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\title{Some Remarks on Indestructibility and Hamkins' Lottery Preparation
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal, lottery preparation,
indestructibility, non-reflecting stationary
set of ordinals}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank Joel Hamkins
and Dorshka Wylie for helpful discussions on the subject
matter of this paper. The author also wishes to
express his sincere gratitude to
the referee for a thorough reading of two versions
of the
manuscript and numerous helpful comments
and suggestions, many of which have been
incorporated into this version of the paper.}\\
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{July 19, 2000\\
(revised July 21, 2002
and January 26, 2003)}
\begin{document}
\maketitle
\begin{abstract}
In this paper,
we first prove several general
theorems about strongness, supercompactness,
and indestructibility,
along the way giving some new applications of
Hamkins' lottery preparation forcing to
indestructibility.
We then show that it is consistent,
relative to the existence of cardinals
$\gk < \gl$ so that $\gk$ is $\gl$
supercompact and $\gl$ is inaccessible,
for the least strongly compact
cardinal $\gk$ to be the least strong cardinal
and to have its strongness, but not
its strong compactness, indestructible under
$\gk$-strategically closed forcing.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{H4}, Hamkins introduced a general method
of forcing indestructibility he calls the lottery preparation.
Besides acting as an alternative to the Laver
preparation \cite{L} for making the supercompactness of
a cardinal $\gk$ indestructible under $\gk$-directed
closed forcing, the lottery preparation also can
be used to force various forms of indestructibility
for strong and strongly compact cardinals.
Readers are advised to consult Section 4 of
\cite{H4} for further details.
The purpose of this paper is to expand upon
the work of \cite{H4}, by
proving several general theorems about
strongness, supercompactness,
and indestructibility, and by
proving a theorem about indestructibility
as related to the least strong and strongly
compact cardinal.
More descriptively,
in the list of results we give below,
Theorem \ref{t1} shows how
to force indestructibility
for the class of supercompact cardinals via a
variant of Hamkins' lottery preparation
making no reference to either a
Laver function or a fast
function.\footnote{The relevant
definitions of these
functions are given in
\cite{L} and \cite{H4}, but for convenience,
we will repeat them below.}
Theorem \ref{t2}, which is essentially
Theorem 1 of \cite{A98}
proven without the use of a Laver
function and augmented
by Hamkins' gap forcing results of
\cite{H1}, \cite{H2}, and \cite{H3},
employs a slightly generalized version of the
forcing used in the proof of Theorem \ref{t1}.
It is a corollary of Theorem \ref{t1}.
As is the case with Theorem 1 of \cite{A98},
the partial ordering for Theorem \ref{t2}
forces a property
(the coincidence of the strongly compact
and supercompact cardinals, except
possibly at measurable limit points)
due originally to Kimchi and Magidor \cite{KM}.
Theorem \ref{t3} provides
an example of a model in
which the supercompact and strong cardinals
all exhibit a certain
level of indestructibility.
Theorem \ref{t4} is an augmentation
of Theorem \ref{t3} in the same sense that
Theorem \ref{t2} is an augmentation of
Theorem \ref{t1}, since in the
model for Theorem \ref{t4}, the aforementioned
property of \cite{KM} is attained as well.
The proof of Theorem \ref{t4}
builds upon the proofs of
Theorems \ref{t2} and \ref{t3}.
Theorem \ref{t5}
obtains a model in which not only is the
least strongly compact cardinal $\gk$ the least
strong cardinal (a theorem first proven in
\cite{AC2}), but the strongness of $\gk$ is
indestructible under $\gk$-strategically closed
forcings.
(See \cite{AH1}, \cite{ASt}, \cite{GS},
and \cite{H4} for additional
results on strongness and indestructibility, and
\cite{A98}, \cite{A99}, \cite{A00}, \cite{AG},
\cite{AH1}, \cite{H4}, and \cite{ANew}
for additional results on strong
compactness and indestructibility.)
Before stating our theorems,
we first make a preliminary
definition.
For any ordinal $\gd$, define
for the rest of this paper $\gs_\gd$
as the smallest regular cardinal
greater than or equal to the
supremum of the supercompact
cardinals below $\gd$, or
$\go$ if there are no
supercompact cardinals below $\gd$.
It will be the case that
$\gs_\gd$ is absolute between
$V$ and any relevant generic
extension $V^\FP$.
Keeping this in mind,
Theorems \ref{t1} - \ref{t5}
may therefore now be explicitly stated
as follows.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + ${\mathfrak K}$ is
the class of supercompact cardinals''. There is
then a partial ordering $\FP \subseteq V$,
a version of Hamkins' lottery preparation
making no reference to either a Laver function or
a fast function, so that in
$V^\FP$, every
supercompact cardinal $\gk$ is indestructible
under $\gk$-directed closed forcing.
In addition, in $V^\FP$,
${\mathfrak K}$ is the
class of supercompact cardinals.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models ``$ZFC + ${\mathfrak K}$ is
the class of supercompact cardinals''. There is
then a partial ordering $\FP \subseteq V$,
a version of Hamkins' lottery preparation
making no reference to either a Laver function or
a fast function, so that in
$V^\FP$, every
supercompact cardinal $\gk$ is indestructible
under $\gk$-directed closed forcing, and the
supercompact and strongly compact cardinals
coincide, except possibly at measurable limit points.
In addition, in $V^\FP$,
${\mathfrak K}$ is the
class of supercompact cardinals.
Finally, in $V^\FP$,
any strongly compact cardinal $\gk$ which is a
measurable limit of strongly
compact cardinals has its strong compactness
indestructible under $\gk$-directed closed
forcings which preserve $\gk$'s measurability.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH + ${\mathfrak K}^*$
is the class of strong cardinals +
${\mathfrak K}$ is
the class of supercompact cardinals''.
%For each $\gd$, define $\gs_\gd$
%as the smallest regular cardinal
%greater than or equal to
%the supremum of the supercompact
%cardinals below $\gd$.
There is
then a partial ordering $\FP \subseteq V$,
a version of Hamkins' lottery preparation, so that
in $V^\FP$, every strong
cardinal $\gk$ is indestructible under
$\gs_\gk$-directed closed forcings
which are also $\gk$-strategically
closed, and every supercompact
cardinal $\gk$ is indestructible under
$\gk$-directed closed forcings
which are also $\gk$-strategically closed.
In addition, in
$V^\FP$, ${\mathfrak K}^*$ is the class of
strong cardinals, and
${\mathfrak K}$ is the
class of supercompact cardinals.
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + GCH +
${\mathfrak K}^*$ is the class of strong
cardinals + ${\mathfrak K}$ is
the class of supercompact cardinals''.
Let
$\Gamma$ be either ${(\sup(\K))}^+$ if
$\K$ is a set, or the class
of all ordinals if $\K$ is
a proper class.
There is
then a partial ordering $\FP \subseteq V$,
a version of Hamkins' lottery preparation, so that
in $V^\FP$, every strong
cardinal $\gk$ is indestructible under
$\gs_\gk$-directed closed forcings
which are also $\gk$-strategically
closed, every supercompact
cardinal $\gk$ is indestructible under
$\gk$-directed closed forcings
which are also $\gk$-strategically closed,
and the supercompact and strongly compact
cardinals coincide, except possibly
at measurable limit points.
In addition, in
$V^\FP$, ${\mathfrak K}^* \cap \Gamma$
is the class of
strong cardinals, and
${\mathfrak K}$ is the
class of supercompact cardinals.
Finally, in $V^\FP$,
any strongly compact cardinal $\gk$ which is a
measurable limit of strongly
compact cardinals has its strong compactness
indestructible under $\gk$-directed closed
forcings which preserve $\gk$'s measurability.
\end{theorem}
\begin{theorem}\label{t5}
Suppose $V \models ``$ZFC + GCH + $\gk < \gl$
are the smallest cardinals
so that $\gk$ is $\gl$ supercompact and
$\gl$ is inaccessible''.
There is then a partial ordering
$\FP \in V$, $|\FP| = \gk$, so that
$\ov V = V^\FP_\gl \models ``$ZFC +
$\gk$ is both the least strongly compact
and least strong cardinal + The strongness of
$\gk$, but not its strong compactness,
is indestructible under $\gk$-strategically
closed forcings''.
\end{theorem}
We take this opportunity to make some
additional comments on
Theorems \ref{t1} - \ref{t4} above.
Partial orderings for forcing the
indestructibility of a supercompact cardinal
$\gk$ under $\gk$-directed closed partial orderings,
such as those given in \cite{L}, \cite{A98}, or
\cite{H4}, generally make use of either some form of
Laver function $f$ or fast function
$g$.\footnote{Although Hamkins' lottery
preparation of \cite{H4} can be given
with respect to functions other than
Laver or fast functions, in order to force
indestructibility via a lottery preparation,
Hamkins' arguments require a certain
sufficient condition
deducible from Laver or fast
functions, namely what Hamkins
calls the Menas property.
