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\title{Diamond, Square, and Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
strong cardinal, diamond, square,
level by level equivalence between strong
compactness and supercompactness}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank
the referee for numerous helpful
comments and suggestions,
which have considerably improved
the presentation of the material
contained herein.
The author also wishes to thank
Andreas Blass, the corresponding editor,
for a useful suggestion, and
Grigor Sargsyan for a very
helpful conversation on the
subject matter of this paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{January 19, 2004\\
(revised May 8, 2004)}
\begin{document}
\maketitle
\begin{abstract}
We force and construct
a model
in which level by level
equivalence between strong
compactness and supercompactness
holds, along with certain
additional combinatorial properties.
In particular, in this model,
$\diamondsuit_\gd$ holds for
every regular uncountable
cardinal $\gd$, and
below the least supercompact
cardinal $\gk$, $\square_\gd$
%holds on a set having measure
%$1$ with respect to a certain
%normal measure over $\gk$.
holds on a stationary subset of $\gk$.
There are no restrictions in
our model
%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}
In \cite{AS97a}, Shelah and the
author introduced the notion
of level by level equivalence
between strong compactness
and supercompactness by proving
the following theorem.
\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 measurable limit
of cardinals $\gd$ which are $\gl$
supercompact''.
\end{theorem}
%As was mentioned in \cite{A02},
We will say that
any model witnessing the conclusions of
Theorem \ref{t0} is a model for
level by level equivalence between
strong compactness and supercompactness.
%We will also say that $\gk$ is a witness
%to level by level equivalence between
%strong compactness and supercompactness
%iff for every regular $\gl > \gk$,
%$\gk$ 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.
In terminology used by Woodin,
this theorem
%to describe the main theorem of \cite{AS97a},
can be classified as an
``inner model theorem proven via forcing.''
This is since the model constructed
satisfies a pleasant property
one usually associates with an
inner model, namely GCH,
along with a property one might
perhaps expect if a ``nice'' inner model
containing supercompact
cardinals ever were to be
constructed, namely
level by level equivalence
between strong compactness
and supercompactness. Still, since
inner models for measurable,
strong, and Woodin cardinals
all satisfy additional combinatorial
properties, such as square and diamond,
one may ask if this is also possible
in a model in which level by level
equivalence between strong
compactness and supercompactness holds.
The purpose of this paper is to
provide an affirmative answer
to this question.
Specifically, we prove the
following theorem.
\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 + $\K$ is
the class of supercompact cardinals
(so $\gk$ is the least
supercompact cardinal) +
Level by level equivalence
between strong compactness and
supercompactness holds +
%$\gk$ is the least supercompact cardinal +
%For unboundedly in $\gk$ many cardinals $\gd < \gk$,
For every $\gd \in A$ where $A$
is a certain stationary subset of $\gk$,
$\square_\gd$ holds + For
every regular uncountable cardinal $\gd$,
$\diamondsuit_\gd$ holds''.
\end{theorem}
We note that the examples
of square
given in Theorem \ref{t1}
cannot be extended beyond
the least supercompact cardinal.
This is since, as indicated
in the discussion given in
Sections 5 and 9 of \cite{CFM},
%a weak square sequence (let alone
%a square sequence) can't exist
certain weaker versions of square
sequences which are implied by
the existence of square sequences can't exist
above a strongly compact cardinal.
Hence, if $\gl$ is supercompact,
since $\square_\gd$ must fail
for any cardinal $\gd > \gl$,
by reflection, there are
unboundedly in $\gl$ many cardinals
$\gd < \gl$ for which
$\square_\gd$ fails.
Thus, the occurrences of square found in
Theorem \ref{t1} are close to
optimal, since
there cannot be a final
segment $\mathfrak F$ of cardinals
below any supercompact cardinal
on which $\square_\gd$ holds for every
$\gd \in \mathfrak F$.
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 some ordinal
$\gd_\ga$, 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 $\dot x$ or
$\check x$, 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 $p \in \FP$.
$\FP$ 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.
In addition, if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
We take this opportunity to recall briefly
the combinatorial notions of diamond
and square, which were
introduced by Jensen in
\cite{Je}. If $\gk$ is a regular uncountable
cardinal, $\diamondsuit_\gk$ is
the principle stating that
there exists a sequence of sets
$\la S_\ga : \ga < \gk \ra$ such that
$S_\ga \subseteq \ga$,
with the additional property that
for every $X \subseteq \gk$,
$\{\ga < \gk : X \cap \ga = S_\ga\}$ is
a stationary subset of $\gk$.
If $\gk$ is an arbitrary
uncountable cardinal,
$\square_\gk$ is the principle
stating that there exists
a sequence of sets
$\la C_\ga : \ga < \gk^+$ and
$\ga$ is a limit ordinal$\ra$ such that
$C_\ga$ is a closed, unbounded subset
of $\ga$ so that
if ${\rm cof}(\ga) < \gk$,
then $C_\ga$ has order type below $\gk$,
with the additional property that
for any limit point
$\gb \in C_\ga$, $C_\ga \cap \gb = C_\gb$.
It is possible to force $\diamondsuit_\gk$
and $\square_\gk$ using reasonably defined
partial orderings.
To force $\square_\gk$, we use
the partial ordering $\FP(\gk)$,
whose definition is essentially
given in Definition 6.1 of \cite{CFM}.
Specifically, $\FP(\gk)$ is the
partial ordering whose elements
all have the form
$\la C_\ga : \ga \le \gb < \gk^+$ and
$\ga$ is a limit ordinal$\ra$ such that
$C_\ga$ is a closed, unbounded subset
of $\ga$ so that if ${\rm cof}(\ga) < \gk$,
then $C_\ga$ has order type below $\gk$,
with the additional property that
for any limit point
$\gg \in C_\ga$, $C_\ga \cap \gg = C_\gg$.
