $V[G][H]$.
Note that $M[G][H][H']$ remains
$\gl$ closed with respect to
$V[G][H][H'] = V[G][H]$.
Then, as the number of dense open subsets
of $\add(j(\gk), j(\gk^{++}))$ in
$M[G][H][H']$ is $j(\gk^{+++})$, which
by GCH and the fact that
$\gl \ge \gk^{++}$ has size $\gl^+$ in
$V[G][H]$, and as $\add(j(\gk), j(\gk^{++}))$
is $\gl^+$-directed closed in both
$M[G][H][H']$ and $V[G][H]$, we can once again
use the standard diagonalization arguments
to construct in $V[G][H]$ an
$M[G][H][H']$-generic object $H''$ containing
a master condition for $j''H$.
We can now fully lift $j$ to
$j : V[G][H] \to M[G][H][H'][H'']$ in
$V[G][H]$, thereby showing that
$V[G][H] \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ was arbitrary, this completes the
proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $V \models
``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact
and $\gl$ is regular'', then
$V^\FP \models ``\gd$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
By Lemma \ref{l1}, Lemma \ref{l2}
is true if $\gd = \gk$.
By the results of
\cite{LS}, since $\FP$ can
be defined so that
$|\FP| = \gk^{++}$,
Lemma \ref{l2} is true if
$\gd > \gk$. It therefore
suffices to prove Lemma \ref{l2}
when $\gd < \gk$, which we
assume for the duration of the
proof of this lemma.
Let $A = \{\gg < \gd : \gg$
is an inaccessible 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 at most $\gd$, and
$\dot \FQ$ is a term for the
rest of $\FP$, i.e., the portion
of $\FP$ whose field contains
ordinals above $\gd$. Since
$\gd < \gk$, it must be the
case that $\gl$ is below the
least $V$-strong cardinal $\gz$ above
$\gd$. This is because otherwise,
$V \models ``\gd$ is ${<}\gz$
supercompact and $\gz$ is strong'',
so by the proof of Lemma 2.4 of
\cite{AC2},
$V \models ``\gd$ is supercompact'',
a contradiction to the fact that
$\gk$ is the least $V$-supercompact cardinal. As
$\forces_{\FP_A} ``\dot \FQ$ is
$\gz$-directed closed and
$\gz$ is inaccessible'',
to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact'',
it hence suffices to show that
$V^{\FP_A} \models ``\gd$ is $\gl$
supercompact''.
If $|\FP_A| < \gd$, then by the
results of \cite{LS},
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
We may therefore assume that
$|\FP_A| \ge \gd$, i.e.,
by the definition of
$\FP$ and $\FP_A$, $|\FP_A| = \gd^{++}$.
If $\gl \ge \gd^{++}$, then the
argument given in the proof of
Lemma \ref{l1} may be used to show that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
Thus, we assume that $\gl = \gd^+$.
When $\gl = \gd^+$, the argument given on
pages 119--120 of \cite{AS97a},
pages 88--90 of \cite{A02}, or
pages 832--833 of \cite{AH4}
(which is originally due to Magidor
and is also found earlier in
\cite{JMMP}, \cite{JW}, and \cite{Ma2})
can be used to show that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
For the convenience of readers, we give this
argument here as well.
Write
$\FP_A = \FP_\gd \ast \dot \add(\gd, \gd^{++})$.
Let $G = G_0 \ast G_1$ be
$V$-generic over $\FP_A$. Fix
$j : V \to M$ an elementary embedding
withnessing the $\gl = \gd^+$
supercompactness of $\gd$ generated
by a supercompact ultrafilter
${\cal U}$ over $P_\gd(\gl)$.
We then have
$j(\FP_A) = \FP_A \ast \dot \add(\gd, \gd^{++})
\ast \dot \FQ \ast \dot
\add(j(\gd), j(\gd^{++}))$.
%where $\dot \FR_1$ is a term for
%$\add(j(\gd), j(\gd^{++}))$ as computed in
%$M^{\FP_A \ast \dot \add(\gd, \gd^{++})
%\ast \dot \FR_0}$.
Therefore, by using the
standard diagonalization argument
mentioned in Lemma \ref{l1}, since
$M[G_0][G_1]$ remains $\gl$ closed
with respect to $V[G_0][G_1]$ and
$V \models {\rm GCH}$, it is possible
working in $V[G_0][G_1]$ to construct an
$M[G_0][G_1]$-generic object $G_2$ over
$\FQ$ and lift $j$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$.
It is then as before the case that
$M[G_0][G_1][G_2]$ remains $\gl$
closed with respect to
$V[G_0][G_1]$.
We construct now in $V[G_0][G_1]$ an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gd), j(\gd^{++}))$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_1 \rest \ga = \{p \rest \ga : p \in G_1\}$. Clearly,
$V[G_0][G_1] \models ``|G_1 \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) : p \in G_1 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \ga$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2] \models ``$GCH holds
for all cardinals at or above $j(\gd)$'',
$M[G_0][G_1][G_2] \models ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2]$ is a
maximal antichain of $\add(j(\gd), j(\gd^{++}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{++}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a supercompact ultrafilter over
$P_\gd(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra \in
V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G_0][G_1][G_2]$.
Working in $V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$ of
elements of $\add(j(\gd), j(\gd^{++}))$ such that
$\forall \ga \in (\gd, \gd^{++}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_3 = \{p \in \add(j(\gd), j(\gd^{++})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2]$ is
$\gd^+$ closed with respect to
$V[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2]$ is closed under
$\gd^+$ sequences with respect to
$V[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb : \gb < \eta \ra$ such that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_3$ is
$M[G_0][G_1][G_2]$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gd(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gd(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gd(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{++})$ be such that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{++})$ is such that for some
$\ga \in (\gd, \gd^{++})$, $p = p \rest \ga$,
$G_3$ is such that if
$p \in G_1$, $j(p) \in G_3$.