The version of the lottery
preparation we will give in
the proofs of Theorems \ref{t1}
and \ref{t2} will use a function
satisfying this property as well.}
Neither of these functions is really canonical,
in the sense that a Laver function is defined by
making arbitrary choices of counterexamples to
a certain property and a fast function is forced.
The form of the lottery preparation
given in Theorems \ref{t1} and \ref{t2}
has as its advantage not only
that it can be given
%completely canonically
with respect to a canonical function
%that satisfies what Hamkins calls on page 116 of
%\cite{H4} the ``supercompact Menas property'',
without first either having to specify a fixed
Laver function or force a fast function,
but that unlike the lottery preparation
given in \cite{H4},
it also forces indestructibility
for all supercompact cardinals.
Also, by its definition,
$\gs_\gd \le \gd$ whenever $\gd$
is a regular cardinal,
with equality achieved only when
$\gd$ is a
limit of supercompact cardinals.
This means that
in Theorems \ref{t3} and \ref{t4},
every strong cardinal $\gk$ and
every supercompact
cardinal $\gk$ is indestructible
under $\gk$-directed closed
forcings which are also
$\gk$-strategically closed.
If, however, $\gk$ is a strong
cardinal which isn't a limit
of supercompact cardinals, then
$\gk$ exhibits a somewhat stronger
form of indestructibility, since
$\gs_\gk < \gk$.
Before giving the proofs of our theorems, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb], [\ga, \gb), (\ga, \gb]$, and $(\ga,
\gb)$ are as
in standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will
abuse notation somewhat and use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If we also have that $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an Easton support iteration of length $\gk$
so that at stage $\ga$, a non-trivial forcing is done
based on the ordinal $\gd_\ga$, then we will say that
$\gd_\ga$ is in the field of $\FP$.
We will frequently (although not necessarily always)
write such an iteration as
$\FP_\ga \ast \dot \FP^\ga$, where
$\FP_\ga$ is the portion of $\FP$
through stage $\ga$, and
$\dot \FP^\ga$ is a term for the
portion of $\FP$ defined from
stage $\ga$ onwards.
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 $V$.
We recall for the benefit of readers Hamkins'
definition from Section 3 of \cite{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the lottery sum is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and
then forces with it.\footnote{The
terminology ``lottery sum'' is due
to Hamkins, although the concept
of the lottery sum of partial
orderings has been around for quite
some time and has been referred to
at different junctures via the names
``disjoint sum of partial orderings'',
``side-by-side forcing'', and
``choosing which partial ordering to
force with generically''.}
If $\gk$ is a regular cardinal and $\FP$ is
a 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 $\gk$-closed if for every sequence
$\langle p_\ga: \ga < \gd \le \gk \rangle$
of elements of $\FP$ so that
$\beta < \gamma < \gd$ implies $p_\beta \le p_\gamma$ (an increasing
chain of length
$\gd$), there is some $p \in \FP$ (an upper bound to this chain)
so that $p_\ga \le p$ for all $\ga < \gd$.
$\FP$ is ${<}\gk$-closed if $\FP$ is $\delta$-closed for all cardinals
$\gd < \gk$.
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$. $\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$ \FP$ is ${<} \gk$-strategically closed
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
Note that trivially, if $\FP$ is $\gk$-closed, then $\FP$ is
$\gk$-strategically
closed and ${\prec} \gk^+$-strategically closed.
Also, if $\FP$ is either $\gk$-closed or
$\gk$-strategically closed, then $\FP$ is
$\gk$-distributive.
Suppose that
$\go \le \gk < \gl$
are regular cardinals.
A partial ordering $\FP_{\gk, \gl}$ that will be used
throughout the course of this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP_{\gk, \gl}$ is
defined as
$\{ p$ : For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\gb \in S_p$ implies
$\gb > \gk$ and cof$(\gb) = \gk \}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP_{\gk, \gl}$ (see
\cite{Bu} or \cite{A99b}), in $V[G]$,
if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G \} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced, the
bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH
have been preserved.
It is also virtually immediate that $\FP_{\gk, \gl}$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu} or \cite{A99b}) that
$\FP_{\gk, \gl}$
is ${\prec} \gl$-strategically 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.
We would specifically like to
point out, however, that a
measurable limit point of the
class of strongly compact cardinals
is a cardinal $\gk$ which is both
measurable and a limit of strongly
compact cardinals.
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 $\gl > \gk$.
At different points in
this paper, we will use Solovay's Theorem 4.8 of
\cite{SRK} and the succeeding
remarks of \cite{SRK}. These tell us that
if $\gd < \gg$ are regular cardinals
and $\gg$ contains a non-reflecting
stationary set of ordinals of
cofinality $\gd$, then no cardinal
in the interval
$(\gd, \gg]$ is strongly compact.
This has as an immediate consequence
that if $\gd$ is a regular cardinal
and unboundedly many cardinals
$\gg \in (\gd, \gk)$ contain
non-reflecting stationary sets
of ordinals of cofinality $\gd$,
then there are no strongly
compact cardinals in the interval
$(\gd, \gk)$.
We recall for the benefit of readers
the definitions of Laver function and
fast function. If
$V \models ``\gk$ is supercompact'',
then as defined by Laver in \cite{L},
a Laver function is a function
$f : \gk \to V_\gk$ so that for every
$x \in V$ and every cardinal
$\gl \ge |{\rm TC}(x)|$, there is an
elementary embedding
$j : V \to M$ generated by a
supercompact ultrafilter
${\cal U}$ over $P_\gk(\gl)$ so that
$j(f)(\gk) = x$.
By Laver's work of \cite{L}
(see also \cite{A98}), the function $f$
can be assumed to be total,
and non-trivially defined only
on the measurable cardinals.
A generalized version of a Laver
function will be constructed in
Section \ref{s3}.
The notion of fast function is
originally due to Woodin and is
exposited by Hamkins in \cite{H4}.
Suppose $\gk$ is an inaccessible
limit of inaccessible cardinals.
Then the fast function partial ordering
$\FF$ is defined as
$\{f : \gk \to \gk$ : $f$ is a
partial function with $\dom(f)$
a subset of the inaccessible cardinals
below $\gk$ so that if $\gd \in \dom(f)$, then
$f '' \gd \subseteq \gd$ and
$|f \rest \gd| < \gd\}$, ordered by inclusion.
The generic function added by forcing with
$\FF$ is called a fast function.
Readers are urged to consult \cite{H4} for
more information on the properties of this
partial ordering.
Finally, we mention that we are also assuming some
familiarity with the basics of extender technology
and the transference of generic objects via
elementary embeddings.
The section on background material of \cite{C}
is extremely useful in this regard.
We will freely in Sections \ref{s3} and \ref{s4}
use notation, definitions, and terminology
found here.
Readers may also consult \cite{MS} for
additional information concerning extenders.
\section{Indestructibility and Supercompactness}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V$ and ${\mathfrak K}$
be as in the hypotheses of Theorem \ref{t1}. Let
${\mathfrak D} = \la \gd_\ga : \ga < \Omega \ra$, where
$\Omega$ is either the class of all ordinals
if there is a proper class of strong cardinals
or the appropriate ordinal otherwise,
enumerate in increasing order all strong cardinals.
The partial ordering $\FP$ with which we force
is the Easton support iteration which first
adds a Cohen subset to $\omega$ and then, for each
$\gd \in {\mathfrak D}$, forces with the lottery sum
of all $\gd$-directed closed partial orderings
having rank below the least
$V$-strong cardinal above $\gd$.
If there is no strong cardinal above $\gd$
in $V$, then
we force with the trivial partial ordering $\{0\}$.
Note that this differs
somewhat from the definition of the
lottery preparation with respect to
the function
$$f(\gd) = {\hbox{\rm The least $V$-strong
cardinal above $\gd$}}$$
found on page 127 of \cite{H4},
as will become clear when the
partial ordering used in the proof
of Theorem \ref{t3} is given.
We observe that
by the definition of $\FP$, $V^\FP \models {\rm ZFC}$.
Fix $\gk \in {\mathfrak K}$. Write
$\FP = \FP_\gk \ast \dot \FP^\gk$.
%where $\FP_\gk$ is the portion of $\FP$
%through stage $\gk$ and $\dot \FP^\gk$ is a term for
%the portion of $\FP$ defined from stage $\gk$ onwards.
By the specification of $\FP$,
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is $\gk$-directed closed''.
Let
$\FQ \in V^{\FP_\gk}$ be so that
$V^{\FP_\gk} \models ``\FQ$ is
$\gk$-directed closed''.
