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\title{Identity Crises and Strong Compactness III:
Woodin Cardinals
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Woodin cardinal, strongly compact
cardinal, strong cardinal, supercompact cardinal,
non-reflecting stationary set of ordinals}}
\author{Arthur W.~Apter\thanks{The
first author's research was
partially supported by
PSC-CUNY Grant 66489-00-35
and a CUNY Collaborative
Incentive Grant.} \\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu\\
\\
Grigor Sargsyan\\
Group in Logic and the Methodology of Science\\
University of California\\
Berkeley, California 94720 USA\\
http://math.berkeley.edu/$\sim$grigor\\
grigor@math.berkeley.edu}
\date{October 25, 2004\\
(revised February 25, 2005)}
\begin{document}
\maketitle
\begin{abstract}
We show that it is consistent,
relative to $n \in \go$
supercompact cardinals,
for the strongly compact
and measurable Woodin cardinals
to coincide precisely.
In particular, it is consistent
for the first $n$ strongly
compact cardinals to be
the first $n$ measurable
Woodin cardinals, with no
cardinal above the
$n^{\rm th}$ strongly compact
cardinal being measurable.
In addition, we show that it is
consistent, relative to
a proper class of supercompact
cardinals, for the strongly
compact cardinals and the
cardinals which are both
strong cardinals and Woodin
cardinals to coincide precisely.
We also show how the techniques
employed can be used
to prove additional
theorems about possible relationships
between Woodin cardinals
and strongly compact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
As is well-known,
%for $n \in \go$, the
%first $n$ strongly compact
the class of strongly compact
cardinals suffers from a severe
identity crisis.
Readers may consult \cite{Ma}, \cite{A80},
\cite{KM}, \cite{A95}, \cite{AC1},
\cite{AC2}, and \cite{A03}
for results obtained in this regard.
We mention only that none
of these articles says anything
concerning possible relationships
between strongly
compact cardinals and Woodin cardinals.
In particular, since it is known
that the first, second,
etc$.$ Woodin cardinals can't be
measurable
(and in fact,
can't even be weakly compact ---
see \cite{J}, page 384),
it is impossible for the classes
of strongly compact and Woodin
cardinals to coincide precisely.
One may still ask, however,
whether there are other possible
relationships between strongly
compact and Woodin cardinals.
In particular, is it possible
for the classes of strongly
compact and {\it measurable}
Woodin cardinals to coincide
precisely? Is it possible for
the class of strongly compact
cardinals to coincide precisely
with the class of cardinals all of
whose members are both strong
cardinals and Woodin cardinals?
%for $n \in \go$, if the first $n$ strongly compact and
%{\it measurable} Woodin cardinals can precisely coincide.
The purpose of this paper is to
provide affirmative answers to
the above questions.
Specifically, we prove the following theorems.
\begin{theorem}\label{t1}
Let $n \in \go$ be fixed but
arbitrary. Suppose
$V \models ``$ZFC + $\gk_1 < \cdots < \gk_n$
are supercompact''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk_1, \ldots,
\gk_n$ are both the first $n$ measurable
Woodin and strongly compact cardinals +
The strongly compact and measurable
Woodin cardinals coincide precisely''.
\end{theorem}
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + There is a
proper class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + There is a
proper class of strongly compact
cardinals + No strongly compact cardinal
$\gk$ is $2^\gk = \gk^+$ supercompact +
For all cardinals $\gk$, $\gk$ is
strongly compact iff $\gk$ is both a
strong cardinal and a Woodin cardinal''.
\end{theorem}
In fact, our methods are general enough
that they will yield further results
concerning the possible relationships
between strongly compact and Woodin cardinals.
We will return to this issue at
the end of Section \ref{s1}.
%different junctures throughout the course of the paper.
We conclude Section \ref{s0}
with some preliminary information.
We mention that our notation and
terminology are standard.
Exceptions to this will be
duly 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 we also have that $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an
%Easton support
iteration of length $\gk$
so that at stage $\ga$, a nontrivial forcing is done
based on the ordinal $\gd_\ga$, then we will say that
$\gd_\ga$ is in the field of $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$,
and $i_G(\dot x)$ will be the interpretation
of $\dot x$ using $G$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a regular cardinal,
$\add(\gk, 1)$ is the standard
partial ordering for adding a
single Cohen subset of $\gk$.
If $\FP$ is an arbitrary partial ordering,
$\FP$ is $\gk$-distributive if for every sequence
$\la D_\ga : \ga < \gk \ra$ of dense open
subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$ is dense open.
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
every two elements
$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
$\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is
$\gk^+$-directed closed, then $\FP$ is
$\gk$-strategically closed.
Also, if $\FP$ is
$\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$\FP$ is ${\prec}\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (again choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Suppose $\gk < \gl$ are regular cardinals.
A partial ordering $\FQ$ that will be used
in this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FQ =
\{s : s$ is a bounded subset of
$\gl$ consisting of ordinals of cofinality
$\gk$ so that for every $\ga < \gl$,
$s \cap \ga$ is non-stationary in $\ga\}$,
ordered by end-extension.
Two things which can be shown
(see \cite{Bu})
are that
$\FQ$ is $\gd$-strategically
closed for every $\gd < \gl$, and
if $G$ is $V$-generic over $\FQ$,
in $V[G]$,
%if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced.
%the bounded subsets of $\gl$ are the same as those in $V$,
%and cardinals, cofinalities, and GCH
%have been preserved.
It is also virtually immediate that $\FQ$
is $\gk$-directed closed.
We mention that we are assuming familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{K} or \cite{SRK}
for further details.
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 $\gk$ so that
$j(\gk) > |V_\gl|$ and $V_\gl \subseteq M$.
As always, $\gk$ is strong if $\gk$ is $\gl$
strong for every $\gl > \gk$.
Since the notion of Woodin cardinal
is perhaps not quite as familiar
as the large cardinals
mentioned in the preceding paragraph,
we finish this section with
the definition of this concept and
a brief discussion of some of the relevant
facts.
We refer readers to Section 26 of \cite{K},
pages 360--365
%(from which we are drawing what follows)
for additional details.
The cardinal $\gk$ is Woodin iff for any
$f : \gk \to \gk$, there is an
$\ga < \gk$ with $f''\ga \subseteq \ga$
such that there is an elementary
embedding $j : V \to M$ having
critical point $\ga$ with the additional
property that $V_{j(f)(\ga)} \subseteq M$.
By Exercise 26.10 of \cite{K}, $\gk$
must be a regular cardinal
(and in fact must actually be a
Mahlo cardinal).
Also, by Theorem 26.14 of \cite{K}
(due to Woodin), the elementary
embedding $j$ in the above definition
can be assumed to
be witnessed by an extender
${\cal E} \in V_\gk$ having the
additional property that
$j(f)(\ga) = f(\ga)$.
This has as an immediate consequence
that the Woodinness of $\gk$ is
witnessed in any model containing the
true $V_{\gk + 1}$, so Theorem 26.14
of \cite{K} actually
implies that if $\gk$ is $2^\gk$
supercompact, $\gk$ is a measurable
Woodin cardinal and $\{\gd < \gk : \gd$
is both a measurable and a Woodin
cardinal$\}$ is unbounded in $\gk$.
However, by Lemma 2.1 of
\cite{AC2} and the succeeding remarks,
if $V \models ``\gk$ is
a strong cardinal'' and
$j : V \to M$ witnesses (at least)
the $2^\gk$ supercompactness of $\gk$,
$M \models ``\gk$ is a strong cardinal''.
This means that if $\gk$ is supercompact,
$\{\gd < \gk : \gd$ is both a strong and
a Woodin cardinal$\}$ is unbounded in $\gk$.
Therefore, if $\gk$ is supercompact,
by increasing the strength of
the embedding $j$ to witness larger
degrees of supercompactness, the
same sort of reflection argument
actually shows that
$\{\gd < \gk : \gd$ is a cardinal which is
measurable, Woodin, and $\gd^{+ \go}$
strong$\}$, $\{\gd < \gk : \gd$ is a
cardinal which is $\gd^{+ 4}$ supercompact and
$\gd^{+ 17}$ strong$\}$, etc., is unbounded
in $\gk$.
\section{The Proof of Theorem \ref{t1}}\label{s1}
%when {\bf $n = 1$}}\label{s1}
\begin{pf}
We begin with a discussion of the
proof of Theorem \ref{t1} when
$n = 1$.