Thus, working in $V[G_0][G_1]$,
we have shown that $j$ lifts to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
i.e.,
$V[G_0][G_1] \models ``\gd$ is $\gl = \gd^+$
supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3}
follows closely
the proof of Lemma 3.2 of \cite{A02}.
Suppose
$V^\FP \models ``\gd < \gl$ are
regular cardinals such that
$\gd$ is $\gl$ strongly compact and
$\gd$ isn't a measurable limit of
cardinals $\gg$ which are
$\gl$ supercompact''.
%As we observed in the first
%paragraph of the proof of
%Theorem \ref{t1}, forcing with
%$\FP$ preserves all cardinals
%and cofinalities.
By Lemma \ref{l2}, any cardinal
$\gg$ such that $\gg$ is
$\gl$ supercompact in $V$ remains
$\gl$ supercompact in $V^\FP$.
This means that
$V \models ``\gd < \gl$ are
regular cardinals such that
%$\gd$ is $\gl$ strongly compact and
$\gd$ isn't a measurable limit of
cardinals $\gg$ which are
$\gl$ supercompact''.
Recall from the first paragraph
of the proof of Theorem \ref{t1}
that we can write $\FP$ as
$\FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed''.
Further, it is easily seen that
if $\gg$ is inaccessible in $V$,
then any subset of $\gg$ in
$V^\FP$ of size below $\gg$
has in $V$ a name of size
below $\gg$.
Therefore, in the terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ is a
``mild forcing with respect to
$\gd$ admitting a gap at $\ha_1$'',
so by the results of
\cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gd$ is $\gl$ supercompact'',
so another application of Lemma \ref{l2}
yields that
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l1} - \ref{l3} complete
the proof of Theorem \ref{t1}.
\end{proof}
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
The proof of Theorem \ref{t2}
uses a partial ordering $\FP$
whose definition is similar to
the definition of the partial
ordering employed in the proof of
Theorem 8 of \cite{AH4}.
Note, however, that this partial
ordering is defined in the context
of a universe containing a
restricted number of large
cardinals, whereas the definition
we are about to present is given
in the context of a universe
in which the class of supercompact
cardinals $\K$ can have arbitrary structure.
Let
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals + $\gk$ is the
least supercompact cardinal''.
As in the proof of Theorem \ref{t1},
without loss of generality, we first
assume that if necessary, a preliminary
forcing has been done so that GCH and
level by level equivalence between
strong compactness and supercompactness
also hold in $V$.
This allows us to define $\FP$
as the Easton support iteration
of length $\gk$ which begins
by adding a Cohen real.
At an
inaccessible cardinal $\gd < \gk$
which is a limit of strong cardinals,
assuming
$V^{\FP_\gd} \models ``$There is
level by level equivalence
between strong compactness
and supercompactness'', we
force with the lottery sum of all
$\gd$-directed closed partial orderings
having rank below the least $V$-strong
cardinal above $\gd$ which preserve this
level by level equivalence between
strong compactness and supercompactness.
At all other stages, the forcing is
trivial. By the results of \cite{LS},
since $\FP$ may be defined so
as to have size $\gk$, in
$V^\FP$, $\K - \{\gk\}$ is the
class of supercompact cardinals
above $\gk$, and level by level
equivalence between strong
compactness and supercompactness
holds for every measurable cardinal
above $\gk$ which isn't a witness
to the Menas exception for some $\gl$.
In addition, since GCH holds in $V$ and
$|\FP| = \gk$, in $V^\FP$, GCH holds
for all cardinals at and above $\gk$.
\begin{lemma}\label{l5}
If $\gg \le \gk$ is such that
$V \models ``\gg$ is $\gl$
supercompact for $\gl > \gg$
a regular cardinal'', then
$V^{\FP_\gg} \models ``\gg$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
%The proof of Lemma \ref{l5} is
%very similar to the proof of
%Lemma \ref{l1} and begins
%in much the same way.
Let $\gl > \gg$ be a regular cardinal,
and let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gg$
generated by a supercompact ultrafilter
over $P_\gg(\gl)$ such that
$M \models ``\gg$ isn't $\gl$ supercompact''.
We may assume that $\gg$ is a
limit of non-trivial stages of forcing,
since otherwise, $\FP_\gg$
is forcing equivalent to a partial
ordering having size below $\gg$,
and the result follows by the
work of \cite{LS}.
As in the proof of Lemma \ref{l1},
$M \models ``\gg$ is an inaccessible
limit of strong cardinals'', and
$M \models ``$No cardinal
$\gd \in (\gg, \gl]$ is strong''.
Further, since trivial forcing
preserves level by level equivalence
between strong compactness and
supercompactness, the trivial
partial ordering $\{\emptyset\}$
may be selected by the lottery
held at stage $\gg$ in $M^{\FP_\gg}$.
%= M^\FP$.
This means
$j(\FP)$ is forcing equivalent to
$\FP \ast \dot \FQ$,
where the first ordinal in
the field of $\dot \FQ$ is
above $\gl$.
This allows us to use a simpler form
of the argument given in the
proof of Lemma \ref{l1} (which
doesn't require the construction of
a master condition and only
requires the standard diagonalization
argument to construct a generic
object for $\FQ$) to show that
$V^\FP \models ``\gg$ is $\gl$
supercompact''.
