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\title{Indestructibility, Instances of
Strong Compactness, and Level by Level Inequivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
indestructibility,
non-reflecting stationary set of ordinals,
level by level inequivalence between
strong compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.}
\thanks{The author wishes to thank
the referee for helpful comments,
suggestions, and corrections which
have been incorporated into the
current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{November 24, 2008\\
(revised July 2, 2010)}
\begin{document}
\maketitle
\begin{abstract}
Suppose $\gl > \gk$ is measurable.
We show that if $\gk$ is either
indestructibly supercompact or
indestructibly strong, then
$A = \{\gd < \gk \mid \gd$ is
measurable, yet $\gd$
is neither $\gd^+$ strongly compact
nor a limit of measurable cardinals$\}$
must be unbounded in $\gk$.
The large cardinal
hypothesis on $\gl$ is necessary,
as we further demonstrate by
constructing via forcing two models in which
$A = \emptyset$. The first
of these contains a
supercompact cardinal $\gk$ and is such that
no cardinal $\gd > \gk$ is measurable,
$\gk$'s supercompactness
is indestructible under $\gk$-directed
closed, $(\gk^+, \infty)$-distributive
forcing, and
every measurable cardinal $\gd < \gk$
is $\gd^+$ strongly compact.
The second of these contains a strong cardinal $\gk$
and is such that
no cardinal $\gd > \gk$ is measurable,
$\gk$'s strongness
is indestructible under ${<}\gk$-strategically
closed, $(\gk^+, \infty)$-distributive forcing, and
level by level inequivalence between strong
compactness and supercompactness holds.
The model from the first of our
forcing constructions is used to show that
it is consistent, relative to a supercompact cardinal,
for the least cardinal $\gk$ which is both strong and
has its strongness indestructible under
$\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing
to be the same as the least supercompact cardinal,
which has its supercompactness indestructible under
$\gk$-directed closed, $(\gk^+, \infty)$-distributive
forcing.
It further follows as a corollary of
the first of our forcing constructions that
it is possible to build a model
containing a supercompact cardinal $\gk$ in which
no cardinal $\gd > \gk$ is measurable,
$\gk$ is indestructibly supercompact, and
every measurable cardinal $\gd < \gk$ which
is not a limit of measurable cardinals is
$\gd^+$ strongly compact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with some key definitions.
For any non-supercompact measurable cardinal $\gd$,
let $\gth_\gd$ be the smallest cardinal greater than
$\gd$ such that $\gd$ is not $\gth_\gd$ supercompact.
Suppose now that $V$ is a model of ZFC in which if
$\gd$ is measurable but not supercompact, then
$\gd$ is $\gth_\gd$ strongly compact.
$V$ is said to witness {\em level by level
inequivalence between strong compactness and
supercompactness}.
The cardinal $\gk$ is {\em indestructibly supercompact}
if $\gk$'s supercompactness is preserved after forcing
with a $\gk$-directed closed partial ordering.
The cardinal $\gk$ is {\em indestructibly strong}
if $\gk$'s strongness is preserved after forcing
with a ${<}\gk$-strategically closed,
$(\gk, \infty)$-distributive partial ordering.
The degrees of indestructibility just mentioned
are essentially standard
%(see, e.g., \cite{L}, \cite{H4}, and \cite{GS})
(see, e.g., \cite{L, H4, GS})
for supercompactness and strongness.\footnote{For
a strong cardinal $\gk$, \cite[Theorem 4.10]{H4}
and the work of \cite[Section 2]{GS}
force indestructibility under $\gk$-strategically
closed partial orderings. However, as our proof of Lemma
\ref{l4} will show, the techniques of
\cite[Theorem 4.10]{H4} actually
force indestructibility under ${<}\gk$-strategically closed,
$(\gk, \infty)$-distributive partial orderings.}
It is a remarkable fact that the
structure of the universe above a
strong or supercompact cardinal $\gk$
with suitable indestructibility properties
can affect what happens at large cardinals below
$\gk$, assuming the universe is sufficiently rich.
On the other hand, these effects can be mitigated
if the universe contains relatively few large cardinals.
For instance,
in \cite[Theorems 5 and 6]{AH4}, it was shown that
%in \cite{AH4}, it was shown (see Theorems 5 and 6) that
if $\gk < \gl$ are such that
$\gk$ is either indestructibly supercompact
or indestructibly strong and
$\gl$ is $2^\gl$ supercompact, then
$B = \{\gd < \gk \mid \gd$
is $\gd^+$ strongly compact
yet $\gd$ is not $\gd^+$
supercompact$\}$
must be unbounded in $\gk$.
However, if the universe
contains a fairly small number of large cardinals,
$\gk$ is either strong
or supercompact, and $\gk$'s supercompactness
or strongness exhibits enough indestructibility,
then \cite[Theorem 8]{AH4} and
\cite[Theorem 1]{A03} indicate that $B$ may be empty.
This sort of occurrence has been
further examined in \cite{A07, A08, A09}.
%\cite{A07}, \cite{A08}, and \cite{A09}.
The purpose of this paper is to continue
studying this phenomenon, but in the context of
investigating the degree of strong compactness
each measurable cardinal can manifest in
universes containing either a supercompact
or a strong cardinal with various
indestructibility properties. Specifically,
we prove the following theorems.
\begin{theorem}\label{tm1}
Suppose $\gl > \gk$ is measurable
and $\gk$ is either indestructibly
supercompact or indestructibly strong.
Then $A = \{\gd < \gk \mid \gd$ is measurable,
yet $\gd$ is neither $\gd^+$
strongly compact nor a limit of
measurable cardinals$\}$ is unbounded in $\gk$.
\end{theorem}
\begin{theorem}\label{tm2}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
In addition, the following hold in $V^\FP$:
\begin{enumerate}
\item $\gk$'s supercompactness
is indestructible under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing.
\item $\gk$ is the least strongly compact cardinal.
\item Every measurable cardinal
$\gd < \gk$ is $\gd^+$ strongly compact.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{tm3}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is the
least cardinal which is
both strong and $\gk^+$ supercompact +
No cardinal $\gd > \gk$ is measurable''.
There is then a partial ordering $\FP \in V$
such that
$V^\FP \models ``$ZFC + $\gk$ is strong +
%the least strong cardinal +
No cardinal $\gd > \gk$ is measurable''.
In addition, the following hold in $V^\FP$:
\begin{enumerate}
\item $\gk$'s strongness
is indestructible under
${<}\gk$-strategically closed, %$\gk$-directed closed
$(\gk^+, \infty)$-distributive forcing.
\item $\gk$ is the least strong cardinal.
\item Level by level inequivalence
between strong compactness and supercompactness holds.
\end{enumerate}
\end{theorem}
As a corollary to Theorem \ref{tm2}
and its proof,
we have the following two theorems.
\begin{theorem}\label{tm4}
It is consistent, relative to the existence
of a supercompact cardinal, for the least
strong cardinal $\gk$ whose strongness is
indestructible under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing
to be the same as the least supercompact cardinal,
which has its supercompactness
indestructible under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing.
\end{theorem}
\begin{theorem}\label{tm5}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
In addition, the following hold in $V^\FP$:
\begin{enumerate}
\item $\gk$ is indestructibly supercompact.
\item $\gk$ is the least strongly compact cardinal.
\item Every measurable cardinal
$\gd < \gk$ which is not a limit of
measurable cardinals is $\gd^+$ strongly compact.
\end{enumerate}
\end{theorem}
We take this opportunity to make a few
remarks concerning Theorems \ref{tm1} -- \ref{tm5}.
The limited amount of indestructibility forced
in Theorems \ref{tm2} and \ref{tm3} is due to
the necessity in our proofs
of preserving a nontrivial degree
of strong compactness.
However, if we weaken the requirement
in Theorem \ref{tm2} of all measurable
cardinals $\gd < \gk$ being $\gd^+$ strongly
compact to only measurable cardinals
$\gd < \gk$ which are not themselves
limits of measurable cardinals being
$\gd^+$ strongly compact, then it is possible for
$\gk$ to be a fully indestructible supercompact cardinal.
Also, as our proof will show,
the degrees of indestructibility mentioned
in the statement of Theorem \ref{tm1}
can be weakened. In particular, $\gk$'s
supercompactness can be indestructible
under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing, and
$\gk$'s strongness can be indestructible under
${<}\gk$-strategically closed,
$(\gk^+, \infty)$-distributive forcing.
This provides a nice balance between
Theorem \ref{tm1} and Theorems
\ref{tm2} and \ref{tm3}.
Further, note that the hypotheses used in Theorem \ref{tm3}
are much weaker than the existence of a supercompact
cardinal, yet much stronger than the existence
of a strong cardinal. The first of these facts
is implied by
\cite[Lemma 2.1]{AC2} and its proof,
since if $\gk$ is
supercompact, then $\gk$ is a limit of cardinals
$\gd$ which are both $2^\gd$ supercompact and strong.
