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%
% ------------------------------------------------------------------------------
%
%\title{On Partial Level by Level Equivalence between
% Strong Compactness and Strongness in
% the Context of Supercompactness
%\title{More on the Level by Level Equivalence
% between Strong Compactness and Strongness
\title{Supercompactness and
Partial Level by Level Equivalence between
Strong Compactness and Strongness
% in the Context of 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,
level by level equivalence between strong
compactness and strongness.}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{February 26, 2003\\
(revised July 13, 2004)}
\begin{document}
\maketitle
\begin{abstract}
We force and construct a model
containing supercompact cardinals
in which, for any
measurable cardinal $\gd$
and any ordinal
$\ga$ below the least beth
fixed point above $\gd$,
if $\gd^{+ \ga}$ is regular,
$\gd$ is $\gd^{+ \ga}$ strongly
compact iff $\gd$ is $\gd + \ga + 1$
strong, except possibly if
$\gd$ is a limit of cardinals
$\gg$ which are $\gd^{+ \ga}$
strongly compact.
The choice of the least beth fixed
point above $\gd$ as our bound
on $\ga$ is arbitrary,
and other bounds are possible.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{A03},
the following theorem was proven.
%on the level by
%level equivalence between strong
%compactness and strongness was proven.
\begin{theorem}\label{t0}
Let
$V \models ``$ZFC + $\gk$ is
supercompact + There is no pair
of cardinals $\gd < \gl$ such that
$\gd$ is $\gl$ supercompact and
$\gl$ is measurable''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH +
There is no pair of cardinals
$\gd < \gl$ such that $\gd$ is
$\gl$ supercompact and $\gl$
is measurable + $\gk$ is both
the least strongly compact and
least strong cardinal (so $\gk$
isn't $2^\gk$ supercompact) +
No cardinal $\gl > \gk$ is measurable +
For $\gd < \gk$, if
$\gd^{+ \ga}$ is regular, then
$\gd$ is $\gd^{+ \ga}$ strongly
compact iff $\gd$ is $\gd + \ga + 1$
strong''.
\end{theorem}
This theorem provides a counterpoint
to the main result of \cite{AS97a},
which is as follows.
\begin{theorem}\label{t0a}
Let
$V \models ``$ZFC + $\K \neq \emptyset$
is the class of
supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\K$ is
the class of supercompact cardinals +
For every pair of regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except
possibly if $\gk$ is a limit
of cardinals $\gd$ which are $\gl$
supercompact''.
\end{theorem}
Whenever the conclusions
of Theorem \ref{t0a} are true,
we will say that
level by level equivalence between
strong compactness and supercompactness
holds. Whenever the conclusions
of Theorem \ref{t0} are true,
we will say that level by level
equivalence between strong compactness
and strongness holds.
Observe that in any model witnessing
the conclusions of Theorem \ref{t0a},
the Kimchi-Magidor property \cite{KM}
holds, i.e., the strongly compact
and supercompact cardinals coincide,
except possibly at measurable
limit points.
Notice that in the model
constructed for Theorem \ref{t0},
there are no supercompact cardinals.
In fact, the number of large
cardinals in the universe
witnessing the conclusions of
Theorem \ref{t0} is severely restricted.
This raises the following questions:
Is it possible to get a model in which
there is level by level equivalence
between strong compactness and strongness
and in which there are supercompact cardinals?
More generally,
is it possible to get a model in which
there is level by level equivalence
between strong compactness and strongness
and in which there is more than one
strongly compact cardinal?
The purpose of this paper is to
provide a partial affirmative answer
to the first of the preceding questions.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
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 + The strongly compact
and supercompact cardinals coincide,
except possibly at measurable limit points +
For any measurable cardinal $\gd$
and any $\ga$ below the least
beth fixed point above $\gd$,
if $\gd^{+ \ga}$ is regular,
$\gd$ is $\gd^{+ \ga}$ strongly
compact iff $\gd$ is $\gd + \ga + 1$
strong, except possibly if $\gd$
is a limit of cardinals $\gg$ which are
$\gd^{+ \ga}$ strongly compact''.
\end{theorem}
The choice of the least
beth fixed point above
$\gd$ as the bound on
$\ga$ in Theorem \ref{t1}
is done purely as a matter
of convenience.
Larger bounds on $\ga$
are also possible.
This will be discussed
in greater detail at the
end of this paper.
We note that by Lemma 1.1 of
\cite{A01} and the succeeding
remark,
it is impossible for
the least measurable cardinal $\gd$
which is a limit
of cardinals $\gg$ which are
either $\gd^+$ strongly compact
or $\gd^+$ supercompact to be
$\gd + 2$ strong.
This generalizes a result of
Menas \cite{Me}, who also showed
that this cardinal $\gd$ must be
%By a result of Menas \cite{Me},
%however, this cardinal must be
$\gd^+$ strongly compact.
Thus, when there are large
enough cardinals in the universe,
it is impossible for there to
be a precise level by level
equivalence between strong
compactness and strongness.
(By Lemma 1.2 of \cite{A03},
a precise level by level
equivalence between
strong compactness and
strongness in the sense that
$\gd$ is $\gd^{+ \ga}$
strongly compact iff
$\gd$ is $\gd + \ga + 1$
strong for arbitrary $\ga$
is impossible if
there are supercompact cardinals
in the universe.)
%This is as a result of the
%same phenomenon discovered
%by Menas in \cite{Me}, who
%showed that a measurable
%limit of strongly compact
%cardinals is strongly compact
%but need not be supercompact.
We observe also that in any model
witnessing the conclusions
of Theorem \ref{t1},
level by level equivalence
between strong compactness
and supercompactness must fail.
To see this,
note that by either Proposition
26.11 of \cite{K} or Lemma 2.1
of \cite{AC2}, if
$\gd$ is $2^\gd$ supercompact,
$\{\gg < \gd : \gg$ is superstrong
with target $\gd\}$ is unbounded
in $\gg$.
Any such $\gg$ will of course
be $\gg + \ga$ strong for every
$\ga$ below the least beth
fixed point above $\gg$.