The ordering is given by
%inclusion, i.e.,
$p \le q$ iff $p$ is a subsequence of $q$.
%$p \subseteq q$.
%$\dom(p) \subseteq \dom(q)$.
%and $q \rest \dom(p) = p$.
Note that the definition of
$\FP(\gk)$ automatically implies that
the ordering is the same as
end-extension.
%$\dom(p) \subseteq \dom(q)$ automatically follows from this definition.
It is intuitively the case that
$\FP(\gk)$ is the partial ordering
consisting of initial segments
of a $\square_\gk$ sequence,
ordered by end-extension.
Assuming GCH, $\card{\FP(\gk)} = \gk^+$.
It can be shown (see Lemma 6.1 of
\cite{CFM}) that $\FP(\gk)$
is $\gk$-strategically closed.
This has an immediate consequence that
forcing with $\FP(\gk)$ over a model
of GCH preserves cardinals,
cofinalities, and GCH.
Since Lemma 6.1 of \cite{CFM}
requires no GCH assumptions,
forcing with $\FP(\gk)$ always
adds a
bona fide $\square_\gk$ sequence.
To force $\diamondsuit_\gk$,
we simply add a Cohen subset
to $\gk$.
That this partial ordering
works is essentially given
%in Exercise 22.12 of \cite{J}.
in Exercise 15.23, page 263 of \cite{J}.
For completeness, we give
now a sketch of the
proof of this fact, following
the outline of \cite{J}.
\begin{lemma}\label{l0}
{\bf (Folklore)} Suppose $\gk$ is a regular
uncountable cardinal, and $\FP$
is the usual partial ordering
for adding a Cohen subset
to $\gk$. Then $\diamondsuit_\gk$
holds in $V^\FP$.
\end{lemma}
\begin{sketch}
For simplicity, we assume GCH.
Let $\FP^*$ consist of all sequences
$\la S_\ga : \ga < \gb < \gk \ra$
such that $S_\ga \subseteq \ga$
for all $\ga < \gb$, ordered by
subsequence as in the definition of
$\FP(\gk)$ given above.
Since by its definition,
%end-extension. Since by its definition,
$\FP^*$ is $\gk$-directed
closed, and by GCH, $\FP^*$
has cardinality $\gk$,
it must be isomorphic to $\FP$.
Hence, without loss of generality,
for the rest of the proof of
Lemma \ref{l0}, we assume we are
forcing with $\FP^*$.
Suppose that
$p \forces ``\dot C$ is a
closed, unbounded subset of $\gk$
and $\dot X \subseteq \gk$''.
%Using the $\gk^+$-directed closure of $\FP^*$,
By the definition of $\FP^*$
and its closure properties, let
$\la q_n : n < \go \ra$ be
an increasing sequence of elements of
$\FP^*$ such that each
$q_n$ extends $p$, each $q_n$ has the form
$\la S_\ga : \ga \le \gb_n \ra$,
$q_n \forces ``\gg_n \in \dot C$''
for some ordinal $\gg_n$,
$\gb_n > \gg_n$,
%the sequence $\la \gg_n : n < \go \ra$ is
%strictly increasing,
$\gg_0 < \gb_0 < \gg_1 < \gb_1 < \cdots <
\gg_n < \gb_n < \cdots (n \in \go)$,
and $q_n$ completely
determines every element of the set
$\dot X \cap \gg_n$.
By construction,
$\bigcup_{n < \go} \gg_n =
\bigcup_{n < \go} \gb_n$.
Call their common supremum $\gb$.
Again by construction,
$q' = \la S_\ga : \ga < \gb \ra$
is such that
$q'$ completely determines every
element of the set $\dot X \cap \gb$,
a set which we call $S_\gb$.
$q = \la S_\ga : \ga \le \gb \ra$ is
then such that $q \ge p$ and
$q \forces ``\gb \in \dot C$ and
$\dot X \cap \gb = S_\gb$''.
This completes the sketch of
the proof of Lemma \ref{l0}.
\end{sketch}
A result which will be used in the proof
of Theorem \ref{t1} is
a corollary of Theorems 3 and 31
and Corollary 14 of
Hamkins' paper \cite{H5}.
This theorem is a generalization of
Hamkins' Gap Forcing Theorem and
Corollary 16 of
\cite{H2} and \cite{H3}
(and we refer readers to \cite{H2},
\cite{H3}, and \cite{H5} for
further details).
We therefore state the theorem
we will be using 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
$|\FQ| \le \gd$,
$\FQ$ is non-trivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H5},
$\FP$ {\rm admits a closure point at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{H3},
$\FP$ is {\rm mild}
with respect to a cardinal $\gk$
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2}, \cite{H3}, \cite{H5},
and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\rm amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Theorems 3 and 31
and Corollary 14 of
\cite{H5} we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a closure point
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
Finally, we mention that we are assuming
familiarity with the
large cardinal notions of measurability, strongness,
%superstrongness,
strong compactness, and supercompactness.
Interested readers may consult
\cite{J}, \cite{K}, or \cite{SRK}
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 Proofs of Theorem \ref{t1}
and Two Preliminary Theorems}\label{s1}
Before proving
Theorem \ref{t1}, we first establish
two preliminary theorems which
will be key to its proof.
We state these theorems now.
\begin{theorem}\label{t2a}
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 +
Level by level equivalence
between strong compactness and
supercompactness holds +
For every regular uncountable cardinal
$\gk$, $\diamondsuit_\gk$ holds''.
