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%
% ------------------------------------------------------------------------------
%
\title{Characterizing Strong
Compactness via Strongness
% A Model
% Containing Supercompact Cardinals
% with a
% Precise Characterization of the
% Strongly Compact Cardinals
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, strong cardinal,
non-reflecting stationary set of ordinals}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank Joel Hamkins
for helpful conversations on the
subject matter of this paper.
The author also wishes to thank
the Rutgers Logic Seminar for having
provided a stimulating research
environment.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{June 25, 2002\\
(revised January 6, 2003)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model in which
the strongly compact cardinals
can be non-trivially characterized
via the statement
``$\gk$ is strongly compact iff
$\gk$ is a measurable limit of
strong cardinals''.
If our ground model contains large
enough cardinals, there will be
supercompact cardinals in the universe
containing this characterization of the
strongly compact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In models in which there are
strongly compact cardinals, a precise, uniform,
regular characterization of these cardinals
in terms of better behaved or better
understood cardinals may or may not be found.
%Examples of universes in which
%the strongly compact cardinals are
%well behaved are plentiful.
Exemplifying universes in which
the strongly compact cardinals
exhibit ``good behavior'',
in the model constructed
by Kimchi and Magidor in \cite{KM}
(see \cite{A98} and \cite{AS97a} for
strengthenings of the result of \cite{KM}),
the strongly compact cardinals are either
supercompact or measurable limits of
supercompact cardinals.
Also, there is
Magidor's theorem (unpublished by
him, but proven as Theorem 1 of
\cite{AC1}) in which the model
constructed has the precise, uniform,
and non-trivial characterization of
the strongly compact cardinals via
the statement
%the strongly
%compact cardinals are uniformly
%characterized via the statement
``$\gk$ is strongly compact iff
$\gk$ is a measurable cardinal''.
Finally, in the model constructed for
Theorem 1 of
\cite{AC2}, the strongly compact cardinals
are precisely, uniformly,
and non-trivially characterized
via the statement
``$\gk$ is strongly compact iff
$\gk$ is a strong cardinal''.
To exemplify universes in which the
strongly compact cardinals exhibit
``bad behavior'',
consider the models
constructed in Theorems 1 and 2 of \cite{A97}
and the Main Theorem of \cite{AH3}.
In these theorems,
the non-supercompact strongly
compact cardinals and the supercompact
cardinals are distinguishable via
fixed ground model classes,
except possibly at
measurable limit points.
As opposed to the results mentioned in the
preceding paragraph, there is no precise
or uniform way of characterizing the strongly compact
cardinals in terms of measurability,
strongness, or supercompactness.
It is unfortunately the case, however,
that in the final two results mentioned
in the next to last paragraph,
the number of large cardinals
in the universe is severely restricted.
In fact, in the model for Theorem 1 of \cite{AC1},
there can only be finitely many
measurable cardinals, and in the model for Theorem 1
of \cite{AC2}, there is no measurable
limit of strongly compact cardinals.
In particular, there are no supercompact
cardinals present in the universes witnessing
the truth of either of these theorems.
The purpose of this paper is to provide
a new model which contains supercompact cardinals
if the ground model is large enough
%in which the number of large
%cardinals in the universe is not subject
%to any severe restrictions and in which
in which all of the strongly compact cardinals
can be precisely, uniformly, and
non-trivially characterized via a
single statement.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC +
$\K \neq \emptyset$ is the class of
supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC +
$\gk$ is
strongly compact iff $\gk$ is a
measurable limit of strong cardinals +
The strongly
compact cardinals are the elements of
$\K$ together with their measurable
limit points''.
Further, in $V^\FP$,
any $\gk \in \K$ which was a supercompact
limit of supercompact cardinals in $V$
remains supercompact.
\end{theorem}
The structure of this paper is as follows.
Section \ref{s1} contains our introductory
comments and preliminary remarks concerning
notation and terminology.
Section \ref{s2} contains a discussion
of the partial ordering essential to
the proof of Theorem \ref{t1}.
Section \ref{s3} gives the proof of
Theorem \ref{t1} and has a
concluding discussion.
Before giving the proof of
Theorem \ref{t1}, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will 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 $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a cardinal and $\FP$ is
a partial ordering,
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
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 $\FP_{\gk, \gl}$ 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, $\FP_{\gk, \gl} =
\{s : s$ 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.
%\{ p$ : For some
%$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
%function of $S_p$, a subset of $\ga$ not stationary at its
%supremum nor having any initial segment which is stationary
%at its supremum, so that $\gb \in S_p$ implies
%$\gb > \gk$ and cof$(\gb) = \gk \}$,
%ordered by $q \ge p$ iff $q \supseteq p$,
%and $S_p = S_q \cap \sup (S_p)$,
%i.e., $S_q$ is an end extension of $S_p$.
Two things which can be shown
(see \cite{Bu} or \cite{A01})
are that
$\FP_{\gk, \gl}$ is $\gd$-strategically
closed for every $\gd < \gl$, and
if $G$ is $V$-generic over $\FP_{\gk, \gl}$,
in $V[G]$,
%if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced.
%the bounded subsets of $\gl$ are the same as those in $V$,
%and cardinals, cofinalities, and GCH
%have been preserved.
It is also virtually immediate that $\FP_{\gk, \gl}$
is $\gk$-directed closed.
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 $\gl > \gk$.
\section{The Building Blocks of $\FP$}\label{s2}
In this section, we define the forcing conditions
that will be employed as building blocks in the
partial ordering $\FP$ we will use in the
proof of Theorem \ref{t1}.
Towards this end, we fix a ground model
$V_0 \models ``$ZFC + $2^\gk = \gk^+$ + $\gk$ is
supercompact and isn't a limit of
supercompact cardinals''.
Let $\xi$ be either the successor of the
supremum of the supercompact cardinals
below $\gk$ or $\go$ if
$\gk$ is the least supercompact
cardinal, and let $\eta$ be
the least strong cardinal
above $\xi$ in $V_0$.
By Lemma 2.1 of \cite{AC2} and the
succeeding remarks,
$\eta \in (\xi, \gk)$.
Our goal will be to define a partial ordering
$\FP_\gk \in V$,
where $V$ is a generic extension of $V_0$
via the still to be defined partial ordering
$\FP^{0, \gk}$,
so that
$V^{\FP_\gk} \models ``$ZFC +
$\gk$ is strongly compact +
There are no strongly compact cardinals
in the interval $[\xi, \gk)$ +
There are no strong cardinals in the
interval $[\xi, \eta]$ +
There are unboundedly many in $\gk$
strong cardinals in the interval
$(\eta, \gk)$ + No cardinal
$\gd \in [\xi, \gk)$ is both
measurable and a limit of strong cardinals''.
We begin by defining
$\FP^{0, \gk} \in V_0$ so that
$V = V^{\FP^{0, \gk}}_0 \models ``$ZFC +
%There are no strong cardinals in the interval
%$[\xi, \eta]$ +
$2^\gd = \gd^+$ for every strong cardinal
$\gd \in (\eta, \gk]$ + $\gk$ is supercompact
and isn't a limit of supercompact cardinals''.
To do this, let
$S = \{\gd \in (\eta, \gk) : \gd$ is a strong
cardinal so that
$2^\gd > \gd^+\}$.
$\FP^{0, \gk}$ is then taken as the Easton
support iteration which begins by
adding a non-reflecting stationary set of
ordinals of cofinality $\xi$ to $\eta$ and
then adds a Cohen subset of $\gd^+$
for every $\gd \in S$.
Note that by its definition,
$\FP^{0, \gk}$ is $\xi$-directed closed
and therefore preserves the supercompactness
of any sufficiently indestructible
supercompact cardinal below $\xi$.
\begin{lemma}\label{n1}
$V = V^{\FP^{0, \gk}}_0 \models ``\gk$ is
supercompact and isn't a limit of
supercompact cardinals''.
\end{lemma}
\begin{proof}
If $|\FP^{0, \gk}| < \gk$, then
by the L\'evy-Solovay
results \cite{LS},
$V^{\FP^{0, \gk}}_0 \models ``\gk$ is
supercompact and isn't a limit of
supercompact cardinals''. We therefore assume
without loss of generality that
$|\FP^{0, \gk}| = \gk$.
Let
$\gl \ge 2^\gk$ be an arbitrary cardinal, and let
$\gg = |2^{[\gl]^{{<}\gk}}|$.
Let
$j : V_0 \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$
so that
$M \models ``\gk$ isn't $\gg$ supercompact''.
By Lemma 2.1 of \cite{AC2} and the
succeeding remarks,
$M \models ``\gk$ is a strong cardinal'',
and by the fact $M^\gg \subseteq M$,
$M \models ``2^\gk = \gk^+$''.
Further, it is also the case that
$M \models ``$No cardinal
$\gd \in (\gk, \gg]$ is strong''. This is
since otherwise, by the closure properties of $M$,
$M \models ``\gk$ is $\gs$
supercompact for every $\gs < \gd$ and
$\gd \in (\gk, \gg]$ is strong'', so by
the proof of Lemma 2.4 of \cite{AC2},
$M \models ``\gk$ is $\gg$ supercompact'',
a contradiction.