Thus, in a universe in which
the conclusions of Theorem \ref{t1}
hold, below the least cardinal
$\gk$ which is $2^\gk = \gk^+$
supercompact, there will be many
cardinals $\gg$ which are
$\gg^{+ \ga}$ strongly compact
but not $\gg^{+ \ga}$ supercompact,
where $\ga$ is any ordinal below
the least beth fixed point above $\gg$.
Before presenting the proof
of our theorem, 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 $\gk$
such 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 $p \in \FP$.
$\FP$ 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 $\gk$ is a Mahlo cardinal.
A partial ordering $\FP(\gk)$
to be used in the
proof of Theorem \ref{t1} is the partial
ordering for adding a non-reflecting
stationary set of ordinals of a certain
type to $\gk$.
Specifically, $\FP(\gk) =
\{ p$ : For some
$\ga < \gk$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, such that if $\gb < \sup(S_p)$ is inaccessible,
then $S_p - S_p \cap \gb$ is composed of
ordinals of cofinality at least $\gb\}$,
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 $S_p$.
By Lemmas 1.1 - 1.3 of \cite{A99b} and the
succeeding remarks, we have the following facts.
%about $\FP(\gk)$.
\begin{enumerate}
\item\label{i1} $\FP(\gk)$ is
${\prec}\gk$-strategically closed.
\item Forcing with $\FP(\gk)$ adds a non-reflecting
stationary set of ordinals to $\gk$.
\item For any inaccessible cardinal
$\gd < \gk$, the partial ordering
$\FP(\gk/\gd) = \{p \in \FP(\gk)$ : $p$ is either
the characteristic function of the empty set or
$p$ is such that $S_p$ contains
an ordinal above $\gd\}$ with the inherited
partial ordering is dense in $\FP(\gk)$ and is
$\gd$-directed closed.
\end{enumerate}
\noindent In addition, since $\gk$ is Mahlo,
it easily follows that
$|\FP(\gk)| = \gk$.
By (\ref{i1}) above therefore,
if GCH holds in our ground model,
it then easily follows that forcing with
$\FP(\gk)$ preserves GCH.
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{K}
for further details.
We note explicitly that the cardinal
$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
supercompact for every cardinal $\gd < \gl$.
Also, unlike \cite{K}, we will say that
the cardinal $\gk$ is $\gl$ strong for
$\gl > \gk$ if there is
$j : V \to M$ an elementary embedding
having critical point $\gk$
such that
$j(\gk) > |V_\gl|$ and
$V_\gl \subseteq M$.
If $|V_\gl|$ is regular,
then we may assume that
$M^\gk \subseteq M$ as well.
In addition, it is easily seen
that any cardinal $\gk$ which is
$\gk^{+ \ga}$ supercompact is
$\gk + \ga + 1$ strong.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + $\K$ is the
class of supercompact cardinals''.
Without loss of generality,
by first forcing GCH and then
forcing with the
partial ordering of
\cite{AS97a}, we may also assume that
GCH and level by level equivalence
between strong compactness and
supercompactness hold in $V$.
We are now in a position to
define the partial ordering $\FP$
that will be used in the proof
of Theorem \ref{t1}.
Let $\mathfrak D$ be the collection
of $V$-measurable cardinals $\gd$
such that for some $\ga$ below
the least beth fixed point above
$\gd$, in $V$, $\gd^{+ \ga}$ is regular,
$\gd$ is $\gd + \ga + 1$ strong,
yet $\gd$ isn't $\gd^{+ \ga}$
supercompact.
$\FP$ is then taken as the
Easton support iteration which
begins by adding a Cohen subset
of $\go$ and then
adds, to every $\gd \in \mathfrak D$,
a non-reflecting stationary set of
ordinals using the partial ordering
$\FP(\gd)$ defined in Section \ref{s1}.
By the usual Easton arguments,
regardless if $\FP$ is a set or a
proper class,
$V^\FP \models {\rm ZFC}$.
Further, it is easily shown
by an induction similar to the
one given in the proof of Lemma 8
of \cite{AS97a} (see also \cite{A03}) that
$V^\FP \models {\rm GCH}$ and forcing
with $\FP$ preserves cardinals and
cofinalities.
This means that henceforth,
without fear of ambiguity, we
will write $\gd^{+ \ga}$
without specifying explicitly
whether we are working in
$V$, $V^\FP$, or some
$\ov V$ with
$V \subseteq \ov V \subseteq V^\FP$.
\begin{lemma}\label{l1}
$V^\FP \models ``\K$ is the class of
supercompact cardinals''.
\end{lemma}
\begin{proof}
The argument we give is
very similar to the one
presented in the proof of
Lemma 3.1 of \cite{A99b}.
Write
$\FP = \FP' \ast \dot \FP''$, where
$|\FP'| = \omega$ and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically
closed''.
In the terminology of
\cite{H1}, \cite{H2}, and
\cite{H3}, $\FP$
``admits a gap at $\ha_1$'', so by the results of
\cite{H1}, \cite{H2}, and \cite{H3}, any
supercompact cardinal in $V^\FP$ had to have been
supercompact in $V$.
This means the proof of Lemma \ref{l1} will be
complete once we have shown that
$V^\FP \models ``$If $\gk \in {\mathfrak K}$, then
$\gk$ is supercompact''.
To do this, fix $\gk \in {\mathfrak K}$.
Let $A$ be an arbitrary
(possibly empty) set of $V$-measurable
cardinals above $\gk$. Take
$\eta = \max(\sup(A), \gk)$, and define
$\eta^*$ as the least
$V$-measurable cardinal above $\eta$
if this cardinal exists, or the
class of all ordinals if no
cardinal above $\eta$ is
measurable in $V$. Let
$\gl \in (\eta, \eta^*)$
be any successor cardinal
above the least beth fixed point above
$\eta$. Let
$\gg = |2^{{[\gl]}^{< \gk}}|$, and fix
$j : V \to M$ an elementary embedding witnessing the
$\gg$ supercompactness of $\gk$.