The proof of Theorem \ref{t1} will be complete
once we have shown that
%$\gk \in {\mathfrak K}$ and $\FQ \in V^{\FP_\gk}$ so that
%$V^{\FP_\gk} \models ``\FQ$ is $\gk$-directed closed'',
$V^{\FP_\gk \ast \dot \FQ} \models
``\gk$ is supercompact''.\footnote{At
first glance, it may appear as
though the proof of Theorem \ref{t1}
is not complete if indestructibility
is only shown for set forcing.
However, as Laver says at the end
of \cite{L}, ``It is a corollary of
the indestructibility of the
supercompactness of $\gk$ under
$\gk$-directed closed orderings,
that $\gk$ is also supercompact
in any $V^P$ such that $P$ is
a $\gk$-directed closed Easton
or upward Easton class partial
ordering.''
It is this phenomenon that
tells us indestructibility
under set forcing suffices
for the proof of Theorem \ref{t1},
as $\FP^\gk$ is in
$V^{\FP_\gk}$ what Laver
refers to as an ``upward
Easton class partial ordering.''
We will use this fact implicitly
throughout the rest of the
paper when relevant,
in the proofs of
Theorems \ref{t2} - \ref{t4}.}
Fix such a $\FQ$. Let
$\gl > \max(2^\gk, |{\rm TC}(\dot \FQ)|)$
be given, and let
$\gg = |2^{\gl^{< \gk}}|$. Let
$j : V \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$
so that
$M \models ``\gk$ isn't $\gg$ supercompact''.
We first observe that
$M \models ``$No cardinal
$\gd \in (\gk, \gg]$ is strong'',
since otherwise, $\gk$ would be
supercompact up to a strong cardinal
and hence would be supercompact.
(See the proof of Lemma 2.4 of
\cite{AC2} for a demonstration of this fact.)
Note now that
Lemma 2.1 of \cite{AC2} and the
succeeding remark yield that since
$\gg \ge 2^\gk$,
$M \models ``\gk$ is strong''. Thus,
$M \models ``$The least strong cardinal
$\gd > \gk$ is also above $\gg$'', which means
that in $M$, $\dot \FQ$ is a term for a
$\gk$-directed closed partial ordering
having rank below the least strong cardinal
above the strong cardinal $\gk$.
Therefore, in $M^{\FP_\gk}$, $\FQ$ is part of
the lottery sum of all $\gk$-directed closed
partial orderings having rank below the least
$M$-strong cardinal above $\gk$.
This means we can at stage $\gk$ in $M$
choose a term
$\dot p_0$ for a
condition $p_0$ in the lottery sum
which ensures $\FQ$ is chosen as the partial
ordering with which we force.
Hence, above a condition mentioning $\dot p_0$,
$j(\FP_\gk \ast \dot \FQ)$ is forcing equivalent to
$\FP_\gk \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$,
where $\dot \FR$ is a term in $M$ for the portion of
$j(\FP_\gk \ast \dot \FQ)$ up to $j(\gk)$ above the
stage $\gk$ forcing.
We are now in a position to be able to use
the standard reverse Easton techniques to
show that
$V^{\FP_\gk \ast \dot \FQ} \models ``\gk$ is
$\gl$ supercompact''. Specifically, let
$G_0$ be $V$-generic over $\FP_\gk$ and
$G_1$ be $V[G_0]$-generic over $\FQ$.
The usual arguments show that
$M[G_0][G_1]$ remains $\gg$ closed with
respect to $V[G_0][G_1]$.
Thus, by the definitions of
$\FP_\gk$ and $\FR$,
since every ordinal in the field of $\FR$
is above $\gg$,
$\FR$ is $\gg^+$-directed closed in
$M[G_0][G_1]$ and hence by closure in
$V[G_0][G_1]$ as well.
Also, since we are dealing with
the usual sort of Easton support iterations throughout,
if $G_2$ is $V[G_0][G_1]$-generic over $\FR$,
$M[G_0][G_1][G_2]$ remains $\gg$ closed with respect to
$V[G_0][G_1][G_2]$, and
$j$ lifts in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$.
This means we can now by closure
and the definition of $j(\FQ)$ find in
$V[G_0][G_1][G_2]$ a master condition $q$ for
$j''G_1$ and let $G_3$ be a
$V[G_0][G_1][G_2]$-generic object over $j(\FQ)$
containing $q$.
Again since we are dealing with Easton
support iterations,
$j$ lifts in $V[G_0][G_1][G_2][G_3]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$, so
$V[G_0][G_1][G_2][G_3] \models ``\gk$ is
$\gl$ supercompact''.
As $G_2 \ast G_3$ is
$V[G_0][G_1]$-generic over $\FR \ast j(\dot \FQ)$,
a partial ordering which is $\gg^+$-directed
closed in $V[G_0][G_1]$,
$V[G_0][G_1] \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ was arbitrary,
$V^{\FP_\gk \ast \dot \FQ} \models ``\gk$ is
supercompact''.
The proof of Theorem \ref{t1} will be complete
once we have shown that
$V^\FP \models ``{\mathfrak K}$ is the
class of supercompact cardinals''.
To do this, write
$\FP = \FP' \ast \dot \FP''$, where
$|\FP'| = \omega$ and
$\forces_{\FP'} ``\dot \FP''$ is
$\ha_1$-strategically closed''.
In Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ ``admits a gap at $\ha_1$'',
so by the results of \cite{H1}, \cite{H2},
and \cite{H3}, any cardinal supercompact in
$V^\FP$ had to have been supercompact in $V$.
Since the preceding paragraphs show that
all $V$-supercompact cardinals remain
supercompact in $V^\FP$,
${\mathfrak K}$ is the class of supercompact
cardinals in both $V$ and $V^\FP$.
This proves Theorem \ref{t1}.
\end{proof}
We note that in the proof just given,
the fact that $\FR$ is $\gg^+$-directed
closed is not necessary to complete
the argument.
If we only knew $\FR$ were
$\gg$-strategically closed,
the line of reasoning remains valid.
We will use this in the proof
of Theorem \ref{t2} that follows.
As Hamkins has pointed out to us,
the choice of ${\mathfrak D}$ as an
enumeration of all of the strong cardinals
in the universe can be modified.
Since any supercompact cardinal $\gk$ is
a limit of strong cardinals
(a proof of this fact is given in
Lemma 2.1 of \cite{AC2} and the
succeeding remark),
it is possible, e.g., to choose
${\mathfrak D}$ as an enumeration of
all of the regular limits of strong
cardinals and then define $\FP$ above
in the same manner as before.
We will exploit this fact in the proof
of Theorem \ref{t2}, which we give now.
\begin{proof}
Let $V$ and ${\mathfrak K}$
be as in the hypotheses of Theorem \ref{t2}.
Let
${\mathfrak D} = \la \gd_\ga : \ga < \Omega \ra$, where
$\Omega$ is either the class of all ordinals
if ${\mathfrak K}$ is a proper class
or is $\sup({\mathfrak K})$ otherwise,
enumerate in increasing order all
regular limits of strong cardinals
below $\Omega$.
Note that if $\mathfrak K$ is a proper
class, $\mathfrak D$ enumerates all
regular limits of strong cardinals in
the universe.
The partial ordering $\FP$ with which we force
will have the form
$\FP_\Omega \ast \dot \FP^\Omega$.
$\FP_\Omega$ is the Easton support iteration
of length $\Omega$ which first
adds a Cohen subset to $\omega$ and then, for each
$\gd \in {\mathfrak D}$, forces with the lottery sum
of all $\gd$-directed closed partial orderings
having rank below the least
$V$-strong cardinal $\gd'$ above
$\gd$ followed by the partial ordering which adds a
non-reflecting stationary set of ordinals of cofinality
$\sigma_{\gd'}$ to $\gd'$.
If $\mathfrak K$ is a proper class, then
$\dot \FP^\Omega$ is a term for the
trivial partial ordering $\{0\}$.
If $\mathfrak K$ is a set, then
%the partial ordering $\FP$ with which we force is
%$\FP_\Omega \ast \dot \FP^\Omega$, where
%$\FP_\Omega$ is the same as the partial
%ordering defined when $\mathfrak K$ is
%a proper class, and
$\dot \FP^\Omega$ is a term for the
Easton support iteration which adds
to each $V$-measurable cardinal
$\gd > \Omega$ a non-reflecting
stationary set of ordinals of
cofinality $\Omega^+$.
As in the proof of Theorem \ref{t1},
$V^\FP \models {\rm ZFC}$.
In addition, the definition of $\FP$
(see the argument given in
the first paragraph on page 153 of
\cite{A98}) yields that each
$\gs_\gd$ remains regular in $V^\FP$.
Now, for $\gk \in {\mathfrak K}$,
%let $\FP_\gk$ and $\dot \FP^\gk$ be as above.
write $\FP = \FP_\gk \ast \dot \FP^\gk$.
Since the definition of $\FP$ ensures that
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is $\gk$-directed closed'',
it suffices to show
$V^{\FP_\gk} \models ``$The supercompactness
of $\gk$ is indestructible under forcing with
$\gk$-directed closed partial orderings''.