Therefore, the preceding means we can write
$j(\FP^{0, \gk}) = \FP^{0, \gk} \ast \dot \FQ$, where
the first ordinal to which $\dot \FQ$ is
forced to add a Cohen subset is above $\gg$.
Let $G$ be $V_0$-generic over $\FP^{0, \gk}$, and
let $H$ be $V_0[G]$-generic over $\FQ$. Since
the last sentence of the preceding paragraph
implies that
$j '' G \subseteq G \ast H$,
in $V_0[G][H]$, $j$ lifts to
$j : V_0[G] \to M[G][H]$. In addition, since
standard arguments show that
$M[G]$ remains $\gg$ closed with respect to
$V_0[G]$,
again by the last sentence of the preceding
paragraph,
$\FQ$ is $\gg^+$-directed closed
in $V_0[G]$. By the definition of $\gg$,
this means the supercompact ultrafilter
${\cal U}$ over
${(P_\gk(\gl))}^{V_0[G]}$ in
$V_0[G][H]$ given by
$x \in {\cal U}$ iff
$\la j(\ga) : \ga < \gl \ra \in j(x)$
is so that
${\cal U} \in V_0[G]$. Hence,
$V^{\FP^{0, \gk}}_0 \models ``\gk$ is $\gl$
supercompact''. As $\gl$ was
arbitrary,
$V^{\FP^{0, \gk}}_0 \models ``\gk$ is
supercompact''.
To show that
$V^{\FP^{0, \gk}}_0 \models ``\gk$ isn't
a limit of supercompact cardinals'',
let $\gz$ be the least
(singular) strong limit cardinal above $\eta$
in $V_0$. Write
$\FP^{0, \gk} = \FQ \ast \dot \FR$, where
$\FQ =
\FP_{\xi, \eta}$,
$|\FQ| < \gz$ and
$\forces_{\FQ} ``\dot \FR$ is
$\gz$-strategically closed''.
In Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FQ$ ``admits a gap at $\gz$'', so
by the Gap Forcing Theorem of
\cite{H2} and \cite{H3},
any supercompact cardinal above $\eta$ in
$V^{\FP^{0, \gk}}_0$ must
have been supercompact in $V_0$. Thus,
$V^{\FP^{0, \gk}}_0 \models ``\gk$ isn't
a limit of supercompact cardinals''.
This completes the proof
of Lemma \ref{n1}.
\end{proof}
\begin{lemma}\label{n2}
$V = V^{\FP^{0, \gk}}_0 \models ``2^\gd = \gd^+$
for every strong cardinal $\gd \in
(\eta, \gk]$''.
\end{lemma}
\begin{proof}
To prove Lemma \ref{n2},
we begin by arguing in analogy to
the last paragraph of the proof of
Lemma \ref{n1}. Let
$\gd \in (\eta, \gk]$ be a
strong cardinal in $V^{\FP^{0, \gk}}_0$.
Working in $V_0$,
let $\gz$ once again be the least
(singular) strong limit cardinal above $\eta$.
Write
$\FP^{0, \gk} = \FQ \ast \dot \FR$, where
$\FQ = \FP_{\xi, \eta}$,
$|\FQ| < \gz$ and
$\forces_{\FQ} ``\dot \FR$ is
$\gz$-strategically closed''.
Since
$\FQ$ ``admits a gap at $\gz$'',
by the Gap Forcing Theorem of
\cite{H2} and \cite{H3}, $\gd$ must
have been a strong cardinal in $V_0$.
Rewrite
$\FP^{0, \gk} =
\FP_{\xi, \eta} \ast \dot
\FS \ast \dot \FT$, where
$\dot \FS$ is forced to add Cohen subsets to the
successors of elements of $S \cap \gd^+$ and
$\dot \FT$ is a term for the rest of
$\FP^{0, \gk}$.
(If $\gd = \gk$, then
$\dot \FT$ is a term for the
trivial partial ordering $\{\emptyset\}$.) If
$V_0 \models ``2^\gd > \gd^+$'', then
standard density arguments
(see Exercise 19.7, page 183 of \cite{J})
show that
$V^{\FP_{\xi, \eta} \ast \dot \FS}_0
\models ``2^\gd = \gd^+$''. Since
$\forces_{\FP_{\xi, \eta}\ast \dot \FS}
``\dot \FT$ is $\gd^+$-directed
closed'',
$V^{\FP_{\xi, \eta} \ast \dot \FS \ast \dot \FT}_0 =
V^{\FP^{0, \gk}}_0 = V \models
``2^\gd = \gd^+$''. If
$V_0 \models ``2^\gd = \gd^+$'',
$|\FP_{\xi, \eta} \ast \dot \FS| \le \gd$, so
$V^{\FP_{\xi, \eta} \ast \dot \FS}_0
\models ``2^\gd = \gd^+$''.
Since once again
$\forces_{\FP_{\xi, \eta} \ast \dot \FS}
``\dot \FT$ is $\gd^+$-directed
closed'',
$V^{\FP_{\xi, \eta} \ast \dot \FS \ast \dot \FT}_0 =
V^{\FP^{0, \gk}}_0 = V \models
``2^\gd = \gd^+$''.
This completes the proof of Lemma \ref{n2}.
\end{proof}
Working now in $V$, let
$\la \gd_\ga : \ga < \gk \ra$ be the
continuous, increasing enumeration of the
cardinals in the interval
$(\eta, \gk)$ which are either
strong cardinals or measurable limits
of strong cardinals.
For $\ga$ an arbitrary ordinal, define
$\ga^-$ as the immediate ordinal
predecessor of $\ga$ if $\ga$ is a
successor ordinal, and $0$ if $\ga$
is either a limit ordinal or $0$.
For each $\ga < \gk$, let
$\gg_\ga = {(\bigcup_{\gb < \ga} \gd_\gb)}^+$,
where if $\ga = 0$, $\gg_\ga =
{(2^{\eta^+})}^+$.
Also, for each $\ga < \gk$, define
$\theta_\ga$ as the least cardinal
so that
$V \models ``\gd_\ga$ isn't $\theta_\ga$
supercompact''.
By the choice of $\gk$ and Lemma \ref{n1},
$\theta_\ga$ is well-defined
for every $\ga < \gk$.
Further, since if $\gd$ is $\gg$
supercompact for every
$\gg < \gd'$ and $\gd'$ is
strong, $\gd$ is supercompact
(see the proof of Lemma 2.4 of
\cite{AC2}), and since
$\gd_{\ga + 1}$ is strong,
it must be the case that
$\theta_\ga < \gd_{\ga + 1}$.
We now define the partial ordering
$\FP_\gk =
\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
as the Easton support
iteration of length $\gk$ satisfying the
following properties:
\begin{enumerate}
\item\label{i1} $\FP_0$ adds a
Cohen subset of $\eta^+$.
\item\label{i2} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ isn't a measurable limit
of strong cardinals,
$\dot \FQ_\ga$ is a term for the
Gitik-Shelah partial ordering of \cite{GS}
for the cardinal $\gd_\ga$
whose first non-trivial forcing is
done at a measurable cardinal above
$\rho_\ga = \max(\theta_{\ga^-},
\gg_\ga, \xi)$
defined using only component partial orderings
that are at least $\rho_\ga$-strategically
closed and $\xi$-directed
closed.\footnote{It is also possible
to let
$\dot \FQ_\ga$ be a term for
Hamkins' partial ordering of
Theorem 4.10 of \cite{H4}
for the cardinal $\gd_\ga$,
assuming the same restrictions on components
as when using the partial ordering of
\cite{GS} and that the fast function
forcing employed in Hamkins' definition
is $\xi$-directed closed.}
Under these restrictions, the
realization of $\dot \FQ_\ga$
makes the strongness of
$\gd_\ga$ indestructible under forcing with
$\gd_\ga$-strategically closed partial orderings
which are at least $\xi$-directed closed.
\item\label{i3} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ is a measurable limit
of strong cardinals,
$\dot \FQ_\ga$ is a term for the
partial ordering
$\FP_{\xi, \gd_\ga}$.
\end{enumerate}
The intuition behind the above definition of
$\FP_\gk$ is quite simple.
$\FP_\gk$ has been explicitly designed so as
to preserve all
sufficiently indestructible supercompact
cardinals below $\xi$ and all
isolated strong cardinals in the
interval $(\eta, \gk)$ while destroying
all ground model
measurable limits of strong cardinals
in the interval
$(\eta, \gk)$ and creating no new
measurable limits of strong cardinals
in the interval $(\eta, \gk)$.
In addition, the definition just given
ensures that forcing with $\FP_\gk$ will
both preserve the strong compactness of
$\gk$ and destroy the strong compactness
of any cardinal $\gd \in (\eta, \gk)$.
This, in tandem with the definition of
$\FP^{0, \gk}$, will show that forcing with
$\FP^{0, \gk} \ast \dot \FP_\gk$
creates a generic extension as described
in the last sentence of the first
paragraph of this section.
\begin{lemma}\label{l1}
$V^{\FP_\gk} \models ``\gk$ is strongly compact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} uses
ideas of Magidor
for preserving strong compactness under
an iteration of forcing to add
non-reflecting stationary sets of ordinals
and is similar to proofs for the preservation
of strong compactness given in \cite{A01},
\cite{AC1}, \cite{AC2}, and \cite{AH3}.