Write
$\FP = \FP^0 \ast \dot \FP^1 \ast \dot \FP^2$, where
$\FP^0$ is the portion of $\FP$ defined through stage
$\gk$, $\dot \FP^1$ is a term for the portion of $\FP$
defined between stages $\gk$ and $\gl$, and
$\dot \FP^2$ is a term for the rest of $\FP$.
By the definition of $\FP$, it will be the case that
$\forces_{\FP^0 \ast \dot \FP^1} ``\dot \FP^2$ is
$\gg$-strategically closed''.
Thus, since $\gl$ may be chosen
arbitrarily large,
to prove Lemma \ref{l1},
it will suffice to show that
$V^{\FP^0 \ast \dot \FP^1} \models ``\gk$ is $\gl$
supercompact''.
If this is not the case, then let
$p = \la p_0, \dot p_1 \ra \in \FP^0 \ast \dot \FP^1$
be such that
$p \forces ``\gk$ isn't $\gl$ supercompact''.
By using Lemma 1.2 of \cite{A99b} if necessary to find
the necessary terms to extend
coordinatewise, we assume without loss of generality
that each non-trivial coordinate of
$p_1$ is a term for a condition in the appropriate
$\FP(\gd / \gk)$.
Let $G_0$ be $V$-generic over $\FP^0$ such that
$p_0 \in G_0$.
Working in $V[G_0]$ and once again using
Lemma 1.2 of \cite{A99b}, let $\FP^3$ be the Easton
support iteration of partial orderings which,
for every $V$-measurable cardinal
$\gd \in (\gk, \gl)$ which is
an element of $\mathfrak D$,
add non-reflecting
stationary sets of ordinals using
$\FP(\gd / \gk)$.
%Again without loss of generality, we assume that
%every element of $\FP^3$ extends $p_1$.
Note now that if $G_1$ is $V[G_0]$-generic over
$\FP^3$ and $p_1 \in G_1$, then $G_1$ must also
generate a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
To see this, it clearly suffices to show that
$G_1$ meets all dense open subsets of $\FP^1$
above $p_1$.
If $D$ is such a set,
then let $D_1 = \{q \in \FP^3 : q$ extends some
element of $D\}$.
$D_1$ is clearly open.
If $q \in \FP^3$, then $q \in \FP^1$,
so by density, there is $q' \ge q$, $q' \in D$.
By using Lemma 1.2 of \cite{A99b} if necessary to
find a term which is forced to extend
each term denoting a
non-trivial coordinate of $q'$ to
a term for an
element of the appropriate $\FP(\gd / \gk)$,
we obtain $q'' \ge q' \ge q$,
$q'' \in D_1$.
Thus, $G_1$ meets $D_1$ and hence meets $D$, so
$G_1$ generates a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
By the definition of
$\FP$ and the closure properties of $M$,
$j(\FP^0 \ast \dot \FP^1) =
\FP^0 \ast \dot \FP^1 \ast \dot \FQ \ast \dot \FR$,
where $\dot \FQ$ is a term for the portion of
$j(\FP^0 \ast \dot \FP^1)$ defined in $M$ between
stages $\gl$ and $j(\gk)$, and
$\dot \FR$ is a term for $j(\dot \FP^1)$, i.e.,
the portion of
$j(\FP^0 \ast \dot \FP^1)$ defined in $M$ between
stages $j(\gk)$ and $j(\gl)$.
If $G_1$ is $V[G_0]$-generic over $\FP^3$ and
$p_1 \in G_1$, then
by the preceding paragraph,
$G_1$ generates a $V[G_0]$-generic
filter $G^*_1$ over $\FP^1$.
We can therefore take $G_2$ as a
$V[G_0][G^*_1]$-generic object over $\FQ$ and
use the usual Easton arguments to infer that
$M[G_0][G^*_1][G_2]$ remains
$\gg$ closed with respect to
$V[G_0][G^*_1][G_2]$ and that $j$ lifts
in $V[G_0][G^*_1][G_2]$ to
$j : V[G_0] \to M[G_0][G^*_1][G_2]$.
Further, since
$G_1 \subseteq G^*_1$ and $G_1$ is $V[G_0]$-generic over
a partial ordering (namely $\FP^3$) that is
$\gk$-directed closed in $V[G_0]$,
$j''G_1$ generates in $V[G_0][G^*_1][G_2]$ a
compatible set of conditions of cardinality
smaller than $\gg < j(\gk)$ in a partial ordering
(namely $j(\FP^3)$) that is $j(\gk)$-directed
closed in $M[G_0][G^*_1][G_2]$.
Therefore, by the fact
$M[G_0][G^*_1][G_2]$ is
$\gg$ closed with respect to
$V[G_0][G^*_1][G_2]$, we can let
$r$ be a master condition for
$j''G_1$ and take $G_3$ to be a
$V[G_0][G^*_1][G_2]$-generic object over
$j(\FP^3)$ containing $r$.
By elementarity, it will be the case that
$G_3$ generates a $V[G_0][G^*_1][G_2]$-generic
object $G^*_3$ over
$\FR = j(\FP^1)$.
As usual, we will then have that in
$V[G_0][G^*_1][G_2][G^*_3]$, $j$ lifts to
$j : V[G_0][G^*_1] \to M[G_0][G^*_1][G_2][G^*_3]$, so
$\gk$ is $\gl$ supercompact in $V[G_0][G^*_1][G_2][G^*_3]$.
Since $\FQ \ast \dot \FR$ is $\gg$-strategically closed in
$V[G_0][G^*_1]$, it will be the case that
$\gk$ is $\gl$ supercompact in $V[G_0][G^*_1]$.
This, however, contradicts that
$p = \la p_0, p_1 \ra \in G_0 \ast G^*_1$ and
$p \forces ``\gk$ isn't $\gl$ supercompact''.
This contradiction completes the proof of
Lemma \ref{l1}.
\end{proof}
We remark that the proof of Lemma \ref{l1}
actually shows that any cardinal $\gd$
which is in $V$ both regular and
a limit of cardinals
$\gk$ which are ${<}\gd$ supercompact
remains in $V^\FP$
regular and a limit of cardinals
$\gk$ which are ${<}\gd$ supercompact.