The second of these facts is also implied by
\cite[Lemma 2.1]{AC2}, since if $\gk$ is
$2^\gk$ supercompact and strong, then $\gk$ is a
limit of strong cardinals.
In addition, the proof of \cite[Theorem 3]{A09}
provides the existence of a model
%where no cardinal is supercompact up to a Mahlo cardinal
in which the least strong cardinal $\gk$
whose strongness is indestructible under
$\gk$-directed closed,
$(\gk, \infty)$-distributive forcing
is the same as the
least supercompact cardinal, which has its
supercompactness indestructible under
$\gk$-directed closed,
$(\gk, \infty)$-distributive forcing.\footnote{Both
this theorem and Theorem \ref{tm4} show that with a
certain amount of indestructibility, it is
relatively consistent for the least strong cardinal
to be the least supercompact cardinal. By
\cite[Lemma 2.1]{AC2}, this is certainly impossible
without indestructibility added on.}
In this model, there are no Mahlo cardinals
above $\gk$.
The model witnessing the conclusions of
Theorem \ref{tm4} will be the model of
Theorem \ref{tm2}, and hence will be such that
there are may be Mahlo cardinals, weakly
compact cardinals, Ramsey cardinals, etc$.$
above the supercompact cardinal $\gk$.
Thus, by slightly weakening the amount
of indestructibility forced, it is possible
to get a witnessing model with
%a richer large cardinal structure
potentially more large cardinals
above the supercompact cardinal in question.
%The structure of this paper is as follows.
%Section \ref{s1} contains our introductory comments and
%a preliminary discussion of notation and terminology.
%Section \ref{s2} contains proofs of Theorems
%\ref{tm1}, \ref{tm2}, and \ref{tm4}.
%Section \ref{s3} contains a proof of Theorem \ref{tm3}
%and our concluding remarks.
%Section \ref{s4} contains our concluding remarks.
We conclude Section \ref{s1}
with a very brief discussion of
some preliminary material.
We presume a basic knowledge
of large cardinals and forcing.
A good reference in this
regard is \cite{J}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
When $G$ is $V$-generic over $\FP$,
we abuse notation slightly and
take both $V[G]$ and
$V^\FP$ as being the generic
extension of $V$ by $\FP$.
We also abuse notation slightly by
occasionally confusing terms with the
sets they denote, especially for
ground model sets and variants of the generic object.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$(\ga, \gb]$, $[\ga, \gb)$, and
$(\ga, \gb)$ are as in standard interval notation.
For $\gk$ a measurable cardinal, the
normal measure ${\cal U}$ over $\gk$ has
{\em trivial Mitchell rank} if for
$j : V \to M$ the elementary embedding
generated by ${\cal U}$,
$M \models ``\gk$ is not measurable''.
Suppose $\gk$ is a regular cardinal.
The partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D$ of
conditions of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
$\FP$ is {\em $\gk$-strategically closed}
if in the two person game in which the
players construct an increasing sequence
$\la p_\ga \mid \ga \le \gk \ra$,
where player I plays odd stages and
player II plays even stages,
player II has a strategy ensuring the game
can always be continued.
$\FP$ is {\em ${\prec}\gk$-strategically closed}
if in the two person game in which the
players construct an increasing sequence
$\la p_\ga \mid \ga < \gk \ra$,
where player I plays odd stages and
player II plays even stages,
player II has a strategy ensuring the game
can always be continued.
$\FP$ is {\em ${<}\gk$-strategically closed}
if $\FP$ is $\gd$-strategically closed
for every $\gd < \gk$.
$\FP$ is {\em $(\gk, \infty)$-distributive}
if the intersection of $\gk$ many
dense open subsets of $\FP$ is dense open.
It therefore follows that
any partial ordering $\FP$ which is
$\gk$-directed closed is also
${<}\gk$-strategically closed, and any
partial ordering which is $\gk$-strategically
closed is $(\gk, \infty)$-distributive.
It further follows that forcing with
any partial ordering $\FP$
which is $(\gk^+, \infty)$-distributive
preserves either the $\gk^+$ strong compactness or
$\gk^+$ supercompactness of $\gk$,
since forcing with $\FP$ adds no new
subsets of $P_\gk(\gk^+)$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorems \ref{tm2} -- \ref{tm5}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''. %$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ is {\em 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, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$
that admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proofs of Theorems \ref{tm1},
\ref{tm2}, \ref{tm4}, and \ref{tm5}}\label{s2}
We begin with the proof of Theorem \ref{tm1}.
\begin{proof}
We follow the proof of \cite[Theorem 2]{A07}.
Suppose $\gl > \gk$ is the least
measurable cardinal and
$\gk$ is either indestructibly supercompact
or indestructibly strong.
Force to add a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl^+$.
After this forcing, which is
$\gk$-directed closed and
$\gl$-strategically closed (and
hence is also ${<}\gk$-strategically closed
and $(\gk^+, \infty)$-distributive),
$\gl$ remains the least measurable
cardinal above $\gk$.
In particular, after the forcing,
$\gl$ is a measurable cardinal which
is not a limit of measurable cardinals.
(See \cite{AC2}
for a more complete discussion of
the partial ordering which adds
a non-reflecting stationary
set of ordinals of cofinality
$\gg$ to a regular cardinal
$\gr > \gg$.)
By \cite[Theorem 4.8]{SRK},
after the forcing,
because stationary reflection
for a set of ordinals of cofinality $\gk < \gl$
is violated at $\gl^+$,
$\gl$ is not $\gl^+$ strongly compact.
Since $\gk$ is suitably indestructible,
by reflection,
$A = \{\gd < \gk \mid \gd$ is measurable,
yet $\gd$ is neither $\gd^+$ strongly compact
nor a limit of measurable cardinals$\}$ is
unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact adding a non-reflecting stationary
set of ordinals of cofinality
$\gk$ is $\gk$-directed closed that
$A$ is unbounded in $\gk$ in the ground model.
\end{proof}
Having completed the proof of Theorem \ref{tm1},
we turn now to the proof of Theorem \ref{tm2}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
Without loss of generality, by first
doing a preliminary forcing if necessary,
we assume in addition that $V \models {\rm GCH}$.
The partial ordering $\FP$
used to establish Theorem \ref{tm2}
will be given as $\FP^0 \ast \dot \FP^1$.
To define $\FP^0$, let
$C = \{\gd < \gk \mid \gd$ is a measurable
cardinal which is not $\gd^+$ supercompact$\}$.
$\FP^0$ is then the reverse Easton iteration
of length $\gk$ which begins by adding a
Cohen subset of $\go$ and then adds,
to every $\gd \in C$, a non-reflecting
stationary set of ordinals of cofinality $\go$.
As in the proof of \cite[Theorem 2]{A01},
$\ov V = V^{\FP^0} \models ``$Every measurable
cardinal $\gd$ is $\gd^+$ strongly compact +
$\gk$ is supercompact''.
In addition, since $\FP^0$ may be defined
so that $\card{\FP^0} = \gk$,
by the results of \cite{LS},
$\ov V \models ``$No cardinal $\gd > \gk$ is measurable''.
Work now in $\ov V$.
Let $f$ be a Laver function
\cite{L} for $\gk$, i.e.,
$f : \gk \to V_\gk$ is such that
for every $x \in {\ov V}$ and every
$\gl \ge \card{{\rm TC}(x)}$, there is
an elementary embedding
$j : {\ov V} \to M$ generated by a
supercompact ultrafilter over
$P_\gk(\gl)$ such that
$j(f)(\gk) = x$.
Our next partial ordering $\FP^1$ is the
%which is used to establish Theorem \ref{tm2} is the
reverse Easton iteration of
length $\gk$ which begins by
adding a Cohen subset of
$\go$ and then (possibly) does nontrivial
forcing only at
cardinals $\gd < \gk$ which
are at least $\gd^+$ supercompact in ${\ov V}$.
At such a stage $\gd$, if
$f(\gd) = \dot \FQ$ and
$\forces_{\FP_\gd} ``\dot \FQ$
is a $\gd$-directed
closed, $(\gd^+, \infty)$-distributive partial
ordering having rank below
the least $\ov V$-measurable cardinal
greater than $\gd$'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$.
If this is not the case, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$,
where $\dot \FQ$ is a term for trivial forcing.
%we perform trivial forcing.
\begin{lemma}\label{l1}
$V^\FP = {\ov V}^{\FP^1} \models ``\gk$'s supercompactness
is indestructible under $\gk$-directed
closed, $(\gk^+, \infty)$-distributive forcing''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A07}.