This completes the proof of
Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$Level
by level equivalence
between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
The proof we give parallels
to a certain extent the
proof of Lemma 8.2 of \cite{AH4},
taking into account that there
are no restrictions on the
number of large cardinals in $V$.
We assume inductively that
Lemma \ref{l4} is true for
all stages of forcing
$\gg < \gd \le \gk$
towards showing that Lemma \ref{l4}
is true in $V^{\FP_\gd}$.
Suppose $\gd$ is a successor ordinal with
$\gd = \gd' + 1$.
By our inductive assumptions,
level by level equivalence between
strong compactness and supercompactness
holds in $V^{\FP_{\gd'}}$.
Without loss
of generality, we assume that a non-trivial
forcing is done at stage $\gd'$, for
otherwise, Lemma \ref{l4} is trivially
true in $V^{\FP_\gd}$.
However, if this is the case, then by
the definition of $\FP$, the forcing
selected at stage $\gd'$ preserves
level by level equivalence between
strong compactness and supercompactness,
meaning that Lemma \ref{l4} is true
in $V^{\FP_\gd}$.
We therefore assume that
$\gd$ is a limit ordinal.
Fix $\eta$
such that $\eta$ is a measurable
cardinal in $V^{\FP_\gd}$.
If $\eta > \gd$, then by the results
of \cite{LS},
the degrees of
strong compactness or supercompactness
$\eta$ witnesses in $V^{\FP_\gd}$
are precisely the same as those witnessed by $\eta$
in $V$. This means $\eta$ isn't
a counterexample to level by level
equivalence between strong
compactness and supercompactness in
$V^{\FP_\gd}$.
If $\eta = \gd$, we assume that
$\eta$ is a limit of non-trivial
stages of forcing.
This is since otherwise,
as in Lemma \ref{l5},
$\FP_\gd$ is forcing equivalent
to a partial ordering having size below $\gd$,
meaning that as we just observed, $\eta$
isn't a counterexample to level by level
equivalence between strong compactness
and supercompactness in $V^{\FP_\gd}$.
We assume now that
$V^{\FP_\gd} \models ``\gl > \eta$ is
regular and $\eta$ is $\gl$ strongly compact''.
Note that by the reasoning given in
the second paragraph of the proof of Lemma \ref{l3},
%by its definition,
both $\FP$ and $\FP_\gd$ are in the
terminology of \cite{H2} and \cite{H3}
``mild forcings with respect to
$\eta = \gd$ admitting a gap at $\ha_1$''.
%meaning that any set of ordinals
%of size below $\gd$ in $V^{\FP_\gd}$
%has a name of size below $\gd$ in $V$.
Thus, by the results of \cite{H2} and
\cite{H3}, as before,
any degree of
strong compactness or supercompactness
$\eta$ witnesses in $V^{\FP_\gd}$
and $V^\FP$
had to have been witnessed by $\eta$
in $V$. Hence,
$V \models ``\gl > \eta$ is
regular and $\eta$ is $\gl$ strongly compact''.
If
$V \models ``\eta$ isn't a witness to the
Menas exception at $\gl$'', then by
level by level equivalence between
strong compactness and supercompactness in $V$,
$V \models ``\eta$ is $\gl$ supercompact''.
By Lemma \ref{l5},
$V^{\FP_\gd} \models ``\eta = \gd$ is
$\gl$ supercompact''.
Thus, the proof of the case $\eta = \gd$
will be done once we have shown that
$\eta$ isn't a witness to the Menas
exception at $\gl$ in $V$.
%We therefore assume to the contrary that
%$\eta$ is a witness to the Menas exception
%at $\gl$ in $V$.
However, to see that this is so, note that
since $\eta$ is in $V$ a limit of strong
cardinals and $\gl > \eta$, this
means that for some
$\gz < \rho < \eta < \gk$, in $V$,
$\gz$ is $\rho$ supercompact and
$\rho$ is strong.
This yields as
we indicated in the proof of Lemma
\ref{l2} that $\gz$ is supercompact.
This immediately contradicts that
$\gk$ is the least supercompact
cardinal in $V$ and gives the
desired result.
%also shows
%that $\eta$ isn't in $V$ a witness
%to the Menas exception, i.e.,
%$\eta$ isn't in $V$ a limit of
%cardinals which are $\gl$ supercompact.
To complete the proof of
Lemma \ref{l4}, we hence assume that
$\eta < \gd$.
Let $\gg$ be the supremum of
the non-trivial stages of
forcing at or below $\eta$, and write
$\FP_\gd = \FP_{\gg + 1} \ast \dot
\FP^{\gg + 1} =
\FP_\gg \ast \dot
\FQ_\gg \ast \dot \FP^{\gg + 1}$,
where $\dot \FQ_\gg$ is a term
for the forcing done at stage $\gg$, and
$\dot \FP^{\gg + 1}$ is a term for the
remainder of $\FP_\gd$.
Note that since the first ordinal
in the field of $\dot \FP^{\gg + 1}$ is
an inaccessible limit of strong
cardinals in $V$,
$\forces_{\FP_\gg \ast \dot \FQ_\gg}
``\dot \FP^{\gg + 1}$ is ${(2^\eta)}^+$-directed
closed''. This means that
$V^{\FP_\gg \ast \dot \FQ_\gg} \models
``\eta$ is a measurable cardinal''.
By the definition of $\FP$,
%If $\gg < \eta$, $\dot \FQ_\gg$
%is a term for the trivial partial
%ordering, and if $\gg = \eta$,
$\dot \FQ_\gg$ is a term for
a partial ordering such that
forcing with this partial
ordering preserves level by level
equivalence between strong compactness
and supercompactness.