Let $\gl > 2^\gk$ be a
singular strong limit cardinal of cofinality at least $\gk$,
and let
$k_1 : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$ so that
$M \models
``\gk$ isn't $\gl$ supercompact''.
%By the choice of $\gl$,
%$M \models ``\gk$ is $\gg$ supercompact for every
%$\gg < \gl$''. Also,
$\gl$ is large enough so that we
may assume by selecting a normal ultrafilter of
trivial Mitchell rank over $\gk$ that
$k_2 : M \to N$ is an embedding witnessing the
measurability of $\gk$ definable in $M$ so that
$N \models ``\gk$ isn't measurable''.
In addition, as $\gl > 2^\gk$,
Lemma 2.1 of \cite{AC2} and the succeeding
remarks imply that in both $V$ and $M$,
$\gk$ is a strong cardinal which is a
limit of strong cardinals,
and in fact, in both $V$ and $M$, $\gk$
carries a normal measure concentrating on
strong cardinals.
Suppose that
$k : V \to N$ is an elementary embedding
definable in $V$ with
critical point $\gk$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
so that $x \subseteq y$ and
$N \models ``|y| < k(\gk)$''.
By Theorem 22.17 of \cite{K},
$k$ witnesses the $\gl$
strong compactness of $\gk$
in the sense that the existence
of this sort of embedding implies that
$\gk$ is $\gl$ strongly compact.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of $\gk$.
We show that $j$ lifts to
$j : V^{\FP_\gk} \to N^{j(\FP_\gk)}$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of $\gk$ in
$V^{\FP_\gk}$, this proves Lemma \ref{l1}.
To do this, write
$j(\FP_\gk)$ as
$\FP_\gk \ast \dot \FQ \ast \dot \FR$, where
$\dot \FQ$ is a term for the portion of
$j(\FP_\gk)$ between $\gk$ and $k_2(\gk)$ and
$\dot \FR$ is a term for the rest of
$j(\FP_\gk)$, i.e., the part above $k_2(\gk)$.
Note that since
$N \models ``\gk$ isn't measurable'',
forcing with the realization of $\dot \FQ$
doesn't add a non-reflecting stationary set
of ordinals of cofinality $\xi$ to $\gk$.
Thus, the ordinals at which the realization of
$\dot \FQ$ does a non-trivial forcing
%is composed of all $N$-measurable limits of strong cardinals
lie in the interval
$(\gk, k_2(\gk)]$ (the endpoint
$k_2(\gk)$ is included since
$k_2(\gk)$ is a measurable limit of
strong cardinals in $N$),
and the ordinals at which the realization
of $\dot \FR$ does a non-trivial forcing
lie in the interval
$(k_2(\gk), k_2(k_1(\gk)))$.
Let $G_0$ be $V$-generic over $\FP_\gk$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR$. Since $\FP_\gk$ is an
Easton support iteration of length $\gk$,
a direct limit is taken at stage $\gk$,
and no forcing is done at stage $\gk$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
This means that
$j : V \to N$ lifts to
$j : V[G_0] \to N[G_0][G_1][G_2]$ in
$V[G_0]$.
To build $G_1$, note that since $k_2$
is generated by an
ultrafilter
%${\cal U}$
over $\gk$ and
since in both $V$ and $M$, $2^\gk = \gk^+$,
$|k_2(\gk^+)| = |k_2(2^\gk)| =
|\{ f : f : \gk \to \gk^+$ is a function$\}| =
|{[\gk^+]}^\gk| = \gk^+$. Thus, as
$N[G_0] \models ``|\wp(\FQ)| = k_2(2^\gk)$'', we can let
$\la D_\ga : \ga < \gk^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of
$\FQ$ present in $N[G_0]$.
For the purpose of the construction of
$G_1$ to be given below, we further
assume without loss of generality that
for every dense open subset
$D \subseteq \FQ$ found in $N[G_0]$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$.
Since the $\gk$ closure of $N$ with respect to either
$M$ or $V$ implies the least ordinal
at which $\FQ$ does a non-trivial forcing
is above $\gk^+$, the definition of $\FQ$ implies that
%as the Easton support iteration which adds
%a non-reflecting stationary set of ordinals of
%cofinality $\xi$ to each $N[G_0]$-measurable limit
%of strong cardinals in the interval
%$(\gk, k_2(\gk)]$ implies that
$N[G_0] \models ``\FQ$ is
${\prec} \gk^+$-strategically closed''.
By the fact the standard arguments show that
forcing with the $\gk$-c.c$.$ partial ordering
$\FP_\gk$ preserves that $N[G_0]$ remains
$\gk$-closed with respect to either
$M[G_0]$ or $V[G_0]$,
$\FQ$ is ${\prec} \gk^+$-strategically closed
in both $M[G_0]$ and $V[G_0]$ as well.
We can now construct $G_1$ in either
$M[G_0]$ or $V[G_0]$ as follows.
Players I and II play a game of length
$\gk^+$. The initial pair of moves is
generated by player II choosing the
trivial condition $q_0$ and player
I responding by choosing
$q_1 \in D_1$.
Then, at an even stage $\ga + 2$,
player II picks
$q_{\ga + 2} \ge q_{\ga + 1}$ by
using some fixed strategy
${\cal S}$, where $q_{\ga + 1}$
was chosen by player I to be so that
$q_{\ga + 1} \in D_{\ga + 1}$ and
$q_{\ga + 1} \ge q_\ga$.
If $\ga$ is a limit ordinal, player II uses
${\cal S}$ to pick $q_\ga$ extending each
$q_\gb$ for $\gb < \ga$.
By the ${\prec} \gk^+$-strategic closure of
$\FQ$ in both $M[G_0]$ and $V[G_0]$,
the sequence
$\la q_\ga : \ga < \gk^+ \ra$
as just described exists.
By construction,
$G_1 = \{p \in \FQ : \exists \ga <
\gk^+ [q_\ga \ge p]\}$ is our
$N[G_0]$-generic object over $\FQ$.
It remains to construct in $V[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR$.
To do this, we first observe that as
$M \models ``\gk$ is a measurable limit of
strong cardinals'',
we can write
$k_1(\FP_\gk)$ as
$\FP_\gk \ast \dot \FS \ast \dot \FT$, where
$\forces_{\FP_\gk} ``\dot \FS =
\dot \FP_{\xi, \gk}$'', and
%adds a
%non-reflecting stationary set of ordinals of
%cofinality $\xi$ to $\gk$'', and
$\dot \FT$ is a term for the rest of
$k_1(\FP_\gk)$.
Note now that
by clause (\ref{i2}) in the
definition of $\FP_\gk$,
the ordinals at which the
realization of $\dot \FT$
does a non-trivial forcing
lie in the interval
$(\gl, k_1(\gk))$, which implies that in
$M$,
$\forces_{\FP_\gk \ast \dot \FS}
``\dot \FT$ is ${\prec} \gl^+$-strategically
closed''. Further,
since $\gl$ is a singular strong limit
cardinal of cofinality at least $\gk$,
$|{[\gl]}^{< \gk}| = \gl$.
By Solovay's theorem \cite{So} that
GCH must hold at any singular strong
limit cardinal above a strongly compact
cardinal, $2^\gl = \gl^+$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter
%${\cal U}$
over $P_\gk(\gl)$,
$|k_1(\gk^+)| = |k_1(2^\gk)| =
|2^{k_1(\gk)}| =
|\{ f : f : P_\gk(\gl) \to \gk^+$ is a function$\}| =
|{[\gk^+]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{F} for the
first published account of term forcing or
\cite{C}, Section 1.2.5, page 8; the notion
is originally due to Laver) associated with
$\dot \FT$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP_\gk \ast \dot \FS$ and
$\forces_{\FP_\gk \ast \dot \FS} ``\tau \in
\dot \FT$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP_\gk \ast \dot \FS} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT$ 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 $\dot \FT$ is also forced
%to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M \models ``|\FT^*| = k_1(\gk)$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP_\gk \ast \dot \FS} ``\dot \FT$ is
${\prec}\gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V$ as well.
To show that
we may restrict the number of terms
so that
$M \models ``|\FT^*| = k_1(\gk)$'',
%we recall that in the official definition of $\FT^*$,
%the basic idea is to include only the canonical terms.
we observe that since
$\forces_{\FP_\gk \ast \dot \FS}
``|\dot \FT| = k_1(\gk)$'',
there is a set
$\{\tau_\alpha : \alpha \gk} \FP_\gd$
and
$\FP_{{<}\gk} = \prod_{\gd \in \K, \gd < \gk} \FP_\gd$.
By the definition of each $\FP_\gd$,
$\FP^\gk$ is $\gk^+$-directed closed, so since
$V \models ``\gk$ is a supercompact limit of
supercompact cardinals and the supercompactness
of any supercompact cardinal $\gd \le \gk$ is
indestructible under $\gd$-directed closed forcing'',
$V^{\FP^\gk} \models ``\gk$ is a supercompact
limit of supercompact cardinals''.