We will treat this as a special
case of the more general situation
of arbitrary finite $n$.
As such, we will actually be proving the
following theorem when $n = 1$.
\begin{theorem}\label{t1a}
Let $V \models ``$ZFC + $\gk$
is supercompact''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk$
is both the least strongly
compact and least measurable
Woodin cardinal''.
\end{theorem}
We will give two proofs
of Theorem \ref{t1a},
one which is valid if
the large cardinal structure
of both our ground model
and generic extension is
restricted, and one which is
valid regardless of the large
cardinal structure of either
our ground model or generic extension.
Each of these proofs will have its
advantages and disadvantages.
The advantage of the proof to be
given when there is a limited
number of large cardinals is that
it can be iterated and will yield
a proof of Theorem \ref{t1} for
arbitrary $n \in \go$.
The disadvantage is that we
must deal with a severely limited
large cardinal structure.
The proof that works regardless
of the large cardinal structure of
either our ground model or generic
extension unfortunately can't be
iterated, and thus has this as its
major disadvantage.
Keeping this in mind, we begin with
the proof when there is only
one supercompact cardinal in our
ground model with no measurable
cardinals above it.
Towards this end, let
$V \models ``$ZFC + $\gk$ is
supercompact + No cardinal
$\gl > \gk$ is measurable''.
Without loss of generality,
by doing a preliminary forcing
if necessary, we assume in
addition that $V \models {\rm GCH}$.
We present now our partial ordering $\FP$,
which will be a reverse Easton iteration
of length $\gk$ that begins
with $\FP_0 = \add(\go, 1)$.
Let $\gd_{-1} = \go$, and assume next that
$\FP_\ga$ for $0 \le \ga < \gk$ has been defined.
Take as an inductive hypothesis that
$\card{\FP_\ga} < \gk$.
Let $\gd_\ga < \gk$ be the least cardinal
greater than $\sup(\{\gd_\gb : \gb < \ga\})$
such that
$\forces_{\FP_\ga} ``\gd_\ga$ is a
measurable Woodin cardinal''.
Note that $\gd_\ga$ always exists,
%since it is well-known (see \cite{K})
since as we mentioned earlier,
any cardinal $\gd$ which is
$2^\gd$ supercompact has unboundedly
many in $\gd$ measurable Woodin
cardinals below it, which means
by our inductive hypothesis and the
results of \cite{LS} and \cite{HW} that
$\forces_{\FP_\ga} ``$There are
unboundedly many in $\gk$ measurable
Woodin cardinals below $\gk$''.
We may now define
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ adds a
non-reflecting stationary set of ordinals of
cofinality $\go$ to $\gd_\ga$''.
It is easily verified that
this definition preserves the inductive
hypothesis.
\begin{lemma}\label{l1}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is a measurable
Woodin cardinal''.
\end{lemma}
\begin{proof}
Suppose to the contrary that
$\gd < \gk$ is such that
$V^\FP \models ``\gd$ is a
measurable Woodin cardinal''.
%Let $\ga$ be the supremum of
%the cardinals at or below $\gd$
%to which non-reflecting
%stationary sets of ordinals
%were added by $\FP$, and write
Write $\FP = \FP_\ga \ast \dot \FP^\ga$,
where $\FP_\ga$ is the
portion of $\FP$ which adds non-reflecting
stationary sets of ordinals of
cofinality $\go$ to cardinals
at or below $\gd$.
By the definition of $\FP$, it
must be the case that
$\forces_{\FP_\ga} ``\dot \FP^\ga$
is $2^\gd$-strategically closed'',
from which we may infer that
$\forces_{\FP_\ga} ``\gd$ is a
measurable Woodin cardinal''.
Since any cardinal containing
a non-reflecting stationary
set of ordinals of cofinality $\go$
can't be measurable (in fact,
such a cardinal can't even be
weakly compact), it must be
the case that $\FP_\ga$
doesn't add a non-reflecting
stationary set of ordinals of
cofinality $\go$ to $\gd$.
Therefore, by the definition of
$\FP_\ga$,
it must be true that $\gd = \gd_\ga$,
from which we may immediately infer that
$\FP_\ga$ does indeed add a non-reflecting
stationary set of ordinals of
cofinality $\go$ to $\gd$.
This contradiction completes
the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
If $V^\FP \models ``$Unboundedly
in $\gk$ many cardinals $\gd < \gk$
contain non-reflecting stationary
sets of ordinals of cofinality $\go$'',
then we may infer that
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
This is since by Theorem 4.8 of
\cite{SRK} and the succeeding
remarks, if $\gd$ contains
a non-reflecting stationary set
of ordinals of cofinality $\eta < \gd$,
then no cardinal in the half-open interval
$(\eta, \gd]$ is $\gd$ strongly compact.
Thus, to prove Lemma \ref{l2}, it
suffices to show that
$V^\FP \models ``$Unboundedly
in $\gk$ many cardinals $\gd < \gk$
contain non-reflecting stationary
sets of ordinals of cofinality $\go$''.
To demonstrate this last fact,
fix a cardinal $\gd < \gk$.
As in Lemma \ref{l1}, write
$\FP = \FP_\ga \ast \dot \FP^\ga$,
where $\FP_\ga$ is the portion of
$\FP$ which adds non-reflecting
stationary sets of ordinals of
cofinality $\go$ at or below $\gd$.
Again as in Lemma \ref{l1}, it is
the case that
$\forces_{\FP_\ga} ``\dot \FP^\ga$ is
$2^\gd$-strategically closed'',
which means that
$V^\FP = V^{\FP_\ga \ast \dot \FP^\ga} \models
``\gd$ contains a non-reflecting
stationary set of ordinals of
cofinality $\go$'' iff
$V^{\FP_\ga} \models
``\gd$ contains a non-reflecting
stationary set of ordinals of
cofinality $\go$''.
This means that the proof of
Lemma \ref{l2} will be complete
once we have shown that
$\FP$ adds a non-reflecting
stationary set of ordinals
of cofinality $\go$ to
unboundedly in $\gk$ many
cardinals $\gd < \gk$.
However, for any $\gb < \gk$,
by the definition of $\FP$,
$\card{\FP_\gb} < \gk$.
Hence, by the results of
\cite{LS} and \cite{HW},
since in $V$ there are unboundedly
in $\gk$ many measurable
Woodin cardinals below $\gk$,
$\forces_{\FP_\gb} ``$There are
unboundedly in $\gk$ many
measurable Woodin cardinals in
the open interval $(\gb, \gk)$''.
This means by the
definition of $\FP$ that for any
$\gb < \gk$, there is a cardinal
at or above $\gb$ to which
$\FP$ adds a non-reflecting
stationary set of ordinals of
cofinality $\go$.
This completes the
proof of Lemma \ref{l2}.
\end{proof}
We note that the arguments given in
Lemmas \ref{l1} and \ref{l2} show that
if the cofinality of the ordinals
in the non-reflecting stationary
set of ordinals added by $\FP$
is changed to some uncountable $\gd > \go$, then
$V^\FP \models ``$No cardinal in the
open interval $(\gd, \gk)$ is either
a measurable Woodin cardinal or is
strongly compact''.
This observation will be key in the proof
of Theorem \ref{t1} for arbitrary
finite $n$.
\begin{lemma}\label{l3}
$V^\FP \models ``\gk$ is a
measurable Woodin cardinal''.
\end{lemma}
\begin{proof}
Let $j : V \to M$ be
an elementary embedding
witnessing the $2^\gk = \gk^+$
supercompactness of $\gk$.
%such that $M \models ``\gk$ isn't $2^\gk$ supercompact''.
This allows us to write
$j(\FP) = \FP \ast \dot \FQ$,
where the first ordinal in the field of
$\dot \FQ$
%is forced to add a non-reflecting stationary set of
%ordinals of cofinality $\go$
is at or above $\gk$.
If $M^{\FP} \models ``\gk$ is
a measurable Woodin cardinal'',
then since
$M^{2^\gk} \subseteq M$,
it is also true that
$V^{\FP} \models ``\gk$ is
a measurable Woodin cardinal'',
and the proof of Lemma \ref{l3}
is complete.
We suppose therefore that this
is not the case.
This means that the first ordinal
in the field of $\dot \FQ$
%is forced to add a non-reflecting stationary
%set of ordinals of cofinality $\go$
is above $\gk^{++}$.
Let $G$ be $V$-generic over $\FP$.