%In either case,
We therefore know that in
$V^{\FP_\gg \ast \dot \FQ_\gg}$, for
$\gl \ge \eta$ a regular cardinal,
$\eta$ is $\gl$ strongly compact
iff $\eta$ is $\gl$ supercompact,
except possibly if $\eta$ is a
limit of cardinals which are
$\gl$ supercompact.
Thus, in order to show that
level by level equivalence between
strong compactness and supercompactness
holds in $V^{\FP_\gd}$, we need to
demonstrate that forcing with
$\FP^{\gg + 1}$ over
$V^{\FP_\gg \ast \dot \FQ_\gg}$
preserves the fact that
for $\gl > \eta$ a regular cardinal,
$\eta$ is $\gl$ strongly compact
iff $\eta$ is $\gl$ supercompact,
except possibly if $\eta$ is a
limit of cardinals which are
$\gl$ supercompact.
To do this, we first assume that
$\eta$ is in
$V^{\FP_\gg \ast \dot \FQ_\gg}$
a limit of cardinals which are
$\gl$ supercompact where $\gl$
is regular, $\eta$
is $\gl$ strongly compact, and
$\eta$ isn't $\gl$ supercompact, i.e., that
$\eta$ is a witness to the
Menas exception at $\gl$. As in the
second paragraph of the proof of
Lemma \ref{l2}, since $\eta < \gk$,
the least $\theta$ for which
$\eta$ isn't $\theta$ supercompact in
$V$ must be below the least $V$-strong
cardinal $\rho$ above $\eta$, and any
cardinal below $\rho$ which exhibits
a particular degree of supercompactness
in $V$
must have that degree of supercompactness
below $\rho$ as well. Further, since
we can write
$\FP_{\gg + 1} = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed'', the
results of \cite{H1}, \cite{H2},
and \cite{H3} tell us that
$\theta^* \le \theta$, where
$\theta^*$ is the least cardinal in
$V^{\FP_{\gg + 1}}$ for which
$\eta$ isn't $\theta^*$ supercompact.
In addition, this factorization of
$\FP_{\gg + 1}$ and the results of
\cite{H1}, \cite{H2}, and \cite{H3}
yield that any cardinal which
%is $\gl$ supercompact in
exhibits a particular degree of
supercompactness in $V^{\FP_{\gg + 1}}$
%had to have been $\gl$ supercompact in $V$,
had to exhibit that same degree of
supercompactness in $V$ as well,
meaning automatically that
%any cardinal
%$\gz < \eta$
%below $\eta$ which is $\gl$
%supercompact in $V^{\FP_{\gg + 1}}$
%must automatically have that
$\gl < \rho$ as there are
cardinals below $\rho$ which are
$\gl$ supercompact in $V^{\FP_{\gg + 1}}$.
Therefore, since the first ordinal
in the field of $\FP^{\gg + 1}$
is above $\rho$ and
$\forces_{\FP_{\gg + 1}}
``\dot \FP^{\gg + 1}$ is
$\rho$-directed closed and
$\rho$ is inaccessible'', in
$V^{\FP_{\gg + 1} \ast \dot \FP^{\gg + 1}} =
V^{\FP_\gd}$, $\eta$ is a limit of
cardinals which are $\gl$ supercompact,
$\gl$ is regular, $\eta$ is
$\gl$ strongly compact, yet $\eta$
isn't $\gl$ supercompact.
This means that $\eta$ isn't a
counterexample to level by level
equivalence between strong compactness
and supercompactness in
$V^{\FP_\gd}$.
Finally, we assume that
$\eta$ is $\gl$
strongly compact in $V^{\FP_{\gg + 1}}$,
$\gl$ is regular, and $\eta$ isn't
a witness to the Menas exception at $\gl$.
Since level by level equivalence
between strong compactness and
supercompactness holds in
$V^{\FP_{\gg + 1}}$, $\eta$ is
$\gl$ supercompact in $V^{\FP_{\gg + 1}}$
as well.
Using the notation and results of
the preceding paragraph, we then have that
$\gl < \rho$,
$\rho$ is beyond the degree of either
$\eta$'s strong compactness or
supercompactness in $V^{\FP_{\gg + 1}}$,
the first ordinal in the field of
$\FP^{\gg + 1}$ is above $\rho$, and
$\forces_{\FP_{\gg + 1}}
``\dot \FP^{\gg + 1}$ is
$\rho$-directed closed and
$\rho$ is inaccessible''.
This means that
$V^{\FP_{\gg + 1} \ast \dot \FP^{\gg + 1}} =
V^{\FP_\gd} \models ``\eta$ is $\gl$
supercompact and level by level
equivalence between strong compactness
and supercompactness holds for $\eta$''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
We remark that as an immediate
corollary of Lemmas \ref{l5} and \ref{l4},
we have that
$V^\FP \models ``\gk$ is the
least supercompact cardinal''.
\begin{lemma}\label{l6}
$V^{\FP \ast \dot \add(\gk, \gk^{++})} \models
``\K$ is the class of supercompact cardinals
(so $\gk$ is the least
supercompact cardinal), and level by level
equivalence between strong compactness and
supercompactness holds''.
\end{lemma}
\begin{proof}
We begin by writing
$\FP \ast \dot \add(\gk, \gk^{++})
= \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed''.
Since $\FP \ast \dot \add(\gk, \gk^{++})$
therefore ``admits a gap at $\ha_1$'',
by the results of \cite{H1}, \cite{H2},
and \cite{H3}, any degree of supercompactness
exhibited by a cardinal in
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$ had
to have been exhibited in $V$.