Thus, the proof of Lemma \ref{l5} will be
complete once we have shown
$V^{\FP^\gk \times \FP_{{<}\gk}} = V^\FP \models
``\gk$ is supercompact''.
Note that the measurable limits of
strong cardinals in
$V^{\FP^\gk}$ below $\gk$ and the measurable limits
of strong cardinals in $V$ below $\gk$ must be
precisely the same. To see this,
we first observe that since
$\FP^\gk$ is $\gk^+$-directed closed,
the measurable cardinals below $\gk$ in
$V^{\FP^\gk}$ and $V$ are precisely the same.
Next, we show that the strong cardinals below
$\gk$ in $V^{\FP^\gk}$ and $V$ are
precisely the same.
To do this, assume that
$V^{\FP^\gk} \models ``\gd < \gk$ is strong''.
As $\FP^\gk$ is $\gk^+$-directed closed,
this means that
$V \models ``\gd$ is $\ga$ strong for every
ordinal $\ga \in (\gd, \gk)$''. Since
$V \models ``\gk$ is supercompact and
hence strong'', by the second paragraph of
Lemma 2.1 of \cite{AC2},
$V \models ``\gd$ is strong''.
Hence, the strong cardinals in
$V^{\FP^\gk}$ below $\gk$ must be a subset
of the strong cardinals in $V$ below $\gk$.
Then, if
$V \models ``\gd < \gk$ is strong'',
by the directed closure properties of
$\FP^\gk$,
$V^{\FP^\gk} \models ``\gd$ is $\ga$
strong for every ordinal
$\ga \in (\gd, \gk)$ and $\gk$ is
supercompact'', so as we just observed,
$V^{\FP^\gk} \models ``\gd$ is strong''.
Further, by the work just done, in both
$V^{\FP^\gk}$ and $V$,
$\gd < \gk$ is strong iff
$\gd$ is $\ga$ strong for every
$\ga \in (\gd, \gk)$,
and the cardinals below $\gk$
which are $\ga$ strong for every
$\ga \in (\gd, \gk)$ are precisely
the same in both
$V^{\FP^\gk}$ and $V$.
It is also the case that
$\gd < \gk$ is supercompact in
$V^{\FP^\gk}$ iff $\gd$ is
supercompact in $V$. We have
already observed in the first
paragraph of the proof of this
lemma that any cardinal
$\gd \le \gk$ which is supercompact
in $V$ is also supercompact in
$V^{\FP^\gk}$. If $\gd < \gk$ is
supercompact in $V^{\FP^\gk}$, then as
$\FP^\gk$ is $\gk^+$-directed closed,
$\gd$ is $\ga$ supercompact in $V$
for every $\ga < \gk$. Since $\gk$
is supercompact in $V$ and hence is
strong in $V$, $\gd$ must be
supercompact in $V$ as well.
We now know that the strong cardinals
and the measurable limits
of strong cardinals below $\gk$ in
$V^{\FP^\gk}$ and $V$ are precisely the same
and that these cardinals are precisely the same
as the cardinals $\gd < \gk$ which are
either $\ga$ strong for every
$\ga \in (\gd, \gk)$ or are measurable
limits of cardinals $\gg$ which are
$\ga$ strong for every
$\ga \in (\gg, \gk)$.
We also know by the preceding paragraph
that the supercompact cardinals below
$\gk$ in $V^{\FP^\gk}$ and $V$ are
identical with one another.
In addition, the $\gk^+$-directed closure of
$\FP^\gk$ implies that
${(V_\gk)}^{V^{\FP^\gk}} =
{(V_\gk)}^V$.
This means we can view
$\FP_{{<}\gk}$ as a class partial ordering
defined in $V_\gk$, so by the work of \cite{GS}
or \cite{H4},
$\FP_{{<\gk}}$ has the same definition and
properties regardless
if it's defined in $V$ or $V^{\FP^\gk}$.
Returning to our proof,
define
$\ov V = V^{\FP^\gk}$.
Let
$\gl > \gk$ be an arbitrary cardinal, and let
$\gg = |2^{[\gl]^{{<}\gk}}|$.
Let
$j : \ov V \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$
(in $\ov V$) so that
$M \models ``\gk$ isn't $\gg$ supercompact''.
Take $\gk'$ as the least supercompact cardinal
above $\gk$ in $M$ and
$\eta_{\gk'}$ as the least strong cardinal
above $\gk$ in $M$.
$\gk'$ and $\eta_{\gk'}$ exist,
since by elementarity,
$M \models ``j(\gk) > \gk$ is a supercompact limit
of supercompact cardinals''.
By the definition of $\eta_{\gk'}$
and the fact that as in the proof of
Lemma \ref{n1},
$M \models ``$No cardinal
$\gd \in (\gk, \gg]$ is strong'',
in $M$,
$\FP_{\gk'}$ is
$\gz$-strategically closed
for every $\gz < \eta_{\gk'}$, and
$\eta_{\gk'} > \gg$.
In particular, in $M$,
$\FP_{\gk'}$ is $\gg$-strategically closed.
Also, the first non-trivial forcing
in the definition of $\FP_{\gk'}$ is
done at an ordinal above $\gg$
(namely $\eta_{\gk'}$).
Thus, if we write
$j(\FP_{{<}\gk}) = \FP_{{<}\gk} \times \FQ$ and
let $G$ be $\ov V$-generic over
$\FP_{{<}\gk}$ and $H$ be $\ov V[G]$-generic over
$\FQ$, in $\ov V[G \times H]$, $j$ lifts to
$\ov j : \ov V[G] \to M[G \times H]$ via the definition
$\ov j(i_G(\tau)) = i_{G \times H}(j(\tau))$. Since
$M \models ``\FQ$ is $\gg$-strategically closed'',
the fact
$M^\gg \subseteq M$ implies
$\ov V \models ``\FQ$ is $\gg$-strategically
closed'' yields that for any cardinal
$\gd \le \gg$,
$\ov V[G]$ and $\ov V[G \times H] =
\ov V[H \times G]$ contain the same
subsets of $\gd$.
This means the supercompact ultrafilter
${\cal U}$ over
${(P_\gk(\gl))}^{\ov V[G]}$ in
$\ov V[G \times H]$ given by
$x \in {\cal U}$ iff
$\la j(\ga) : \ga < \gl \ra \in \ov j(x)$
is so that
${\cal U} \in \ov V[G]$. Hence,
$V^{\FP^\gk \times \FP_{{<}\gk}} = V^\FP \models
``\gk$ is $\gl$ supercompact''.
As $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$ is strongly
compact iff $\gk$ is a measurable
limit of strong cardinals +
The strongly compact cardinals
are the elements of $\K$ together
with their measurable limit points''.
\end{lemma}
\begin{proof}
Fix $\gk \in \K$.
By Lemma \ref{l5}, if $\gk$ is a
supercompact limit of supercompact
cardinals, we know $\gk$ remains
supercompact in $V^\FP$.
Also, as we observed during the
proof of Lemma \ref{l1},
$\gk$ must be a limit of strong cardinals.
We may
therefore assume $\gk$ isn't a
supercompact limit of supercompact
cardinals and write
$\FP = \FP^\gk \times \FP_\gk \times \FP_{{<}\gk}$,
where
$\FP^\gk$ and $\FP_{{<}\gk}$ are as in
Lemma \ref{l5}.
As before,
by the definition of each $\FP_\gd$,
$\FP^\gk$ is $\gk^+$-directed closed. Since
$V \models ``$The supercompactness of $\gk$
is indestructible under $\gk$-directed closed
forcing + $2^\gk = \gk^+$'',
$V^{\FP^\gk} \models ``\gk$ is supercompact and
$2^\gk = \gk^+$''.
By the analysis given in the proof of
Lemma \ref{l5},
%since $\FP^{0, \gk}$ adds Cohen subsets
%to cardinals strictly below $\gk$,
we can view
$\FP_\gk$ as a class partial ordering
defined in $V_\gk$ and infer that
%so by the work of \cite{GS},
$\FP_\gk$ has the same definition and
properties regardless
if it's defined in $V$ or $V^{\FP^\gk}$.
Hence, by Lemmas \ref{l1} - \ref{l4a},
$V^{\FP^\gk \times \FP_\gk} \models
``\gk$ is strongly compact, $\gk$ is a
limit of strong cardinals, and there
are neither strongly compact cardinals
nor measurable limits of strong cardinals in
the interval $[\xi_\gk, \gk)$'', where
$\xi_\gk$ is the cardinal $\xi$ of
Section \ref{s2} defined with respect to $\gk$.
Since
$|\FP_{{<}\gk}| < 2^{\xi_\gk}$ in either
$V$, $V^{\FP^\gk}$, or
$V^{\FP^\gk \times \FP_\gk}$,
the results of \cite{LS}
and Woodin's results of \cite{HW}
stating that the strongness of a cardinal
is preserved after a small forcing
has been done and no new strong
cardinals are created after a small
forcing has been done
imply that
$V^{\FP^\gk \times \FP_\gk \times \FP_{{<}\gk}} =
V^\FP \models
``\gk$ is strongly compact, $\gk$ is a
limit of strong cardinals, and there
are neither strongly compact cardinals
nor measurable limits of strong cardinals in
the interval $[\xi_\gk, \gk)$''.