Using GCH, standard arguments,
as given, e.g., in the construction of
the generic object $G_1$ in the
proof of Lemma 2.4 of \cite{AC2}, allow
us to build in $V[G]$ an
$M[G]$-generic object $H$ for
$\FQ$. We may hence lift $j$ in $V[G]$ to
$j : V[G] \to M[G][H]$, which means that
$\gk$ is $2^\gk$ supercompact
in $V[G]$ (and consequently is also a measurable
Woodin cardinal in $V[G]$).
%Thus, in $V[G]$, there are unboundedly
%in $\gk$ many measurable Woodin cardinals
%below $\gk$, which contradicts Lemma \ref{l1}.
This completes the proof of
Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``\gk$ is strongly compact''.
\end{lemma}
\begin{proof}
Let $\gl \ge 2^\gk$ be a fixed but arbitrary
regular cardinal, and let
$j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness
of $\gk$ generated by a supercompact
ultrafilter over $P_\gk(\gl)$ such that
$M \models ``\gk$ isn't $\gl$ supercompact''.
By the choice of $\gl$, there is
$\U \in M$ a normal measure over $\gk$ and
$k : M \to N$ an elementary embedding
generated via the ultrapower by $\U$ such that
$N \models ``\gk$ isn't measurable''.
Let $i = k \circ j$.
Write $i(\FP) = \FP \ast \dot \FQ \ast \dot \FR$
and
$j(\FP) = \FP \ast \dot \FS \ast \dot \FT$,
where the field of $\dot \FQ$ is a
subset of the closed interval $[\gk, k(\gk)]$,
the field of $\dot \FR$ is
composed of ordinals in the open interval
$(k(\gk), k(j(\gk))$, $\dot \FS$
is a term for the forcing done at
stage $\gk$ in $M$, and $\dot \FT$
is a term for the forcing done between
stages $\gk$ and $j(\gk)$ in $M$.
By Lemma \ref{l3}, in $V$,
$\forces_{\FP} ``\gk$ is a measurable Woodin
cardinal''. Therefore, since $M^\gl \subseteq M$ and
$\gl \ge 2^\gk$, it is the case that in $M$,
$\forces_{\FP} ``\gk$ is a measurable Woodin
cardinal'' as well.
This means that $\dot \FS$ is a term for
the partial ordering adding a non-reflecting
stationary set of ordinals of cofinality $\go$
to $\gk$. Further, since
$V \models ``$No cardinal above $\gk$ is measurable'',
$M \models ``$No cardinal in the half-open interval
$(\gk, \gl]$ is measurable''. Hence, as
$\card{\FP \ast \dot \FS} = \gk$, the results of
\cite{LS} and \cite{HW}
imply that no cardinal in the half-open interval
$(\gk, \gl]$ is both measurable
and Woodin in $M^{\FP \ast \dot \FS}$.
We may thus infer that $\dot \FT$ is
a term for a partial ordering that does
only trivial forcing for cardinals in
the half-open interval $(\gk, \gl]$, since the first
cardinal in the field of
$\dot \FT$ to which a non-reflecting
stationary set of ordinals of cofinality
$\go$ is added must be both
measurable and Woodin in $M^{\FP \ast \dot \FS}$.
Work now in $N$. By the definition of $\FP$,
write $\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is
$\ha_1$-strategically closed''.
In the terminology of \cite{H2} and \cite{H3},
$\FP$ ``admits a gap at $\ha_1$'', so
by Hamkins' gap forcing results of
\cite{H2} and \cite{H3}, any cardinal
measurable in $N^\FP$ had to have been
measurable in $N$. Thus, since
$N \models ``\gk$ isn't measurable'',
$N^\FP \models ``\gk$ isn't measurable'' as well.
This means that in $N^\FP$, the partial ordering
$\FQ$ does not add a non-reflecting stationary
set of ordinals of cofinality $\go$ to $\gk$.
However, since by elementarity and the fact that
in $M$, $\forces_\FP ``\gk$ is a measurable Woodin cardinal'',
in $N$, $\forces_{k(\FP)}
``k(\gk)$ is a measurable Woodin cardinal''.
Because in $N$, $k(\FP) = {(i(\FP))}_{k(\gk)}$,
this means that
the field of $\dot \FQ$ is actually composed of
ordinals in the half-open interval
$(\gk, k(\gk)]$.
We can now use the argument given in Lemma 4 of
\cite{AC1}, Lemma 2.4 of \cite{AC2}, and Lemma 2.3 of
\cite{A03} (originally due to Magidor but
unpublished by him) to show that
$V^\FP \models ``\gk$ is $\gl$ strongly compact''.
For a complete proof, we refer readers of
this paper to one of the aforementioned articles.
However, for completeness and comprehensibility,
we provide an outline of the method of
reasoning used.
Let $G_0$ be $V$-generic over $\FP$. Since
$N \models ``\gk$ isn't measurable'',
$N$ is an ultrapower of $M$ via a
normal measure over $\gk$, and
GCH holds at $\gk$ in $V$, it is possible
once again to use the standard diagonalization
techniques employed in the proof of Lemma 2.4 of
\cite{AC2} in the construction of the
generic object $G_1$
%mentioned in the proof of Lemma \ref{l3}
to build in $V[G_0]$ an $N[G_0]$-generic
object $G_1$ for $\FQ$. Since GCH holds at
$\gl$ in $V$ and no cardinal
$\gd \in (\gk, \gl]$ is in the field of
$\dot \FT$, we can once again use the
standard diagonalization techniques to
construct in $V[G_0]$ an $M$-generic
object $H$ for the term
forcing partial ordering $\FT^*$
associated with $\dot \FT$
(which is defined with respect to
$\FP \ast \dot \FS$), transfer it using $k$
%lifted to $k : M[G_0] \to N[G_0]$
to an
$N$-generic object $G^*_2$ for $k(\FT^*)$
generated by $k '' H$,
and realize the transferred generic object
using $G_0 \ast G_1$ to obtain an
$N[G_0][G_1]$-generic object
$G_2$ for $\FR$.
$i$ then lifts to
$i : V[G_0] \to N[G_0][G_1][G_2]$, which
witnesses the $\gl$ strong compactness of
$\gk$ in $V[G_0]$.
Since $\gl$ was arbitrary, this completes
the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} - \ref{l4} complete the
proof of Theorem \ref{t1a} in the case
when there is only one supercompact
cardinal with no measurable
cardinals above it.\footnote{We
note that under the circumstances
just described,
Theorem \ref{t1} and Theorem \ref{t1a}
actually mean the same thing.
This is since by the results of
\cite{LS} and \cite{HW},
$V^\FP \models ``$No cardinal $\gl > \gk$
is both measurable and Woodin''.}
\end{pf}
\begin{pf}
We continue with our discussion of the
proof of Theorem \ref{t1a} when $n = 1$.
We turn now to the proof of this theorem
in a universe in which the large cardinal
structure can be arbitrary.
Towards this end, let
$V \models ``$ZFC + $\gk$ is supercompact''.
%As before, without loss of generality,
%by doing a preliminary forcing if necessary,
%we assume in addition that $V \models {\rm GCH}$.
We define our partial ordering $\FP$, which
will be a Magidor style iteration \cite{Ma}
of length $\gk$ of Prikry forcing that
begins with $\FP_0$ as trivial forcing.
As before, let $\gd_{-1} = \go$.
Assume next that
$\FP_\ga$ for $0 \le \ga < \gk$ has been defined.
Take as an inductive hypothesis that
$\card{\FP_\ga} < \gk$.
Let $\gd_\ga < \gk$ be the least cardinal
greater than $\sup(\{\gd_\gb : \gb < \ga\})$
such that
$\forces_{\FP_\ga} ``\gd_\ga$ is a
measurable Woodin cardinal''.
As we had earlier, $\gd_\ga$ always exists.
We may now define
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\forces_{\FP_\ga} ``\dot \FQ_\ga$
is Prikry forcing over $\gd_\ga$ defined
with respect to some normal measure
$\mu_\ga$ over $\gd_\ga$''.
Once again, it is easily verified that
this definition preserves the inductive
hypothesis.
\begin{lemma}\label{l5}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is a measurable
Woodin cardinal''.
\end{lemma}
\begin{proof}
We mimic the proof of Lemma \ref{l1}.
Suppose to the contrary that
$\gd < \gk$ is such that
$V^\FP \models ``\gd$ is a
measurable Woodin cardinal''.