Thus, in $V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
$\gk$ is less than or equal to the least
supercompact cardinal. Further, since
$|\FP \ast \dot \add(\gk, \gk^{++})| = \gk^{++}$,
we know that in
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
$\K - \{\gk\}$ is the class of supercompact
cardinals above $\gk$.
Also, as we observed in the proof of
Lemma \ref{l4}, since $\FP$ ``admits
a gap at $\ha_1$'' and hence by the
results of \cite{H1}, \cite{H2}, and
\cite{H3} forces
no new degrees of supercompactness,
in $V^\FP$, the degree of supercompactness
of any $\eta < \gk$ must be bounded
below $\gk$.
Therefore, since level by level
equivalence between strong compactness and
supercompactness holds in $V^\FP$ and
$\add(\gk, \gk^{++})$ is
%${(\add(\gk, \gk^{++}))}^{V^\FP}$ is
$\gk$-directed closed, in
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
level by level equivalence between
strong compactness and supercompactness
holds below $\gk$.
By the results of \cite{LS}, forcing
over $V^\FP$ with
$\add(\gk, \gk^{++})$
%${(\add(\gk, \gk^{++}))}^{V^\FP}$
will preserve that level by level
equivalence between strong compactness
and supercompactness holds above
$\gk$. Thus, the proof of Lemma \ref{l6}
will be complete once we have shown that
$V^{\FP \ast \dot \add(\gk, \gk^{++})} \models
``\gk$ is supercompact''.
As in the proof of Lemma \ref{l4},
the proof we give of this fact
parallels to
a certain extent a proof from
\cite{AH4} (namely Corollary 10), taking
into account that there are no
restrictions on the number of
large cardinals in $V$.
Suppose $\gl > \gk$ is a
sufficiently large regular cardinal,
and $j : V \to M$ is an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$.
If $\add(\gk, \gk^{++})$ is an allowable
choice in the lottery held at stage $\gk$ in
$M^{\FP_\gk} = M^\FP$, then the argument
given in the proof of Lemma \ref{l1}
(this time using the construction of the
master condition) remains valid and allows
us to infer that
$V^{\FP \ast \dot \add(\gk, \gk^{++})} \models
``\gk$ is supercompact''.
Thus, to prove Lemma \ref{l6}, it suffices
to demonstrate this as a definitive fact.
Since $V$ and $M$ are elementarily equivalent,
the proof of Lemma \ref{l6} will therefore
be complete once we have shown that at any
non-trivial stage of forcing
$\gd < \gk$, $\add(\gd, \gd^{++})$ is an
allowable choice in the lottery held
at stage $\gd$ in $V^{\FP_\gd}$.
To see that this is indeed so, fix
$\gd < \gk$ as a non-trivial
stage of forcing, meaning that
$\gd$ is in $V$ an inaccessible limit
of strong cardinals. Assume that
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})} \models
``\gd$ is $\gl$ strongly compact and
$\gl > \gd$ is regular''.
Assume further inductively that for any
non-trivial stage of forcing $\gg < \gd$,
$\add(\gg, \gg^{++})$ is an allowable
choice in the stage $\gg$ lottery held in
$V^{\FP_\gg}$.
We first note that as in
Lemma \ref{l4}, $\gd$ can't be in $V$
a measurable limit of cardinals which
are $\gl$ supercompact, i.e., $\gd$
can't witness the Menas exception
at $\gl$ in $V$.
In addition, by writing
$\FP_\gd \ast \dot \add(\gd, \gd^{++}) =
\FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed'',
$\FP_\gd \ast \dot \add(\gd, \gd^{++})$ is a
``mild forcing with respect to $\gd$
admitting a gap at $\ha_1$''.
This means that by the
results of \cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''
as well.
Since level by level equivalence between
strong compactness and supercompactness
holds in $V$ and $\gd$ isn't in
$V$ a witness to the Menas exception at $\gl$,
$V \models ``\gd$ is $\gl$ supercompact''.
To show that level by level equivalence
between strong compactness and supercompactness
holds at $\gd$ in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$,
it thus suffices to show that
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})} \models
``\gd$ is $\gl$ supercompact''.
To do this, we first note that the non-trivial
stages of forcing below $\gd$ must be
unbounded in $\gd$. Otherwise,
$\FP_\gd$ is forcing equivalent to
a partial ordering having size below $\gd$,
so by a result of Hamkins \cite{H5},
$\gd$ is no longer measurable in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$.
Thus, if we fix $j : V \to M$ an
elementary embedding witnessing the
$\gl$ supercompactness of $\gd$,
by the fact $V$ and $M$ are elementarily
equivalent and our inductive assumptions,
$\add(\gd, \gd^{++})$ is an allowable
choice in the stage $\gd$ lottery
held in $M^{\FP_\gd}$.
By opting for
$\add(\gd, \gd^{++})$ at
the stage $\gd$ lottery
held in $M^{\FP_\gd}$,
we can then use the argument given
in the proof of Lemma \ref{l2},
which is applicable for any regular $\gl > \gd$,
and show that $\gd$ is $\gl$ supercompact
in $V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$.
Thus, forcing with $\add(\gd, \gd^{++})$
over $V^{\FP_\gd}$ preserves level by
level equivalence between strong
compactness and supercompactness at $\gd$.
Then, by the argument given in the
next to last paragraph
of the proof of Lemma \ref{l4},
%which remains valid if the partial ordering
%used is either $\FP_\gd$ or
%$\FP_\gd \ast \dot \add(\gd, \gd^{++})$,
any degree of supercompactness exhibited
by a cardinal $\eta$ below $\gd$ in
$V^{\FP_\gd}$ has to be below
$\gd$ as well, meaning by the fact
$\add(\gd, \gd^{++})$ is $\gd$-directed
closed that $\eta$ witnesses
the same degree of supercompactness in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$.