Thus, if $\gk \in \K$, regardless if
$\gk$ is a supercompact limit
of supercompact cardinals,
$V^\FP \models ``\gk$ is strongly compact and
is a limit of strong cardinals''.
Further, by a result of Menas \cite{Me},
any measurable limit of strongly compact
cardinals is itself strongly compact,
so in $V^\FP$,
any strongly compact cardinal which is a
measurable limit of elements of
$\K$ is a measurable limit of strong cardinals.
This means the proof of Lemma \ref{l6}
will be complete once we have shown
that any cardinal which isn't either an
element of $\K$ or a measurable limit
of elements of $\K$ can't be
either strongly compact or a measurable
limit of strong cardinals
in $V^\FP$.
However, if $\gd$ is such a cardinal
and $\gk$ is the least element of
$\K$ above $\gd$
($\gk$ exists since $\K$
is a proper class),
then it must be the case that
$\gd \in [\xi_\gk, \gk)$.
This is since by hypothesis,
$\gd$ isn't a measurable limit of
elements of $\K$, and
$\xi_\gk$ is both the successor of
the supremum of the supercompact
cardinals below $\gk$ and the
successor of the supremum of the
supercompact cardinals below $\gd$.
By the work just done, this is impossible.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l5} and \ref{l6} complete
the proof of Theorem \ref{t1}.
\end{proof}
We note that in $V^\FP$, any
$\gk \in \K$ which isn't a
supercompact limit of supercompact
cardinals becomes a non-supercompact
strongly compact cardinal.
This is since as we observed
in the proof of Lemma \ref{l6},
such a cardinal has a final
segment below it containing
no measurable limits of
strong cardinals, which by
Lemma 2.1 of \cite{AC2} and
the succeeding remarks implies that
$V^\FP \models ``\gk$ isn't $2^\gk$
supercompact''.
Also, any $\gk$ a measurable limit of
elements of $\K$ which isn't
supercompact in $V$ remains a
non-supercompact strongly compact
cardinal in $V^\FP$.
This is since the forcing used to
build $V^\FP$, including any
preliminary forcing that might
need to be done to create $V$,
is in the terminology of
\cite{H2} and \cite{H3} a
``mild forcing with respect to
$\gk$ admitting a gap below the
least inaccessible cardinal'',
so by the Gap Forcing Theorem of
\cite{H2} and \cite{H3},
$\gk$ does not become supercompact
in $V^\FP$.
We conclude Section \ref{s3} and this
paper by outlining how to prove
Theorem \ref{t1} if $\K$ is a set.
Under this assumption, let
$\Omega = \sup(\K)$. Working in the
$V$ of Theorem \ref{t1}, define
$\FP_\Omega$ as the Easton support iteration
which begins by adding a Cohen subset of
$\Omega^+$ and then adds a non-reflecting stationary set
of ordinals of cofinality
$\Omega^+$ to every measurable cardinal
above $\Omega$ (if there are any).
Note that $\FP_\Omega$ is
$\Omega^+$-directed closed.
Let $\gz$ be the least strong limit
cardinal above $\Omega$.
Since we can write
$\FP_\Omega = \FQ \ast \dot \FQ'$, where
$|\FQ| < \gz$ and
$\forces_{\FQ} ``\dot \FQ'$ is
$\gz$-strategically closed'', the Gap Forcing
Theorem of \cite{H2} and \cite{H3} together
with the definition of $\FP_\Omega$ ensure that
$V^{\FP_\Omega} \models ``$No cardinal above
$\Omega$ is measurable''. Therefore, since for the
$\FP$ of Theorem \ref{t1},
$|\FP| < \gz$ in either $V$ or $V^{\FP_\Omega}$,
%regardless of the exact definition of $\FP_\Omega$,
by the results of \cite{LS},
$V^{\FP_\Omega \times \FP} \models ``$No
cardinal above $\Omega$ is measurable'' as well.
This means that if we replace $\K$ with
$\K' = \K \cup \{\Omega\}$, define $\FP$ as
the Easton support product
$\prod_{\gk \in \K'} \FP_\gk$, and
restrict our attention to cardinals below
$\Omega$ in Lemmas \ref{l5} and \ref{l6},
the proofs of these lemmas
are essentially the same as before.
%The proof of Lemma \ref{l5} is also the
%same as before.
This completes our outline of how to
prove Theorem \ref{t1} when
$\K$ is a set instead of a proper class.
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\end{document}
\begin{graveyard}
\ast \dot \FR_\ga$, where
$\forces_{\FP_\ga \ast \dot \FP_{\eta, \gd_\ga}}
``\dot \FR_\ga$ is a term for the
Gitik-Shelah partial ordering of \cite{GS}
defined using component partial orderings
that are at least $\gs_\ga$-strategically
closed which makes the strongness of
$\gd_\ga$ indestructible under
$\gd_\ga$-strategically closed partial orderings''.
%the supremum of the
%supercompact cardinals below $\gk$,
%together with their measurable limit points.
%(where $\gd_0$ is the least $V$-strong cardinal).
as in Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''.
Thus, the field of
$\dot \FT$ is composed of all
$M$-measurable limits of
strong cardinals in the interval
%$\gl$ is a singular strong limit cardinal of
%cofinality at least $\gk$,
By Solovay's theorem \cite{So} that
GCH must hold at any singular strong
limit cardinal above a strongly compact cardinal,
$2^\gl = \gl^+$.
The proof of Lemma \ref{l1} uses
ideas of Magidor
for preserving strong compactness under
an iteration of forcing to add
non-reflecting stationary sets of ordinals
and is similar to proofs for the preservation
of strong compactness given in \cite{A01},
\cite{AC1}, and \cite{AC2}.
Let $\gl \ge 2^\gk$
be an arbitrary successor cardinal,
%singular strong limit cardinal of cofinality at least $\gk$,
and let
$k_1 : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$ so that
$M \models
``\gk$ isn't $\gl$ supercompact''.
%By the choice of $\gl$,
%$M \models ``\gk$ is $\gg$ supercompact for every
%$\gg < \gl$''. Also,
$\gl$ is large enough so that we
may assume by selecting a normal ultrafilter of
trivial Mitchell rank over $\gk$ that
$k_2 : M \to N$ is an embedding witnessing the
measurability of $\gk$ definable in $M$ so that
$N \models ``\gk$ isn't measurable''.
In addition, as $\gl \ge 2^\gk$,
Lemma 2.1 of \cite{AC2} and the succeeding
remark imply that in both $V$ and $M$,
$\gk$ is a strong limit of strong cardinals,
and in fact, in both $V$ and $M$, $\gk$
carries a normal measure concentrating on
strong cardinals.
Suppose that
$k : V \to N$ is an elementary embedding
definable in $V$ with
critical point $\gk$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
so that $x \subseteq y$ and
$N \models ``|y| < k(\gk)$''.
By Theorem 22.17 of \cite{K},
$k$ witnesses the $\gl$
strong compactness of $\gk$
in the sense that the existence
of this sort of embedding implies that
$\gk$ is $\gl$ strongly compact.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of $\gk$.
We show that $j$ lifts to
$j : V^{\FP_\gk} \to N^{j(\FP_\gk)}$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of $\gk$ in
$V^{\FP_\gk}$, this proves Lemma \ref{l1}.
To do this, write
$j(\FP_\gk)$ as
$\FP_\gk \ast \dot \FQ \ast \dot \FR$, where
$\dot \FQ$ is a term for the portion of
$j(\FP_\gk)$ between $\gk$ and $k_2(\gk)$ and
$\dot \FR$ is a term for the rest of
$j(\FP_\gk)$, i.e., the part above $k_2(\gk)$.
Note that since
$N \models ``\gk$ isn't measurable'',
forcing with the realization of $\dot \FQ$
doesn't add a non-reflecting stationary set
of ordinals of cofinality $\eta$ to $\gk$.
Thus, the ordinals at which the realization of
$\dot \FQ$ does a non-trivial forcing
%is composed of all $N$-measurable limits of strong cardinals
lie in the interval
$(\gk, k_2(\gk)]$ (the endpoint
$k_2(\gk)$ is included since
$k_2(\gk)$ is a measurable limit of
strong cardinals in $N$),
and the ordinals at which the realization
of $\dot \FR$ does a non-trivial forcing
lie in the interval
$(k_2(\gk), k_2(k_1(\gk)))$.
Let $G_0$ be $V$-generic over $\FP_\gk$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR$. Since $\FP_\gk$ is an
Easton support iteration of length $\gk$,
a direct limit is taken at stage $\gk$,
and no forcing is done at stage $\gk$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
This means that
$j : V \to N$ extends to
$j : V[G_0] \to N[G_0][G_1][G_2]$ in
$V[G_0]$.