Let $\FQ \in {\ov V}^{\FP^1}$ be such that
${\ov V}^{\FP^1} \models ``\FQ$ is
$\gk$-directed closed and
$(\gk^+, \infty)$-distributive''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{{\FP^1}} ``\dot \FQ$ is
$\gk$-directed closed and
$(\gk^+, \infty)$-distributive''.
Suppose $\gl \ge
\max(\gk^{+}, \card{{\rm TC}(\dot \FQ)})$
is an arbitrary cardinal, and let
$\gg = 2^{\card{[\gl]^{< \gk}}}$. Take
$j : {\ov V} \to M$ as an elementary
embedding witnessing the $\gg$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gg)$ such that
$j(f)(\gk) = \dot \FQ$. Since
${\ov V} \models ``$No cardinal $\gd$ above
$\gk$ is measurable'', $\gg \ge
2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
$M \models ``\gk$ is $\gk^+$ supercompact and
no cardinal $\gd$ in the %half-open
interval $(\gk, \gg]$ is measurable''. Hence,
the definition of ${\FP^1}$ implies that
$j({\FP^1} \ast \dot \FQ) = {\FP^1} \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first stage at which
$\FR$ does nontrivial forcing
is well above $\gg$.
%$2^{[\gg]^{< \gk}}$.
Laver's original argument from \cite{L} now applies
and shows
${\ov V}^{{\FP^1} \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be ${\ov V}$-generic over
${\FP^1} \ast \dot \FQ \ast \dot \FR$,
lift $j$ in ${\ov V}[G_0][G_1][G_2]$ to
$j : {\ov V}[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
${\ov V}[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ)$ containing $p$, lift $j$
again in ${\ov V}[G_0][G_1][G_2][G_3]$ to
$j : {\ov V}[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gg^+$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gl))}^{{\ov V}[G_0][G_1]}$
generated by $j$ is actually a member of
${\ov V}[G_0][G_1]$.)
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of
Lemma \ref{l1}.
\end{proof}
Since trivial forcing is both
$\gk$-directed closed and $(\gk^+, \infty)$-distributive,
Lemma \ref{l1} implies that
$ {\ov V}^{\FP^1} = V^\FP \models ``\gk$ is supercompact''.
Also, because $\FP^1$ may be defined so that
$\card{\FP^1} = \gk$, as before,
%standard arguments in tandem with the results of \cite{LS} show that
$V^\FP = \ov V^{\FP^1} \models
``$No cardinal $\gd > \gk$ is measurable''.
\begin{lemma}\label{l2}
$V^\FP = {\ov V}^{\FP^1} \models ``$Every measurable
cardinal $\gd < \gk$ is $\gd^+$ strongly compact''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd < \gk$ is measurable''.
We show that
$V^\FP \models ``\gd$ is $\gd^+$ strongly compact''.
To do this, work in $\ov V$. Write
$\FP^1 = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically
closed''.\footnote{Although not directly relevant to
this proof, it is actually true that
$\forces_{\FP'} ``\dot \FP''$ is $\ha_2$-directed closed''.}
By Theorem \ref{tgf}, $\gd$ had to have been
measurable in $\ov V$.
This means by the fact $\ov V = V^{\FP^0}$ that
$\ov V \models ``\gd$ is $\gd^+$ strongly compact''.
%Further, if we now let $\gr$
%be the least $V$-inaccessible cardinal
%above $\gd$, then by the definition of
%$\FP^0$, $\gr$ is the least inaccessible
%cardinal above $\gd$ in $\ov V$.
If we write $\FP^1 =
\FP^1_\gd \ast \dot \FQ^*$, by the definition of
$\FP^1$,
$\forces_{\FP^1_\gd} ``\dot \FQ^*$ is
$(\gd^+, \infty)$-distributive''.
Thus, to show that
$V^\FP = {\ov V}^{\FP^1} \models ``\gd$ is
$\gd^+$ strongly compact'', it suffices
to show that ${\ov V}^{\FP^1_\gd} \models
``\gd$ is $\gd^+$ strongly compact''.
For this, we consider two cases.
\bigskip\noindent Case 1: $\ov V \models ``\gd$
is $\gd^+$ supercompact''. In this case,
if $\card{\FP^1_\gd} < \gd$, then by the
results of \cite{LS},
${\ov V}^{\FP^1_\gd} \models ``\gd$ is
$\gd^+$ supercompact''. If not, it must be true that
$\card{\FP^1_\gd} \ge \gd$, so by the
definition of $\FP^1_\gd$, $\card{\FP^1_\gd} = \gd$.
We then let $j_0 : \ov V \to M$ be an elementary
embedding witnessing the $\gd^+$ supercompactness
of $\gd$ generated by a supercompact
ultrafilter over $P_\gd(\gd^+)$
such that $M \models ``\gd$ is not
$\gd^+$ supercompact''. By the definition of
$\FP^1_\gd$, $j_0(\FP^1_\gd) = \FP^1_\gd \ast \dot \FQ$, where
the first ordinal at which $\FQ$
does nontrivial forcing is well above $\gd^+$.
Standard arguments, as given, e.g., in the proof of
\cite[Lemma 8.1]{AH4} now show that $j_0$ lifts in
$\ov V^{\FP^1_\gd}$ to
$j_0 : \ov V^{\FP^1_\gd} \to M^{j_0(\FP^1_\gd)}$, i.e.,
$\ov V^{\FP^1_\gd} \models ``\gd$ is $\gd^+$ supercompact''.
(If $G$ is $\ov V$-generic over $\FP^1_\gd$, then working
in $\ov V[G]$, we may use the usual diagonalization
argument to build an $M[G]$-generic filter $H$ over $\FQ$
with ${j_0} '' G \subseteq G \ast H$.\footnote{An
outline of this argument is as follows.
$M[G]$ remains $\gd^+$-closed
with respect to $\ov V[G]$, which means that
$\FQ$ is ${\prec}\gd^{++}$-strategically closed
in both $M[G]$ and $\ov V[G]$. Since
$\ov V \models {\rm GCH}$ and $M$ is given
via an ultrapower by a supercompact
ultrafilter over $P_\gd(\gd^+)$, we may let
$\la D_\ga \mid \ga < \gd^{++} \ra \in \ov V[G]$
enumerate the dense open subsets of $\FQ$
present in $M[G]$. We may then use the
fact that $\FQ$ is ${\prec}\gd^{++}$-strategically
closed in $\ov V[G]$ to meet each $D_\ga$ and
thereby construct $H$.}
We then have
that $j_0$ lifts in $\ov V[G]$ to $j_0 : \ov V[G] \to M[G][H]$.)
\bigskip\noindent Case 2: $\ov V = V^{\FP^0} \models ``\gd$
is not $\gd^+$ supercompact''. In this case, write
$\FP^0 = \FQ' \ast \dot \FQ''$, where
$\card{\FQ'} = \go$, $\FQ'$ is nontrivial, and
$\forces_{\FQ'} ``\dot \FQ''$ is $\ha_1$-strategically closed''.
By Theorem \ref{tgf}, since $\ov V \models ``\gd$
is measurable'', $\gd$ had to have been
measurable in $V$. However, by the definition of
$\FP^0$, since every $V$-measurable cardinal $\gg$
which is not $\gg^+$ supercompact in $V$ contains
a non-reflecting stationary set of ordinals of
cofinality $\go$ in $\ov V$,
such a $\gg$ is not weakly compact in $\ov V$.
Consequently, it must in fact be the case that
$V \models ``\gd$ is $\gd^+$ supercompact''.
Now, let $\gr$ be the least inaccessible cardinal
above $\gd$ in $V$.
If we write
$\FP^0 = \FP^0_\gd \ast \dot \FQ'''$, we have that
$\forces_{\FP^0_\gd} ``\dot \FQ'''$ is $\gr$-strategically
closed and $\gr$ is the least inaccessible cardinal
above $\gd$''. Thus, $\FP^1_\gd$ is in fact
definable in $V^{\FP^0_\gd}$, and
to show that ${\ov V}^{\FP^1_\gd} =
V^{\FP^0 \ast \dot \FP^1_\gd} \models
``\gd$ is $\gd^+$ strongly compact'',
it suffices to show that
$V^{\FP^0_\gd \ast \dot \FP^1_\gd} \models
``\gd$ is $\gd^+$ strongly compact''.
For this, we use
exactly the same argument as given in
\cite[Lemma 2.3, Claim 1 of Case 2]{A02}.
We will not provide as many details
as in \cite{A02}, although we will give
a reasonably complete proof. %sketch of the proof.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gd^+$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gd^+)$ such that $M \models ``\gd$
is not $\gd^+$ supercompact''. Since GCH holds
in both $V$ and $M$, we may let
$k : M \to N$ be an elementary embedding
generated by a normal measure ${\cal U} \in M$
having trivial Mitchell rank.