\end{theorem}
\begin{theorem}\label{t3a}
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 + $\K$ is
the class of supercompact cardinals
(so $\gk$ is the least supercompact
cardinal) +
Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$ where $A$
is a certain stationary subset of $\gk$,
$\square_\gd$ holds''.
\end{theorem}
We turn now to the proof of
the first of these theorems,
Theorem \ref{t2a}.
\bigskip
\begin{proof}
Let
$V \models ``$ZFC + $\K \neq
\emptyset$ is the class of
supercompact cardinals''.
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 our partial
ordering $\FP = \la \la \FP_\ga,
\dot \FQ_\ga \ra : \ga \in {\rm Ord} \ra$ as the
proper class Easton support iteration
such that $\FP_0$ is the trivial
partial ordering $\{\emptyset\}$, and
$\dot \FQ_\ga$ is a
term for
%the trivial partial ordering
$\{\emptyset\}$ except if $\ga$
is a $V$-regular infinite cardinal
(including $\go$). Under these
circumstances, $\dot \FQ_\ga$
is a term for the usual partial
ordering which adds a Cohen subset
to $\ga$.
%which adds a Cohen subset to each
%$V$-regular cardinal (including $\go$).
Standard arguments (see \cite{J}) then
show that
$V^\FP \models $ ZFC + GCH
and that $V$ and $V^\FP$ have
the same cardinals and cofinalities.
\begin{lemma}\label{l1}
$V^\FP \models ``$For every
regular uncountable cardinal $\gk$,
$\diamondsuit_\gk$ holds''.
\end{lemma}
\begin{proof}
Let $\gk$ be a regular uncountable
cardinal in either
$V$ or $V^\FP$. Write
$\FP = \FP_{\gk + 1}
\ast \dot \FP^{\gk + 1}$.
By the definition of $\FP$ and
Lemma \ref{l0}, $\diamondsuit_\gk$
holds in $V^{\FP_{\gk + 1}}$. Since
$\forces_{\FP_{\gk + 1}} ``\dot \FP^{\gk + 1}$
is $\gk^+$-directed closed'', forcing
with $\FP^{\gk + 1}$ over
$V^{\FP_{\gk + 1}}$ adds no new
subsets of $\gk$. In particular,
$\diamondsuit_\gk$ holds in
$V^{\FP_{\gk + 1} \ast \dot \FP^{\gk + 1}} =
V^\FP$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $V \models
``\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact
and $\gl$ is regular'', then
$V^\FP \models ``\gk$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$\gk$ and $\gl$ are as
in the hypotheses of Lemma \ref{l2}.
Write $\FP = \FP_{\gl + 1} \ast
\dot \FP^{\gl + 1}$. Since
$\forces_{\FP_{\gl + 1}} ``\dot
\FP^{\gl + 1}$ is $\gl^+$-directed
closed'' and GCH holds in
$V$, $V^{\FP_{\gl + 1}}$, and $V^\FP$,
${({[\gl]}^{< \gk})}^{V^\FP} =
{({[\gl]}^{< \gk})}^{V^{\FP_{\gl + 1}}} = \gl$, and
the subsets of ${[\gl]}^{< \gk}$ are
the same in both $V^\FP$ and
$V^{\FP_{\gl + 1}}$.
Thus, it suffices to show that
$V^{\FP_{\gl + 1}} \models ``\gk$
is $\gl$ supercompact''.
To do this,
we use a standard technique
(employed, e.g., in the
argument that $\FP$ may be
used to show the relative
consistency of GCH with a
class of supercompact cardinals).
Specifically, let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$. Write
$\FP_{\gl + 1} = \FP_\gk \ast \dot \FQ$ and
$j(\FP_{\gl + 1}) =
\FP_\gk \ast \dot \FQ \ast \dot \FR
\ast j(\dot \FQ)$.
As $M^\gl \subseteq M$,
it is the case that
in both $V$ and $M$,
$\forces_{\FP_\gk} ``\dot \FQ$ is
$\gk$-directed closed''.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be
$V[G]$-generic over
$\FQ$.
The usual diagonalization argument
(as given, e.g., in the construction
of the generic object $G_1$ in
Lemma 2.4 of \cite{AC2})
may be used to build in
$V[G][H]$ an $M[G][H]$-generic
object $H'$ over $\FR$ and lift $j$ to
$j : V[G] \to M[G][H][H']$ in $V[G][H]$.
We may then once again
use the usual diagonalization argument
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''.
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
$V^\FP \models ``\gk < \gl$ are
regular cardinals such that
$\gk$ is $\gl$ strongly compact and
$\gk$ isn't a measurable limit of
cardinals $\gd$ 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
$\gd$ such that $\gd$ is
$\gl$ supercompact in $V$ remains
$\gl$ supercompact in $V^\FP$.
This means that
$V \models ``\gk < \gl$ are
regular cardinals such that
%$\gk$ is $\gl$ strongly compact and
$\gk$ isn't a measurable limit of
cardinals $\gd$ which are
$\gl$ supercompact''.
Note that it is possible to
write $\FP =
\FP_{\go + 1} \ast \dot \FQ$, where
$|\FP_{\go + 1}| = \go$,
$\FP_{\go + 1}$ is non-trivial, and
$\forces_{\FP_{\go + 1}} ``\dot \FQ$ is
$\go$-strategically closed''.
Further, by the definition of
$\FP$, it is easily seen that
$\FP$ is mild with respect to $\gk$.
Therefore, by Theorem \ref{t3},
$V \models ``\gk$ is $\gl$ strongly compact''.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gk$ is $\gl$ supercompact'',
so another application of Lemma \ref{l2}
yields that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l3a}
$V^\FP \models ``\K$ is the
class of supercompact cardinals''.