Write $\FP = \FP_\ga \ast \dot \FP^\ga$,
where $\FP_\ga$ is the
portion of $\FP$ which adds
Prikry sequences to cardinals
at or below $\gd$.
By the definition of $\FP$, it
must be the case that
$\forces_{\FP_\ga} ``\dot \FP^\ga$
doesn't add any new subsets of $2^\gd$'',
from which we may infer that
$\forces_{\FP_\ga} ``\gd$ is a
measurable Woodin cardinal''.
Since any cardinal containing
a Prikry sequence must have
cofinality $\go$ and hence can't be
either measurable or Woodin,
it must be
the case that $\FP_\ga$
doesn't add a Prikry sequence to $\gd$.
Therefore, by the definition of
$\FP_\ga$,
it must be true that $\gd = \gd_\ga$,
from which we may immediately infer that
$\FP_\ga$ does indeed add a
Prikry sequence to $\gd$.
This contradiction completes
the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l2},
there are unboundedly in $\gk$ many
cardinals $\gd < \gk$ which are in
the field of $\FP$, which in this
case means that in $V^\FP$,
there are unboundedly in $\gk$
many cardinals $\gd < \gk$
which contain Prikry sequences.
However, since by Theorem 11.1
of \cite{CFM}, adding a Prikry
sequence also adds a
non-reflecting stationary set
of ordinals of cofinality $\go$,
again as in the proof of Lemma \ref{l2},
%weak square sequence,
%and since, as can be inferred from
%the proof of Theorem 5.4 of \cite{CFM},
%a weak square sequence can't
%exist above a strongly compact cardinal,
in $V^\FP$, there are no strongly
compact cardinals below $\gk$.
This completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``\gk$ is a
measurable Woodin cardinal''.
\end{lemma}
\begin{proof}
We begin by mimicing the
proof of Lemma \ref{l3}.
Let $j : V \to M$ be
an elementary embedding
witnessing the $2^\gk$
supercompactness of $\gk$.
This allows us to write
$j(\FP) = \FP \ast \dot \FQ$,
where the first ordinal
in the field of $\dot \FQ$ is
at or above $\gk$.
If $M^{\FP} \models ``\gk$ is
a measurable Woodin cardinal'',
then since
$M^{2^\gk} \subseteq M$,
it is also true that
$V^{\FP} \models ``\gk$ is
a measurable Woodin cardinal'',
and the proof of Lemma \ref{l7}
is complete.
We suppose therefore that this
is not the case.
This means that the first ordinal
in the field of $\dot \FQ$
is above $2^\gk$.
Let now $|\ \ \ |$ be the distance
function of \cite{Ma}.
Define a term $\dot \U$ in $V$ by
$p \forces ``\dot A \in \dot \U$'' iff
$p \forces ``\dot A \subseteq
{(P_\gk(2^\gk))}^{V^\FP}$'' and there is
$q \in j(\FP)$ such that
$q \ge j(p)$,
%($q$ extends $p$),
$|j(p) - q| = 0$,
$j(p) \rest \gk = q \rest \gk = p$, and
$q \forces ``\la j(\ga) : \ga < 2^\gk \ra
\in j(\dot A)$''.
By the same proof as in the Lemma of \cite{A00},
$\dot \U$ is a well-defined term for a
supercompact ultrafilter over
$P_\gk(2^\gk)$ in $V^\FP$. Thus, since
$V^\FP \models ``\gk$ is $2^\gk$ supercompact'',
%not only does
$V^\FP \models ``\gk$ is a measurable
Woodin cardinal''.
%but there are unboundedly in $\gk$ many measurable
%Woodin cardinals below $\gk$. This, however,
%contradicts Lemma \ref{l5}.
This completes the proof of Lemma \ref{l7}.
\end{proof}
Magidor's methods of \cite{Ma} show that
$V^\FP \models ``\gk$ is strongly compact''.
(This is since the work of \cite{Ma} demonstrates
that if any set of measurable cardinals
below a strongly compact cardinal $\gl$
is destroyed via Magidor's notion of
iterated Prikry forcing, then the strong
compactness of $\gl$ is preserved.)
This observation, together with
Lemmas \ref{l5} - \ref{l7}, complete the
proof of Theorem \ref{t1a} when $n = 1$
and the large cardinal structure of
both the ground model and the generic
extension is arbitrary.
\end{pf}
We note that the proof just presented
requires no GCH assumptions on our
ground model.
This is in contrast to the earlier
proof of Theorem \ref{t1a} given
for a universe containing a restricted
number of large cardinals, which
requires instances of GCH in order to
show that strong compactness is preserved.
\begin{pf}
%We conclude our discussion of the proof of
%Theorem \ref{t1} by considering the case
%of arbitrary finite $n$.
We are now ready to give the proof of
Theorem \ref{t1} for arbitrary finite $n$.
Towards this end, fix $n \in \go$, $n > 1$.
Suppose that
$V \models ``$ZFC + $\gk_1 < \cdots <
\gk_n$ are supercompact +
No cardinal $\gl > \gk_n$ is measurable''.
Without loss of generality, as in the proof
given in \cite{AC1} for arbitrary
finite $n$ of the Theorem 1 of that paper,
we also assume that
$V \models ``$Each $\gk_i$ for $i = 1,
\ldots, n$ is Laver indestructible \cite{L} +
$2^{\gk_i} = \gk^+_i$ for $i = 1, \ldots, n$''.
Take $\gk_0 = \go$.
For $i = 1, \ldots, n$, we define a
partial ordering $\FP_i$ in a manner
analogous to the definition of the
partial ordering $\FP$ given at
the beginning of this section.
$\FP_i$ will be a reverse Easton iteration
of length $\gk_i$ that begins with
$\FP_{i, 0} = \add(\gk_{i - 1}, 1)$ and
$\gd_{i, -1} = \gk_{i - 1}$. Then, as earlier,
assume that
$\FP_{i, \ga}$ for $0 \le \ga < \gk_i$ has been defined.
Take as an inductive hypothesis that
$\card{\FP_{i, \ga}} < \gk_i$.
Let $\gd_{i, \ga} < \gk_i$ be the least cardinal
greater than $\sup(\{\gd_{i, \gb} : \gb < \ga\})$
such that
$\forces_{\FP_{i, \ga}} ``\gd_{i, \ga}$ is a
measurable Woodin cardinal''.
As before, $\gd_{i, \ga}$ always exists.
We may now define
$\FP_{i, \ga + 1} = \FP_{i, \ga} \ast \dot \FQ_{i, \ga}$, where
$\forces_{\FP_{i, \ga}} ``\dot \FQ_{i, \ga}$ adds a
non-reflecting stationary set of ordinals of
cofinality $\gk_{i - 1}$ to $\gd_{i, \ga}$''.
It is easily verified that
this definition preserves the inductive
hypothesis.
Our partial ordering $\FP$ used in the
proof of Theorem \ref{t1} for arbitrary
finite $n$ is then
the product partial ordering
$\FP_1 \times \cdots \times \FP_n$.
We do now a ``downwards induction''
to show that $V^\FP$ is as desired.
By the arguments given in the proofs
of Lemmas \ref{l1} - \ref{l4},
which remain valid regardless of the
cofinality of the ordinals in the
non-reflecting stationary sets added
by the appropriate forcing,
$V^{\FP_n} \models ``\gk_n$ is both
a strongly compact and a Woodin cardinal,
and no cardinal in the open interval
$(\gk_{n - 1}, \gk_n)$ is either
strongly compact or both measurable
and Woodin''.
Since by its definition,
$\card{\FP_1 \times \cdots \times \FP_{n - 1}}
< \gk_n$, by the results of \cite{LS}
and \cite{HW},
these facts remain true in
$V^{\FP_n \times \FP_{n - 1} \times \cdots
\times \FP_1} = V^\FP$.
Suppose now by induction that $1 \le i < n$ and that
$V^{\FP_n \times \cdots \times \FP_{i + 1}} \models
``\gk_{i + 1}, \ldots, \gk_n$ are both strongly
compact and Woodin cardinals, and no cardinal
in the open intervals
$(\gk_i, \gk_{i + 1}), \ldots, (\gk_{n - 1}, \gk_n)$
is either strongly compact or both measurable
and Woodin''. Since by its definition,
$\FP_n \times \cdots \times \FP_{i + 1}$ is
$\gk_i$-directed closed, by indestructibility,
$V^{\FP_n \times \cdots \times \FP_{i + 1}} \models
``\gk_i$ is supercompact''.