To see this, let $\gd > \gl > \gk$
be such that
$V \models ``\gk$ is ${<}\gd$
supercompact and $\gd$ is regular''.
Work in $\ov V = V_\gd$, and take
$\FQ = \FP_\gd$.
Fix $j : \ov V \to M$ an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$.
The same proof as just given shows that
${\ov V}^\FQ \models ``\gk$ is $\gl$ supercompact''.
Since writing
$\FP = \FP_\gd \ast \dot \FP^\gd$ tells us
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
${<}\gd$-strategically closed'',
we can now immediately infer that
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models
``\gd$ is regular and
$\gk$ is ${<}\gd$ supercompact''.
As $\gk$ and $\gl$ were arbitrary, in
$V^\FP$, $\gd$ is regular and is
a limit of ${<}\gd$ supercompact cardinals.
We note also that the proof of Lemma \ref{l1}
does not necessarily show, however,
that if $\gl > \gd$ and $\gd$ is in
$V$ both regular and a limit of cardinals
$\gk$ which are $\gl$ supercompact, then
$\gd$ remains in $V^\FP$ a limit of
cardinals $\gk$ which are $\gl$ supercompact.
To see this, let
$\gk < \gd$ be $\gl$ supercompact
in $V$, and fix
$j : V \to M$ an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$.
It is possible that the closure properties
of $M$ with respect to $V$ are not enough
to guarantee that $V$ and $M$ make the
same decision as to whether a non-reflecting
stationary set of ordinals must be
added to $\gd$.
If the decisions differ, then the proof
of Lemma \ref{l1} suitably modified
will not remain valid.
\begin{lemma}\label{l2}
$V^\FP \models ``$The strongly
compact and supercompact cardinals
coincide, except possibly at measurable limit points''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd$ is
strongly compact''.
Write
$\FP = \FP_\gd \ast \dot \FP^\gd$.
%where $\FP_\gd$ is the partial
%ordering $\FP$ defined through
%stage $\gd$, and $\dot \FP^\gd$
%is a term for the rest of $\FP$.
Using this factorization,
it is easy to see by
$\FP$'s definition that
$\FP$ is, in the terminology
of \cite{H2} and \cite{H3},
``mild with respect to $\gd$.''
This means that any set of
ordinals $x$ in $V^\FP$ of size
below $\gd$ has a
``nice'' name $\tau$ in
$V$ of size below $\gd$, i.e.,
there is a set $y$ in $V$,
$|y| < \gd$, such that any
ordinal forced by a condition
in $\FP$ to be in $\tau$
is an element of $y$.
Since we have already seen that
$\FP$ ``admits a gap at $\ha_1$'',
by the results of \cite{H2} and
\cite{H3}, $\gd$ had to have been
strongly compact in $V$.
Thus, in $V$, $\gd$ is either an
element of $\K$ or a measurable
limit of elements of $\K$, so
by Lemma \ref{l1}, in $V^\FP$,
$\gd$ is either supercompact or
a measurable limit of supercompact
cardinals.
This completes the proof of
Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$For
any measurable cardinal
$\gd$ and any $\ga$
below the least beth
fixed point above $\gd$,
if $\gd^{+ \ga}$ is
regular and
$\gd$ is $\gd + \ga + 1$
strong, then
$\gd$ is $\gd^{+ \ga}$
strongly compact''.
\end{lemma}
\begin{proof}
We begin by observing
we may assume
without loss of generality
that $\ga > 0$.
This is since if
$\gd$ is $\gd + 0 + 1 = \gd + 1$ strong,
$\gd$ is measurable, so
by definition,
$\gd$ is both $\gd^{+ 0} = \gd$
strongly compact and $\gd$ supercompact.
Suppose now that $\ga > 0$,
$\ga$ is below the least
beth fixed point above $\gd$, and
$V^\FP \models ``\gd$ is
$\gd + \ga + 1$ strong and
$\gd^{+ \ga}$ is regular''.
Since we have already observed that
$\FP$ ``admits a gap at $\ha_1$'',
by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
$\gd$ had to have been $\gd + \ga + 1$
strong in $V$ as well.
Therefore, $\gd$ was $\gd^{+ \ga}$
supercompact in $V$ also, since otherwise,
the definition of $\FP$ tells us
we can write
$\FP = \FP_\gd \ast \dot \FP(\gd)
\ast \dot \FP^\gd$.
%the definition of $\FP$ tells us that
We may then infer that
$\forces_{\FP_\gd} ``\dot \FP(\gd)$
adds a non-reflecting stationary set
of ordinals to $\gd$'' and
$\forces_{\FP_\gd \ast \dot \FP(\gd)}
``\dot \FP^\gd$ is $\eta$-strategically
closed for $\eta$ the least inaccessible
above $\gd$''.
This means that
$V^{\FP_\gd \ast \dot \FP(\gd) \ast \dot \FP^\gd} =
V^\FP \models ``\gd$ contains a non-reflecting
stationary set of ordinals and hence
isn't weakly compact'', a contradiction
to the fact that
$V^\FP \models ``\gd$ is
$\gd + \ga + 1$ strong''.
Thus, we can actually write
$\FP = \FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd}
``\dot \FP^\gd$ is $\eta$-strategically
closed for $\eta$ the least inaccessible
above $\gd$''.
Hence, since $\ga$ is below
the least beth fixed point
above $\gd$, to show that
$V^\FP \models ``\gd$ is $\gd^{+ \ga}$
strongly compact'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gd^{+ \ga}$
strongly compact''.
To show $V^{\FP_\gd} \models ``\gd$ is
$\gd^{+ \ga}$ strongly compact'', we use an
argument of Magidor for the preservation
of strong compactness.
Although the essentials of this argument can
be found in \cite{A99b} and \cite{AC2}
(as well as elsewhere),
for completeness and for the benefit of
readers, we give the argument here as well.