\end{lemma}
\begin{proof}
By Lemma \ref{l2}, if
$\gk$ is supercompact in $V$,
then $\gk$ is supercompact in $V^\FP$.
Further, by the factorization of
$\FP$ as $\FP_{\go + 1} \ast \dot \FQ$ given
in Lemma \ref{l3} and
an application of Theorem \ref{t3},
any cardinal which is supercompact in
$V^\FP$ had to have been supercompact
in $V$. Thus, $\K$ is precisely the
class of supercompact cardinals in $V^\FP$.
This completes the proof of Lemma \ref{l3a}.
\end{proof}
Lemmas \ref{l1} - \ref{l3a} complete
the proof of Theorem \ref{t2a}.
\end{proof}
%\section{The Proof of Theorem \ref{t2}}\label{s2}
Having completed the proof of
Theorem \ref{t2a},
we turn now to the proof of Theorem \ref{t3a}.
\bigskip
\begin{proof}
Let
$V \models ``$ZFC + $\K \neq
\emptyset$ is the class of
supercompact cardinals + $\gk$ is the
least supercompact cardinal''.
As in the proof of Theorem \ref{t2a},
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 our partial
ordering $\FP = \la \la \FP_\ga,
\dot \FQ_\ga \ra : \ga < \gk \ra$ as the
Easton support iteration of length $\gk$
which begins by adding a Cohen subset
to $\go$, i.e.,
$\FP_0$ is the trivial partial ordering
$\{\emptyset\}$, and
$\FQ_0$ is the partial ordering
for adding a Cohen subset to $\go$.
At all other stages $\ga > 0$,
$\dot \FQ_\ga$ is a
term for the trivial partial ordering
$\{\emptyset\}$ except if $\ga$
is in $V$ a non-measurable
regular limit of strong cardinals.
Under these
circumstances, $\dot \FQ_\ga$
is a term for the partial
ordering $\FP(\ga)$ of
Section \ref{s0} which adds a
$\square_{\ga}$ sequence.
%where $\ga^*$ is the least
%$V$-strong cardinal above $\ga$.
Standard arguments once again
show that
$V^\FP \models $ GCH
and that $V$ and $V^\FP$ have
the same cardinals and cofinalities.
\begin{lemma}\label{l4}
$V^\FP \models
``$For every $\gd < \gk$ which is in
$V$ a non-measurable regular
limit of strong cardinals,
$\square_\gd$ holds''.
\end{lemma}
\begin{proof}
By Lemma 2.1 of \cite{AC2}
and the succeeding remarks,
there are in $V$ unboundedly
in $\gk$ many cardinals below $\gk$
which are measurable limits of
strong cardinals.
From this, it easily follows that
there are unboundedly in $\gk$
many cardinals below $\gk$ which
are in $V$ non-measurable regular
limits of strong cardinals.
%For any $\ga < \gk$,
%we may therefore let
%$\gd \in (\ga, \gk)$ be a cardinal
We may therefore let $\gd < \gk$
be such a cardinal.
%be a cardinal which is in $V$ a.
%non-measurable regular limit of strong cardinals.
%Let $\gd$ be such a cardinal, and write
Write $\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$.
By the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\square_{\gd}$
holds and $\dot \FP^{\gd + 1}$ is
$\eta$-strategically closed for
$\eta$ the least inaccessible
cardinal above $\gd$''.
This has as an immediate consequence that
$\square_{\gd}$ holds in
$V^{\FP_{\gd + 1} \ast \dot \FP^{\gd + 1}} =
V^\FP$.
%which further implies that
%for unboundedly in $\gk$ many cardinals
%$\gg < \gk$, $\square_\gg$ holds.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l6}
The set $A = \{\gd < \gk : \gd$
is in $V$ a non-measurable
regular limit of strong cardinals$\}$
is a stationary subset of $\gk$
in $V^\FP$.
\end{lemma}
\begin{proof}
Since $V \models ``\gk$
is supercompact'', Lemma 2.1 of
\cite{AC2} and the succeeding
remarks imply that
in $V$, $\gk$ is a limit
of strong cardinals.
Therefore, if $\mu \in V$
is any normal measure over
$\gk$ concentrating on
non-measurable cardinals, it
easily follows via a simple
ultrapower and reflection argument that
%for $A = \{\gd < \gk: \gd$ is in
%$V$ a non-measurable regular
%limit of strong cardinals$\}$,
$A \in \mu$.
Thus, $A$ is a stationary
subset of $\gk$ in $V$.
However, since by its
definition, the partial
ordering $\FP$ is $\gk$-c.c.,
by Exercise H2, page 247 of \cite{Ku},
$A$ remains a stationary
subset of $\gk$ in $V^\FP$.
This completes the proof
of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l5}
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}
Since the definition of
$\FP$ ensures that
$\card{\FP} = \gk$,
the L\'evy-Solovay results
\cite{LS} yield that Lemma \ref{l5}
is true when $\gd > \gk$.
If $\gd = \gk$ and $\gl > \gk$
is a regular cardinal, let
$j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$ such that
$M \models ``\gk$ isn't $\gl$ supercompact''.
As in the proof of Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''
(since otherwise, $\gk$ is supercompact
up to a strong cardinal and hence
is fully supercompact).
Further, since $\gl \ge 2^\gk$
by GCH, Lemma 2.1 of \cite{AC2}
%and the succeeding remarks
implies that in both $V$
and $M$, $\gk$ is a measurable limit
of strong cardinals.