Also, the fact
$\FP_n \times \cdots \times \FP_{i + 1}$ is
$\gk_i$-directed closed implies that
$V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$
have the same bounded subsets of $\gk_i$.
However, the definition of
$\FP_i$ is such that it
retains in $V^{\FP_n \times \cdots \times \FP_{i + 1}}$
the forcing properties it
had in $V$. This means that
the arguments of Lemmas \ref{l1} - \ref{l3}
remain valid and show that
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i} \models
``\gk_i$ is a measurable Woodin cardinal, and no
cardinal in the open interval
$(\gk_{i - 1}, \gk_i)$ is either strongly compact or
both measurable and Woodin''.
Further, for any $j$ such that
$1 \le j \le n$, by writing
$\FP_j = \FP_{j, 0} \ast \dot \FQ$, since
$\FP_{j, 0}$ adds a Cohen subset of $\gk_{j - 1}$,
$\forces_{\FP_{j, 0}} ``\dot \FQ$ is
$\gk^+_{j - 1}$-strategically closed'', and
$\card{\FP_j} = \gk_j$, it is easy to
see that forcing with $\FP_j$ preserves
GCH at both $\gk_{j - 1}$ and $\gk_j$.
It is then inductively the case that
$V^{\FP_n \times \cdots \times \FP_{i + 1}} \models
``$GCH holds at $\gk_{i + 1}$'',
so since in addition,
$V^{\FP_n \times \cdots \times \FP_{i + 1}} \models
``$There are no
measurable Woodin cardinals in the open interval
$(\gk_i, \gk_{i + 1})$'', as above,
the argument given in Lemma \ref{l4} remains
valid and shows that
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i} \models
``\gk_i$ is $\gk_{i + 1}$ strongly
compact''.\footnote{To expand on this
further, suppose that
$j : V^{\FP_n \times \cdots \times \FP_{i + 1}}
\to M$ is in analogy to Lemma \ref{l4}, i.e.,
$j$ is an elementary embedding witnessing
the $\gk_{i + 1}$ supercompactness of $\gk_i$
such that
$M \models ``\gk_i$ isn't
$\gk_{i + 1}$ supercompact''.
As before, suppose that
$k : M \to N$ is an elementary
embedding generated by a normal
measure over $\gk_i$ such that
$N \models ``\gk_i$ isn't measurable''.
The arguments of Lemma \ref{l4}
then proceed
%exactly as earlier,
almost virtually without change.
We do explicitly note, however, that
the argument given
in Lemma \ref{l4} requires that for the
version of $\dot \FT$ appropriate to the
current situation, no ordinal in the
half-open interval $(\gk_i, \gk_{i + 1}]$
is an element of the field of $\dot \FT$.
This is true if in $M$, there are
no measurable Woodin cardinals in the
half-open interval $(\gk_i, \gk_{i + 1}]$.
By the $\gk_{i + 1}$ closure
of $M$, there are no measurable Woodin cardinals
in $M$
in the open interval $(\gk_i, \gk_{i + 1})$.
However, since
$M \models ``\gk_i$ isn't
$\gk_{i + 1}$ supercompact'',
$M \models ``\gk_{i + 1}$
isn't measurable''. This is since otherwise,
$M \models ``\gk_i$ is $\gd$ supercompact for
every $\gd < \gk_{i + 1}$ and
$\gk_{i + 1}$ is measurable'', so
$M \models ``\gk_i$ is $\gk_{i + 1}$
supercompact'', a contradiction.
Thus, in $M$, there are no measurable Woodin
cardinals in the half-open interval
$(\gk_i, \gk_{i + 1}]$.}
Therefore, since
$\card{\FP_i} < \gk_{i + 1}$ in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1}}$,
by the results of \cite{LS},
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i} \models
``\gk_{i + 1}$ is strongly compact''.
Hence, by a theorem of DiPrisco \cite{DH}, as
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i} \models
``\gk_i$ is $\gk_{i + 1}$ strongly compact and
$\gk_{i + 1}$ is strongly compact'',
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i} \models
``\gk_i$ is strongly compact''.
Thus, since
$\card{\FP_{i - 1} \times \cdots \times \FP_1} < \gk_i$
in both $V$ and
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i}$,
the results of \cite{LS} and \cite{HW}
allow us to infer that
$V^{\FP_n \times \cdots \times \FP_{i + 1} \times \FP_i
\times \FP_{i - 1} \times \cdots \times \FP_1}
= V^\FP \models
``\gk_i, \ldots, \gk_n$ are both strongly
compact and Woodin cardinals, and no cardinal
in the open intervals
$(\gk_{i - 1}, \gk_i), \ldots, (\gk_{n - 1}, \gk_n)$
is either strongly compact or both measurable
and Woodin''.
From this, we may now infer that
$V^\FP \models ``\gk_1, \ldots, \gk_n$
are both the first $n$ strongly compact
and measurable Woodin cardinals''. Since
$\card{\FP} = \gk_n$ and
$V \models ``$No cardinal $\gl > \gk_n$
is measurable'', by the results of \cite{LS},
$V^\FP \models ``$No cardinal $\gl > \gk_n$
is measurable'' as well. We therefore now know that
$V^\FP \models ``$The strongly compact and
measurable Woodin cardinals coincide precisely''.
This completes our discussion of the proof of
Theorem \ref{t1} for arbitrary finite $n$.
\end{pf}
We conclude Section \ref{s1}
by making several remarks.
The first is that the techniques
of this section are quite general
and allow for a diverse number
of different theorems in the
spirit of Theorems \ref{t1}
and \ref{t1a}. For instance,
if we wished to prove versions of
Theorems \ref{t1} and \ref{t1a}
with the property ``measurable
Woodin cardinal'' replaced with
the property ``measurable Woodin
cardinal $\gk$ which is also $\gk^{+ \go}$
strong''\footnote{To be more
explicit, we wish as before to
construct three models.
In the first two,
the least strongly compact
cardinal $\gk$ is also the least
cardinal which has the
additional properties of being
measurable, Woodin, and $\gk^{+ \go}$
strong. In the last model, for some
$n \in \go$, there are exactly
$n$ strongly compact cardinals, which
are precisely those cardinals $\gk$ which are
also measurable, Woodin, and $\gk^{+ \go}$
strong.},
then this is easily accomplished
by a slight modification of the definitions
of our partial orderings.
Instead of destroying cardinals which
are forced to be both measurable and Woodin,
we destroy cardinals $\gd$ which
are forced to be
Woodin and $\gd^{+ \go}$ strong.
Simple reworkings of our arguments then
yield the desired results.
In particular, when proving the
appropriate analogues of Lemmas
\ref{l3} and \ref{l7}, we
choose supercompactness embeddings
into transitive inner models $M$
which are at least $2^{\gk^{+ \go}}$ closed,
and then essentially follow
almost verbatim the remainder
of the arguments presented in each of
the relevant lemmas.
What is key to this analysis is that
the property we have given in the
preceding paragraph can be determined
``locally'' in two senses.
One is that if the property is
forced by the partial ordering
to hold at a cardinal below
the length of the iteration, then
an initial segment
of the partial ordering
forces this property to hold as well
(which is critical to the proofs of
Lemmas \ref{l1} and \ref{l5}).
The other is that if
the particular property is witnessed
in a sufficiently closed transitive
inner model $M$, then it must be
true in $V$ as well
(which is critical to
the proofs of Lemmas \ref{l3} and \ref{l7}).
Since this sort of ``localness'' is
witnessed by many other potential properties
(another example would be given by
a cardinal $\gd$ being both
$\gd^{+ 4}$ supercompact and
$\gd^{+ 17}$ strong), there are
a number of different versions of
Theorems \ref{t1} and \ref{t1a} which
consequently can be proven.
Another interesting aspect of
the methods of this section is
that they can be used when
Hamkins' gap forcing results of
\cite{H2} and \cite{H3} (and
the generalizations thereof given
in \cite{H4}) fail.
For instance, since Hamkins' techniques
of \cite{H2}, \cite{H3}, and \cite{H4}
do not tell us that a gap forcing
as defined in any of these papers
creates no new cardinals $\gd$ which are
$\gd^{+ \go}$ strong, we are forced to
rely on the ideas used in the proof of
Lemma \ref{l1} if we want to change
the property from ``measurable Woodin cardinal''
to something on the order of
``measurable Woodin cardinal $\gd$ which
is also $\gd^{+ \go}$ strong''.