To do this, let
$\FQ \in V^{\FP_\gk}$ be so that
$V^{\FP_\gk} \models ``\FQ$ is $\gk$-directed closed''.
If we fix $\gl$, $\gg$, and $j$ as before,
then since
$M \models ``\gk$ is a regular limit of strong cardinals'',
we will once again have that $\FQ$ is an
allowable choice at stage $\gk$ in $M^{\FP_\gk}$.
The only difference now is that after
forcing in $M^{\FP_\gk}$ with $\FQ$,
a non-reflecting stationary set of ordinals of
cofinality $\sigma_{\gk'}$ will be added to the least
$M$-strong cardinal $\gk'$ above $\gk$.
This forcing followed by the analogue of the
forcing $\FR$ of Theorem \ref{t1}, however, is
$\gg$-strategically closed, and as we remarked earlier,
this is sufficient to allow the proof of indestructibility
given in Theorem \ref{t1} to go through as before. Thus,
$V^{\FP_\gk} \models ``$The supercompactness
of $\gk$ is indestructible under forcing with
$\gk$-directed closed partial orderings''.
And, now that we know this, we can once again
infer as in Theorem \ref{t1} that
$V^\FP \models ``{\mathfrak K}$ is the class of
supercompact cardinals''.
Assuming now that
$V^\FP \models ``$The supercompact and strongly
compact cardinals coincide, except possibly at measurable
limit points'', the same proof as given in Lemma 4
of \cite{A98} will show that
$V^\FP \models ``$Any strongly compact cardinal
$\gk$ which is a measurable limit of
strongly compact cardinals has its strong
compactness indestructible under
$\gk$-directed closed forcings which preserve
$\gk$'s measurability''.
Thus, the proof of Theorem \ref{t2} will be
complete once we have shown the following.
\begin{lemma}\label{l1}
$V^\FP \models ``$The supercompact and strongly
compact cardinals coincide, except possibly
at measurable limit points''.
\end{lemma}
\begin{proof}
We begin by noting that if
$\mathfrak K$ is a set, then
$V^\FP \models ``$No cardinal
$\gd > \Omega$ is measurable''.
To see this,
note that since $\FP$
admits a gap at $\ha_1$, by the
results of \cite{H1}, \cite{H2},
and \cite{H3}, any cardinal
measurable in $V^\FP$ had to have
been measurable in $V$.
Therefore, since the definition of
$\FP$ if $\mathfrak K$ is a set
ensures that
$V^\FP \models ``$All $V$-measurable
cardinals $\gd > \Omega$ contain
non-reflecting stationary sets of
ordinals of cofinality $\Omega^+$
and hence are no longer weakly compact'',
$V^\FP \models ``$No cardinal
$\gd > \Omega$ is measurable''.
Hence, regardless
if ${\mathfrak K}$
is a set or proper class,
if $\rho$ is a strongly compact cardinal
in $V^\FP$ which is neither an element of
${\mathfrak K}$ nor a measurable limit
point of ${\mathfrak K}$, then
$\rho \in (\sigma_\rho, \gk)$, where
$\gk$ is the least element of
${\mathfrak K}$ above $\rho$.
Note that by the choice of $\gk$,
$\gk$ isn't a limit of
supercompact cardinals.
Let $\gd \in (\sigma_\rho, \gk)$ be a
regular limit of strong cardinals in $V$. Write
$\FP = \FP_\gd \ast \dot \FQ_0 \ast
\dot \FQ_1 \ast \dot \FP^{\gd + 1}$, where
%$\FP_\gd$ is the portion of $\FP$ through
%stage $\gd$,
$\dot \FQ_0$ is a term for the lottery sum of
$\gd$-directed closed partial orderings in
$V^{\FP_\gd}$ having rank below the least
$V$-strong cardinal $\gd'$ above $\gd$ and
$\dot \FQ_1$ is a term for the partial ordering
which adds a non-reflecting stationary set of
ordinals of cofinality
$\gs_{\gd'} = \sigma_\rho$ to $\gd'$.
%and $\dot \FP^\gd$ is a term for the rest of $\FP$.
By the definition of $\FQ_0$,
$V^{\FP_\gd \ast \dot \FQ_0} \models
``\gd'$ is a regular cardinal'', meaning by the
definition of $\FQ_1$ that
$V^{\FP_\gd \ast \dot \FQ_0 \ast \dot \FQ_1} \models
``\gd'$ contains a non-reflecting stationary set of
ordinals of cofinality $\sigma_\rho$''. Since
$V^{\FP_\gd \ast \dot \FQ_0 \ast \dot \FQ_1} \models
``\FP^{\gd + 1}$ is $\gd'$-strategically closed'',
$V^{\FP_\gd \ast \dot \FQ_0 \ast \dot \FQ_1
\ast \dot \FP^{\gd + 1}} = V^\FP \models
``\gd'$ contains a non-reflecting stationary set of
ordinals of cofinality $\sigma_\rho$''.
As there are unboundedly many in $\gk$
$V$-regular limits of $V$-strong cardinals in
$(\sigma_\rho, \gk)$,
$V^\FP \models ``$Unboundedly many in $\gk$ cardinals
$\eta \in (\sigma_\rho, \gk)$ contain non-reflecting
stationary sets of ordinals of cofinality
$\sigma_\rho$''.
By Solovay's Theorem 4.8 of \cite{SRK}
and the succeeding remarks,
$V^\FP \models ``$No cardinal
$\eta \in (\sigma_\rho, \gk)$ is strongly compact''.
This contradiction completes the proof of both
Lemma \ref{l1} and Theorem \ref{t2}.
\end{proof}
\end{proof}
Hamkins has pointed out that the function
$$\gd \mapsto {\hbox{\rm The least inaccessible
above the $V$-failure
of supercompactness of $\gd$}}$$
defined
on the strong cardinals in the case of Theorem \ref{t1}
or the regular limits of strong cardinals in the
case of Theorem \ref{t2}
can be used as a replacement for the function
$$\gd \mapsto {\hbox{\rm
The least $V$-strong cardinal above $\gd$}}$$
in the proofs of these theorems.
By the work of \cite{H4}, changes to the
domains of the functions used in the proofs
of Theorems \ref{t1} and \ref{t2} are also possible,
assuming that these changes are made so that for the
function $f$ being used in the definition,
on a large enough subset of $f$'s domain,
$f''\gd \subseteq V_\gd$.
\section{Indestructibility, Strongness, and Supercompactness}\label{s3}
Before giving the proofs of
Theorems \ref{t3} and \ref{t4},
we will need the following fact,
which shows the existence of a
``universal Laver function'' for all
strong and supercompact cardinals
in the universe.
Readers should consult \cite{KM} and Lemma 1 of \cite{A98}
for the existence of the corresponding
sort of function for supercompact cardinals,
and \cite{H03} for a generalized version
of this fact.
\begin{lemma}\label{l2}
Let ${\mathfrak K}^*$ be the class of strong cardinals, and
let ${\mathfrak K}$ be the class of supercompact cardinals.
Let $\Omega$ be $\sup(\K^*)$ if $\K^*$
is a set or the class of all ordinals if
$\K^*$ is a proper class.
There is then a total function
$f : \Omega \to V$ so that for any
$\gk \in {\mathfrak K}^*$,
$f \rest \gk : \gk \to V_\gk$.
Further, if $\gk \in {\mathfrak K}^*$,
then for every $x \in V$ and every
ordinal $\gl > \max(\gk, |{\rm TC}(x)|)$,
there is an elementary embedding
$j_{\gk, \gl, x} : V \to M$
generated by a $(\gk, |V_\gl|)$-extender
${\cal E}_{\gk, \gl, x}$
%of width $\gk$
witnessing the $\gl$ strongness of $\gk$ so that
$j_{\gk, \gl, x}(f)(\gk) = x$.
If $\gk \in {\mathfrak K}$, then for every
$x \in V$ and every cardinal
$\gl \ge \max(\gk, |{\rm TC}(x)|)$,
there is an elementary embedding
$j^*_{\gk, \gl, x} : V \to M$
generated by a supercompact ultrafilter
${\cal U}_{\gk, \gl, x}$ over $P_\gk(\gl)$
witnessing the $\gl$ supercompactness of $\gk$ so that
$j^*_{\gk, \gl, x}(f)(\gk) = x$.
\end{lemma}
\begin{proof}
We use an
argument analogous to the one given in
Lemma 1 of \cite{A98}.
By a result of Felgner \cite{Fe}
(see also \cite{G}), we assume that the
Axiom of Choice holds globally and can
be applied to proper classes.
We define $f$ by induction on $\Omega$.
If $\gd$ is not a measurable cardinal, then we let
$f(\gd) = 0$.
If $\gd$ is a measurable cardinal, then we consider
three cases.