It is the case that if $i : V \to N$ is an
elementary embedding having critical point $\gd$
and for any $x \subseteq N$ with $\card{x} \le \gd^+$,
there is some $y \in N$ such that
$x \subseteq y$ and $N \models ``\card{y} < i(\gd)$'',
then $i$ witnesses the $\gd^+$ strong compactness
of $\gd$. Using this fact, it is easily verifiable that
the elementary
embedding $i = k \circ j$ witnesses the
$\gd^+$ strong compactness of $\gd$ in $V$.
We show that this
embedding lifts in $V^{\FP^0_\gd \ast \dot \FP^1_\gd}$
to an elementary embedding
$i : V^{\FP^0_\gd \ast \dot \FP^1_\gd}
\to N^{i(\FP^0_\gd \ast \dot \FP^1_\gd)}$.
The lifted embedding
witnesses the $\gd^+$ strong compactness of $\gd$
in $V^{\FP^0_\gd \ast \dot \FP^1_\gd}$,
thereby completing the proof of Lemma \ref{l2}.
To do this, write $i(\FP^0_\gd \ast \dot \FP^1_\gd) =
(\FP^0_\gd \ast \dot \FQ^0 \ast \dot \FR^0) \ast
(\dot \FP^1_\gd \ast \dot \FQ^1 \ast \dot \FR^1)$, where
for $n = 0,1$, the
$\dot \FQ^n$ are terms for the portions of
$i(\FP^0_\gd \ast \dot \FP^1_\gd)$ between
$\gd$ and $k(\gd)$ and the
$\dot \FR^n$ are terms for the rest of
$i(\FP^0_\gd \ast \dot \FP^1_\gd)$,
i.e., the parts above $k(\gd)$.
Note that by the definition of $\FP^0_\gd$, since
$N \models ``\gd$ is inaccessible but is not measurable'',
only trivial forcing will occur at stage
$\gd$ in $N$ in the definition of
$i(\FP^0_\gd \ast \dot \FP^1_\gd)$.
Thus, the ordinals at which $\FQ^0$
does nontrivial forcing
are composed of an unbounded subset of
$N$-measurable cardinals in the interval
$(\gd, k(\gd)]$. As
$M \models ``\gd$ is measurable but
is not $\gd^+$ supercompact'',
by the definition of $\FP^0_\gd$,
nontrivial forcing occurs at stage $\gd$
in $M$ in the definition of
$j(\FP^0_\gd)$.
This means that by elementarity,
$\FQ^0$ does
nontrivial forcing at stage $k(\gd)$. Further,
the ordinals at which $\FR^0$ does
nontrivial forcing
are also composed of an unbounded subset of the
$N$-measurable cardinals in the interval
$(k(\gd), k(j(\gd)))$.
Let $G = G_0 \ast G_1$ be $V$-generic over
$\FP^0_\gd \ast \dot \FP^1_\gd$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G^0_1$ over
$\FQ^0$ and an
$N[G_0][G^0_1]$-generic object $G^0_2$ over
$\FR^0$.
We then construct in
$V[G_0][G_1] = V[G]$ an
$N[G_0][G^0_1][G^0_2][G_1]$-generic object
$G^1_1$ over $\FQ^1$ and an
$N[G_0][G^0_1][G^0_2][G_1][G^1_1]$-generic object
$G^1_2$ over $\FR^1$.
Our construction will guarantee that
${i} '' (G_0 \ast G_1)
\subseteq (G_0 \ast G^0_1 \ast G^0_2) \ast
(G_1 \ast G^1_1 \ast G^1_2)$.
This means that
$i : V \to N$ lifts to
$i : V[G_0][G_1] \to N[G_0][G^0_1][G^0_2]
[G_1][G^1_1][G^1_2]$ in $V[G] = V[G_0][G_1]$,
meaning that
$V^{\FP^0_\gd \ast \dot \FP^1_\gd} \models
``\gd$ is $\gd^+$ strongly
compact''.\footnote{To
see that the lifted embedding satisfies the
cover property mentioned above, suppose that
$\gg$ is an ordinal with the property that
$p \forces ``\dot x \subseteq \gg$ is
%N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$ is
such that $\card{\dot x} \le \gd^+$''.
Note that this is unambiguous, since
$\FP^0_\gd \ast \dot \FP^1_\gd$ satsifies
$\gd$-c.c. For $\ga < \gd^+$, let $\dot x_\ga$
be a term for the $\ga^{\rm th}$ member of
$\dot x$. Let
$y_\ga = \{\gb \mid \exists q \ge p[q
\forces ``\gb = \dot x_\ga$''$]\}$, and define
$y' = \bigcup_{\ga < \gd^+} y_\ga$.
Since
$\FP^0_\gd \ast \dot \FP^1_\gd$ satsifies
$\gd$-c.c., $\card{y'} \le \gd^+$.
By the fact $i : V \to N$ is an elementary
embedding witnessing the $\gd^+$ strong
compactness of $\gd$, let $y \in N$,
$y' \subseteq y$ be such that
$N \models ``\card{y} < i(\gd)$''.
%It is then the case that
%$p \forces ``\dot x \subseteq \check y$ and
%$\check y \in N$''.
%Since $N \models ``\card{y} < i(\gd)$'' and
Since $N \subseteq
N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$,
$y$ is the desired set covering $x$ in
$N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$.}
To obtain $G^0_1$, note that since $k$
is generated by an
ultrafilter over $\gd$ and
since GCH holds in both $V$ and $N$,
$\card{k(\gd^+)} = \card{k(2^\gd)} =
|\{ f \mid f : \gd \to \gd^+$ is a function$\}| =
\card{{[\gd^+]}^\gd} = \gd^+$. Thus, as
$N[G_0] \models ``|\wp(\FQ^0)| = k(2^\gd)$'',
we can let
$\la D_\ga \mid \ga < \gd^+ \ra$ enumerate in $V[G_0]$
%either $V[G_0]$ or $M^{**}[G_0]$
the dense open subsets of
$\FQ^0$ found in $N[G_0]$.
%so that for every dense open set
%$D \in \la D_\ga \mid \ga < \gd^+ \ra$,
%$D = D_\gb$ for some odd ordinal $\gb$.
Since
$N[G_0] \models ``\FQ^0$ is
${\prec}\gd^+$-strategically closed'',
the argument given in Case 1 above
for the construction of the generic object $H$
is applicable here as well and allows us to build
$G^0_1$ in $V[G_0]$ which is
$N[G_0]$-generic over $\FQ^0$ in the same manner.
We next construct in $V[G_0]$ the
desired $N[G_0][G^0_1]$-generic object
$G^0_2$ over $\FR^0$.
To do this, we first note that as
$M \models ``\gd$ is measurable but is
not $\gd^+$ supercompact'',
we can write
$j(\FP^0_\gd)$ as
$\FP^0_\gd \ast \dot \FS^0 \ast \dot \FT^0$, where
$\dot \FS^0$ is a term for the partial ordering
adding a non-reflecting stationary set of ordinals
of cofinality $\go$ to $\gd$, and
$\dot \FT^0$ is a term for the rest of
$j(\FP^0_\gd)$. In addition, the ordinals
at which $\FT^0$ does
nontrivial forcing are
composed of an unbounded subset of
the $M$-measurable cardinals
in the interval $(\gd^+, j(\gd))$.
This implies that in $M$,
$\forces_{\FP^0_\gd \ast \dot \FS^0}
``\dot \FT^0$ is ${\prec}\gd^{++}$-strategically
closed''. Further,
$|{[\gd^+]}^{< \gd}| = \gd^+$, and since
$V \models {\rm GCH}$,
$2^{\gd^+} = \gd^{++}$.
Therefore, as $j$ is
generated by a supercompact ultrafilter over
$P_\gd(\gd^+)$,
$|{(j(\gd))}^+| =
|j(\gd^+)| = |j(2^\gd)| =
|2^{j(\gd)}| =
|\{ f \mid f : P_\gd(\gd^+) \to \gd^+$ is a function$\}| =
|{[\gd^+]}^{\gd^+}| = \gd^{++}$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{F} for the first published account of
term forcing or \cite[Section 1.2.5, page 8]{C};
the notion is originally due to Laver) associated with
$\dot \FT^0$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^0_\gd \ast \dot \FS^0$ and
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau \in
\dot \FT^0$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau \ge \sigma$''.
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 in \cite{A02},
this can be done in such a way that
$M \models ``|\FT^*| = j(\gd)$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\dot \FT^0$ is
${\prec}\gd^{++}$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gd^{++}$-strategically closed in $M$ and, since
$M^{\gd^+} \subseteq M$, in $V$ as well. Since
$M \models ``2^{j(\gd)} = {(j(\gd))}^+ =
j(\gd^+)$'', this means we can let
$\la D_\ga \mid \ga < \gd^{++} \ra \in V$ enumerate
the dense open subsets of $\FT^*$ found in $M$ and
use the same argument employed in the construction of
$G^0_1$ to build in $V$ an $M$-generic object
$G^*$ over $\FT^*$.