This is critical in showing that all
cardinals $\gd$ below the
length of the iteration which are
measurable, Woodin, and $\gd^{+ \go}$
strong are destroyed.
(Since Hamkins' gap forcing results
do not hold for Prikry iterations,
we must use the ideas of this paper,
specifically those given in Lemma \ref{l5},
when proving either Theorem
\ref{t1a} as originally stated
or any generalizations thereof
when there are no restrictions
on the large cardinal structure of
the universe.)
We will comment on this further towards
the end of the paper.
Finally, for the same reasons as
given in \cite{A95} and \cite{AC1},
we are restricted to proving
Theorem \ref{t1} only for finite
values of $n$.
A more detailed discussion of why this
is so may be found at the end
of \cite{A95}.
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + There is a proper
class of supercompact cardinals''.
Take ${\cal K}$ as the class of
supercompact cardinals in $V$.
As in \cite{AC2}, by ``cutting off'' the
universe if necessary, we assume in addition
without loss of generality that there are
no inaccessible limits of
supercompact cardinals.
Further, as in \cite{AC2}, we also assume that
for $\FR = \add(\go, 1) \ast \dot \FR'$,
$V$ has been generically extended to a model
$V_1 = V^\FR$ in which ${\cal K}$
is the class of supercompact
cardinals\footnote{The fact that
${\cal K}$ remains the class of supercompact
cardinals in $V_1$ follows from the
gap forcing results of \cite{H2} and \cite{H3}
and the fact that all $V$-supercompact
cardinals remain supercompact in $V_1$.
This is in direct analogy to the argument
given in \cite{AC2}.}
such that
$V_1 \models ``$GCH + Every supercompact cardinal
$\gk$ has its supercompactness indestructible
under $\gk$-directed closed set or class
partial orderings preserving GCH''.
Work now in $V_1$.
For each supercompact cardinal $\gk$, let
$\gg_\gk$ be the successor of the
supremum of the supercompact cardinals
below $\gk$ (with $\gg_{\gk_0} = \go$
for $\gk_0$ the least supercompact
cardinal). By our assumptions on $V$
and their corresponding implications
for $V_1$,
$\gg_\gk < \gk$.
This allows us to define the partial
ordering $\FP^\gk$ as the reverse Easton
iteration of length $\gk$ which begins
by adding a Cohen subset of $\gg_\gk$
and then adds a non-reflecting stationary
set of ordinals of cofinality $\gg_\gk$
to each cardinal in the open interval
$(\gg_\gk, \gk)$ which is in $V$
(the original ground model) both
a strong cardinal and a Woodin cardinal.
The partial ordering $\FP$ which is
used to construct the model witnessing
the conclusions of Theorem \ref{t2} is
then defined as the Easton support product
$\prod_{\gk \in {\cal K}} \FP^\gk$.
Standard arguments show that
$V^\FP_1 \models ``$ZFC + GCH''.
We establish now some basic properties
of each partial ordering $\FP^\gk$.
We take as definitions that
$\FP^{> \gk} = \prod_{\gl \in
{\cal K}, \gl > \gk} \FP^\gl$ and
$\FP^{< \gk} = \prod_{\gl \in
{\cal K}, \gl < \gk} \FP^\gl$,
and as a notational convention that
in Lemmas \ref{l8} - \ref{l11}
that follow, $\gk$ always denotes
a $V$- or $V_1$-supercompact cardinal.
Note that once again, standard arguments
show that for any $\gk \in {\cal K}$,
$V^{\FP^{> \gk}}_1 \models
``$ZFC + GCH'' and
$V^{\FP^{> \gk} \times \FP^\gk}_1 \models
``$ZFC + GCH''.
\begin{lemma}\label{l8}
%$V^{\FP^{> \gk} \times \FP_\gk}_1
$V^\FP_1
\models ``$No cardinal in
the open interval $(\gg_\gk, \gk)$ is
both a strong cardinal and a Woodin cardinal''.
\end{lemma}
\begin{proof}
Write $\FR \ast \dot \FP =
\FR \ast (\dot \FP^{> \gk} \times \dot \FP^\gk
\times \dot \FP^{< \gk})$
in the form
$\add(\go, 1) \ast (\dot \FR'
\ast (\dot \FP^{> \gk} \times \dot \FP^\gk
\times \dot \FP^{< \gk}))
= \FS_0 \ast \dot \FS_1$.
Since $\card{\FS_0} = \go$,
$\FS_0$ is nontrivial, and
$\forces_{\FS_0} ``\dot \FS_1$ is
$\ha_1$-strategically closed'',
by the results of \cite{H2} and \cite{H3},
any cardinal in
$V^{\FS_0 \ast \dot \FS_1} =
V^\FP_1$ which is
both a strong cardinal and a Woodin
cardinal had to have been both a
strong cardinal and a Woodin cardinal in $V$.
Further, since by its definition,
$\FP^{> \gk}$ is $\gk$-directed closed in $V_1$,
the partial ordering $\FP^\gk$ retains
the same forcing properties in
$V^{\FP^{> \gk}}_1$ as it had in $V_1$.
This means that in
$V^{\FP^{> \gk} \times \FP^\gk}_1$,
%= V^{\FS_0 \ast \dot \FS_1}$,
no cardinal
lying in the open interval $(\gg_\gk, \gk)$
which was in $V$ both
a strong cardinal and a Woodin cardinal
remains both a strong cardinal and
a Woodin cardinal,
since each such cardinal contains a
non-reflecting stationary set of
ordinals of cofinality $\gg_\gk$
and hence is no longer weakly compact.
As $\card{\FP^{< \gk}} < \gg_\gk$ in both
$V_1$ and $V^{\FP^{> \gk} \times \FP^\gk}_1$,
by the results of \cite{HW},
no cardinal
lying in the open interval $(\gg_\gk, \gk)$
which was in $V$ both
a strong cardinal and a Woodin cardinal
remains both a strong cardinal and
a Woodin cardinal in
$V^{\FP^{> \gk} \times \FP^\gk \times
\FP^{< \gk}}_1 = V^\FP_1$.
Thus, there can be no cardinals in the
open interval $(\gg_\gk, \gk)$ which are in
$V^\FP_1$ both a strong cardinal and a
Woodin cardinal.
This completes the proof of Lemma \ref{l8}.
\end{proof}
\begin{lemma}\label{l9}
$V^\FP_1 \models ``\gk$ isn't
$2^\gk = \gk^+$ supercompact''.
\end{lemma}
\begin{proof}
By Lemma \ref{l8}, in $V^\FP_1$,
no cardinal in the open interval
$(\gg_\gk, \gk)$ is both a strong
cardinal and a Woodin cardinal.
However, if $\gk$ were $2^\gk$
supercompact, then there would
have to be unboundedly many in
$\gk$ cardinals below $\gk$
which are both Woodin cardinals
and strong cardinals.
This completes the proof of Lemma \ref{l9}.
\end{proof}
\begin{lemma}\label{l10}
$V^\FP_1 \models ``$No cardinal in the
open interval $(\gg_\gk, \gk)$ is strongly compact''.
\end{lemma}
\begin{proof}
As we remarked in the proof of Lemma \ref{l8},
$\FP^\gk$ retains in $V^{\FP^{> \gk}}_1$ the
forcing properties it had in $V_1$.
Further, since $\gk$ is supercompact in $V$,
there are unboundedly many in $\gk$ cardinals
in the open interval $(\gg_\gk, \gk)$
which are both strong cardinals and Woodin cardinals
in $V$.
Therefore, by the definition of $\FP^\gk$,
$V^{\FP^{> \gk} \times \FP^\gk}_1 \models
``$Unboundedly many in $\gk$ cardinals in the open
interval $(\gg_\gk, \gk)$ contain non-reflecting
stationary sets of ordinals of cofinality
$\gg_\gk$''.
Consequently, as in the proof of Lemma \ref{l2}
and the succeeding remarks,
$V^{\FP^{> \gk} \times \FP^\gk}_1 \models
``$No cardinal in the open interval
$(\gg_\gk, \gk)$ is strongly compact''.
Since $\card{\FP^{< \gk}} < \gg_\gk$,
by the results of \cite{LS},
$V^{\FP^{> \gk} \times \FP^\gk \times \FP^{< \gk}}_1
= V^\FP_1 \models ``$No cardinal in the open interval
$(\gg_\gk, \gk)$ is strongly compact''.