This means we can write
$j(\FP) = \FP \ast \dot \FQ$,
%where in both $V$ and $M$,
where $\gk \not\in {\rm field}(\dot \FQ)$,
in both $V$ and $M$,
$\forces_\FP ``\dot \FQ$ is
$\gl$-strategically closed'', and
the first ordinal in the field of
$\dot \FQ$ greater than
or equal to $\gk$ is well above $\gl$.
Thus, a simplified version of
the arguments of Lemma \ref{l2}
(which doesn't require the construction
of a generic object containing a
master condition) may now be used
to show that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
To complete the proof of Lemma \ref{l5},
we now assume that
$\gd < \gk$ and
$V \models ``\gd$ is $\gl$ supercompact
where $\gl > \gd$ is regular''.
Note that $\gl < \gd^*$, where
$\gd^*$ is the
least $V$-strong cardinal above $\gd$.
This is since
otherwise, $\gd$ is supercompact
up to a strong cardinal and hence
is fully supercompact, which
contradicts that $\gk$ is the
least supercompact cardinal in $V$.
Further,
since $\gd$ is a measurable cardinal,
we can write
$\FP = \FP_\gd \ast \dot \FP^\gd$,
where $\gd \not\in {\rm field}(\dot \FP^\gd)$,
and the first ordinal in the field of
$\dot \FP^\gd$ is above $\gd^*$. Consequently,
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\eta$-strategically closed for
$\eta$ the least inaccessible cardinal
above $\gl$'', so to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl$ supercompact''.
If $\card{\FP_\gd} < \gd$, this
follows from the results of \cite{LS}.
If $\card{\FP_\gd} = \gd$, this follows
from the argument given in the preceding
paragraph.
This completes the proof of Lemma \ref{l5}.
\end{proof}
By replacing the factorization of
the $\FP$ of Theorem \ref{t1}
given in Lemma \ref{l3} with
the factorization of the
current $\FP$ as
$\FP_1 \ast \dot \FQ$ and
changing all references to
Lemma \ref{l2} to Lemma \ref{l5},
the proofs of Lemmas \ref{l3} and
\ref{l3a} go through exactly as before.
This shows that level by level
equivalence between strong
compactness and supercompactness
holds in $V^\FP$ and that
$\K$ is the class of supercompact
cardinals in $V^\FP$.
Hence, since
$V \models ``\gk$ is the least
supercompact cardinal'' and
$V^\FP \models ``\K$ is
the class of supercompact cardinals'',
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
This completes the proof of
Theorem \ref{t3a}.
\end{proof}
Now that we have completed the
proofs of the two preliminary results,
Theorems \ref{t2a} and \ref{t3a},
we turn our attention to the
proof of the main result,
Theorem \ref{t1}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\K \neq \emptyset$
is the class of supercompact cardinals
+ $\gk$ is the least supercompact
cardinal''. As in the proofs of
Theorems \ref{t2a} and \ref{t3a},
without loss of generality,
we assume that GCH and level by
level equivalence between strong
compactness and supercompactness
also hold in $V$.
If we first force with the
partial ordering used in
the proof of Theorem \ref{t3a}
and subsequently follow this by
the partial ordering used
in the proof of Theorem \ref{t2a},
then we have added diamond
sequences for every regular
uncountable cardinal
without destroying the square
sequences initially added.
This is since the notion of
closed, unbounded set is upwards
absolute for any partial
ordering that preserves
cardinals and cofinalities.
This follows from the fact that if
$C$ is a ground model closed,
unbounded set and
$p \forces
``\la \dot \gb_\ga : \ga < \gl \ra$ is
an increasing sequence of elements
of $C$ whose supremum is $\gb$'', then
in $V$, $\{\gg < \gb : \gg \in C\}$
will contain an increasing sequence
of elements of $C$ whose supremum,
which is of course a member of $C$,
must be $\gb$.
Further, Theorem \ref{t3} can be
applied twice to show that
$\K$ remains the class of
supercompact cardinals,
and the partial orderings
of Theorems \ref{t2a} and
\ref{t3a} both preserve GCH
and the level by level equivalence
between strong compactness
and supercompactness.
Finally, since the $\FP$ of
Theorem \ref{t2a} may be written as
$\FP_\gk \ast \dot \FP^\gk$ where
$\FP_\gk$ is $\gk$-c.c$.$ and
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is
$\gk$-directed closed'',
Exercise H2, page 247 of \cite{Ku}
once again allows us to infer that
the forcing of Theorem \ref{t2a}
preserves stationary subsets of $\gk$.
This completes the proof of
Theorem \ref{t1}.
\end{proof}
\section{Concluding Remarks}\label{s2}
In conclusion to this paper, we
make two remarks.
The first is that the proof of
Theorem \ref{t3a} yields another
proof of Magidor's theorem of
\cite{Ma} that it is consistent,
relative to the existence of
a supercompact cardinal, for
the least supercompact cardinal
to be the least strongly compact
cardinal.
To see this, suppose
$V \models ``\gk$ is supercompact'',
and let $\FP$ be the partial
ordering used in the proof of
Theorem \ref{t3a} defined
with respect to $\gk$.
The proof of Lemma \ref{l4},
which remains valid without
any use of GCH, shows that
in $V^\FP$, there are unboundedly
in $\gk$ many cardinals $\gd < \gk$
at which $\square_\gd$ holds.
As we indicated in the paragraph
immediately following the
statement of Theorem \ref{t1},
this implies that no cardinal
below $\gk$ is strongly compact in $V^\FP$.