Note that since $N$ is given
by an ultrapower of $M$ via a normal ultrafilter
${\cal U} \in M$ over $\gd$,
\cite[Fact 2, Section 1.2.2]{C} tells us that
$k '' G^*$ generates an $N$-generic object
$G^*_2$ over $k(\FT^*)$. By elementarity,
$k(\FT^*)$ is the term forcing in $N$
defined with respect to
$k(j(\FP^0_\gd)_{\gd + 1}) =
\FP^0_\gd \ast \dot \FQ^0$.
Therefore, since
$i(\FP^0_\gd) = k(j(\FP^0_\gd)) =
\FP^0_\gd \ast \dot \FQ^0 \ast
\dot \FR^0$,
$G^*_2$ is $N$-generic over
$k(\FT^*)$, and $G_0 \ast G^0_1$ is
$k(\FP^0_\gd \ast \dot \FS^0)$-generic over
$N$, \cite[Fact 1, Section 1.2.5]{C}
tells us that for
$G^0_2 = \{i_{G_0 \ast G^0_1}(\tau) \mid \tau \in
G^*_2\}$, $G^0_2$ is $N[G_0][G^0_1]$-generic over
$\FR^0$.
Working in $V[G_0][G_1]$, we build the
generic objects $G^1_1$ and $G^1_2$.
To construct $G^1_1$, we note that
by the strategic closure properties
of the partial orderings over which
$G^0_1$ and $G^0_2$ are generic,
$N[G_0][G^0_1][G^0_2]$ remains
$\gd$-closed with respect to
$V[G_0][G^0_1][G^0_2]$ = $V[G_0]$.
Therefore, since $\FP^1_\gd$ is
$\gd$-c.c.,
$N[G_0][G^0_1][G^0_2][G_1]$ remains
$\gd$-closed with respect to
$V[G_0][G^0_1][G^0_2][G_1] = V[G_0][G_1]$.
Further, since $M \models ``\gd$ is not
$\gd^+$ supercompact'', $N \models ``k(\gd)$
is not $k(\gd^+)$ supercompact'', and so any
ordinals at which $\FQ^1$
does nontrivial forcing lie in the interval
$(\gd, k(\gd))$.
This means we can construct
the $N[G_0][G^0_1][G^0_2][G_1]$-generic
object $G^1_1$ over $\FQ^1$ in
$V[G_0][G_1]$ in the same way $G^0_1$
was constructed in $V[G_0]$.
To build $G^1_2$,
we once again work in $M$.
Write $j(\FP^0_\gd \ast \dot \FP^1_\gd)$ as
$\FP^0_\gd \ast \dot \FS^0 \ast \dot \FT^0
\ast \dot \FP^1_\gd \ast \dot \FT^1$, where
$\dot \FS^0$ and $\dot \FT^0$ are as before, and
$\dot \FT^1$ is a term for the portion of
$j(\FP^0_\gd \ast \dot \FP^1_\gd)$ defined in
$M$ between stages $\gd$ and $j(\gd)$.
Note that since $M \models ``\gd$
is not $\gd^+$ supercompact'',
all ordinals at which $\FT^1$
does nontrivial forcing lie
in the interval $(\gd^+, j(\gd))$.
Let $\FT^{**}$ be
the term forcing partial ordering
associated with $\FT^1$, i.e.,
$\tau \in \FT^{**}$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd$ and
$\forces_{\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd} ``\tau \in
\dot \FT^1$'', ordered by
$\tau \ge \sigma$ iff
$\forces_{\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd}
``\tau \ge \gs$''.
A similar analysis to that given for
the term forcing $\FT^*$, using the
observations made in the preceding paragraphs,
now allows us to construct in $V$ an
$M$-generic object $G^{**}$ for $\FT^{**}$,
infer that $k '' G^{**}$ generates an $N$-generic
object $G^{**}_2$
for the relevant term forcing partial
ordering in $N$, and working in
$V[G_0][G_1]$, use
$G^{**}_2$ to create an
$N[G_0][G^0_1][G^0_2][G_1][G^1_1]$-generic object
$G^1_2$ over $\FR^1$.
Thus, in $V[G_0][G_1]$,
$i : V \to N$ lifts to
$i : V[G_0][G_1] \to N[G_0][G^0_1][G^0_2][G_1]
[G^1_1][G^1_2]$.
This means that
$V^{\FP^0_\gd \ast \dot \FP^1_\gd}
\models ``\gd$ is $\gd^+$
strongly compact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l2a}
$V^\FP = \ov V^{\FP^1} =
V^{\FP^0 \ast \dot \FP^1} \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
By the definition of $\FP^0$,
$\ov V = V^{\FP^0} \models ``$Unboundedly
in $\gk$ many regular cardinals $\gg < \gk$
contain non-reflecting stationary sets of
ordinals of cofinality $\go$''. Therefore,
by \cite[Theorem 4.8]{SRK} and the succeeding
remarks, $\ov V = V^{\FP^0} \models ``$No
cardinal $\gd < \gk$ is strongly compact''.
Then, to show that $V^\FP = \ov V^{\FP^1} \models
``$No cardinal $\gd < \gk$ is strongly compact'',
suppose that $V^\FP = \ov V^{\FP^1}
\models ``\gd < \gk$ is measurable''.
Let $\FP^1 = \FP' \ast \dot \FP''$ be the factorization
of $\FP^1$ given in Lemma \ref{l2}.
As in Lemma \ref{l2}, by this factorization,
Theorem \ref{tgf} implies that $\gd$
had to have been measurable in $\ov V$.
However, by its definition, $\FP^1$ is
mild with respect to $\gd$, so if
$V^\FP = \ov V^{\FP^1} \models ``\gd$
is strongly compact'', then
$\ov V \models ``\gd$ is strongly compact'' as well.
Since we have already seen that this does not occur,
$V^\FP = \ov V^{\FP^1} \models ``\gd$
is not strongly compact''.
This completes the proof of Lemma \ref{l2a}.
\end{proof}
Lemmas \ref{l1} -- \ref{l2a} and the
intervening remarks complete the proof of
Theorem \ref{tm2}.
\end{proof}
\begin{pf}
Theorem \ref{tm4} now follows as an immediate
corollary to Theorem \ref{tm2}, with the model
$V^\FP$ constructed for Theorem \ref{tm2}
as its witness. To see this, it suffices to show that
no strong cardinal $\gd < \gk$ has its strongness
indestructible under $\gd$-directed closed,
$(\gd^+, \infty)$-distributive forcing.
However, if this were the case, then by Theorem \ref{tm1},
there would be measurable cardinals $\gg < \gd$
which were not $\gg^+$ strongly compact. By the
construction of $V^\FP$, this is impossible.
The proof of Theorem \ref{tm4} is now completed
by the observation that since $\gk$'s supercompactness
is indestructible under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing,
its strongness is as well.
\end{pf}
\begin{pf}
Theorem \ref{tm5} follows as a corollary to
the proof of Theorem \ref{tm2}.
The definition of the partial ordering $\FP$ used in the
proof of Theorem \ref{tm5} is the same as in
Theorem \ref{tm2}, with the exception that at
nontrivial stages of forcing $\gd$ in the definition
of $\FP^1$, we allow $\dot \FQ$ to be a term for
an arbitrary $\gd$-directed closed partial
ordering in $V^{\FP_\gd}$ having rank below
the least $\ov V$-measurable cardinal
greater than $\gd$. In other words,
at a nontrivial stage of forcing $\gd$,
$\dot \FQ$ is such that
$\forces_{\FP_\gd} ``\dot \FQ$ is a $\gd$-directed
closed partial ordering having rank below the
least $\ov V$-measurable cardinal greater than $\gd$''.
With this definition, the proof that
$V^\FP \models ``\gk$ is indestructibly supercompact''
is exactly the same as in Lemma \ref{l1}.
Further, the proof that
$V^\FP \models ``$No cardinal $\gd < \gk$ is
strongly compact'' is precisely the same as in Lemma \ref{l2a}.
Since just as in the proof of Theorem \ref{tm2},
$V^\FP \models ``$No cardinal
$\gd > \gk$ is measurable'', the proof of
Theorem \ref{tm5} will be complete once we have shown that
$V^\FP \models ``$If $\gd < \gk$ is a measurable
cardinal which is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ strongly compact''.
To do this, let $\gd < \gk$ be such that
$\ov V \models ``\gd$ is a measurable cardinal
which is not a limit of measurable cardinals''.
By the definition of $\FP^1$, we may write
$\FP^1 = \FP^1_\gd \ast \dot \FQ^*$, where
$\card{\FP^1_\gd} < \gd$ and
$\forces_{\FP^1_\gd} ``\dot \FQ^*$ is
$\gr$-directed closed for $\gr$ the
least inaccessible cardinal above $\gd$''.