Hence, again by the fact
$\add(\gd, \gd^{++})$ is $\gd$-directed
closed, regardless if $\eta$ is a witness
to the Menas exception at some $\gl$
or to level by
level equivalence between strong
compactness and supercompactness in
$V^{\FP_\gd}$, this property of
$\eta$ is preserved in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$.
Finally, as
$\FP_\gd \ast \dot \add(\gd, \gd^{++})$
has small cardinality with respect to
any measurable cardinal $\eta$ above $\gd$,
the results of \cite{LS} tell us
that forcing with
$\FP_\gd \ast \dot \add(\gd, \gd^{++})$
over $V$ preserves either level by level
equivalence between strong compactness
and supercompactness at $\eta$ or that
$\eta$ is a witness to the Menas exception
at some $\gl$.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l5} - \ref{l6} and the
intervening remarks complete the
proof of Theorem \ref{t2}.
\end{proof}
We note that the proof of Lemma \ref{l6}
actually shows that in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$,
$\gd$ isn't a witness to the Menas
exception at any $\gl$.
This is since after forcing with
$\add(\gd, \gd^{++})$ over
$V^{\FP_\gd}$, $\gd$ isn't measurable
if the non-trivial stages of forcing
below $\gd$ are bounded in $\gd$,
or if the non-trivial stages of forcing
below $\gd$ are unbounded in $\gd$,
for $\gl > \gd$ a regular cardinal
in $V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$,
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})} \models
``\gd$ is $\gl$ strongly compact iff
$\gd$ is $\gl$ supercompact''.
\section{Concluding Remarks}\label{s3}
We conclude this paper by beginning with
%We take this opportunity to make a
a few remarks concerning Theorem \ref{t2}.
First, as in our comments immediately
following the statements of
Theorems \ref{t1} and \ref{t2},
in the model
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
$\gk$ carries $2^{2^\gk}$ many normal
measures concentrating on sets on
which GCH fails.
This is since if
$j : V^{\FP \ast \dot \add(\gk, \gk^{++})} \to M$
is an elementary embedding witnessing
(at least) the measurability of $\gk$
and $\mu$ is the normal measure associated with $j$,
then as before,
$M \models ``\gk$ is an inaccessible limit
of strong cardinals at which GCH
fails'', so once more, by reflection,
$\{\gd < \gk : \gd$ is an inaccessible
limit of strong cardinals at which
GCH fails$\} \in \mu$.
It is also the case that in both
$V^{\FP}$ and
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
the failures of GCH that exist below
$\gk$ are quite complicated. To see this, let
$j : V \to M$ be an elementary embedding
witnessing the $\gk^{++}$ supercompactness
of $\gk$ such that
$M \models ``\gk$ isn't $\gk^{++}$ supercompact''.
We have already seen that at stage $\gk$
of the lottery held in $M^\FP = M^{\FP_\gk}$,
$\add(\gk, \gk^{++})$ is an allowable choice,
as forcing with this partial ordering will
preserve the level by level equivalence
between strong compactness and supercompactness.
Further, by the same argument as given in
%the next to last paragraph of the proof of
%Lemma \ref{l4},
Lemma \ref{l6}, $\gk$ isn't a witness to
the Menas exception at any $\gl$ in
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}$,
and by the same argument
as given in the proof of Lemma \ref{l4},
$\gk$ isn't a witness to the Menas
exception at any $\gl$ in $M$.
Therefore, by writing
$\FP_\gk = \FP_0 \ast \dot \FQ$
in $M$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed'', we can
once again apply the results of
\cite{H1}, \cite{H2}, and \cite{H3}
to infer that in
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}$,
$\gk$ is neither $\gk^{++}$ strongly
compact nor $\gk^{++}$ supercompact.
However, if we let, e.g.,
$\dot \FR$ be a term for the usual
Easton partial ordering which, in
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}$,
forces $2^\gd = \gd^{+17}$ for every
regular cardinal $\gd$ between
${(2^{\gk^{++}})}^+ = \gk^{+4}$
and the least Ramsey
cardinal above $\gk$, since forcing
with $\FR$ over
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}$
won't add a strongly compact ultrafilter for
$P_\gk(\gk^{++})$, the arguments we just
gave remain valid when forcing over
$M^{\FP_\gk}$ with
$\add(\gk, \gk^{++}) \ast \dot \FR$
instead of just $\add(\gk, \gk^{++})$
to show that in
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++}) \ast \dot \FR}$,
$\gk$ isn't a witness to the Menas exception
at any $\gl$ and
level by level equivalence between strong
compactness and supercompactness holds at $\gk$.
Since the same arguments as given in the proof
of Lemma \ref{l6} then again show that in
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++}) \ast \dot \FR}$,
level by level equivalence between strong
compactness and supercompactness holds
for every other measurable cardinal,
$\add(\gk, \gk^{++}) \ast \dot \FR$ is a valid
choice in the stage $\gk$ lottery held in
$M^{\FP_\gk}$.
This reflects below $\gk$
to show that for unboundedly in $\gk$ many
non-trivial stages of forcing
$\gd$, the partial ordering
selected by the lottery held at
stage $\gd$ in $V^{\FP_\gd}$ is
$\add(\gd, \gd^{++}) \ast \dot \FR_\gd$,
where $\dot \FR_\gd$ is a term for
the analogous Easton partial ordering
to the one just mentioned.
Of course, there is nothing special
about the definition of $\FR$ used
above, and other reasonable variations
will also work to yield further
different kinds of failures of GCH
below $\gk$ in both
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$ and
$V^\FP$.