Since by Solovay's theorem of
\cite{So}, we can always find
an arbitrarily large
cardinal $\gl > \gk$ at
which GCH holds, e.g., a singular
strong limit cardinal of cofinality
at least $\gk$, we can use
the proofs of Lemmas \ref{l2} and
\ref{l5}
(which remain valid if
$\gl$ is singular and has
cofinality at least $\gk$) to show that
$V^\FP \models ``\gk$ is supercompact''.
Thus, $V^\FP \models ``\gk$ is both
the least strongly compact and
least supercompact cardinal''.\footnote{In
fact, if we redefine the $\FP$
just discussed so that at any
non-trivial stage of forcing,
the component partial ordering
is the one for adding
a non-reflecting stationary set of
ordinals of cofinality $\go$,
%a non-reflecting stationary set of
%ordinals of cofinality $\go$ is added,
then we have still another proof
of the aforementioned theorem of Magidor.
This follows by the
arguments of this paragraph,
%just given,
together with the fact
that by Theorem 4.8 of
\cite{SRK} and the succeeding remarks,
if a cardinal $\gl$ contains a non-reflecting
stationary set of ordinals of
cofinality $\go$, then no cardinal
less than or equal to $\gl$ is
strongly compact.}
%We note that Theorem \ref{t1}
%provides an example of a model
%for level by level equivalence
%between strong compactness
%and supercompactness exhibiting
We finish by mentioning that
it is unclear at this juncture
if other combinatorial properties
than those given in Theorem \ref{t1}
are possible in a model in which
level by level equivalence between
strong compactness and supercompactness
holds.
We therefore pose
this as our concluding question.
\begin{thebibliography}{99}
%\bibitem{A00} A.~Apter, ``A New Proof of
%a Theorem of Magidor'',
%{\it Archive for Mathematical Logic 39},
%2000, 209--211.
%\bibitem{A99} A.~Apter, ``Aspects of Strong
%Compactness, Measurability, and Indestructibility'',
%to appear in the
%{\it Archive for Mathematical Logic}.
%\bibitem{A99flm} A.~Apter, ``Forcing the Least
%Measurable to Violate GCH'',
%{\it Mathematical Logic Quarterly 45}, 1999, 551--560.
\bibitem{A02} A.~Apter,
``On Level by Level Equivalence and
Inequivalence between Strong
Compactness and Supercompactness'',
{\it Fundamenta Mathematicae 171},
2002, 77--92.
%\bibitem{A99a} A.~Apter, ``On the Level by Level Equivalence
%between Strong Compactness and Strongness'', to appear in
%the {\it Journal of the Mathematical Society
%of Japan}.
%\bibitem{A97} A.~Apter, ``Patterns of Compact
%Cardinals'', {\it Annals of Pure and
%Applied Logic 89}, 1997, 101--115.
%\bibitem{A00a} A.~Apter, ``Some Remarks on Indestructibility
%and Hamkins' Lottery Preparation'', submitted for publication
%to the {\it Archive for Mathematical Logic}.
%\bibitem{A99c} A.~Apter, ``Some Structural Results
%Concerning Supercompact Cardinals'',
%{\it Journal of Symbolic Logic 66}, 2001, 1919--1927.
%\bibitem{A99b} A.~Apter, ``Strong Compactness,
%Measurability, and the
%Class of Supercompact Cardinals'',
%{\it Fundamenta Mathematicae 167}, 2001, 65--78.
%\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness'',
%{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness II: Strong Cardinals'',
{\it Archive for Mathematical Logic 40}, 2001, 25--38.
%\bibitem{AH2} A.~Apter, J.~D.~Hamkins,
%``Indestructible Weakly Compact Cardinals
%and the Necessity of Supercompactness
%for Certain Proof Schemata'',
%{\it Mathematical Logic Quarterly 47}, 2001, 563--571.
%\bibitem{AH4} A.~Apter, J.~D.~Hamkins,
%``Indestructibility and the Level-by-Level Agreement
%between Strong Compactness and Supercompactness'',
%{\it Journal of Symbolic Logic 67}, 2002, 820--840.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong
Equality between Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349},
1997, 103--128.
%\bibitem{AS97b} A.~Apter, S.~Shelah, ``Menas' Result is
%Best Possible'',
%{\it Transactions of the American Mathematical Society 349},
%1997, 2007--2034.
%\bibitem{Bu} J.~Burgess, ``Forcing'', in:
%J.~Barwise, editor,
%{\it Handbook of Mathematical Logic},
%North-Holland, Amsterdam, 1977, 403--452.
%\bibitem{C} J.~Cummings, ``A Model in which GCH Holds at
%Successors but Fails at Limits'',
%{\it Transactions of the American Mathematical
%Society 329}, 1992, 1--39.
\bibitem{CFM} J.~Cummings, M.~Foreman,
M.~Magidor, ``Squares, Scales, and
Stationary Reflection'', {\it Journal of
Mathematical Logic 1}, 2001, 35--98.
%\bibitem{FMS} M.~Foreman, M.~Magidor, and
%S.~Shelah, ``Martin's Maximum, Saturated
%Ideals, and Non-Regular Ultrafilters:
%Part I'', {\it Annals of Mathematics 127},
%1988, 1--47.
%\bibitem{H1} J.~D.~Hamkins,
%``Destruction or Preservation As You
%Like It'',
%{\it Annals of Pure and Applied Logic 91},
%1998, 191--229.
\bibitem{H5} J.~D.~Hamkins, ``Extensions with
the Approximation and Cover Properties
Have No New Large Cardinals'',
to appear in
{\it Fundamenta Mathematicae}.
\bibitem{H2} J.~D.~Hamkins, ``Gap Forcing'',
{\it Israel Journal of Mathematics 125}, 2001, 237--252.