Since $\ov V \models ``\gd$ is $\gd^+$
strongly compact'', by the results of \cite{LS},
$\ov V^{\FP^1_\gd} \models ``\gd$ is
$\gd^+$ strongly compact''. Thus,
$\ov V^{\FP^1_\gd \ast \dot \FQ^*} =
\ov V^{\FP^1} = V^\FP \models ``\gd$ is
$\gd^+$ strongly compact''. We will have
consequently finished the proof of
Theorem \ref{tm5} once we have shown that if
$\ov V^{\FP^1} = V^\FP \models ``\gd$ is a
measurable cardinal which is not a limit
of measurable cardinals'', then
$\ov V \models ``\gd$ is a
measurable cardinal which is not a limit
of measurable cardinals''.
For this, by the direct analogue of the
factorization of $\FP^1$ given in Lemma \ref{l2}
and Theorem \ref{tgf}, if
$\ov V^{\FP^1} \models ``\gd$ is a measurable
cardinal'', then
$\ov V \models ``\gd$ is measurable'' as well.
We must therefore show that if
$\ov V^{\FP^1} \models ``\gd$ is not a
limit of measurable cardinals'', then
$\ov V \models ``\gd$ is not a
limit of measurable cardinals''. However, if
$\ov V \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals'',
then in particular,
$\ov V \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals $\gg$
such that $\gg$ is not a limit of measurable cardinals''.
%which are themselves not limits of measurable cardinals''.
By our remarks in the preceding paragraph,
such $\gg$ are preserved to $\ov V^{\FP^1}$, i.e.,
$\ov V^{\FP^1} = V^\FP \models ``\gd$ is
a measurable cardinal which is a limit of
measurable cardinals''. This contradiction
completes the proof of Theorem \ref{tm5}.
\end{pf}
\section{The Proof of Theorem \ref{tm3}}\label{s3}
Having completed the proofs of Theorems \ref{tm1},
\ref{tm2}, \ref{tm4}, and \ref{tm5}, we turn now to the
proof of Theorem \ref{tm3}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH + $\gk$ is
the least cardinal which is
both strong and $\gk^+$ supercompact + No
cardinal $\gd > \gk$ is measurable''.
As with the proof of Theorem \ref{tm2},
the partial ordering $\FP$
used to establish Theorem \ref{tm3}
will be given as $\FP^0 \ast \dot \FP^1$.
$\FP^0$ is defined as before, i.e., let
$C = \{\gd < \gk \mid \gd$ is a measurable
cardinal which is not $\gd^+$ supercompact$\}$.
$\FP^0$ is then the reverse Easton iteration
of length $\gk$ which begins by adding a
Cohen subset of $\go$ and then adds,
to every $\gd \in C$, a non-reflecting
stationary set of ordinals of cofinality $\go$.
Once again,
$\ov V = V^{\FP^0} \models ``$Every measurable
cardinal $\gd$ is $\gd^+$ strongly compact''.
We have now the following lemma.
\begin{lemma}\label{l3}
$\ov V \models ``\gk$ is strong''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3} is very similar
to the one given in \cite[Lemma 2.5]{AC2}.
We consequently quote from this argument,
making the appropriate changes where necessary.
We use for the proof of this lemma
the same notation and terminology
as in \cite[Lemma 2.5]{AC2}, which is taken from
the introductory section of \cite{C}.
Fix $\gz > \gk^+$ an arbitrarily large ordinal, and let
$\gl > \gz$ be the least cardinal such that
$\gl = \ha_\gl$ and $\gl$ has
cofinality $\gk$. Let
$j : V \to M$ be an elementary embedding witnessing
the $\gl$ strongness of $\gk$ generated by a
$(\gk, \gl)$-extender of width $\gk$
such that
$M \models ``\gk$ is not $\gl$ strong'',\footnote{As
in \cite[page 4]{C}, an $(\ga, \gb)$-extender ${\cal E} =
\la E_a \mid a \in {[\gb]}^{< \go} \ra$ is
said to have {\em width $\gg$} if for all
$a \in {[\gb]}^{< \go}$, $E_a$ is an
$\ga$-additive ultrafilter over $[\gg]^{\card{a}}$.}
and let
$i : V \to N$ be the elementary embedding
witnessing the measurability of $\gk$
generated by the normal ultrafilter
${\cal U} = \{x \subseteq \gk \mid
\gk \in j(x)\}$.
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 $\gk^+$.
Observe that
$M \models ``$No cardinal
$\rho \in (\gk, \gl)$ is measurable'',
for if this were false, then since
$V_\gl \subseteq M$,
$V \models ``\rho$ is measurable''.
Because $V \models ``$No cardinal
$\gd > \gk$ is measurable'', this is impossible.
As the choice of $\gl$
implies that $M \models ``\gl$ is singular'',
this means that in $M$, the least measurable cardinal
$\gd > \gk$ is such that $\gd > \gl$.
For any ordinal $\ga$, define
$\sigma_\ga$ as the least ordinal
greater than $\ga$ such that $\ga$ is not
$\sigma_\ga$ strong if such an
ordinal exists, and
$\sigma_\ga = 0$ otherwise. Define
$f : \gk \to \gk$ as
$f(\ga) =$ The least inaccessible
cardinal greater than $\sigma_\ga$.
By our choice of $\gl$ and the
preceding paragraph,
$\gk < \gl < j(f)(\gk) < \gd$,
where $\gd$ is the least measurable
cardinal in $M$ greater than $\gk$, i.e.,
the least element above $\gk$
to which $j(\FP^0)$ adds a non-reflecting
stationary set of ordinals of cofinality $\go$.
Note now that
$M = \{j(g)(a) \mid a \in {[\gl]}^{< \omega}$,
$\dom(g) = {[\gk]}^{|a|}$,
$g : {[\gk]}^{|a|} \to V\} =
\{k(i(g))(a) \mid a \in {[\gl]}^{< \omega}$,
$\dom(g) = {[\gk]}^{|a|}$,
$g : {[\gk]}^{|a|} \to V\}$.
By defining $\gg = i(f)(\gk)$, we have
$k(\gg) = k(i(f)(\gk)) = j(f)(\gk) > \gl$.
This means
$j(g)(a) = k(i(g))(a) =
k(i(g) \rest {[\gg]}^{|a|})(a)$ for
$a \in {[\gl]}^{< \omega}$. Hence,
$M = \{k(h)(a) \mid a \in {[\gl]}^{< \omega}$,
$h \in N$, $\dom(h) = {[\gg]}^{|a|}$,
$h : {[\gg]}^{|a|} \to N\}$.
By elementarity, we must have
$N \models ``\gk$ is not strong and
$\gk < \gg = i(f)(\gk) < \gd_0$ =
The least measurable cardinal in $N$
greater than $\gk$ =
The least element above $\gk$
to which $i(\FP^0)$ adds a non-reflecting
stationary set of ordinals of cofinality $\go$'', since
$M \models ``k(\gk) = \gk$ is not strong and
$k(\gk) = \gk < k(\gg) = k(i(f)(\gk)) =
j(f)(\gk) < k(\gd_0) = \gd$''.
Thus, $k$ can be assumed to be generated
by an $N$-extender of width
$\gg \in (\gk, \gd_0)$.
Write $i(\FP^0) = \FP^0 \ast
\dot \FQ^0$, where $\dot \FQ^0$
is a term for the portion of
$i(\FP^0)$ which does nontrivial forcing
on ordinals in the interval
$[\gk, i(\gk))$.
By elementarity and the choice of $\gl$,
$N \models ``\gk$ is $\gk^+$ supercompact'', since
$M \models ``k(\gk) = \gk$ is $k(\gk^+) = \gk^+$
supercompact''. Therefore,
%Since $N \models ``\gk$ is not a strong cardinal'',
the ordinals on which $\FQ^0$ does nontrivial forcing
actually lie in the interval
$(\gk, i(\gk))$, or more precisely,
in the interval $[\gd_0, i(\gk))$.
This means that if $G_0$ is
$V$-generic over $\FP^0$, the
argument from Lemma \ref{l2} for the
construction of the generic object
$G^0_1$ 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 $k(\gk) = \gk$, $k$ lifts
in $V[G_0]$ to
$k : N[G_0] \to M[G_0]$.
By \cite[Fact 3, Section 1.2.2]{C},
$k : N[G_0] \to M[G_0]$ can also be
assumed to be generated by an
extender of width
$\gg \in (\gk, \gd_0)$.
In analogy to the preceding paragraph, write
$j(\FP^0) = \FP^0 \ast \dot \FQ^1$.