Let
$\gl = \gd^{+ \ga}$, and let
$k_1 : V \to M$ be an
elementary embedding witnessing the $\gl$
supercompactness of $\gd$ such that
$M \models ``\gd$ isn't $\gl$ supercompact''.
Since $M \models ``\gd$ is measurable'',
we may choose a normal ultrafilter of
Mitchell order $0$ over $\gd$ such that
$k_2 : M \to N$ is an
elementary embedding witnessing the
measurability of $\gd$ definable in $M$ with
$N \models ``\gd$ isn't measurable''.
It is the case that if
$k : V \to N$ is an elementary embedding with
critical point $\gd$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
such that $x \subseteq y$ and
$N \models ``|y| < k(\gd)$'',
then $k$ witnesses the $\gl$
strong compactness of $\gd$.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of $\gd$.
We show that $j$ lifts to
$j : V^{\FP_\gd} \to N^{j(\FP_\gd)}$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of $\gd$ in
$V^{\FP_\gd}$, this proves Lemma \ref{l3}.
To do this, write
$j(\FP_\gd)$ as
$\FP_\gd \ast \dot \FQ^0 \ast \dot \FR^0$, where
$\dot \FQ^0$ is a term for the portion of
$j(\FP_\gd)$ between $\gd$ and $k_2(\gd)$ and
$\dot \FR^0$ is a term for the rest of
$j(\FP_\gd)$, i.e., the part above $k_2(\gd)$.
Note that since
$N \models ``\gd$ isn't measurable'',
$\gd \not\in {\rm field}(\dot \FQ^0)$.
Thus, the field of $\dot \FQ^0$
is composed of all $N$-measurable cardinals
$\gg \in (\gd, k_2(\gd)]$
for which in $N$,
for some $\gb$ below
the least beth fixed point above
$\gg$, $\gg^{+ \gb}$ is regular,
$\gg$ is $\gg + \gb + 1$ strong, yet
$\gg$ isn't $\gg^{+ \gb}$ supercompact.
This means
$k_2(\gd) \in {\rm field}(\dot \FQ^0)$, since
by either Proposition 26.11 of
\cite{K} or Lemma 2.1 of \cite{AC2},
$M \models ``\gd$ is superstrong'' yet
$M \models ``\gd$ isn't
$\gd^{+ \ga}$ supercompact and
$\ga$ is below the least
beth fixed point above $\gd$'',
so by elementarity,
$N \models ``$There is some $\gb$
below the least beth fixed point
above $j(\gd)$ for which
$j(\gd)$ is $j(\gd) + \gb + 1$
strong yet $j(\gd)$ isn't
${(j(\gd))}^{+ \gb}$ supercompact''.
Also, the field of $\dot \FR^0$ is composed of all
$N$-measurable cardinals
$\gg \in (k_2(\gd), k_2(k_1(\gd)))$
for which in $N$,
for some $\gb$ below
the least beth fixed point above
$\gg$, $\gg^{+ \gb}$ is regular,
$\gg$ is $\gg + \gb + 1$ strong, yet
$\gg$ isn't $\gg^{+ \gb}$ supercompact.
Let $G_0$ be $V$-generic over $\FP_\gd$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ^0$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR^0$. Since $\FP_\gd$ is an
Easton support iteration of length $\gd$,
a direct limit is taken at stage $\gd$,
and no forcing is done at stage $\gd$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
This means that
$j : V \to N$ lifts to
$j : V[G_0] \to N[G_0][G_1][G_2]$
in $V[G_0]$.
To build $G_1$, note that since $k_2$
is generated by an
ultrafilter ${\cal U}$ over $\gd$ and
since in both $V$ and $M$, $2^\gd = \gd^+$,
$|k_2(\gd^+)| = |k_2(2^\gd)| =
|\{ f : f : \gd \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gd| = \gd^+$. Thus, as
$N[G_0] \models ``|\wp(\FQ^0)| = k_2(2^\gd)$'', we can let
$\la D_\gb : \gb < \gd^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of
$\FQ^0$ found in $N[G_0]$.
For the purpose of the construction of
$G_1$ to be given below, we further assume
that for every
dense open subset $D \subseteq \FQ^0$
present in $N[G_0]$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$.
Since the $\gd$ closure of $N$ with respect to either
$M$ or $V$ implies the least element of the field of
$\FQ^0$ is above $\gd^+$, the definition of
$\FQ^0$ as given above implies that
%as the Easton support iteration which adds
%a non-reflecting stationary set of ordinals
%to each $N[G_0]$-measurable
%cardinal $\gg$ in the interval
%$(\gd, k_2(\gd)]$
%which isn't $2^\gg$ supercompact implies that
$N[G_0] \models ``\FQ^0$ is
${\prec}\gd^+$-strategically closed''.
By the fact the standard arguments show that
forcing with the $\gd$-c.c$.$ partial ordering
$\FP_\gd$ preserves that $N[G_0]$ remains
$\gd$-closed with respect to either
$M[G_0]$ or $V[G_0]$,
$\FQ^0$ is ${\prec}\gd^+$-strategically closed
in both $M[G_0]$ and $V[G_0]$.
We can now construct $G_1$ in either
$M[G_0]$ or $V[G_0]$ as follows.
Players I and II play a game of
length $\gd^+$.
The initial pair of moves is generated by
player II choosing the trivial
condition $q_0$ and player I responding by
choosing $q_1 \in D_1$. Then,
at an even stage $\gb + 2$,
%since $\FQ^0$ is
%${\prec} \gd^+$-strategically closed,
player II picks $q_{\gb + 2} \ge
q_{\gb + 1}$ by using some fixed
strategy ${\cal S}$, where
$q_{\gb + 1}$ was chosen by player I
to be such that
$q_{\gb + 1} \in D_{\gb + 1}$ and
$q_{\gb + 1} \ge q_\gb$.
%$p_\ga \in D_\ga$ extending
%$\sup(\la q_\gb : \gb < \ga \ra)$
%(initially, $q_{-1}$ is the empty condition)
%and player II responds by picking
%$q_\ga \ge p_\ga$ (so $q_\ga \in D_\ga$).