\setlength{\parindent}{0pt}
Case 1: $\gd$ is not a strong cardinal and
there is a set $x$ and an ordinal
$\gl > \max(\gd, |{\rm TC}(x)|)$ so that
there is no elementary embedding
$j_{\gd, \gl, x} : V \to M$
generated by a $(\gd, |V_\gl|)$-extender
${\cal E}_{\gd, \gl, x}$
%of width $\gd$
witnessing the $\gl$ strongness of $\gd$ with
$j_{\gd, \gl, x}(f)(\gd) = x$.
If this occurs, then if $\gl$ is the least
such ordinal so that such an $x$ exists,
we let $x$ be a witness and define
$f(\gd) = x$.
Case 2: $\gd$ is a strong cardinal and
there is a set $x$ and a cardinal
$\gl \ge \max(\gd, |{\rm TC}(x)|)$ so that
there is no elementary embedding
$j^*_{\gd, \gl, x} : V \to M$
generated by a supercompact ultrafilter
${\cal U}_{\gd, \gl, x}$ over $P_\gd(\gl)$
witnessing the $\gl$ supercompactness of $\gd$ with
$j^*_{\gd, \gl, x}(f)(\gd) = x$.
If this occurs, then if $\gl$ is the least
such cardinal so that such an $x$ exists,
we let $x$ be a witness and define
$f(\gd) = x$.
Case 3: Neither Case 1 nor Case 2 holds.
Under these circumstances, we let
$f(\gd) = 0$.
\setlength{\parindent}{1.5em}
We first note that if either Case 1 or Case 2
occurs for $\gd$, then the smallest
$\gl \ge \gd$ which bears witness to this
is below the least strong cardinal
$\gk > \gd$
(which exists since the induction has
length $\Omega$).
To see this, assume
$\gl \ge \gk$, and let
$j : V \to M$ be an elementary
embedding witnessing the
$\gg$ strongness of $\gk$ for
$\gg > \gl$ having been chosen large enough so that
$M \models ``\gl$ is a witness to
either Case 1 or Case 2 for
$j(\gd) = \gd$ below $j(\gk) > \gg > \gl > \gk$ which is the
least strong cardinal above $\gd$''. By reflection,
$V \models ``$There is a witness to
either Case 1 or Case 2 for $\gd$ below $\gk$''.
Thus, $f$ is so that
$f \rest \gk : \gk \to V_\gk$ for
$\gk \in {\mathfrak K}^*$.
Let now $\gk \in {\mathfrak K}^*$ be so that
the conclusions of Lemma \ref{l2} fail for $f$,
and let $\gl > \gk$ be the least witness to
this failure. Once more, choose
$\gg > \gl$ to be sufficiently large
(such as the first beth fixed point above $\gl$)
so that
for some $(\gk, |V_\gg|)$-extender
${\cal E}_{\gk, \gg}$ which
%of width $\gk$ which
generates an elementary embedding
$j_\gg : V \to M_\gg$
witnessing the $\gg$ strongness of $\gk$,
$M_\gg \models ``$Case 1 holds for $\gk$ at
$\gl$ and
$j_\gg(f)(\gk) = y$ is a set witnessing this so that
$|{\rm TC}(y)| < \gl$''.
Note that such an extender exists, since it is
always possible to find a
$(\gk, |V_\gg|)$-extender $\cal E$ so that for
$i : V \to N$ the elementary embedding
generated by $\cal E$,
$N \models ``\gk$ isn't a strong cardinal''.
%$\gg$ strong''.
If we let
${\cal E}_{\gk, \gl}$ be the
$(\gk, |V_\gl|)$-extender
%of width $\gk$
which is the restriction of
${\cal E}_{\gk, \gg}$ to $|V_\gl|$ and take
$j_\gl : V \to M_\gl$ as the associated
elementary embedding witnessing the $\gl$ strongness of $\gk$,
we can decompose $j_\gg$ as
$h \circ j_\gl : V \to M_\gg$ with
$h : M_\gl \to M_\gg$ an elementary embedding
so that
$h \rest (\gl + 1)$ is the identity.
Since $|{\rm TC}(y)| \le \gl$, this means
$h(y) = y$.
Thus, in analogy to what is done in
\cite{L} and \cite{A98},
$j_\gl(f)(\gk) = h^{-1}(j_\gg(f)(\gk)) =
h^{-1}(y) = y$. Since
$j_\gl$ is generated by ${\cal E}_{\gk, \gl}$,
${\cal E}_{\gk, \gl} \in M_\gg$, and
$\gg$ has been chosen large enough so that
$M_\gg$ computes $j_\gl(f)(\gk)$ correctly,
this contradicts that
$M_\gg \models ``$Case 1 holds for $\gk$ at $\gl$''.
Hence, $f$ is as desired if
$\gk \in \K^*$.
If $\gk \in \K$, then as we observed during
the proof of Theorem \ref{t1},
by Lemma 2.1 of \cite{AC2} and the
succeeding remarks, for any
$j : V \to M$ an elementary embedding
witnessing (at least) the $2^\gk$
supercompactness of $\gk$,
$M \models ``\gk$ is a strong cardinal''.
This means that for such an embedding,
Case 1 can never hold for $\gk$ in $M$.
Therefore,
we can reason in almost
exactly the same way
as in the preceding paragraph if
we assume $\gk \in {\mathfrak K}$ is so that
the conclusions of Lemma \ref{l2} fail for $\gk$
by replacing Case 1 with Case 2 and using
supercompact ultrafilters instead of extenders
(readers are urged to consult Lemma 1 of
\cite{A98} for further details).
Hence, $f$ is also as desired if $\gk \in \K$.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We note that by the definition of $f$
and the proof given above, if
$\gk \in {\mathfrak K}$, we will have that
$f(\gk) = 0$.
It won't necessarily be the case, however, that
$f(\gk) = 0$ if $\gk \in {\mathfrak K}^* - \K$.
The reason is that even though Case 1 doesn't occur
for a strong cardinal,
by definition, Case 2 will occur
for a strong cardinal if the cardinal isn't supercompact.
We turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Let $V \models ``$ZFC + GCH +
${\mathfrak K}^*$ is the class of strong cardinals +
${\mathfrak K}$ is the class of supercompact cardinals''.
Fix an $f$ as given by Lemma \ref{l2}.
The partial ordering $\FP$ with which we force
is in essence the lottery preparation given
with respect to $f$ as defined on page 127 of
\cite{H4}. More specifically, $\FP$
is the Easton support iteration which first
adds a Cohen subset to $\omega$ and then,
at only those measurable cardinal
stages $\gd$ for which
$f''\gd \subseteq V_\gd$
and $f(\gd)$ is an ordinal, forces with the
partial ordering $\FQ_\gd$.
$\FQ_\gd$ is defined as the
lottery sum in
$V^{\FP_\gd}$ of all partial orderings in
$H({f(\gd)}^+)$ which are both
$\gs_\gd$-directed closed and
$\gd$-strategically closed.
At all other stages, the forcing is trivial.
By the definition of $\FP$,
$V^\FP \models {\rm ZFC}$.
In addition, as in the
proof of Theorem \ref{t2},
the definition of $\FP$ in tandem
with the argument given in the
first paragraph on page 153
of \cite{A98} show that
${(\gs_\gd)}^V = {(\gs_\gd)}^{V^\FP}$.
To show that any
$\gk \in {\mathfrak K}^*$
has the desired indestructibility properties
after forcing with $\FP$, we first write
$\FP = \FP_\gk \ast \dot \FP^\gk$, where
$\dot \FP^\gk$ possibly includes
a term for a non-trivial forcing
defined at stage $\gk$ if $\gk$ is
strong but not supercompact.
%$\FP_\gk$ is $\FP$ through stage $\gk$, and
%$\dot \FP^\gk$ is a term for
%the rest of $\FP$
%(including possibly a non-trivial
%forcing defined at stage $\gk$ if $\gk$
%is strong but not supercompact).
Note that
$V^{\FP_\gk} \models ``\FP^\gk$ is
$\gs_\gk$-directed closed and $\gk$-strategically
closed'' if $\gk$ is a strong cardinal.
Since $f(\gk) = 0$ if $\gk$ is supercompact
(meaning that the forcing at stage
$\gk$ is trivial),
by the definition of
%$\FP$ and
$\gs_\gd$ for $\gd > \gk$, it is
then the case that
$V^{\FP_\gk} \models ``\FP^\gk$ is
$\gk$-directed closed and $\gk$-strategically
closed'' if $\gk$ is a supercompact cardinal.
It therefore suffices to show that the
desired indestructibility properties hold for
$\gk$ after forcing with $\FP_\gk$.
However, an application of (the proof of)
Theorem 4.10 of \cite{H4} or
%if $\gk$ is a strong but non-supercompact cardinal or
an application of (the proof of)
Corollary 4.6 of \cite{H4} yields that
after forcing with $\FP_\gk$,
either the strongness of $\gk$
is indestructible under forcing with
partial orderings which are both
$\gs_\gk$-directed closed
and $\gk$-strategically closed, or the
supercompactness of $\gk$ is indestructible
under forcing with partial orderings which are both
$\gk$-directed closed and $\gk$-strategically closed.