\bibitem{H3} J.~D.~Hamkins, ``Gap Forcing:
Generalizing the L\'evy-Solovay Theorem'',
{\it Bulletin of Symbolic Logic 5}, 1999, 264--272.
%\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
%{\it Annals of Pure and Applied Logic 101},
%2000, 103--146.
%\bibitem{H5} J.~D.~Hamkins, ``Small Forcing Makes
%Any Cardinal Superdestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 51--58.
\bibitem{J} T.~Jech, {\it Set Theory:
The Third Millennium Edition,
Revised and Expanded},
%$3^{\rm rd}$ Millennium Edition,
Springer-Verlag, Berlin and New York, 2003.
\bibitem{Je} R.~Jensen, ``The Fine Structure
of the Constructible Hierarchy'',
{\it Annals of Mathematical Logic 4},
1972, 229--308.
%\bibitem{JMMP} T.~Jech, M.~Magidor,
%W.~Mitchell, K.~Prikry, ``Precipitous Ideals'',
%{\it Journal of Symbolic Logic 45}, 1980, 1--8.
%\bibitem{JW} T.~Jech, W.H.~Woodin,
%``Saturation of the Closed Unbounded Filter on the
%Set of Regular Cardinals'',
%{\it Transactions of the American Mathematical
%Society 292}, 1985, 345--356.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite}, Springer-Verlag,
Berlin and New York, 1994.
\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
between the Concepts of Compactness and Supercompactness'',
circulated manuscript.
\bibitem{Ku} K.~Kunen, {\it Set Theory:
An Introduction to Independence Proofs},
{\bf Studies in Logic and the Foundations of
Mathematics 102}, North-Holland, Amsterdam
and New York, 1980.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
\bibitem{Ma} M.~Magidor, ``How Large is the First
Strongly Compact Cardinal?'', {\it Annals of
Mathematical Logic 10}, 1976, 33--57.
%\bibitem{Ma2} M.~Magidor, ``On the Existence of
%Nonregular Ultrafilters and the Cardinality of
%Ultrapowers'', {\it Transactions of the American
%Mathematical Society 249}, 1979, 97--111.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
\bibitem{So} R.~Solovay, ``Strongly Compact Cardinals
and the GCH'', in: {\it Proceedings of the Tarski
Symposium}, {\bf Proceedings of Symposia in Pure
Mathematics 25}, American Mathematical Society,
Providence, 1974, 365--372.
\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
``Strong Axioms of Infinity and Elementary Embeddings'',
{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}
\begin{graveyard}
$ \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{t2a}
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.
In the proof of Theorem \ref{t2a},
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.
\begin{lemma}\label{l6}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6}
follows closely
the proof of Lemma \ref{l3}.
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''.
By Lemma \ref{l5}, 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
$\gd$ isn't a measurable limit of
cardinals $\gg$ which are
$\gl$ supercompact''.
Note that it is possible to
write $\FP =
\FP_1 \ast \dot \FQ$, where
$|\FP_1| = \go$ and
$\forces_{\FP_1} ``\dot \FQ$ is
$\go$-strategically closed''.
Further, by the definition of
$\FP$, it is easily seen that
$\FP$ is mild with respect to $\gd$.
Therefore, by Theorem \ref{t3a},
$V \models ``\gd$ 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{l5}
yields that
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``\K$ is the
class of supercompact cardinals''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l7}
follows closely the proof
of Lemma \ref{l3a}.
By Lemma \ref{l5}, if
$\gd$ is supercompact in $V$,
then $\gd$ is supercompact in $V^\FP$.
Further, by the factorization of
$\FP$ as $\FP_1 \ast \dot \FQ$ given
in Lemma \ref{l6} and Theorem \ref{t3a},
any cardinal which is supercompact in
$V^\FP$ had to have been supercompact
in $V$. Thus, $\K$ is precisely the
class of supercompact cardinals in $V^\FP$.
This completes the proof of Lemma \ref{l7}.
\end{proof}
Since
$V \models ``\gk$ is the least
supercompact cardinal'',
Lemma \ref{l7} implies that
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
Lemmas \ref{l4} - \ref{l7} therefore
complete the proof of Theorem \ref{t3a}.
\end{proof}
\footnote{$S$
has other interesting uses.
Suppose we take $S$ as the field of
Magidor's notion of iterated
Prikry forcing $\FP^*$ of \cite{Ma}.
As before, $\gk \not\in j(S)$
for any $j : V \to M$ witnessing
at least the $2^\gk$ supercompactness
of $\gk$, and for any
$j : V \to M$ witnessing the $\gl$
supercompactness of $\gk$ such that
$M \models ``\gk$ isn't $\gl$ supercompact'',
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''. Thus,
the argument given in
the proof of the Lemma of \cite{A00}
remains valid and shows that
$V^{\FP^*} \models ``\gk$ is both
the least strongly compact and
least supercompact cardinal''.
This yields still another proof
of Magidor's theorem of \cite{Ma}
mentioned above.}
We mention in addition that as
pointed out by Grigor Sargsyan,
it is possible for the field
of the partial ordering $\FP$
used in the proof of Theorem
\ref{t3a} to be a stationary set.
Specifically, we redefine
$\FP$ as the Easton support
iteration of length $\gk$ which
begins by adding a Cohen subset
to $\go$. $\FP$ is then non-trivial
only at those stages $\ga$ which are
in $V$ strong cardinals which
aren't superstrong.
Under these circumstances, we force
with the partial ordering
$\FP(\ga)$ of Section \ref{s0}.
To see that this works, we first
note that the set
$S = \{\gd < \gk : \gd$ is
strong but isn't superstrong$\}$
is stationary.