To build $G_1$, note that since $k_2$
is generated by an
ultrafilter ${\cal U}$ over $\gk$ and
since in both $V$ and $M$, $2^\gk = \gk^+$,
$|k_2(\gk^+)| = |k_2(2^\gk)| =
|\{ f : f : \gk \to \gk^+$ is a function$\}| =
|{[\gk^+]}^\gk| = \gk^+$. Thus, as
$N[G_0] \models ``|\wp(\FQ)| = k_2(2^\gk)$'', we can let
$\la D_\ga : \ga < \gk^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of
$\FQ$ present in $N[G_0]$.
For the purpose of the construction of
$G_1$ to be given below, we further
assume without loss of generality that
for every dense open subset
$D \subseteq \FQ$ found in $N[G_0]$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$.
Since the $\gk$ closure of $N$ with respect to either
$M$ or $V$ implies the least ordinal
at which $\FQ$ does a non-trivial forcing
is above $\gk^+$, the definition of $\FQ$ implies that
%as the Easton support iteration which adds
%a non-reflecting stationary set of ordinals of
%cofinality $\eta$ to each $N[G_0]$-measurable limit
%of strong cardinals in the interval
%$(\gk, k_2(\gk)]$ implies that
$N[G_0] \models ``\FQ$ is
${\prec} \gk^+$-strategically closed''.
By the fact the standard arguments show that
forcing with the $\gk$-c.c$.$ partial ordering
$\FP_\gk$ preserves that $N[G_0]$ remains
$\gk$-closed with respect to either
$M[G_0]$ or $V[G_0]$,
$\FQ$ is ${\prec} \gk^+$-strategically closed
in both $M[G_0]$ and $V[G_0]$.
We can now construct $G_1$ in either
$M[G_0]$ or $V[G_0]$ as follows.
Players I and II play a game of length
$\gk^+$. The initial pair of moves is
generated by player II choosing the
trivial condition $q_0$ and player
I responding by choosing
$q_1 \in D_1$.
Then, at an even stage $\ga + 2$,
player II picks
$q_{\ga + 2} \ge q_{\ga + 1}$ by
using some fixed strategy
${\cal S}$, where $q_{\ga + 1}$
was chosen by player I to be so that
$q_{\ga + 1} \in D_{\ga + 1}$ and
$q_{\ga + 1} \ge q_\ga$.
If $\ga$ is a limit ordinal, player II uses
${\cal S}$ to pick $q_\ga$ extending each
$q_\gb$ for $\gb < \ga$.
By the ${\prec} \gk^+$-strategic closure of
$\FQ$ in both $M[G_0]$ and $V[G_0]$,
the sequence
$\la q_\ga : \ga < \gk^+ \ra$
as just described exists.
By construction,
$G_1 = \{p \in \FQ : \exists \ga <
\gk^+ [q_\ga \ge p]\}$ is our
$N[G_0]$-generic object over $\FQ$.
It remains to construct in $V[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR$.
To do this, we first observe that as
$M \models ``\gk$ is a measurable limit of
strong cardinals'',
we can write
$k_1(\FP_\gk)$ as
$\FP_\gk \ast \dot \FS \ast \dot \FT$, where
$\forces_{\FP_\gk} ``\dot \FS =
\dot \FP_{\eta, \gk}$, and
%adds a
%non-reflecting stationary set of ordinals of
%cofinality $\eta$ to $\gk$'', and
$\dot \FT$ is a term for the rest of
$k_1(\FP_\gk)$.
Note now that
by clause \ref{i2} in the
definition of $\FP$,
the ordinals at which the
realization of $\dot \FT$
does a non-trivial forcing
lie in the interval
$(\gl, k_1(\gk))$, which implies that in
$M$,
$\forces_{\FP_\gk \ast \dot \FS}
``\dot \FT$ is ${\prec} \gl^+$-strategically
closed''. Further, by our assumption of GCH,
$2^\gl = \gl^+$ and
$|{[\gl]}^{< \gk}| = \gl$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter ${\cal U}$ over
$P_\gk(\gl)$,
$|k_1(\gk^+)| = |k_1(2^\gk)| =
|2^{k_1(\gk)}| =
|\{ f : f : P_\gk(\gl) \to \gk^+$ is a function$\}| =
|{[\gk^+]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{F} for the
first published account of term forcing or
\cite{C}, Section 1.2.5, page 8; the notion
is originally due to Laver) associated with
$\dot \FT$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP_\gk \ast \dot \FS$ and
$\forces_{\FP_\gk \ast \dot \FS} ``\tau \in
\dot \FT$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP_\gk \ast \dot \FS} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT$ 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 $\dot \FT$ is also forced
to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M \models ``|\FT^*| = k_1(\gk)$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP_\gk \ast \dot \FS} ``\dot \FT$ is
${\prec}\gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V$ as well.
Observe that
$M \models ``k_1(\gk)$ is measurable and
$|\FP_\gk \ast \dot \FS| < k_1(\gk)$'' and
$\forces_{\FP_\gk \ast \dot \FS} ``\dot \FT$ is
an Easton support iteration of length $k_1(\gk)$ and
$|\dot \FT| = k_1(\gk)$''.
We can thus let $\dot f$ be a term so that
$\forces_{\FP_\gk \ast \dot \FS}
``\dot f : k_1(\gk) \to \dot \FT$ is
a bijection''.
Since
$M \models ``|\FP_\gk \ast \dot \FS| < k_1(\gk)$'',
for each $\ga < k_1(\gk)$, let
$S_\ga = \{ r^\ga_\gb : \gb < \eta^\ga < k_1(\gk) \}$
be a maximal incompatible set of elements of
$\FP_\gk \ast \dot \FS$ so that for some term
$\tau^\ga_\gb$,
$r^\ga_\gb \forces ``\tau^\ga_\gb = \dot f(\ga)$''.
Define $T_\ga = \{\tau^\ga_\gb : \gb < \eta^\ga \}$ and
$T = \bigcup_{\ga < k_1(\gk)} T_\ga$. Clearly,
$|T| = k_1(\gk)$, so we can let
$\la \tau_\ga : \ga < k_1(\gk) \ra$ enumerate the
members of $T$.
%Each sequence
%$\la \tau^\ga_\gb : \gb < \eta^\ga < k_1(\gk) \ra$
%can be used to define a sequence of terms
$\la \tau_\ga : \ga < k_1(\gk) \ra$ is so that if
$\forces_{\FP_\gk \ast \dot \FS} ``\tau \in \dot \FT$'',
then for some $\ga < k_1(\gk)$,
$\forces_{\FP_\gk \ast \dot \FS} ``\tau = \tau_\ga$''.
Therefore, we can restrict the set of terms we choose so that
we can assume that in $M$,
$|\FT^*| = k_1(\gk)$. Since
$M \models ``2^{k_1(\gk)} = {(k_1(\gk))}^+ =
k_1(\gk^+)$'',
this means we can let
$\la D_\ga : \ga < \gl^+ \ra$
enumerate in $V$ the dense open subsets of $\FT^*$
found in $M$,
so that as before, for every
dense open subset
$D \subseteq \FT^*$ present in $M$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$,
and argue as we did when constructing $G_1$
to build in
$V$ an $M$-generic object $H_2$ over $\FT^*$.
As readers can verify for themselves,
this line of reasoning remains valid,
in spite of the fact $\gl$ is singular.
Note now that since $N$ can be assumed to be given
by an ultrapower of $M$ via a normal ultrafilter
${\cal U} \in M$ over $\gk$,
Fact 2 of Section 1.2.2 of \cite{C}
(see also \cite{F}) tells us that
$k_2 '' H_2$ generates an $N$-generic object
$G^*_2$ over $k_2(\FT^*)$. By elementariness,
$k_2(\FT^*)$ is the term forcing in $N$
defined with respect to
$k_2(k_1(\FP_\gk)_{\gk + 1}) =
\FP_\gk \ast \dot \FQ$.
Therefore, since
$j(\FP_\gk) = k_2(k_1(\FP_\gk)) =
\FP_\gk \ast \dot \FQ \ast
\dot \FR$,
$G^*_2$ is $N$-generic over
$k_2(\FT^*)$, and $G_0 \ast G_1$ is
$k_2(\FP_\gk \ast \dot \FS)$-generic over
$N$, Fact 1 of Section 1.2.5 of \cite{C}
(see also \cite{F}) tells us that for
$G_2 = \{i_{G_0 \ast G_1}(\tau) : \tau \in
G^*_2\}$, $G_2$ is $N[G_0][G_1]$-generic over
$\FR$.
Thus, in $V[G_0]$,
$j : V \to N$ extends to
$j : V[G_0] \to N[G_0][G_1][G_2]$, i.e.,
$V[G_0] \models ``\gk$ is $\gl$ strongly
compact''. As $\gl$ was an arbitrary
successor cardinal above $\gk$,
this completes the proof of Lemma \ref{l1}.