And, now that we know that each element of
${\mathfrak K}^*$ has either
its strongness or supercompactness preserved
after forcing with $\FP$, we can once again
use Hamkins' results of
\cite{H1}, \cite{H2}, and \cite{H3} to infer
in a manner analogous to what was done in
the proof of Theorem \ref{t1}
that in $V^\FP$,
${\mathfrak K}^*$ is precisely the class of
strong cardinals and
${\mathfrak K}$ is precisely the class of
supercompact cardinals.
This completes the proof of Theorem \ref{t3}.
\end{proof}
\begin{pf}
To prove Theorem \ref{t4},
let $V \models ``$ZFC + GCH +
${\mathfrak K}^*$ is the class of strong cardinals +
${\mathfrak K}$ is the class of supercompact cardinals''.
Fix an $f$ as given by Lemma \ref{l2}.
The partial ordering $\FP$ with which we force
is a variation of the lottery preparation
given with respect to $f$ in the spirit
of the proof of Theorem \ref{t2}.
Let $\Gamma$ be as in the
statement of Theorem \ref{t4},
i.e., either
${(\sup(\K))}^+$ if $\K$ is a set, or
the class of all ordinals if
$\K$ is a proper class.
The partial ordering
$\FP$ with which we force
will have the form
$\FP_\Gamma \ast \dot \FP^\Gamma$.
$\FP_\Gamma$ is defined as the Easton
support iteration of length
$\Gamma$ which first
adds a Cohen subset to $\omega$ and then,
at only those measurable cardinal
stages $\gd < \Gamma$ for which
$f''\gd \subseteq V_\gd$ and
$f(\gd)$ is an ordinal
above $\gd$, forces with the
partial ordering $\FQ^0_\gd \ast \dot
\FQ^1_\gd$.
$\FQ^0_\gd$ is defined as the
lottery sum in
$V^{\FP_\gd}$ of all partial orderings in
$H({f(\gd)}^+)$ which are both
$\gs_\gd$-directed closed and
$\gd$-strategically closed.
$\dot \FQ^1_\gd$ is then defined as a
term for the partial ordering
which adds a non-reflecting
stationary set of ordinals of cofinality
$\gs_\gd$ to the least inaccessible
cardinal above $f(\gd)$.
%$\max(\gd, f(\gd))$,
%unless $\gd$
%is supercompact. If this is the
%situation, then
%$\dot \FQ^1_\gd$ is a term for the
%trivial partial ordering $\{0\}$.
At all other stages below
$\Gamma$, the forcing is trivial.
If $\K$ is a proper class, then
$\dot \FP^\Gamma$ is a term for the
trivial partial ordering $\{0\}$.
If $\K$ is a set, then
$\dot \FP^\Gamma$ is a term for the
Easton support iteration which adds
to each $V$-measurable cardinal
$\gd > \Gamma$ a non-reflecting
stationary set of ordinals of
cofinality $\Gamma$.
By the definition of $\FP$,
$V^\FP \models {\rm ZFC}$.
Further, as in the proofs of
Theorems \ref{t2} and \ref{t3},
the definition of $\FP$ in tandem
with the argument given in the
first paragraph on page 153
of \cite{A98} once again show that
${(\gs_\gd)}^V = {(\gs_\gd)}^{V^\FP}$.
%as before, the argument
%given in the first paragraph on
%page 153 of \cite{A98} shows that
%each $\gs_\gd$ remains regular in
%$V^\FP$.
To show that any
$\gk \in {\mathfrak K}^* \cap \Gamma$
has the desired indestructibility properties
after forcing with $\FP$,
we argue in analogy to the proof of Theorem \ref{t3}.
We first write
$\FP = \FP_\gk \ast \dot \FP^\gk$, where
as in the proof of Theorem \ref{t3},
$\dot \FP^\gk$ possibly includes a
term for a non-trivial forcing defined
at stage $\gk$ if $\gk$ is strong but
not supercompact.
%$\FP_\gk$ is $\FP$ through stage $\gk$, and
%$\dot \FP^\gk$ is a term for the rest of $\FP$
%(including possibly a non-trivial
%forcing defined at stage $\gk$ if $\gk$
%is strong but not supercompact).
Note that as earlier,
$V^{\FP_\gk} \models ``\FP^\gk$ is
$\gs_\gk$-directed closed and $\gk$-strategically
closed'' if $\gk$ is a strong cardinal, and
$V^{\FP_\gk} \models ``\FP^\gk$ is
$\gk$-directed closed and $\gk$-strategically
closed'' if $\gk$ is a supercompact cardinal.
These facts are not affected by
the additional term
$\dot \FQ^1_\gd$, since
at any stage $\gd$ at which a
non-trivial forcing is done,
$\dot \FQ^1_\gd$ always denotes
a partial ordering which is both
$\gs_\gd$-directed closed and
$\gd$-strategically closed.
It therefore once more suffices to show that the
desired indestructibility properties hold for
$\gk$ after forcing with $\FP_\gk$.
However, once again, an application of (the proof of)
Theorem 4.10 of \cite{H4} or
%if $\gk$ is a strong but non-supercompact cardinal or
an application of (the proof of)
Corollary 4.6 of \cite{H4} yields that
after forcing with $\FP_\gk$,
either the strongness of $\gk$
is indestructible under forcing with
partial orderings which are both
$\gs_\gk$-directed closed
and $\gk$-strategically closed, or the
supercompactness of $\gk$ is indestructible
under forcing with partial orderings which are both
$\gk$-directed closed and $\gk$-strategically closed.
The additional term $\dot \FQ^1_\gd$ does
not affect the application of the proof of
either Theorem 4.10 or Corollary 4.6 of
\cite{H4}.
This is since $\dot \FQ^1_\gd$ will
denote a partial ordering that is
strategically closed enough to be
absorbed into the partial ordering Hamkins
calls $\FP_{{\rm tail}}$ in \cite{H4}
to allow Hamkins' original proofs to remain
valid.
As in the proof of Lemma \ref{l1},
$V^\FP \models ``$No cardinal
$\gd > \Gamma$ is measurable'' if
$\K$ is a set. Therefore, as
we now know that each element of
${\mathfrak K}^* \cap \Gamma$ has either
its strongness or supercompactness preserved
after forcing with $\FP$, we can once again
use Hamkins' results of
\cite{H1}, \cite{H2}, and \cite{H3} to infer
in a manner analogous to what was done in
the proof of Theorem \ref{t1}
that in $V^\FP$,
${\mathfrak K}^* \cap \Gamma$
is precisely the class of
strong cardinals and
${\mathfrak K}$ is precisely the class of
supercompact cardinals.
As in the proof of Theorem \ref{t2},
if we know that
$V^\FP \models ``$The supercompact
and strongly compact cardinals
coincide, except possibly at
measurable limit points'',
we are able to infer via
the proof of Lemma 4 of \cite{A98} that
$V^\FP \models ``$Any strongly compact
cardinal $\gk$ which is a measurable
limit of strongly compact cardinals
has its strong compactness indestructible
under $\gk$-directed closed forcings
which preserve $\gk$'s measurability''.
This means the proof of Theorem \ref{t4}
is now completed by the proof
of the following lemma.
\begin{lemma}\label{l2b}
$V^\FP \models ``$The supercompact and strongly
compact cardinals coincide, except possibly
at measurable limit points''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l1},
regardless if ${\mathfrak K}$
is a set or proper class,
if $\rho$ is a strongly compact cardinal
in $V^\FP$ which is neither an element of
${\mathfrak K}$ nor a measurable limit
point of ${\mathfrak K}$, then
$\rho \in (\sigma_\rho, \gk)$, where
$\gk$ is the least element of
${\mathfrak K}$ above $\rho$.
Note as in the proof of Lemma \ref{l1}
that by the choice of $\gk$,
$\gk$ isn't a limit of supercompact
cardinals.
Let
$\gd$ be the least
non-trivial stage of forcing
above $\rho$
in the definition of $\FP$, i.e.,
$\gd \in (\rho, \gk)$ is
the least measurable cardinal
for which $f''\gd \subseteq V_\gd$ and
$f(\gd)$ is an ordinal above $\gd$.
By the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``$The least
inaccessible cardinal above $f(\gd)$
%$\max(\gd, f(\gd))$
contains a
non-reflecting stationary set
of ordinals of cofinality
$\gs_\gd = \gs_\gr$''.
Again by the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ is
$\gl$-strategically closed for
$\gl$ the least inaccessible cardinal above
$f(\gd)$'',
%$\max(\gd, f(\gd))$'',
meaning that
$V^{\FP_{\gd + 1} \ast \dot \FP^{\gd + 1}} =
V^\FP \models
``\gd$ contains a non-reflecting stationary
set of ordinals of cofinality $\gs_\gr$''.