However, this still paints an
incomplete picture as to what
sorts of failures of GCH are
possible with level by level
equivalence between strong
compactness and supercompactness.
In particular, what other sorts
of failures of GCH hold below $\gk$ in either
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$ or
$V^\FP$?
Note that since it is unclear if
a partial ordering which preserves
level by level equivalence when forcing over
$V^\FP$ is an allowable choice in the
stage $\gk$ lottery held in $M^\FP$ for
$j : V^\FP \to M$ an elementary embedding
witnessing any degree of supercompactness
of $\gk$, the sorts of failures of GCH
below $\gk$ in either
$V^{\FP \ast \dot \add(\gk, \gk^{++})}$ or
$V^\FP$ seems to be a very difficult
question to answer.
Can there be
level by level equivalence between
strong compactness and supercompactness,
along with failures of GCH above
the least supercompact cardinal?
Is it even possible to have a universe
containing a supercompact
cardinal, regardless of any other
assumptions on the large cardinal
structure of this universe, in which
$2^\gd = \gd^{++}$ for every
regular cardinal and level by level
equivalence between strong compactness
and supercompactness holds?
In general, what sorts of failures
of GCH are compatible with level
by level equivalence between
strong compactness and supercompactness?
These are the questions with which we
conclude this paper.
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\begin{graveyard}
$\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$ \FP$ is ${<}\gk$-strategically closed if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
%Note that trivially, if $\FP$ is ${<}\gk$-closed, then $\FP$ is
%${<}\gk$-strategically
%closed and ${\prec}\gk $-strategically closed. The converse of
%both of these facts is false.
Suppose now that $\gk < \gl$ are regular cardinals.
A partial ordering
$\FP(\gk, \gl)$ that will be used in the proof
of Theorem \ref{t1}
is the partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically,
$\FP(\gk, \gl) =
\{ p$ : For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\gb \in S_p$ implies
$\gb > \gk$ 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 $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP(\gk, \gl)$ (see
\cite{Bu}, \cite{A99b}, or \cite{KM}), 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, and it can be shown
(see \cite{Bu}, \cite{A99b}, or \cite{KM}) that
$\FP(\gk, \gl)$
is ${\prec}\gl$-strategically closed.
$V^\FP \models ``$For every pair of
regular cardinals $\gd < \gl$,
$\gd$ is $\gl$ strongly compact
Since forcing with $\FP$
preserves all cardinals and
cofinalities,
$V \models ``\gd$ and $\gl$
are regular''.
Case 2: $\gl$ is the successor of an inaccessible cardinal.
Once again, write
$\FP = \FP_\gl \ast \dot \FP^\gl$, where
the field of $\FP_\gl$ consists of
ordinals below $\gl$, and $\FP^\gl$ is the rest of
$\FP$, i.e., the field of
$\FP^\gl$ consists of ordinals above $\gl$.
In this instance, it is not the case that
$|\FP_\gl| \le \gl$, since for the $\gd$ so that
$\gl = \gd^+$,
$|\FP_\gl| = \gd^{++} = \gl^+ > \gl$.
However, the arguments given on pages
119--120 of \cite{AS97a} or pages
555-556 of \cite{A99flm} (which are
originally due to Magidor and are
also found earlier in
\cite{JMMP}, \cite{JW}, and
\cite{Ma2}) will yield that
$V^{\FP_\gl} \models ``\gk$ is $\gl = \gd^+$
supercompact''.
For the convenience of readers, we give these
arguments below.
\setlength{\parindent}{1.5em}
Write
$\FP_\gl = \FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++})$,
where the field of $\FQ_0$ consists of
ordinals below $\gk$, and the field of
$\dot \FQ^1$ is composed of all remaining ordinals
in the interval $[\gk, \gl)$. Let $G$ be $V$-generic
over $\FP_\gl$, with
$G_0 \ast G_1 \ast G_2$ the corresponding
factorization of $G$. Fix
$j : V \to M$ an elementary embedding witnessing
the $\gl = \gd^+$ supercompactness of
$\gk$ which is generated by a supercompact
ultrafilter $\U$ over $P_\gk(\gl)$. We then have
$j(\FP_\gl) = \FQ_0 \ast \dot \FQ_1
\ast \dot \add(\gd, \gd^{++}) \ast \dot \FR_0
\ast \dot \FR_1$, where $\dot \FR_1$ is a term for
$\add(j(\gd), j(\gd^{++}))$ as computed in
$M^{\FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++}) \ast \dot \FR_0}$.
Therefore, by using the argument given in Lemma \ref{l1},
since $M[G_0][G_1][G_2]$ remains $\gl$ closed with respect to
$V[G_0][G_1]$ and $V \models {\rm GCH}$,
it is possible working in $V[G_0][G_1]$
to construct an $M[G_0][G_1][G_2]$-generic object
$G_3$ over $\FR_0$ and extend $j$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2]$.
It is then the case that
$M[G_0][G_1][G_2]$ remains $\gl$ closed
with respect to $V[G_0][G_1]$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_2 \rest \ga = \{p \rest \ga : p \in G_2\}$. Clearly,
$V[G_0][G_1] \models ``|G_2 \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) : p \in G_2 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal so that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_2 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_2 \rest \gb$ and $G_2 \rest \ga$, for
$p \rest \gb = q \in G_2 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2] \models ``$GCH holds
for all cardinals $\ge j(\gd)$'',
$M[G_0][G_1][G_2] \models \break ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2]$ is a
maximal antichain of $\add(j(\gd), j(\gd^{++}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{++}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a supercompact ultrafilter over
$P_\gk(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra \in
V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G_0][G_1][G_2]$.