If $\gb$ is a limit ordinal, player II uses
${\cal S}$ to pick
$q_\gb$ extending each $q_\gg$ for $\gg < \gb$.
%$\sup(\la q_\gb : \gb < \ga \ra)$.
By the ${\prec}\gd^+$-strategic closure of
$\FQ^0$ in both $M[G_0]$ and $V[G_0]$,
%player II has a winning strategy for this
%game, so
the sequence $\la q_\gb : \gb < \gd^+ \ra$
as just described exists. By construction,
%can be taken
%as an increasing sequence of conditions with
%$q_\ga \in D_\ga$ for $\ga < \gd^+$. Clearly,
$G_1 = \{p \in \FQ^0 : \exists \gb <
\gd^+ [q_\gb \ge p]\}$ is our
$N[G_0]$-generic object over $\FQ^0$.
It remains to construct in $V[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR^0$.
To do this, we first note that as
$M \models ``\gd$ is superstrong but $\gd$
isn't $\gd^{+ \ga}$ supercompact and
$\ga$ is below the least beth fixed
point above $\gd$'',
we can write
$k_1(\FP_\gd)$ as
$\FP_\gd \ast \dot \FS^0 \ast \dot \FT^0$, where
$\forces_{\FP_\gd} ``\dot \FS^0 = \dot \FP(\gd)$'', and
$\dot \FT^0$ is a term for the rest of
$k_1(\FP_\gd)$.
Note now that
$M \models ``$No cardinal
$\gg \in (\gd, \gl]$ is measurable''.
Thus, the field of
$\dot \FT^0$ is composed
only of $M$-measurable cardinals
in the interval $(\gl, k_1(\gd))$,
%of all $M$-measurable cardinals
%$\gg \in (\gl, k_1(\gd))$
%for which in $M$,
%for some $\ga$ below
%the least beth fixed point above
%$\ga$, $\gg^{+ \ga}$ is regular,
%$\gg$ is $\gg + \ga + 1$ strong, yet
%$\gg$ isn't $\gg^{+ \ga}$ supercompact,
which implies that in $M$,
$\forces_{\FP_\gd \ast \dot \FS^0}
``\dot \FT^0$ is ${\prec}\gl^+$-strategically
closed''. Further, since
$V \models {\rm GCH}$ and $\gl$ is regular,
$|{[\gl]}^{< \gd}| = \gl$ and $2^\gl = \gl^+$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter ${\cal U}$ over
$P_\gd(\gl)$,
%$|k_1(\gl)| =
$|k_1(\gd^+)| =
%|k_1(\gd^{+ \ga})| =
|2^{k_1(\gd)}| =
|\{ f : f : P_\gd(\gl) \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gl| = |{[\gl]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{F} for the first published
account of term forcing or
\cite{C}, Section 1.2.5, page 8;
the notion is originally due
to Laver) associated with
$\dot \FT^0$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP_\gd \ast \dot \FS^0$ and
$\forces_{\FP_\gd \ast \dot \FS^0} ``\tau \in
\dot \FT^0$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP_\gd \ast \dot \FS^0} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT^0$ to be a set,
it is possible to restrict the terms
appearing in it to a sufficiently large
set-sized collection, with the additional
crucial property that any term $\tau$
forced to be in $\dot \FT^0$ is also forced
to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M \models ``|\FT^*| = k_1(\gd)$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP_\gd \ast \dot \FS^0} ``\dot \FT^0$ is
${\prec}\gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V$ as well.
To show that
we may restrict the number of terms
so that
$M \models ``|\FT^*| = k_1(\gd)$'',
%we recall that in the official definition of $\FT^*$,
%the basic idea is to include only the canonical terms.
we observe that since
$\forces_{\FP_\gd \ast \dot \FS^0}
``|\dot \FT^0| = k_1(\gd)$'',
there is a set
$\{\tau_\gb : \gb 0$.
This is since $\gd$ is
measurable iff $\gd$ is
$\gd^{+ 0} = \gd$ strongly
compact iff $\gd$ is $\gd$
supercompact, so any elementary
embedding $j : V \to M$
witnessing the measurability of $\gd$
which is generated by a normal
ultrafilter over $\gd$ also
witnesses that $\gd$ is
$\gd + 0 + 1 = \gd + 1$ strong.
We therefore suppose
$\gd$ is as in the
hypotheses of Lemma \ref{l4}, i.e.,
in $V^\FP$,
$\ga$ is below the least beth
fixed point above $\gd$,
$\gd^{+ \ga}$ is regular,
$\gd$ is $\gd^{+ \ga}$ strongly
compact, yet $\gd$ isn't a limit
of cardinals $\gg$ which are
$\gd^{+ \ga}$ strongly compact.
Assume further towards a contradiction
that in $V$,
$\gd$ is a limit of cardinals
$\gg$ which are $\gd^{+ \ga}$ supercompact.
Since any cardinal $\gg$ which is
$\gd^{+ \ga}$ supercompact is
automatically ${<}\gd$ supercompact,
by the remarks immediately
following the proof of Lemma \ref{l1},
we know that in $V^\FP$,
$\gd$ is a limit of cardinals
$\gg$ which are ${<}\gd$ supercompact.
As $V^\FP \models ``\gd$ is
$\gd^{+ \ga}$ strongly compact'',
by a theorem of DiPrisco \cite{DH},
$V^\FP \models ``$Any cardinal $\gg$
which is either ${<}\gd$ supercompact
or ${<}\gd$ strongly compact is
$\gd^{+ \ga}$ strongly compact''.
Thus, in $V^\FP$, $\gd$ is a
limit of cardinals $\gg$ which are
$\gd^{+ \ga}$ strongly compact,
a contradiction.
We now know that in $V$,
$\gd$ isn't a limit of cardinals
$\gg$ which are $\gd^{+ \ga}$
supercompact.
Also, as in the proof of Lemma \ref{l2},
by writing
$\FP = \FP_\gd \ast \dot \FP^\gd$,
it is readily seen that $\FP$
``admits a gap at $\ha_1$ and
is mild with respect to $\gd$''.