As in the proof of Lemma \ref{l1}, Solovay's
Theorem 4.8 of \cite{SRK}
and the succeeding remarks now immediately yield that
$V^\FP \models ``\rho$ isn't strongly compact''.
This contradiction completes the proof of
both Lemma \ref{l2b} and Theorem \ref{t4}.
\end{proof}
\end{pf}
We observe that it is impossible to
get a model in which
every strong cardinal $\gk$ has its strongness
indestructible under $\gk$-strategically closed
forcing and every supercompact cardinal $\gk$
has its supercompactness indestructible under
$\gk$-directed closed forcing,
if in this model, there is
a supercompact cardinal
having at least one
strong cardinal above it.
%it is
%more than one supercompact
%cardinal in the universe, it is
%impossible to obtain a model in which
%every supercompact cardinal $\gk$ is
%indestructible under $\gk$-directed
%closed forcing and every strong
%cardinal $\gk$ is indestructible
%under $\gk$-strategically closed forcing.
To see this, if in some model $V$,
$\gk$ is supercompact and
$\gd > \gk$ is strong and has its
strongness indestructible under
$\gd$-strategically closed forcings,
then it is possible to add a non-reflecting
stationary set of ordinals of cofinality
$\go$ to $\gd^+$ via the
partial ordering ${\FP_{\go, \gd^+}}$ and preserve the
strongness of $\gd$ in $V^{\FP_{\go, \gd^+}}$.
By reflection in $V^{\FP_{\go, \gd^+}}$,
there will be
unboundedly many in $\gd$
successor cardinals that contain
non-reflecting stationary sets
of ordinals of cofinality $\go$.
In particular, there will be such
a cardinal above $\gk$ in $V^{\FP_{\go, \gd^+}}$.
Since ${\FP_{\go, \gd^+}}$ is $\gd$-strategically
closed, there is a cardinal above
$\gk$ in $V$ containing a non-reflecting
stationary set of ordinals of
cofinality $\go$.
However, by Solovay's Theorem 4.8 of
\cite{SRK} and the succeeding
remarks, this is impossible,
since it is then immediately the
case that
$\gk$ is no longer strongly compact
in $V$.\footnote{In fact, as
the referee has pointed out,
this argument shows that
if a strong cardinal $\gd$ is
indestructible under
$\gd$-strategically closed forcing,
then no smaller cardinal is
strongly compact.
This observation will be used
in the proof of Theorem \ref{t5}.}
The question still remains, though, as to
whether it is possible to
have a universe in which every
strong cardinal $\gk$ is
indestructible under
%$\gk$-strategically closed forcings which are also
$\gs_\gk$-directed closed forcings
which are also $\gk$-strategically
closed, and every
supercompact cardinal $\gk$ is
indestructible under $\gk$-directed
closed forcing.
(Note that in a model in which there
is one supercompact cardinal
with no strong cardinals above it,
this sort of indestructibility does indeed
mean that every strong cardinal $\gk$
has its strongness indestructible under
$\gk$-strategically closed forcing and
every supercompact cardinal $\gk$
has its supercompactness indestructible
under $\gk$-directed closed forcing.)
We conjecture that the answer to this question
is yes, although
since any partial ordering known to the author for
forcing indestructibility requires a certain
uniformity in its definition, this seems like a
difficult question to answer.
We conclude this section by
noting that in
Theorem \ref{t4}, if
$\K$ is a proper class, then
our proof shows that the class
$\K^*$ of strong cardinals is
precisely the same in both
$V$ and $V^\FP$.
It is only when $\K$ is a set
that there may be strong cardinals in
$V$ which are destroyed by the
partial ordering $\FP$.
\section{Indestructibility, Strongness, and Strong Compactness}\label{s4}
We turn now to the proof of Theorem \ref{t5}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk < \gl$ are
the smallest cardinals so that
$\gk$ is $\gl$ supercompact and $\gl$ is inaccessible''.
Let
$\la \gd_\ga : \ga < \gk \ra$
enumerate in increasing order
$\{\gd < \gk : \gd$ is $\gg$ strong for
all $\gg < \gl_\gd\}$, where
$\gl_\gd$ is the least inaccessible
cardinal above $\gd$.
The partial ordering $\FP$ with which
we force is the Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
of length $\gk$ which at stage $0$ adds
a Cohen subset to $\omega$.
At any stage $\ga > 0$, if
$\forces_{\FP_\ga} ``$After forcing with any
$\gd_\ga$-strategically closed partial ordering
of rank below $\gl_{\gd_\ga}$, $\gd_\ga$ remains
$\gg$ strong for all $\gg < \gl_{\gd_\ga}$'', then
$\dot \FQ_\ga$ is a term for $\FP_{\go, \gd_\ga}$.
%the partial ordering
%which adds a non-reflecting stationary set of
%ordinals of cofinality $\omega$ to $\gd_\ga$.
If this is not the case, then
$\dot \FQ_\ga$ is a term for a $\gd_\ga$-strategically
closed partial ordering of rank below
$\gl_{\gd_\ga}$ so that
$\forces_{\FP_\ga} ``$After forcing with
$\dot \FQ_\ga$, there is some $\gg < \gl_{\gd_\ga}$
so that $\gd_\ga$ is no longer $\gg$ strong''.
Note that by the definition of
$\FP$, for any $\ga < \gk$, it is
inductively the case that
$\gl_{\gd_\ga}$ has the same
meaning in $V$, $V^{\FP_\ga}$,
$V^{\FP_{\ga + 1}}$, or $V^\FP$.
Note also that $\FP$ can be defined so that
$|\FP| = \gk$.
\begin{lemma}\label{l3}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is $\gg$ strong for all
$\gg < \gl_\gd$''.
\end{lemma}
\begin{proof}
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$|\FP_0| = \omega$ and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
Since $\FP$ admits a gap at $\ha_1$,
by the work of \cite{H1}, \cite{H2}, and \cite{H3},
any cardinal $\gd < \gk$ in $V^\FP$ which is $\gg$
strong for all $\gg < \gl_\gd$
had to have been
$\gg$ strong for all $\gg < \gl_\gd$ in $V$.
Suppose now towards a contradiction that
$V^\FP \models ``\gd < \gk$ is
$\gg$ strong for all
$\gg < \gl_\gd$''.
By the preceding paragraph, there is some
$\ga < \gk$ so that $\gd = \gd_\ga$.
However, by the definition of $\FP$,
$V^{\FP_{\ga + 1}} \models ``$There is some cardinal
$\gg < \gl_{\gd_\ga}$ so that $\gd_\ga$ isn't $\gg$ strong''.
If we write
$\FP = \FP_{\ga + 1} \ast \dot \FQ^*$, since
$\forces_{\FP_{\ga + 1}} ``\dot \FQ^*$ is
$\gl_{\gd_\ga}$-strategically closed'',
$V^{\FP_{\ga + 1} \ast \dot \FQ^*} = V^\FP \models
``$There is some cardinal $\gg < \gl_{\gd_\ga}$ so that
$\gd_\ga$ isn't $\gg$ strong''.
This contradiction proves Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$After forcing with any
$\gk$-strategically closed partial ordering
of rank below $\gl$, $\gk$ is $\gg$ strong
for all $\gg < \gl$''.
\end{lemma}
\begin{proof}
Fix
$j : V \to M$ an elementary embedding witnessing the
$\gl$ strongness of $\gk$ generated by a
$(\gk, \gl)$-extender.
%of width $\gk$.
Assume that
$\FQ' \in V^\FP$ is a $\gk$-strategically closed
partial ordering of rank below $\gl$
so that for some $\gg' < \gl$,
$V^{\FP \ast \dot \FQ'} \models ``\gk$ isn't
$\gg'$ strong''.
As $j$ is a $\gl$ strongness embedding and
$\gl$ is inaccessible,
$M \models ``$The $\gk$-strategically closed
partial ordering $\FQ' \in M^\FP$ is so that
after forcing with
$\FQ'$, $\gk$ isn't $\gg'$ strong in
$M^{\FP \ast \dot \FQ'}$''. Thus, in $M$,
since $\FP_\gk = \FP$,
at stage $\gk$ in the definition of $j(\FP)$,
$\dot \FQ_\gk$ is a term for a $\gk$-strategically
closed partial ordering having rank below
$\gl$ so that for some $\gg < \gl$,
$M^{\FP_\gk \ast \dot \FQ_\gk} =
M^{\FP \ast \dot \FQ_\gk} \models
``\gk$ isn't $\gg$ strong''.
Therefore, since $V_\gl \subseteq M$,
$V^{\FP \ast \dot \FQ_\gk} \models
``\gk$ isn't $\gg$ strong'' as well.
Let $\dot \FQ = \dot \FQ_\gk$.