By the last sentence of the preceding paragraph
and the fact $\gd_0$ is the least ordinal
at which $\FQ^0$ does
nontrivial forcing,
we can use \cite[Fact 2, Section 1.2.2]{C}
to infer that
$H = \{p \in \FQ^1 \mid \exists q \in k '' G^*_1
[q \ge p]\}$ is $M[G_0]$-generic over
$k(\FQ^0) = \FQ^1$. Thus, $k$ lifts 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}
Since
$M \models ``$No cardinal
$\rho \in (\gk, \gl]$ is measurable'',
the ordinals at which $\FQ^1$ does
nontrivial forcing lie
in the interval $(\gl, j(\gk))$. Thus, as
$V_\gl \subseteq M$,
$V_\gl[G_0] \subseteq M[G_0]$,
and as $\FQ^1$
%does nontrivial forcing at
is only defined on ordinals in the interval
$(\gl, j(\gk))$,
$V_\gl[G_0]$ is the set of all sets of
rank less than $\gl$ in $M[G_0][H]$. Hence,
$j$ is a $\gl$ strongness embedding.
Since both $\gz$ and
$\gl$ may be chosen arbitrarily large,
this completes the proof of Lemma \ref{l3}.
\end{proof}
Work now in $\ov V$.
Before defining $\FP$, we recall a
definition from \cite{H4}.
%As in \cite{H4}, if
If ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra \mid \FP \in {\cal A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. The terminology ``lottery sum''
is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the expressions ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
We complete the definition of $\FP$ by defining
$\FP^1$ as the reverse Easton iteration of length $\gk$
which begins by adding a Cohen subset of $\go$
and then (possibly) does nontrivial forcing only
at cardinals $\gd < \gk$ which are
measurable in $\ov V$. If
$\ov V \models ``\gd$ is $\gd^+$ supercompact'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for the partial ordering
adding a non-reflecting stationary set of ordinals
of cofinality $\go$ to $\gd$. If
$\ov V \models ``\gd$ is not $\gd^+$ supercompact'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for the lottery sum of all
${<}\gd$-strategically closed, $(\gd^+, \infty)$-distributive
partial orderings
in $\ov V^{\FP_\gd}$ which
have rank below the
least $\ov V$-measurable cardinal greater than $\gd$.
We explicitly note that the definition of
$\FP^1$ uses a lottery sum, instead of a Laver
function as earlier, in order to emphasize the
similarities between the proofs of Lemma \ref{l4}
and \cite[Theorem 4.10]{H4}.
\begin{lemma}\label{l5}
$V^\FP \models ``$No cardinal $\gd < \gk$ is strong''.
\end{lemma}
\begin{proof}
We show in turn that
$V^{\FP^0} = \ov V \models ``$No cardinal
$\gd < \gk$ is strong'' and
$V^\FP = {\ov V}^{\FP^1} = V^{\FP^0 \ast \dot \FP^1} \models
``$No cardinal $\gd < \gk$ is strong''.
By the definition of $\FP^0$ and the fact that
$V \models ``\gk$ is the least cardinal which
is both strong and $\gk^+$ supercompact'',
$\ov V \models ``$Any cardinal $\gd < \gk$
which was strong in $V$ is non-measurable,
since $\gd$ was not $\gd^+$ supercompact and therefore
contains a non-reflecting stationary
set of ordinals of cofinality $\go$''. Write
$\FP^0 = \FQ' \ast \dot \FQ''$, where
$\card{\FQ'} = \go$, $\FQ'$ is nontrivial, and
$\forces_{\FQ'} ``\dot \FQ''$ is $\ha_1$-strategically closed''.
By Theorem \ref{tgf}, if $\ov V \models ``\gd$
is strong'', then $V \models ``\gd$ is strong'' as well.
Since we have just seen that this cannot be the case
if $\gd < \gk$,
$V^{\FP^0} = \ov V \models ``$No cardinal
$\gd < \gk$ is strong''.
However, by writing $\FP^1 = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed'',
Theorem \ref{tgf} once again implies that if
$\ov V^{\FP^1} \models ``\gd$ is strong'', then
$\ov V \models ``\gd$ is strong'' as well.
Consequently, since $\ov V \models ``$No cardinal
$\gd < \gk$ is strong'',
$V^\FP = {\ov V}^{\FP^1} = V^{\FP^0 \ast \dot \FP^1} \models
``$No cardinal $\gd < \gk$ is strong''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP = \ov V^{\FP^1} \models ``$Every measurable
cardinal $\gd \le \gk$ is $\gd^+$ strongly compact''.
\end{lemma}
\begin{proof}
By the definition of $\FP^1$, if $\gd < \gk$ and
$\ov V \models ``\gd$ is $\gd^+$ supercompact'', then
$V^\FP = \ov V^{\FP^1} \models ``\gd$ contains a
non-reflecting stationary set of ordinals of
cofinality $\go$ and hence is non-measurable''.
For $\gd < \gk$, this will consequently allow us to
use virtually the same arguments as in the proof
of Case 2 of Lemma \ref{l2}.
We therefore indicate
the only two places where the proofs
are slightly different,
and the necessary modifications that need to be made,
using the same notation as in Lemma \ref{l2}.
These two changes occur in the construction of
the generic objects $G^1_1$ and $G^1_2$ in Case 2,
because of the use of lottery sums in the
%They are necessitated because by the
definition of $\FP^1$.
%it may be the case that $\FT^1$ does nontrivial
%forcing at stage $\gd$ in $M^{\FP^0_\gd \ast \dot \FS^0 \ast
%\dot \FT^0 \ast \dot \FP^1_\gd}$.
However, by forcing if necessary above a condition
opting for trivial forcing at stage $\gd$ in
$M^{\FP^0_\gd \ast \dot \FS^0 \ast
\dot \FT^0 \ast \dot \FP^1_\gd}$,
we will have that
$\FT^1$ is forcing equivalent to a partial ordering
which does only trivial forcing at stage $\gd$ in
$M^{\FP^0_\gd \ast \dot \FS^0 \ast
\dot \FT^0 \ast \dot \FP^1_\gd}$.
Once this has been done, we may use a version of
$\FQ^1$ that does only trivial forcing at stage $k(\gd)$ in
$N^{\FP^0_\gd \ast \dot \FQ^0
\ast \dot \FR^0 \ast \dot \FP^1_\gd}$,
thereby allowing us to construct $G^1_1$ as before.
$G^1_2$ is also built as earlier, using the fact that
the ordinals at which $\FT^1$
does nontrivial forcing lie in the
interval $(\gd^+, j(\gd))$.
Thus, Lemma \ref{l6} is true if $\gd < \gk$.
For $\gd = \gk$, we note that
$\ov V \models ``\gk$ is not
$2^\gk = \gk^+$ supercompact''.
This is since \cite[Lemma 2.1]{AC2}
tells us that any cardinal $\gd$
which is both $2^\gd$ supercompact and strong
must be a limit of strong cardinals.
As this is false by Lemma \ref{l5},
$\ov V \models ``\gk$ is $\gk^+$ strongly compact but
is not $\gk^+$ supercompact''.
The arguments of Case 2 of Lemma \ref{l2}
as modified in the preceding paragraph
are thus applicable and show that
$V^\FP = \ov V^{\FP^1} \models ``\gk$ is
$\gk^+$ strongly compact''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP = \ov V^{\FP^1} \models ``\gk$'s strongness
is indestructible under ${<}\gk$-strategically
closed, $(\gk^+, \infty)$-distributive forcing''.
\end{lemma}
\begin{proof}
Suppose $\FQ \in \ov V^{\FP^1}$ is such that
$\ov V^{\FP^1} \models ``\FQ$ is ${<}\gk$-strategically
closed and $(\gk^+, \infty)$-distributive''.
Take $\dot \FQ$ as a term for $\FQ$ such that
$\forces_{\FP^1}
``\dot \FQ$ is ${<}\gk$-strategically
closed and $(\gk^+, \infty)$-distributive''.
Fix $\gz > \gk^+$ an arbitrarily large ordinal, and assume that
$\gl > \max(\gz, \card{{\rm TC}(\dot \FQ)})$ is
the least cardinal of cofinality $\gk$ such that
$\gl = \ha_\gl$.
Let $j : \ov V \to M$ be an elementary embedding
witnessing the $\gl$ strongness of $\gk$ generated by a
$(\gk, \gl)$-extender of width $\gk$.
We now show that the embedding $j$
lifts in $\ov V^{\FP \ast \dot \FQ}$ to
$j : \ov V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
%that the embedding $j$ lifts
The methods for doing this
are quite similar to those given in the proof of
\cite[Theorem 4.10]{H4} (as well as elsewhere).
%or the argument given in the
%proof of Lemma 4.2 of \cite{A03}.
For the benefit of readers, we give
the argument here as well, taking the
liberty to quote freely from \cite[Theorem 4.10]{H4}.
%We may assume that
Because $j$ is an extender embedding, we have that
$M = \{j(f)(a) \mid a \in {[\gl]}^{< \omega}$,
$f \in \ov V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq \ov V\}$.