By the results of \cite{H2} and
\cite{H3}, $\gd$ is therefore
$\gd^{+ \ga}$ strongly compact
in $V$.
This means,
by level by level equivalence
between strong compactness and
supercompactness, that in $V$,
$\gd$ is $\gd^{+ \ga}$ supercompact.
Hence, as in the proof of
Lemma \ref{l3},
%we can write
%$\FP = \FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\eta$-strategically closed for
$\eta$ the least inaccessible above $\gd$''.
Therefore, since $\ga$ is below the least
beth fixed point above $\gd$,
to show that
$V^\FP \models ``\gd$ is $\gd + \ga + 1$
strong'', it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is
$\gd + \ga + 1$ strong''.
To do this,
we argue now in an analogous
fashion to the proof of
Lemma 2.2 of \cite{A03}.
(See also the proof of Lemma
2.4 of \cite{AC2}.)
We use for the proof
of this lemma notation and terminology from
the introductory section of \cite{C}.
Let $\gl = \ga + 1$.
Let
$j : V \to M$ be an elementary embedding witnessing
the $\gd + \gl$ strongness of $\gd$ generated by a
$(\gd, \gd^{+ \gl})$-extender of width $\gd$
with $j(\gd)$ minimal so that
$M \models ``\gd$ isn't $\gd + \gl$ strong'',
and let
$i : V \to N$ be the elementary embedding
witnessing the measurability of $\gd$
generated by the normal ultrafilter
${\cal U} = \{x \subseteq \gd :
\gd \in j(x)\}$.
We then have the commutative diagram
\commtriangle{V}{N}{M}{i}{k}{j}
where $j = k \circ i$ and
the critical point of $k$ is
above $\gd$.
Observe that
$M \models ``$No cardinal
$\rho \in (\gd, \gd^{+ \gl}]$ is measurable''.
This is since
$V_{\gd + \gl} \subseteq M$ and
$V \models ``$Both $\ga$ and
$\gl = \ga + 1$ are below the
least beth fixed point above $\gd$''.
Also, since
$V_{\gd + \gl} \subseteq M$,
${(\gd^{+ \gl})}^V = {(\gd^{+ \gl})}^M$.
This means in $M$, the least measurable cardinal
$\gd_0 > \gd$ in the field of $j(\FP_\gd)$
is so that $\gd_0 > \gd^{+ \gl}$.
In addition, it is the case that
$\gd \not\in {\rm field}(j(\FP_\gd))$.
This is since, by choice of $\gl$,
$M \models ``$For every $\rho < \ga$
so that $\gd^{+ \rho}$ is regular,
$\gd$ is $\gd^{+ \rho}$ supercompact and
$\gd + \rho + 1$ strong''. As
$M \models ``\gd$ isn't $\gd + \gl$ strong'',
there are no other
degrees of either supercompactness or
strongness that could affect whether
$\gd$ is an element of
${\rm field}(j(\FP_\gd))$.
Define now
$f : \gd \to \gd$ by
$$f(\rho) = {\hbox{\rm The least measurable
cardinal above $\rho$}}.$$
We then have
$\gd < \gd^{+ \gl} < j(f)(\gd) < \gd_0$.
This last inequality is since the least measurable
cardinal $\gg$ above any $\rho$
isn't $\gg + 2$ strong, and by
GCH in both $V$ and $M$,
$\gg$ isn't $2^\gg = \gg^+$
supercompact either.
Thus, $\gg$ is both
$\gg^{+ 0} = \gg$ supercompact and
$\gg + 0 + 1 = \gg + 1$ strong and
shows no further degrees of either
supercompactness or strongness.
Note that
$M = \{j(g)(a) : a \in {[\gd^{+ \gl}]}^{< \omega}$,
$\dom(g) = {[\gd]}^{|a|}$,
$g : {[\gd]}^{|a|} \to V\} =
\{k(i(g))(a) : a \in {[\gd^{+ \gl}]}^{< \omega}$,
$\dom(g) = {[\gd]}^{|a|}$,
$g : {[\gd]}^{|a|} \to V\}$.
By defining $\gg = i(f)(\gd)$, we have
$k(\gg) = k(i(f)(\gd)) = j(f)(\gd) > \gd^{+ \gl}$.
This means
$j(g)(a) = k(i(g))(a) =
k(i(g) \rest {[\gg]}^{|a|})(a)$, i.e.,
$M = \{k(h)(a) : a \in {[\gd^{+ \gl}]}^{< \omega}$,
$h \in N$, $\dom(h) = {[\gg]}^{|a|}$,
$h : {[\gg]}^{|a|} \to N\}$.
By elementarity, we must have
$N \models ``\gd \not\in {\rm field}(i(\FP_\gd))$ and
$\gd < \gg = i(f)(\gd) < \gz$ =
The least element of the field of
$i(\FP_\gd) - \gd$'', since
$M \models ``k(\gd) = \gd$ isn't
in the field of $j(\FP_\gd)$ and
$k(\gd) = \gd < k(\gg) = k(i(f)(\gd)) =
j(f)(\gd) < k(\gz) = \gd_0$''.
Therefore, $k$ is generated
by an $N$-extender of width
$\gg \in (\gd, \gz)$.
Write $i(\FP_\gd) = \FP_\gd \ast
\dot \FQ^0$, where $\dot \FQ^0$
is a term for the portion of
$i(\FP_\gd)$ whose field is composed
of ordinals in the interval
$[\gd, i(\gd))$.
By our previous work,
%Since
%$N \models ``\gd$ isn't a strong cardinal'',
the field of $\dot \FQ^0$ is actually
composed of ordinals in the interval
$(\gd, i(\gd))$, or more precisely,
of ordinals in the interval
$[\gz, i(\gd))$.
This means that if $G_0$ is once again
$V$-generic over $\FP_\gd$, the
argument from Lemma \ref{l3} for the
construction of the generic object
$G_1$ can be applied here as well to
construct in $V[G_0]$ an
$N[G_0]$-generic object $G^*_1$ over
$\FQ^0$. Since
$i '' G_0 \subseteq G_0 \ast G^*_1$,
$i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G^*_1]$,
and since $k '' G_0 = G_0$
and $k(\gd) = \gd$, $k$ lifts in $V[G_0]$ to
$k : N[G_0] \to M[G_0]$.