This completes the proof of Lemma \ref{l10}.
\end{proof}
\begin{lemma}\label{l11}
$V^\FP_1 \models ``\gk$ is
a Woodin cardinal''.
\end{lemma}
\begin{proof}
Since $\FP^{> \gk}$ is
$\gk$-directed closed in $V_1$
and preserves GCH,
by the indestructibility of
$\gk$ in $V_1$ under $\gk$-directed closed
partial orderings preserving
GCH, $V^{\FP^{> \gk}}_1 \models
``\gk$ is supercompact and hence is
a Woodin cardinal''.
Consequently, to establish Lemma \ref{l11},
we show that
$V^{\FP^{> \gk} \times \FP^\gk}_1 \models
``\gk$ is a Woodin cardinal''.
This will suffice, since
$\card{\FP^{< \gk}} < \gk$ in both
$V_1$ and
$V^{\FP^{> \gk} \times \FP^\gk}_1$,
so by the results of \cite{HW},
if
$V^{\FP^{> \gk} \times \FP^\gk}_1 \models
``\gk$ is a Woodin cardinal'', then
$V^{\FP^{> \gk} \times \FP^\gk \times \FP^{< \gk}}_1
= V^\FP_1 \models ``\gk$ is a Woodin cardinal''
as well.
Towards this end,
we begin by defining
for notational simplicity
$V_2 = V^{\FP^{> \gk}}_1$ and
$\FQ = \FP^\gk$.
Take $V_2$ as our ground model.
Let $f_1 \in V^\FQ_2$
be a function
such that $f_1 : \gk \to \gk$.
As $\FQ$ retains in $V_2$
the same forcing properties
it had in $V_1$, $\FQ$ is in $V_2$
a reverse Easton
iteration of length $\gk$ in
$V_2$ and hence is $\gk$-c.c.
This means the function
$f_2 : \gk \to \gk$ defined in $V_2$ as
$f_2(\ga) = \max(\ga, \sup(\{\gb : \exists p \in \FQ
[p \forces ``f_1(\ga) = \gb$''$]\})) + 1$
is well-defined and is such that for all $\ga < \gk$,
$V^\FQ_2 \models ``f_1(\ga) < f_2(\ga)$''.
Let $f : \gk \to \gk$ be given by
$f(\ga) = \gb$ where $\gb$ is
the least $V_2$-strong cardinal above $f_2(\ga)$.
Note that $\gb$ is not a Woodin cardinal,
since if it were, then by Proposition 26.13
of \cite{K}, $A = \{\gg < \gb : \gg$ is
$\gd$ strong for every $\gd < \gb\}$ is
a stationary subset of $\gb$.
However, by the proof of Lemma 2.1 of
\cite{AC2}, every $\gg \in A$ must
be a strong cardinal,
since $V_2 \models ``\gg$ is
$\gd$ strong for every $\gd < \gb$
and $\gb$ is a strong cardinal''.
%meaning there must be a strong
This means there must be a $V_2$-strong
cardinal in the open interval $(f_2(\ga), \gb)$,
which contradicts
the fact that $\gb$ is the least
$V_2$-strong cardinal above $f_2(\ga)$.
Since $\gk$ is a Woodin cardinal in $V_2$,
by Theorem 26.14 of \cite{K}, we can
let $\gl < \gk$ be the least cardinal such that
$f''\gl \subseteq \gl$ and there is an
extender ${\cal E} \in V_\gk$ and an
elementary embedding $j : V_2 \to M$
generated by ${\cal E}$
having critical point $\gl$ such that
$V_{j(f)(\gl)} \subseteq M$ and
$j(f)(\gl) = f(\gl)$.
It is then the case that $\gl$ is
not a Woodin cardinal in $V_2$.
This is because if $\gl$ were
a Woodin cardinal in $V_2$, then
since $f \rest \gl : \gl \to \gl$,
again by Theorem 26.14 of \cite{K},
there is some $\gb < \gl$ such that
$f''\gb \subseteq \gb$ and there is
an extender ${\cal E}' \in V_\gl$ and
an elementary embedding $i : V_2 \to N$
generated by ${\cal E}'$
having critical point $\gb$ such that
$V_{i(f)(\gb)} \subseteq N$ and
$i(f)(\gb) = f(\gb)$.
As $\gl < \gk$, ${\cal E}' \in V_\gk$.
This, however, contradicts the
minimality of $\gl$.
Let $j : V_2 \to M$ be an
elementary embedding
generated by an extender ${\cal E}
\in V_\gk$ having
critical point $\gl$ such that
$V_{j(f)(\gl)} \subseteq M$ and
$j(f)(\gl) = f(\gl)$.
Let $\gl' = f(\gl)$, and let
$\FQ^*$ be the portion of
$\FQ$ defined between $\gl$
and $\gl'$.
We present now an argument using ideas
of Woodin and Gitik-Shelah
\cite{GS}, found in the proofs of
Lemma 4.2 of \cite{A03a} and Theorem
4.10 of \cite{H00}, which shows that
$j$ lifts to
$j : V_2^{\FQ_\gl \ast \dot \FQ^*}
\to M^{j(\FQ_\gl \ast \dot \FQ^*)}$
in a way such that
${(V_{j(f)(\gl)})}^{V_2^{\FQ_\gl \ast \dot \FQ^*}}
\subseteq M^{j(\FQ_\gl \ast \dot \FQ^*)}$.
To begin, since
$V_2 \models ``\gl$ isn't a Woodin cardinal'',
a fact which is absolute between transitive
models containing the same $V_{\gl + 1}$,
the facts $\gl' > \gl$ and $V_{\gl'}
\subseteq M$ imply that
$M \models ``\gl$ isn't a Woodin cardinal'' as well.
We thus know that both
$\gl \not\in {\rm field}(\dot \FQ^*)$ and
$\gl \not\in {\rm field}(j(\FQ))$.
Further, because
$V_2 \models ``$Every member of the
range of $f$ is not a Woodin cardinal'',
by elementarity,
$M \models ``$Every member of the
range of $j(f)$ is not a Woodin cardinal''.
In particular, since $j(f)(\gl) = f(\gl) = \gl'$,
$M \models ``\gl'$ isn't a Woodin cardinal''.
We therefore also know that both
$\gl' \not\in {\rm field}(\dot \FQ^*)$ and
$\gl' \not\in {\rm field}(j(\FQ))$.
Note now that for any cardinal
$\gd \in (\gl, \gl')$,
$V_2 \models ``\gd$ is a strong cardinal'' iff
$M \models ``\gd$ is a strong cardinal''.
To see this, suppose first that
$V_2 \models ``\gd$ is a strong cardinal''.
Since $V_2 \models ``$Every member of
the range of $f$ is a strong cardinal'',
by elementarity,
$M \models ``$Every member of the range of
$j(f)$ is a strong cardinal''. Consequently, as
$j(f)(\gl) = f(\gl) = \gl'$,
$M \models ``\gl'$ is a strong cardinal''.
Thus, as $V_{\gl'} \subseteq M$ and
$V_2 \models ``\gd$ is a strong cardinal'',
$M \models ``\gd$ is $\gg$ strong for every
$\gg < \gl'$ and $\gl'$ is a strong cardinal'',
so as before,
$M \models ``\gd$ is a strong cardinal''.
Then, if
$M \models ``\gd$ is a strong cardinal'',
since $V_{\gl'} \subseteq M$,
$V_2 \models ``\gd$ is $\gg$ strong for every
$\gg < \gl'$ and $\gl'$ is a strong cardinal'',
so again,
$V_2 \models ``\gd$ is a strong cardinal''.
Hence, as $V_{\gl'} \subseteq M$
implies that the Woodin cardinals in
the open interval $(\gl, \gl')$
are precisely the same in both
$V_2$ and $M$,
we therefore have that the
cardinals which are both strong cardinals
and Woodin cardinals and which lie
in the open interval $(\gl, \gl')$
are precisely the same in both $V_2$ and $M$.
Putting the work of
the preceding two paragraphs together,
we can now infer that
$j(\FQ_\gl) = \FQ_\gl \ast
\dot \FQ^* \ast \dot \FR$,
and that the first ordinal in the
field of $\dot \FR$ is above $\gl'$.