Since $\ov V \models ``\gk$ is not $\gk^+$
supercompact'' and $V_\gl \subseteq M$
(which means that $M \models ``$There are
no measurable cardinals in the interval
$(\gk, \gl]$''),
by forcing above the appropriate condition,
we will have that
$j(\FP)$ is forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FR$.
Consequently,
we may infer that the first ordinal at which
$\FR$ does a
lottery sum is above $\gl$.
Since $\gl$ has been chosen to have cofinality $\gk$,
we may assume that $M^\gk \subseteq M$.
This means that if
$G$ is $\ov V$-generic over $\FP$ and
$H$ is $\ov V[G]$-generic over $\FQ$,
$\FR$ is ${\prec} \gk^+$-strategically closed in both
$\ov V[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$.
As in \cite{H4},
by using a suitable coding that allows us
to identify finite subsets of $\gl$
with elements of $\gl$,
by the definition of $M$,
there must be some
$\ga_0 < \gl$ and function $g$ such that
$\dot \FQ = j(g)(\ga_0)$.
%(assuming that $\dot \FQ$ has been chosen reasonably).
Let $N = \{i_{G \ast H}(\dot z) \mid \dot z =
j(f)(\gk, \ga_0, \gl)$ for some function $f \in \ov V\}$.
It is easy to verify that
$N \prec M[G][H]$, that $N$ is closed under
$\gk$ sequences in $\ov V[G][H]$, and that
$\gk$, $\ga_0$, $\gl$, $\FQ$, and $\FR$ are all
elements of $N$.
Further, since
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : \gk \to V_\gk$ in $\ov V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G][H]$ and $\ov V[G][H]$, we can
use the argument for the construction of the
generic object $G^0_1$ given in Lemma \ref{l2} to
build $H' \subseteq \FR$ in $\ov V[G][H]$
which is $N$-generic.
%Let $H' = \{p \in \FR \mid \exists \gs <
%\gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $H'$ is actually
$M[G][H]$-generic over $\FR$.
If $D$ is a dense open subset of
$\FR$ in $M[G][H]$, then
$D = i_{G \ast H}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in \ov V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$,
where $\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that
$j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of
$\FR$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$ and therefore $(\gl, \infty)$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is
definable without the parameters
$\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N$.
Thus, since $H'$ meets every dense open
subset of $\FR$ present in $N$,
$H' \cap i_{G \ast H}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$H' \cap i_{G \ast H}(\dot D) \neq \emptyset$.
This means $H'$ is
$M[G][H]$-generic over $\FR$, so in
$\ov V[G][H]$, as
$j''G \subseteq G \ast H \ast H'$,
$j$ lifts to
$j : \ov V[G] \to M[G][H][H']$ via the definition
$j(i_G(\tau)) = i_{G \ast H \ast H'}(j(\tau))$.
It remains to lift $j$ through the forcing $\FQ$
while working in $\ov V[G][H]$.
To do this, it suffices to show that
$j''H \subseteq j(\FQ)$ generates an
$M[G][H][H']$-generic object $H''$ over $j(\FQ)$.
Given a dense open subset
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = i_{G \ast H \ast H'}(\dot D)$ for some name
$\dot D = j(\vec D)(a)$, where
$a \in {[\gl]}^{< \go}$ and
$\vec D = \la D_\gs \mid \gs \in {[\gk]}^{|a|} \ra$
is a function.
%{[\gk]}^{< \go} \ra$.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $(\gk^+, \infty)$-distributive
(and hence is of course also
$(\gk, \infty)$-distributive),
it follows that
%$D' = \bigcap_{\gs \in {[\gk]}^{< \go}} D_\gs$ is
$D' = \bigcap_{\gs \in {[\gk]}^{|a|}} D_\gs$ is
a dense open subset of $\FQ$. As
$j(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$j''H \cap D \neq \emptyset$. Thus,
$H'' = \{p \in j(\FQ) \mid \exists q \in j''H
[q \ge p]\}$ is our desired generic object, and
$j$ lifts in $\ov V[G][H]$ to
$j : \ov V[G][H] \to M[G][H][H'][H'']$.
This final lifted version of $j$ is
$\gl$ strong since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{\ov V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$.
Since both $\gz$ and $\gl$ can be
made arbitrarily large, this completes the
proof of Lemma \ref{l4}.
\end{proof}
Since trivial forcing is both ${<}\gk$-strategically
closed and $(\gk^+, \infty)$-distributive,
Lemma \ref{l4} implies that
$\ov V^{\FP^1} = V^\FP \models ``\gk$ is strong''.
Also, because $\FP$ may be defined so that
$\card{\FP} = \gk$, as before,
$V^\FP \models ``$No cardinal $\gd > \gk$ is measurable''.
In addition, Lemma \ref{l5} and the argument
found in the last paragraph of Lemma \ref{l6}
imply that $V^\FP = \ov V^{\FP^1} \models ``\gk$
is not $\gk^+$ supercompact''.
\begin{lemma}\label{l7}
$\ov V^{\FP^1} = V^\FP \models ``$If $\gd$ is measurable,
then $\gd$ is not $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose $\ov V^{\FP^1} = V^\FP \models ``\gd$ is
$\gd^+$ supercompact''.
By our remarks in the paragraph immediately
following the proof of Lemma \ref{l4},
we may assume without
loss of generality that $\gd < \gk$. If we then
use the factorization of $\FP^1 = \FP' \ast \dot
\FP''$ given in Lemma \ref{l5}, we may infer
by Theorem \ref{tgf} that since
$\ov V^{\FP^1} = V^\FP \models ``\gd$ is
$\gd^+$ supercompact'', it follows that
$\ov V \models ``\gd$ is $\gd^+$ supercompact'' as well.
However, as we have already observed in the proof of
Lemma \ref{l6},
by the definition of $\FP^1$, if $\gd < \gk$ and
$\ov V \models ``\gd$ is $\gd^+$ supercompact'', then
$V^\FP = \ov V^{\FP^1} \models ``\gd$ contains a
non-reflecting stationary set of ordinals of
cofinality $\go$ and hence is non-measurable''.
This contradiction completes the proof of Lemma \ref{l7}.
\end{proof}
By Lemmas \ref{l7}, \ref{l6}, and the remarks
after the proof of Lemma \ref{l4},
$V^\FP \models ``$Level by level inequivalence holds''.
This observation, together with Lemmas \ref{l3} -- \ref{l4}
and the intervening comments, complete the proof of
Theorem \ref{tm3}.
\end{proof}
Except for the fact that the proofs of
Lemmas \ref{l2} and \ref{l6} require
$(\gd^+, \infty)$-distributivity at
each nontrivial stage of forcing $\gd$
in the definition of the relevant
partial ordering $\FP^1$, there is no
reason prima facie to believe that the amount of
indestructibility forced must include
this additional condition.
We therefore ask if this restriction can be removed.
In addition, is it possible to prove an analogue
of Theorem \ref{tm2} in which level by level
inequivalence between strong compactness and
supercompactness holds?
Further, is it possible to prove an analogue of
Theorem \ref{tm4} for a universe with no
restrictions on its large cardinal structure?
Finally, is it possible to prove analogues of
Theorems \ref{tm2}, \ref{tm3},
and \ref{tm5} in which $\gk$
is neither the least strongly compact nor least
strong cardinal?
These are the questions with which we conclude
this paper.
%\section{Concluding Remarks}\label{s4}
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compact''.\footnote{To
see that the lifted embedding satisfies the
cover property mentioned above, suppose that
$\gg$ is an ordinal with the property that
$p \forces ``\dot x \subseteq \gg$ is
%N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$ is
such that $\card{\dot x} \le \gd^+$''.
Note that this is unambiguous, since
$\FP^0_\gd \ast \dot \FP^1_\gd$ satsifies
$\gd$-c.c. For $\ga < \gd^+$, let $\dot x_\ga$
be a term for the $\ga^{\rm th}$ member of
$\dot x$. Let
$y_\ga = \{\gb \mid \exists q \ge p[q
\forces ``\gb = \dot x_\ga$''$]\}$.
Since $N^\gd \subseteq N$ and
$\FP^0_\gd \ast \dot \FP^1_\gd$ satsifies
$\gd$-c.c., each $y_\ga$ is definable in $N$. Thus,
$\la y_\ga \mid \ga < \gd^+ \ra \in N$
and $y = \bigcup_{\ga < \gd^+} y_\ga \in N$.
It is then the case that
$p \forces ``\dot x \subseteq \check y$ and
$\check y \in N$''. Since $N \models ``\card{y} <
i(\gd)$'' and $N \subseteq
N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$,
$y$ is the desired set covering $x$ in
$N[G_0][G^0_1][G^0_2][G_1][G^1_1][G^1_2]$.}