We show this by first choosing
$j_0 : V \to M$ as an elementary
embedding witnessing (at least) the
$2^\gk$ supercompactness of $\gk$.
As mentioned in Lemma 2.1 of
\cite{AC2},
$M \models ``\gk$ is superstrong''.
%with target $j_0(\gk)$''.
We may therefore pick
$j_1 : M \to N$ as an elementary
embedding witnessing the
superstrongness of $\gk$ such that
$N \models ``\gk$ isn't superstrong''.
Let $j = j_1 \circ j_0$.
The composed embedding
$j : V \to N$ has critical point $\gk$.
Further, since
$M \models ``\gk$ is strong'',
by elementarity,
$N \models ``j_1(\gk)$ is strong''.
In particular, as
$M \models ``N^{j_1(\gk)} \subseteq N$'',
$N \models ``\gk$ is $\gd$ strong for
every $\gd < j_1(\gk)$ and $j_1(\gk)$
is strong''.
Hence, as indicated in Lemma 2.5 of
\cite{AC2},
$N \models ``\gk$ is strong''.
Thus, $\gk \in j(S)$, i.e.,
$S$ is a stationary subset of $\gk$.
%Now that we know that $S$ is stationary,
The proof of
Lemma \ref{l4} goes through as
before to show that after
forcing with our redefined
version of $\FP$, there are
$\square_\gg$ sequences for
unboundedly in $\gk$ many $\gg < \gk$.
Suppose further that
$\gd < \gl$ are cardinals such
that $\gd$ is $\gl$ supercompact and
$\gl$ is regular. Since
$\gl \ge 2^\gd$ by GCH, as
we indicated above, any
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
is such that
$M \models ``\gd$ is superstrong''.
From this, it can be immediately
inferred that
$j(\FP) = \FP \ast \dot \FQ$
has the properties that
$\gd \not\in {\rm field}(\dot \FQ)$,
%$\gd \not\in {\rm field}(j(\FP))$,
$\forces_{\FP} ``\dot \FQ$ is
$\gl$-strategically closed'',
and the first ordinal in the field
of $\dot \FQ$ is well above $\gl$.
This allows the proof of Lemma
\ref{l5} suitably modified
to go through as earlier.
Since we may then conclude as
in the original proof of
Theorem \ref{t3a} that after
forcing with $\FP$, GCH and
level by level equivalence between
strong compactness and supercompactness
hold and $\K$ is the class of
supercompact cardinals, the stationary set
$S$ may be used as the field of
$\FP$ in the proof of
Theorem \ref{t3a}.
It is also possible to combine
the results of Theorems \ref{t2a}
and \ref{t3a}.
\begin{lemma}\label{l2}
If $V \models
``\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact
and $\gl$ is regular'', then
$V^\FP \models ``\gk$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$\gk$ and $\gl$ are as
in the hypotheses of Lemma \ref{l2}.
Write $\FP = \FP_{\gl + 1} \ast
\dot \FP^{\gl + 1}$. Since
$\forces_{\FP_{\gl + 1}} ``\dot
\FP^{\gl + 1}$ is $\gl^+$-directed
closed'' and GCH holds in
$V$, $V^{\FP_{\gl + 1}}$, and $V^\FP$,
${({[\gl]}^{< \gk})}^{V^\FP} =
{({[\gl]}^{< \gk})}^{V^{\FP_{\gl + 1}}} = \gl$, and
the subsets of ${[\gl]}^{< \gk}$ are
the same in both $V^\FP$ and
$V^{\FP_{\gl + 1}}$.
Thus, it suffices to show that
$V^{\FP_{\gl + 1}} \models ``\gk$
is $\gl$ supercompact''.
To do this, let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$. Write
$\FP_{\gl + 1} = \FP_\gk \ast \dot \FQ$ and
$j(\FP_{\gl + 1}) =
\FP_\gk \ast \dot \FQ \ast \dot \FR
\ast j(\dot \FQ)$.
As $M^\gl \subseteq M$,
it is the case that
in both $V$ and $M$,
$\forces_{\FP_\gk} ``\dot \FQ$ is
$\gk$-directed closed''.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be
$V[G]$-generic over
$\FQ$.
Since $\FQ$ is $\gl^+$-c.c$.$
in both $V[G]$ and $M[G]$,
standard arguments show that
$M[G][H]$ remains $\gl$ closed
with respect to $V[G][H]$.
As
$M[G][H] \models ``|\FR| = j(\gk)$'' and
GCH holds in both $V$ and $M$,
there are $2^{j(\gk)} = j(\gk^+)$
dense open subsets of $\FR$
present in $M[G][H]$.
Further,
$|j(\gk^+)| =
|\{f : f : P_\gk(\gl) \to \gk^+$
is a function$\}| = \gl^+$ by GCH,
and $\FR$ is $\gl^+$-directed closed
in both $V[G][H]$ and $M[G][H]$.
We can therefore
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 build in
$V[G][H]$ an $M[G][H]$-generic
object $H'$ over $\FR$ 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
$V[G][H][H'] = V[G][H]$.
Also, the number of dense open subsets
of $j(\FQ)$ in $M[G][H][H']$ is
$2^{j(\gl)} = j(\gl^{+})$, which
by GCH has size $\gl^+$ in
$V[G][H]$. In addition, because
$j(\gk) > \gl^+$ and
$\forces_{\FP_\gk} ``\dot \FQ$ is
$\gk$-directed closed'', $j(\FQ)$
is $\gl^+$-directed closed in both
$M[G][H][H']$ and $V[G][H]$.
We can hence 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 $j''H$,
a set having size $\gl$.
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''.
This completes the
proof of Lemma \ref{l2}.
\end{proof}
\end{graveyard}