\item\label{i2} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ isn't a measurable limit
of strong cardinals,
$\dot \FQ_\ga$ is a term for the
Gitik-Shelah partial ordering of \cite{GS}
defined using component partial orderings
that are at least $\gg_\ga$-strategically
closed and $\eta$-directed closed
which first does a non-trivial
forcing at a measurable cardinal at least
$\gs_{\ga^-}$ and makes the strongness of
$\gd_\ga$ indestructible under forcing with
$\gd_\ga$-strategically closed partial orderings
which are at least $\eta$-directed closed.
and the fact
if
$\gd_\ga \in (\eta_\gk, \gk)$ is a strong
cardinal which isn't a limit of strong cardinals
(where
$\la \gd_\ga : \ga < \gk \ra$, $\gg_\ga$,
and $\gs_\ga$ are as in Section \ref{s2}),
then the definition of $\dot \FQ_\ga$
%$\eta_\gk =
%\bigcup_{\gd < \gk, \gd \in \K} \gd$ if
%$\bigcup_{\gd < \gk, \gd \in \K} \gd$ is
%regular, and
If $\gd$, the supremum of the members of $\K$
below $\gk$, is measurable but isn't
supercompact, then we impose the additional
requirement that the first non-trivial
forcing in the definition of
$\FP_\gk$ is done at a measurable cardinal
above $\gl$, where $\gl$ is the smallest
cardinal so that $\gd$ isn't $\gl$ supercompact,
and that $\FP_\gk$ is defined so as to be at least
$\rho$-strategically closed for
$\rho$ the least inaccessible cardinal above $\gl$.
%By the restrictions placed on the definition of
%$\FP_{\gk'}$
%the first non-trivial forcing in the definition of
cardinal above $\gg$ and is at least
$\rho$-strategically closed for $\rho$
the least inaccessible cardinal above
$\gg$.
Towards this end, let
$\gl > \gk$ be an arbitrary cardinal, and let
$\gg = |2^{[\gl]^{{<}\gk}}|$. Define
$\ov V = V^{\FP^\gk}$. Let
$j : \ov V \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$
(in $\ov V$) so that
$M \models ``\gk$ isn't $\gg$ supercompact''.
Take $\gk'$ as the least supercompact cardinal
above $\gk$ in $M$.
$\gk'$ exists since by elementarity,
$M \models ``j(\gk) > \gk$ is a supercompact limit
of supercompact cardinals''.
By the definition of $\eta_{\gk'}$
if the supremum of the elements of
$j(\K)$ below $\gk'$ is measurable but isn't
supercompact (which, by the choice of $j$, is
the situation under which we are working),
$\FP_{\gk'}$ is
$\eta_{\gk'}$-directed closed and
$\eta_{\gk'} > \gg$.
Also, the first non-trivial forcing
in the definition of $\FP_{\gk'}$ is
done at an ordinal above $\gg$
(namely $\eta_{\gk'}$).
Thus, if we write
$j(\FP_{{<}\gk}) = \FP_{{<}\gk} \times \FQ$ and
let $G$ be $\ov V$-generic over
$\FP_{{<}\gk}$ and $H$ be $\ov V[G]$-generic over
$\FQ$, in $\ov V[G \times H]$, $j$ lifts to
$j : \ov V[G] \to M[G \times H]$ via the definition
$\ov j(i_G(\tau)) = i_{G \times H}(j(\tau))$. Since
$M \models ``\FQ$ is $\eta_{\gk'}$-directed closed
and $\eta_{\gk'} > \gg$'',
the fact
$M^\gg \subseteq M$ implies
$\ov V \models ``\FQ$ is $\eta_{\gk'}^+$-strategically
closed'' yields that for any cardinal
$\gd \le \gg$,
$\ov V[G]$ and $\ov V[G \times H] =
\ov V[H \times G]$ contain the same
subsets of $\gd$.
This means the supercompact ultrafilter
${\cal U}$ over
${(P_\gk(\gl))}^{\ov V[G]}$ in
$\ov V[G \times H]$ given by
$x \in {\cal U}$ iff
$\la j(\ga) : \ga < \gl \ra \in \ov j(x)$
is so that
${\cal U} \in \ov V[G]$. Hence,
$V^{\FP^\gk \times \FP_{{<}\gk}} = V^\FP \models
``\gk$ is supercompact''.
As $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l5}.
Note that the measurable limits of
strong cardinals in
$V^{\FP^\gk}$ below $\gk$ and the measurable limits
of strong cardinals in $V$ below $\gk$ must be
precisely the same. To see this,
we first observe that since
$\FP^\gk$ is $\gk^+$-directed closed,
the measurable cardinals below $\gk$ in
$V^{\FP^\gk}$ and $V$ are precisely the same.
Next, we show that the strong cardinals below
$\gk$ in $V^{\FP^\gk}$ and $V$ are
precisely the same.
To do this, assume that
$V^{\FP^\gk} \models ``\gd < \gk$ is strong''.
As $\FP^\gk$ is $\gk^+$-directed closed,
this means that
$V \models ``\gd$ is $\ga$ strong for every
ordinal $\ga \in (\gd, \gk)$''. Since
$V \models ``\gk$ is supercompact and
hence strong'', by the second paragraph of
Lemma 2.1 of \cite{AC2},
$V \models ``\gd$ is strong''.
Hence, the strong cardinals in
$V^{\FP^\gk}$ below $\gk$ must be a subset
of the strong cardinals in $V$ below $\gk$.
Then, if
$V \models ``\gd < \gk$ is strong'',
by the directed closure properties of
$\FP^\gk$,
$V^{\FP^\gk} \models ``\gd$ is $\ga$
strong for every ordinal
$\ga \in (\gd, \gk)$ and $\gk$ is
supercompact'', so as we just observed,
$V^{\FP^\gk} \models ``\gd$ is strong''.
Further, by the work just done, in both
$V^{\FP^\gk}$ and $V$,
$\gd < \gk$ is strong iff
$\gd$ is $\ga$ strong for every
$\ga \in (\gd, \gk)$,
and the cardinals below $\gk$
which are $\ga$ strong for every
$\ga \in (\gd, \gk)$ are precisely
the same in both
$V^{\FP^\gk}$ and $V$.
We now know that the strong cardinals
and the measurable limits
of strong cardinals below $\gk$ in
$V^{\FP^\gk}$ and $V$ are precisely the same
and that these cardinals are precisely the same
as the cardinals $\gd < \gk$ which are
either $\ga$ strong for every
$\ga \in (\gd, \gk)$ or are measurable
limits of cardinals $\gg$ which are
$\ga$ strong for every
$\ga \in (\gd, \gk)$.
Also, the $\gk^+$-directed closure of
$\FP^\gk$ implies that
${(V_\gk)}^{V^{\FP^\gk}} =
{(V_\gk)}^V$.
Also, by the definition of
$\FP^\gk \times \FP_\gk$,
$V^{\FP^\gk \times \FP_\gk} \models
``$Unboundedly many cardinals in
$(\gg, \gk)$ contain non-reflecting
stationary sets of ordinals of
Further, since as we remarked at the
beginning of the proof of Lemma \ref{l1},
any supercompact cardinal has a normal
measure concentrating on strong cardinals,
$\gk$ is not a supercompact limit of
supercompact cardinals.
cofinality
In analogy to what was done in
Lemma \ref{l7}, write
$\FP^*$ as
However, since the analysis given in
Lemma \ref{l6} shows that
$\FP^* \ast \dot \FP$
is ``a mild forcing admitting
a gap at $\ha_1$'', the Gap Forcing Theorem
of \cite{H2} and \cite{H3} implies that
$\gd$ must be in
Therefore, the
proof of Lemma \ref{l7} will be
complete once we have shown
the L\'evy-Solovay results \cite{LS}
Observe that
$M \models ``k_1(\gk)$ is measurable and
$|\FP_\gk \ast \dot \FS| < k_1(\gk)$'' and
$\forces_{\FP_\gk \ast \dot \FS} ``\dot \FT$ is
an Easton support iteration of length $k_1(\gk)$ and
$|\dot \FT| = k_1(\gk)$''.
We can thus let $\dot f$ be a term so that
$\forces_{\FP_\gk \ast \dot \FS}
``\dot f : k_1(\gk) \to \dot \FT$ is
a bijection''.
Since
$M \models ``|\FP_\gk \ast \dot \FS| < k_1(\gk)$'',
for each $\ga < k_1(\gk)$, let
$S_\ga = \{ r^\ga_\gb : \gb < \eta^\ga < k_1(\gk) \}$
be a maximal incompatible set of elements of
$\FP_\gk \ast \dot \FS$ so that for some term
$\tau^\ga_\gb$,
$r^\ga_\gb \forces ``\tau^\ga_\gb = \dot f(\ga)$''.
Define $T_\ga = \{\tau^\ga_\gb : \gb < \eta^\ga \}$ and
$T = \bigcup_{\ga < k_1(\gk)} T_\ga$.
If $\ga$ is a successor ordinal so that
$\gd_{\ga^-}$ isn't supercompact,
then we impose the additional requirement that
$\dot \FQ_\ga$ is a term for a partial ordering
which first does a non-trivial
forcing at a measurable cardinal above
$\gs_{\ga^-}$ and that for $\gl$ the least
measurable cardinal above $\gs_{\ga^-}$,
$\dot \FQ_\ga$ is forced to be at least
$\gl$-strategically closed.
To exemplify a universe in which the
strongly compact cardinals exhibit
``bad behavior'',
consider the models
constructed in Theorems 1 and 2 of \cite{A97}.