By Fact 3 of Section 1.2.2 of \cite{C},
$k : N[G_0] \to M[G_0]$ is also
generated by an
extender of width
$\gg \in (\gd, \gz)$.
In analogy to the preceding paragraph, write
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ^1$.
By the last sentence of the preceding paragraph
and the fact $\gz$ is the least ordinal in
the field of $\dot \FQ^0$,
we can use Fact 2 of Section 1.2.2 of \cite{C}
to infer that
$H = \{p \in \FQ^1 : \exists q \in k '' G^*_1
[q \ge p]\}$ is $M[G_0]$-generic over
$k(\FQ^1)$. Thus, $k$ lifts in $V[G_0]$ to
$k : N[G_0][G^*_1] \to M[G_0][H]$,
and we get the new commutative diagram
\commtriangle{V[G_0]}{N[G_0][G^*_1]}{M[G_0][H]}{i}{k}{j}
%By the second paragraph of
%the proof of this lemma,
%the field of $\FQ^1$ is
%composed of ordinals in the interval
%$({\gd^{+ \gl}}, j(\gd))$.
As in the proofs of
Lemma 2.2 of \cite{A03} and
Lemma 2.4 of \cite{AC2},
%and Theorems 1.6 and 3.6 of \cite{H4},
since
$V_{\gd + \gl} \subseteq M$ and
$G_0 \in M[G_0][H]$, we can deduce that
${(V[G_0])}_{\gd + \gl} \subseteq M[G_0][H]$,
i.e., that $j$ remains a $\gd + \gl$
strong embedding
%and as the field of $\FQ^1$ is
%composed of ordinals in the interval
%$(\gd^{+ \gl}, j(\gd))$,
%$V_{\gd + \gl}[G_0]$ is the set of all sets of
%rank $< {\gd + \gl}$ in $M[G_0][H]$. Hence,
%$j$ is a $\gl$ strong embedding
after forcing with $\FP_\gd$.
This completes the proof of
Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} - \ref{l4},
along with the intervening remarks,
complete the proof of Theorem \ref{t1}.
\end{proof}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we
make several remarks.
The first is that, as noted
immediately following the statement
of Theorem \ref{t1}, larger
degrees of level by level
equivalence between strong
compactness and strongness are
possible, assuming the definition
of our partial ordering $\FP$
has been suitably modified.
To see this, suppose the
set or class $\mathfrak D$ used in
the definition of $\FP$ is changed
to be the collection of $V$-measurable
cardinals $\gd$ such that for some
$\ga$ below the second $V$-measurable
cardinal above $\gd$, in $V$,
$\gd^{+ \ga}$ is regular,
$\gd$ is $\gd + \ga + 1$ strong, yet
$\gd$ isn't $\gd^{+ \ga}$ supercompact.
$\FP$ is then defined as before, using
this new
$\mathfrak D$ as the collection of
measurable cardinals to which
non-reflecting stationary sets of
ordinals are added.
Because our ground model $V$ satisfies
level by level equivalence between
strong compactness and supercompactness,
no non-reflecting stationary sets of
ordinals will be added to either the
first or second measurable cardinal
above any measurable cardinal.
This is since these cardinals won't
manifest any non-trivial degree of
either strong compactness or
supercompactness.
Thus, as readers may easily
verify for themselves, the
proofs of Lemmas \ref{l1} - \ref{l4}
suitably modified all remain valid,
and produce a model
$V^\FP$ in which for any measurable
cardinal $\gd$ and any $\ga$ below
the second measurable cardinal above
$\gd$, if $\gd^{+ \ga}$ is regular,
$\gd$ is $\gd^{+ \ga}$ strongly
compact iff $\gd$ is $\gd + \ga + 1$
strong, except possibly if $\gd$
is a limit of cardinals $\gg$ which are
$\gd^{+ \ga}$ strongly compact.
There are limits, though, to
the modifications that can be
made to the set or class $\mathfrak D$
that still allow our methods
of proof to go through.
As an example,
we may come to a point
where the different
non-reflecting stationary set
forcings begin to interfere
with one another, such as
if a non-reflecting stationary
set of ordinals has to be
added to some cardinal $\gg$,
where $\gd < \gg < \gl$ and
$\gd$ is $\gl$ supercompact.
This would require a significant
change in proof in the appropriate
analogues of Lemmas \ref{l3} and
\ref{l4}.
Thus, we conclude this paper by
restating the questions we asked
in Section \ref{s1}, i.e.,
how much level by level
equivalence between strong
compactness and strongness is
possible in a model containing
supercompact cardinals, or more
generally, in a model containing
more than one strongly compact cardinal?
In particular, is it possible
to have a model containing more than one
strongly compact cardinal in which
there is unrestricted level by level
equivalence between strong
compactness and strongness, i.e.,
in which for every measurable
cardinal $\gd$, if $\gd^{+ \ga}$
is regular, $\gd$ is $\gd^{+ \ga}$
strongly compact iff $\gd$ is
$\gd + \ga + 1$ strong,
except possibly if $\gd$ is a
measurable limit of cardinals
$\gg$ which are $\gd^{+ \ga}$
strongly compact?
(Recall that this question is answered
in the case of a model containing
one strongly compact cardinal by
Theorem \ref{t0}.)
%the work of \cite{A03}.)
These seem like very difficult
questions to answer, and constructing
models in which these properties
are true would require significantly
different proof methods from those
of this paper.
In particular, in such models,
not only would there not be
level by level equivalence between
strong compactness and supercompactness,
but because every strong cardinal
would have to be strongly compact
and every supercompact cardinal
has to have a normal measure
concentrating on strong cardinals
(see Lemma 2.1 of \cite{AC2} for
a proof of this fact),
the Kimchi-Magidor property
would fail as well, i.e.,
there would be non-supercompact
strongly compact cardinals which
aren't measurable limits of
strongly compact cardinals.
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\end{document}