Working in $V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$ of
elements of $\add(j(\gd), j(\gd^{++}))$ so that
$\forall \ga \in (\gd, \gd^{++}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and so that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_3 = \{p \in \add(j(\gd), j(\gd^{++})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2]$ is
$\gd^+$ closed with respect to
$V[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ so that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2]$ is closed under
$\gd^+$ sequences with respect to
$V[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb : \gb < \eta \ra$ so that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is so that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_3$ is
$M[G_0][G_1][G_2]$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gk(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gk(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gk(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{++})$ be so that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ so that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{++})$ is so that for some
$\ga \in (\gd, \gd^{++})$, $p = p \rest \ga$,
$G_3$ is so that if
$p \in G_2$, $j(p) \in G_3$.
Thus, working in $V[G_0][G_1]$,
we have shown that $j$ extends to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
i.e.,
$V[G_0][G_1] \models ``\gk$ is $\gl = \gd^+$
supercompact''.
Since as in Case 1,
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
${(2^{[\gl]^{< \gk}})}^{+}$-directed closed'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models
``\gk$ is $\gl$ supercompact''.
This completes the proof of Case 2 and Lemma
\ref{m1}.
\end{proof}
A result which will be key in the proof
of Theorem \ref{t1} is Hamkins'
Gap Forcing Theorem of \cite{H2} and \cite{H3}.
We therefore state this theorem now, along
with some associated terminology, quoting
freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| < \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ {\it admits a gap at $\gd$}.
Also, in Hamkins' terminology of
\cite{H2} and \cite{H3}, 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 Gap Forcing Theorem is then
the following.
\begin{theorem}\label{t2}
(Hamkins' Gap Forcing Theorem)
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a gap at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ 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 $V$.
\end{theorem}
Since the partial ordering
$\FQ_\gg$ preserves level by level
equivalence between strong compactness
and supercompactness when forcing over
$V^{\FP_\gg}$,
$\eta$ isn't a witness to the Menas
exception in $V^{\FP_\gg}$
\section{Concluding Remarks}\label{s3}
In conclusion to this paper,
we mention that Theorem \ref{t1} raises
a number of open questions.
Some of these are as follows.
\begin{enumerate}
\item In the model for Theorem \ref{t1},
is there precise level by level equivalence
between strong compactness and supercompactness?
If not,
is there any model for precise level by
level equivalence between strong compactness and
supercompactness in which $2^\gd = \gd^{++}$ for
every inaccessible cardinal $\gd$?
\item Is it possible to get a model witnessing
the conclusions of Theorem \ref{t1} in which
there are different kinds of failures of GCH
at the inaccessible cardinals,
e.g., in which $2^\gd = \gd^{+++}$ for every
inaccessible cardinal $\gd$, or
$2^\gd = \gd^{++}$ at every regular cardinal?
\item In general,
what sorts of failures of GCH are possible
in a model for precise level by level equivalence
between strong compactness and supercompactness?
\end{enumerate}
then by its
definition, $\FP_\gd$ is in the
terminology of \cite{H2} and \cite{H3} a
``mild forcing with respect to
$\gd$ admitting a gap at $\ha_1$''.
%meaning that any set of ordinals
%of size below $\gd$ in $V^{\FP_\gd}$
%has a name of size below $\gd$ in $V$.
Thus, by the results of \cite{H2} and
\cite{H3}, as before,
any degree of
strong compactness or supercompactness
$\eta$ witnesses in $V^{\FP_\gd}$
had to have been witnessed by $\eta$
in $V$. This means once again
$\eta$ isn't
a counterexample to level by level
equivalence between strong
compactness and supercompactness in
$V^{\FP_\gd}$.
To see that this is indeed so, fix
$\gd < \gk$ as a non-trivial
stage of forcing, meaning that
$\gd$ is in $V$ an inaccessible limit
of strong cardinals. Assume that
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})} \models
``\gd$ is $\gl$ strongly compact and
$\gl > \gd$ is regular''.
Assume further inductively that for any $\gg < \gd$,
$\add(\gg, \gg^{++})$ is an allowable
choice in the stage $\gg$ lottery held in
$V^{\FP_\gg}$.
We first note that $\gd$ can't be in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$
a measurable limit of cardinals which
are $\gl$ supercompact, i.e., $\gd$
can't witness the Menas exception in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$.
To see this, assume to the contrary that
$\gd$ is indeed in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$
a measurable limit of cardinals which are
$\gl$ supercompact. Since we can write
$\FP_\gd \ast \dot \add(\gd, \gd^{++}) =
\FP_0 \ast \dot \FQ$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_2$-directed closed'', by the
results of \cite{H1}, \cite{H2},
and \cite{H3}, any cardinal
exhibiting a degree of supercompactness in
%which is $\gl$ supercompact in
$V^{\FP_\gd \ast \dot \add(\gd, \gd^{++})}$
had to have exhibited that same degree of
supercompactness in $V$.
%have been $\gl$ supercompact in $V$.
Since $\gd$ is in $V$ a limit of strong
cardinals and $\gl > \gd$, this
means that for some
$\eta < \rho < \gd < \gk$, in $V$,
$\eta$ is $\rho$ supercompact and
$\rho$ is strong, thus yielding as
we indicated earlier that
$\eta$ is supercompact.
This immediately contradicts that
$\gk$ is the least supercompact
cardinal in $V$ and also shows
that $\gd$ isn't in $V$ a witness
to the Menas exception, i.e.,
$\gd$ isn't in $V$ a limit of
cardinals which are $\gl$ supercompact.
\end{graveyard}