Here, the non-supercompact strongly
compact cardinals and the supercompact
cardinals are distinguishable via a
fixed ground model function $f$, except at
measurable limit points.
As opposed to the results mentioned in the
preceding paragraph, there is no precise
or uniform way of characterizing the strongly compact
cardinals in terms of measurability,
strongness, or supercompactness.
then forcing GCH at each inaccessible
cardinal, and then forcing
which without loss of generality
can be assumed to do a
non-trivial forcing only at stages which
are measurable cardinals,
we may also assume that
$V \models
+ The only
strongly compact cardinals are the
elements of $\K$ or their measurable
limit points +
Let
$\gg$ be the supremum of the elements
of $\K$ below $\gd$, and let
$\gk$ be the least supercompact
cardinal above $\gd$. By our assumptions,
$\gd \in (\gg, \gk)$, and $\gk$ is not
a limit of supercompact cardinals.
This means we can write
$\FP = \FP^\gk \times \FP_\gk \times \FP_{{<}\gk}$,
where
$\FP^\gk$ and $\FP_{{<}\gk}$ are as in
the preceding paragraph, and infer as
in the preceding paragraph that
$V^\FP \models ``$No cardinal in the interval
$(\eta_\gk, \gk)$ is strongly compact''. Further,
since $\gk$ is the least supercompact cardinal above
$\gd$, the supremum of the supercompact cardinals
below $\gd$ must be the same as the supremum of
the supercompact cardinals below $\gk$.
Therefore, by their definitions,
$\gg < \eta_\gk$, so it must be the case that
$\gd \in (\gg, \eta_\gk]$. However, if
$\eta_\gk$ is a successor cardinal,
$\gg^+ = \eta_\gk$, meaning that
$\gd \not\in (\gg, \eta_\gk]$.
And, if $\eta_\gk$ is a limit cardinal,
then by its definition,
$\eta_\gk$ is the least inaccessible
cardinal above
${(\theta_\gg)}^V$,
which immediately implies that
$\gd \in (\gg, {(\theta_\gg)}^V )$.
Note that there are a number of
different universes which can be used
as the ground model $V_0$ for Theorem \ref{t1}.
One example is given by the model constructed
for the Main Theorem of \cite{AS97a}.
start by
adding a Cohen real and then doing its
next non-trivial forcing at a stage
which is a measurable cardinal,
we may also assume that
$V^{\FP^*}_0 = V \models
``$Every $\gk \in \K$ is
fully Laver indestructible \cite{L} +
For every
$\gk \in \K$, $2^\gk = \gk^+$''.
Since by its definition, in the
terminology of \cite{H2} and \cite{H3},
$\FP^*$ is a ``mild forcing admitting a
low gap'', by the Gap Forcing Theorem of
\cite{H2} and \cite{H3},
the only strongly compact cardinals
in $V$ are those which were strongly
compact in $V_0$. Since by the choice of
$V_0$, these cardinals are the elements of
$\K$ or their measurable limit points,
$V \models ``$The only strongly compact
cardinals are the elements of $\K$ or
their measurable limit points''.
$\eta_\gk =
{(\bigcup_{\gd < \gk, \gd \in \K} \gd)^+}$,
unless
$\gz$, the supremum of the members of $\K$
below $\gk$, is measurable but isn't
supercompact. Under these circumstances,
we let $\eta_\gk$ be the the least inaccessible
cardinal above $\theta_\gz$, where
$\theta_\gz$ is as defined in Section \ref{s2}.
These restrictions are possible, since as
we mentioned in Section \ref{s2},
if $\gz$ isn't supercompact,
then $\theta_\gz$ must be below the least strong
cardinal above $\gz$.
By its definition, $\eta_\gk$ is regular, and
$\eta_\gk < \gk$.
if the supremum of the elements of
$j(\K)$ below $\gk'$ is measurable but isn't
supercompact (which, by the choice of $j$, is
the situation under which we are working),
$The strongly compact
cardinals are measurable limits
of strong cardinals and are
the elements of $\K$
together with their measurable limit points''.
For $\gk_0 \in \K$ the least
supercompact cardinal, define
$\eta_{\gk_0} = \ha_1$.
For each $\gk \in \K$,
$\gk > \gk_0$ which isn't a
supercompact limit of supercompact
cardinals, define
$\eta_\gk$ as the least strong
cardinal above
$\sup(\{\gd < \gk : \gd \in \K\})$.
%$\bigcup_{\gd < \gk, \gd \in \K} \gd$.
Towards this end, fix
$\gd$ a cardinal which isn't either an
element of $\K$ or a measurable limit of
elements of $\K$.
By its definition, we can write
$\FP^* \ast \dot \FP$ as
$\FQ_0 \ast \dot \FQ_1$, where
$|\FQ_0| = \go$ and
$\forces_{\FQ_0} ``\dot \FQ_1$ is
$\ha_1$-strategically closed''.
Since this means
$\FP^* \ast \dot \FP$
is ``a mild forcing admitting
a gap at $\ha_1$'', the Gap Forcing Theorem
of \cite{H2} and \cite{H3} once again implies
that the only strongly compact cardinals
are the elements of $\K$ or their
measurable limit points,
as these are the only strongly compact
cardinals in both $V_0$ and $V$.
Thus, $\gd$ can't be strongly compact
in $V^\FP$.
This contradiction completes
the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``$The measurable limits
of strong cardinals are the elements of
$\K$ together with their measurable
limit points''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd$ is a measurable
limit of strong cardinals which is
neither an element of $\K$ nor a
measurable limit of elements of $\K$''.
Let
$\gg$ be the supremum of the elements
of $\K$ below $\gd$, and let
$\gk$ be the least supercompact
cardinal above $\gd$. By our assumptions,
$\gd \in (\gg, \gk)$, and $\gk$ is not
a limit of supercompact cardinals.
This means we can write
$\FP = \FP^\gk \times \FP_\gk \times \FP_{{<}\gk}$,
where
$\FP^\gk$ and $\FP_{{<}\gk}$ are as in
Lemmas \ref{l5} and \ref{l6}. Since as
in the proof of Lemma \ref{l6},
$|\FP_{{<}\gk}| < \gk$ in either
$V$, $V^{\FP^\gk}$, or
$V^{\FP^\gk \times \FP_\gk}$,
Woodin's results of \cite{HW}
stating that small forcing
cannot create new strong cardinals,
together with the results of \cite{LS},
imply that
$V^{\FP^\gk \times \FP_\gk} \models
``\gd$ is a measurable limit of
strong cardinals''.
As in the proof of Lemma \ref{l5}, we know
that the measurable limits of strong
cardinals below $\gk$ in $V$ and $V^{\FP^\gk}$
are precisely the same. Further, as in
the proof of Lemma \ref{l6}, $\FP_\gk$
has the same definition and properties
regardless if it's defined in
$V$ or $V^{\FP^\gk}$.
Therefore, by the proof of Lemma \ref{l4},
$V^{\FP^\gk \times \FP_\gk} \models
``$No cardinal in the interval
$(\eta_\gk, \gk)$ is a measurable
limit of strong cardinals''.
Since $\gk$ is the least supercompact cardinal above
$\gd$, the supremum of the supercompact cardinals
below $\gd$ must be the same as the supremum of
the supercompact cardinals below $\gk$. Hence,
if $\eta_\gk$ is a successor cardinal,
by the definition of $\eta_\gk$,
any inaccessible cardinal in the interval
$(\gg, \gk)$ is actually an element of
the interval $(\eta_\gk, \gk)$.
This means that
$V^{\FP^\gk \times \FP_\gk} \models
``\gd$ isn't a measurable limit of
strong cardinals'', and the proof of
Lemma \ref{l7} is complete.
If, however, $\eta_\gk$ is a limit
cardinal, then by its definition,
$\eta_\gk$ is the least inaccessible
cardinal above
${(\theta_\gg)}^V$.
Under these circumstances,
%we must then have that
the last sentence of the
preceding paragraph implies that
$\gd \in (\gg, \eta_\gk]$,
so by $\eta_\gk$'s definition,
$\gd \in (\gg, {(\theta_\gg)}^V]$.
However, since
$\FP^\gk \times \FP_\gk$ is
$\rho$-strategically closed for every
$\rho < \eta_\gk$, this means
$V^{\FP^\gk \times \FP_\gk} \models ``\gg$
is $\gz$ supercompact for every
$\gz < \gd' < \gd$,
where $\gd'$ is a strong cardinal
and $\gd$ is a measurable limit of
strong cardinals''.
This immediately yields that
$V^{\FP^\gk \times \FP_\gk} \models
``\gg$ is supercompact'', so the
strategic closure properties of
$\FP^\gk \times \FP_\gk$ tell us that
$V \models ``\gg$ is ${(\theta_\gg)}^V$
supercompact'',
a contradiction to the
definition of ${(\theta_\gg)}^V$.
This tells us that
$V^{\FP^\gk \times \FP_\gk} \models
``\gd$ isn't a measurable limit of
strong cardinals''.
This last contradiction completes the
proof of Lemma \ref{l7}.
\end{proof}
By letting
$\FP' = \FP^* \ast \dot \FP$,
\end{graveyard}