We show that in $V^{\FP \ast \dot \FQ}$,
$j$ lifts to a $\gl$ strong embedding
$j : V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
This contradiction will complete the proof of
Lemma \ref{l4}.
The argument that the embedding $j$
lifts is virtually identical to the
argument given in the proof of
Theorem 4.10 of \cite{H4}.
For the benefit of readers, we present
the argument here as well, taking the
liberty to quote freely from it.
We may assume that
$M = \{j(f)(a) : a \in {[\gl]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq V\}$. Since
Since
$j(\FP) = \FP \ast \dot \FQ \ast \dot \FR$,
we know that the first ordinal in the
field of $\dot \FR$ is above $\gl$.
Since we may assume that $M^\gk \subseteq M$,
this means that if
$G$ is $V$-generic over $\FP$ and
$H$ is $V[G]$-generic over $\FQ$,
$\FR$ is ${\prec} \gk^+$-strategically closed in both
$V[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$.
As in \cite{H4}, by using a suitable coding
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
$N = \{i_{G \ast H}(\dot z) : \dot z =
j(f)(\gk, \ga, \gl)$ for some function $f \in V\}$.
It is easy to verify that
$N \prec M[G][H]$, that $N$ is closed under
$\gk$ sequences in $V[G][H]$, and that
$\gk$, $\ga$, $\gl$, $\FQ$, and $\FR$ are all
elements of $N$.
Further, since
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : {[\gk]}^3 \to V_\gk$ in $V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G][H]$ and $V[G][H]$, we can
build $H'$ in $V[G][H]$
%an $N$-generic object $H'$ over $\FR$
as follows. Let
$\la D_\gs : \gs < \gk^+ \ra$ enumerate in
$V[G][H]$ 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 ${\prec} \gk^+$-strategic closure of
$\FR$ in $V[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs : \gs < \gk^+ \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gk^+$.
Let
$H' = \{p \in \FR : \exists \gs <
\gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $H'$ is actually
$M[G][H]$-generic over $\FR$.
If $D$ is a dense open subset of
$\FR$ in $M[G][H]$, then
$D = i_{G \ast H}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$, where
$\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that $j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of $\FR$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$ and therefore $\gl$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is definable without
the parameters $\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N$.
Thus, since $H'$
meets every dense open subset
of $\FR$ present in $N$,
%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
$V[G][H]$, $j$ lifts to
$j : V[G] \to M[G][H][H']$.
It remains to lift $j$ through the forcing $\FQ$
while working in $V[G][H]$.
To do this, it suffices to show that
$j''H \subseteq j(\FQ)$ generates an
$M[G][H][H']$-generic object $H''$ over $j(\FQ)$.
Given a dense open subset
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = i_{G \ast H \ast H'}(\dot D)$
for some name $\dot D = j(\vec D)(a)$
for some $a \in {[\gl]}^{< \go}$
and some function
$\vec D = \la D_\gs : \gs \in {[\gk]}^{|a|} \ra$.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $\gk$-distributive, it follows that
$D' = \bigcap_{\gs \in {[\gk]}^{|a|}} D_\gs$ 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
$j : V[G][H] \to M[G][H][H'][H'']$.
This final lifted version of $j$ is
$\gl$ strong since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$. Therefore,
$V[G][H] \models ``\gk$ is $\gl$ strong'', which
contradicts that $\gg < \gl$ and
$V[G][H] \models ``\gk$ isn't $\gg$ strong''.
This proves Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is $\gl$ strongly compact''.
\end{lemma}
\begin{proof}
Let
$k_1 : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$.
By the closure of $M$ and Lemma \ref{l4}, we
know that in $M$, $\dot \FQ_\gk$ is a term
for $\FP_{\go, \gk}$.
%the partial ordering which adds a
%non-reflecting stationary set of ordinals
%of cofinality $\omega$ to $\gk$.
We can now complete the proof of
Lemma \ref{l5} using an argument of
Magidor, unpublished by him but
also exposited in
%\cite{AC1},
\cite{AC2}, \cite{A99},
\cite{A99b}, and
\cite{AH03} (as well as
elsewhere). Specifically,
$\gl$ is large enough so that we
may assume by choosing a normal ultrafilter of
Mitchell order $0$ 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''.
%It is the case that if
Suppose that
$k : V \to N$ is an elementary embedding 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
$N \models ``|y| < k({\gk})$''.
Under these circumstances,
$k$ witnesses the $\gl$
strong compactness of ${\gk}$
in the sense that $k$ can be
used to generate a fine,
$\gk$-additive ultrafilter over
$P_\gk(\gl)$.
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
$j : V^{\FP} \to N^{j(\FP)}$ in $V^\FP$.
Since this lifted embedding witnesses
the $\gl$ strong compactness of ${\gk}$ in
$V^{\FP}$, this proves Lemma \ref{l5}.
To do this, write
$j(\FP)$ as
$\FP \ast \dot \FQ^{\gk} \ast \dot \FR^{\gk}$, where
$\dot \FQ^{\gk}$ is a term for the portion of
$j(\FP)$ between ${\gk}$ and $k_2({\gk})$ and
$\dot \FR^{\gk}$ is a term for the rest of
$j(\FP)$, i.e., the part above $k_2({\gk})$.
Note that since
$N \models ``{\gk}$ isn't measurable'',
${\gk} \not\in {\rm field}(\dot \FQ^{\gk})$.
Also, since
$M \models ``{\gk}$ is measurable'',
by elementarity,
$N \models ``k_2({\gk})$ is measurable''.
Thus, the field of $\dot \FQ^{\gk}$
is composed of $N$-measurable cardinals
in the interval
$({\gk}, k_2({\gk})]$ (so by elementarity,
a non-reflecting stationary set of ordinals
of cofinality $\go$ is added to
$k_2(\gk)$ when forcing over
$N^{\FP}$ with $\FQ^{\gk}$),
and the field of $\dot \FR^{\gk}$ is composed of
$N$-measurable cardinals in the interval
$(k_2({\gk}), k_2(k_1({\gk})))$.
Let $G_0$ be $V$-generic over $\FP$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ^{\gk}$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR^{\gk}$. Since $\FP$ 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
$N[G_0] \models ``|\wp(\FQ^{\gk})| = k_2(2^{\gk})$'', we can let
$\la D_\ga : \ga < \gk^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of
$\FQ^{\gk}$ present in $N[G_0]$.
Since the ${\gk}$ closure of $N$ with respect to either
$M$ or $V$ implies the least element of the field of
$\FQ^{\gk}$ is above
$\gk^+$, the definitions of $\FP$ and
$\FQ^{\gk}$
imply that
$N[G_0] \models ``\FQ^{\gk}$ is
${\prec} \gk^+$-strategically closed''.
By the fact the standard arguments show that
forcing with the ${\gk}$-c.c$.$ partial ordering
$\FP$ preserves that $N[G_0]$ remains
${\gk}$-closed with respect to either
$M[G_0]$ or $V[G_0]$,
$\FQ^{\gk}$ 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 $V[G_0]$ using the argument given in
Lemma \ref{l4} for the construction of $H'$.
It remains to construct in $V[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR^{\gk}$.
To do this, we first write
$k_1(\FP)$ as
$\FP \ast \dot \FQ_{\gk} \ast \dot \FT^{\gk}$, where
$\dot \FT^{\gk}$ is a term for the portion of
$k_1(\FP)$ above $\gk$.
Since $\gl$ is the least inaccessible cardinal
above $\gk$ in both $V$ and $M$,
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is measurable''.
Thus, the field of
$\dot \FT^{\gk}$ is composed of
$M$-measurable cardinals in the interval
$(\gl, k_1({\gk}))$, which implies that in
$M$,
$\forces_{\FP \ast \dot \FQ_{\gk}}
``\dot \FT^{\gk}$ is ${\prec} \gl^+$-strategically
closed''. Further, as
$\gl$ is inaccessible,
$|{[\gl]}^{< {\gk}}| = \gl$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter ${\cal U}$ over
$P_{\gk}(\gl)$,
GCH in $V$ implies that for any
$\gd \in [\gk, \gl]$,
$|k_1(\gd^+)| = |k_1(2^\gd)| =
|2^{k_1(\gd)}| =
|\{ f : f : P_{\gk}(\gl) \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gl| = \gl^+$.
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
$\dot \FT^\gk$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP \ast \dot \FQ_\gk$ and
$\forces_{\FP \ast \dot \FQ_\gk} ``\tau \in
\dot \FT^\gk$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP \ast \dot \FQ_\gk} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT^\gk$ 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 $\dot \FT^\gk$ 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
$\forces_{\FP \ast \dot \FQ_\gk} ``\dot \FT^\gk$ 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
$\forces_{\FP \ast \dot \FQ_\gk}
``|\dot \FT^\gk| = k_1(\gk)$'',
there is a set
$\{\tau_\alpha : \alpha