Since we may assume
by the regularity of $\gl'$
that $M^\gl \subseteq M$,
this means that if
$G$ is $V_2$-generic over $\FQ_\gl$ and
$H$ is $V_2[G]$-generic over $\FQ^*$,
$\FR$ is ${\prec} \gl^+$-strategically closed in both
$V_2[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl'$-strategically closed in
$M[G][H]$.
We may assume that
$M = \{j(h)(a) : a \in {[\gl']}^{< \omega}$,
$h \in V_2$, and $\dom(h) = {[\gl]}^{|a|}\}$.
Therefore, as in both \cite{H00} and
\cite{A03a}, by using a suitable coding
which allows us to identify finite
subsets of $\gl'$ with elements of $\gl'$,
the definition of $M$ allows us to find some
$\ga < \gl'$ and function $g$ so that
$\dot \FQ^* = j(g)(\ga)$. Let
%(assuming that $\dot \FQ$ has been chosen reasonably). Let
$N = \{i_{G \ast H}(\dot z) : \dot z =
j(h)(\gl, \ga, \gl')$ for some function $h \in V_2\}$.
It is easy to verify that
$N \prec M[G][H]$, that $N$ is closed under
$\gl$ sequences in $V_2[G][H]$, and that
$\gl$, $\ga$, $\gl'$, $\FQ^*$, and $\FR$ are all
elements of $N$.
Further, since
$\FR$ is $j(\gl)$-c.c$.$ in $M[G][H]$ and
by GCH at $\gl$ in $V_2$,
there are only $2^\gl = \gl^+$ many functions
$h : {[\gl]}^3 \to V_\gl$ in $V_2$, there are at most
$\gl^+$ many dense open subsets of $\FR$ in $N$.
Therefore, since $\FR$ is
${\prec} \gl^+$-strategically closed in both
$M[G][H]$ and $V_2[G][H]$, we can
build $H'$ in $V_2[G][H]$
%an $N$-generic object $H'$ over $\FR$
as follows. Let
$\la D_\gs : \gs < \gl^+ \ra$ enumerate in
$V_2[G][H]$ the dense open subsets of
$\FR$ present in $N$ so that
every dense open subset of $\FR$
occurring in $N$ appears at an
odd stage at least once in the
enumeration.
If $\gs$ is an odd ordinal,
$\gs = \tau + 1$ for some $\tau$.
Player I picks
$p_\gs \in D_\gs$ extending $q_\tau$
%$\sup(\la q_\gb : \gb < \gs \ra)$
(initially, $q_{0}$ is the empty condition),
and player II responds by picking
$q_\gs \ge p_\gs$
according to a fixed strategy ${\cal S}$
(so $q_\gs \in D_\gs$).
If $\gs$ is a limit ordinal, player II
uses ${\cal S}$ to pick
$q_\gs$ extending each
$q \in \la q_\gb : \gb < \gs \ra$.
By the ${\prec} \gl^+$-strategic closure of
$\FR$ in $V_2[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs : \gs < \gl^+ \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gl^+$.
Let
$H' = \{p \in \FR : \exists \gs <
\gl^+ [q_\gs \ge p]\}$.
%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,
$\dot D = j(h_0)(\gl, \gl_1, \ldots, \gl_n)$
for some fixed function $h_0 \in V_2$ and
$\gl < \gl_1 < \cdots < \gl_n < \gl'$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(h_0)(\gl, \ga_1, \ldots, \ga_n))$, where
$\gl < \ga_1 < \cdots < \ga_n < \gl'$ is
such that $j(h_0)(\gl, \ga_1, \ldots, \ga_n)$
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 $\gl'$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is definable without
the parameters $\gl_1, \ldots, \gl_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N$.
Thus, since $H'$
meets every dense open subset
of $\FR$ present in $N$,
%is $N$-generic over $\FR$,
$H' \cap i_{G \ast H}(\ov D) \neq \emptyset$,
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
$V_2[G][H]$, $j$ lifts to
$j : V_2[G] \to M[G][H][H']$.
It remains to lift $j$ through the forcing $\FQ^*$
while working in $V_2[G][H]$.
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 $\vec D = \la D_\gs : \gs \in {[\gl]}^{|a|} \ra$
is a function.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ^*$.
Since $\FQ^*$ is $\gl$-distributive, it follows that
$D' = \bigcap_{\gs \in {[\gl]}^{|a|}} D_\gs$ is
also a dense open subset of $\FQ^*$. As
$j(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$j''H \cap D \neq \emptyset$. Thus,
$H'' = \{p \in j(\FQ^*) : \exists q \in j''H
[q \ge p]\}$ is our desired generic object, and
$j$ lifts to
$j : V_2[G][H] \to M[G][H][H'][H'']$.
Since $V_{\gl'} \subseteq M$,
${(V_{\gl'})}^{V_2[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$. Therefore,
$j$ lifts to
$j : V_2^{\FQ_\gl \ast \dot \FQ^*}
\to M^{j(\FQ_\gl \ast \dot \FQ^*)}$
in a way such that
${(V_{j(f)(\gl)})}^{V_2^{\FQ_\gl \ast \dot \FQ^*}}
\subseteq M^{j(\FQ_\gl \ast \dot \FQ^*)}$.
Write
$\FQ = \FQ_\gl \ast \dot \FQ^* \ast \dot \FQ^{**}$. Since
$\forces_{\FQ_\gl \ast \dot \FQ^*}
``\dot \FQ^{**}$ is $\gz$-strategically closed
for $\gz$ the least inaccessible cardinal
above $\gl'$'', the extender
${\cal F}$ witnessing
the aforementioned lift of $j$ in
$V^{\FQ_\gl \ast \dot \FQ^*}_2$ generates in
$V^{\FQ_\gl \ast \dot \FQ^* \ast \dot \FQ^{**}}_2
= V^\FQ_2$ an elementary embedding
$\ov j : V^\FQ_2 \to N^*$ having
critical point $\gl$ such that
${(V_{\ov j(f)(\gl)})}^{V^\FQ_2}
\subseteq N^*$. As for all $\ga < \gk$,
$V^\FQ_2 \models ``f_1(\ga) < f_2(\ga) < f(\ga)$'',
in $V^\FQ_2$, $f_1 '' \gl \subseteq \gl$, and
${(V_{\ov j(f_1)(\gl)})}^{V^\FQ_2}
\subseteq N^*$.
Since $f_1 \in V^\FQ_2$ was arbitrary,
this means that
$V^\FQ_2 \models ``\gk$ is a Woodin cardinal''.
This completes the proof of Lemma \ref{l11}.
\end{proof}
Lemmas \ref{l8} - \ref{l11},
the proof of Theorem 1 of \cite{AC2}
for a proper class of cardinals,
and the proofs of Lemmas 2.4 and 2.5 of
\cite{AC2} now allow us to infer that
$V^\FP_1 \models ``$Each $V_1$-supercompact
cardinal $\gk$
isn't $2^\gk = \gk^+$ supercompact but
is strongly compact,
strong, and Woodin, and there are no strongly compact
cardinals or cardinals which are both
strong cardinals and Woodin cardinals
in the open interval $(\gg_\gk, \gk)$
if $\gk$ is a $V_1$-supercompact cardinal''.
%and for any $V_1$-supercompact cardinal $\gk$,
%no cardinal in the open interval
%$(\gg_\gk, \gk)$ is strongly compact''.
Further, the fact there are no
inaccessible limits of $V_1$-supercompact
cardinals tells us that any cardinal
(other than a $V_1$-supercompact cardinal)
which is in $V^\FP_1$
either strongly compact or both a
strong cardinal and a Woodin cardinal
would have to lie in an open interval
of the form $(\gg_\gk, \gk)$ for some
$V_1$-supercompact cardinal $\gk$.
Since this is impossible,
this completes the proof of Theorem \ref{t2}.
\end{proof}
In conclusion to this paper,
we note that unlike in Section \ref{s1},
we may rely on the gap forcing results
of \cite{H2} and \cite{H3} to
show that after forcing with each component
partial ordering $\FP^\gk$, all cardinals
in the open interval $(\gg_\gk, \gk)$
which are both strong cardinals
and Woodin cardinals are destroyed.
However, the ``localness'' that was
present in the proofs
found in Section \ref{s1}, particularly
for the
preservation of Woodinness,
is not available in
the proof of Theorem \ref{t2} given,
since strongness is a global and not
a local property.
It is for this reason that the
ideas used in Lemma \ref{l11}
are quite different from the ones
employed in Lemmas
\ref{l3} and \ref{l7}.
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\end{document}