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%
\title{The Least Strongly Compact can be
the Least Strong and Indestructible
%\title{The Least Strongly Compact can be Non-Supercompact,
% Strong, and Indestructible
%\title{Indestructibility, Strong Compactness, and Strongness
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, strong cardinal, indestructibility,
Prikry forcing, Radin forcing, closure point,
non-reflecting stationary set of ordinals.}}
\author{Arthur W.~Apter
\thanks{The author wishes to express his heartfelt
thanks and deep appreciation and
gratitude to Mirna
D\v zamonja, for her great patience,
unfailing grace, and unflagging
good humor at having
listened to and suffered through many
incorrect attempts at establishing the
results of this paper.
The author also wishes to thank
Moti Gitik for helpful and stimulationg
discussions
on the subject matter of this paper.}
\thanks{The author is pleased to
offer this paper as his contribution
to the special volume of {\it Annals
of Pure and Applied Logic} being
compiled in honor of the 60th birthday
of Professor James Baumgartner of
Dartmouth College.
As is true with so many people, Jim's
encouragement and support have been
invaluable over the years and have enriched
both the mathematical and personal lives
of those fortunate enough to know him.}
\thanks{The contents of this paper were presented
in a lecture given at the
January 20 - 26, 2002 meeting in
Set Theory held at the Mathematics
Research Institute, Oberwolfach, Germany.
The author wishes to thank the organizers
for having invited him to speak at and
participate in a very stimulating conference.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{December 19, 2003\\
(revised January 9, 2006)}
\begin{document}
\maketitle
\begin{abstract}
We construct two models in which
the least strongly compact
cardinal $\gk$ is also the least
strong cardinal.
In each of these models,
$\gk$ satisfies
indestructibility properties
for both its strong compactness
and strongness.
\end{abstract}
\baselineskip=24pt
%\newpage
\section{Introduction and Preliminaries}\label{s1}
The different possible interrelationships
among strongness, strong compactness,
and indestructibility are quite
varied.
In \cite{GS}, Gitik and Shelah showed that
any strong cardinal $\gk$ can be forced
to be indestructible under what they call
``$\gk^+$-weakly closed forcing notions
satisfying the Prikry condition,'' which
is a version of Laver's result of
\cite{L} for strong cardinals.
Although the definition of this degree
of closure is quite technical
(and we refer readers of this paper
%to \cite{GS} for a more
to later in Section \ref{s1} for a
precise definition),
it encompasses a large number of diverse
partial orderings, such as anything which
is either $\gk$-strategically closed,
$\gk$-closed, or ${\le} \gk$-directed closed,
or iterations of Prikry or Radin forcing
based on cardinals above $\gk$.
Note that henceforth, we will refer
to this sort of indestructibility as
``Gitik-Shelah indestructibility.''
There are also interesting possible
interrelationships between strong
compactness and indestructibility
and strong compactness and strongness.
In \cite{AG}, the author and Gitik
showed that it is consistent,
relative to a supercompact cardinal,
for the least strongly compact
cardinal $\gk$ to be non-supercompact
yet to have its strong compactness
indestructible under ${<} \gk$-directed
closed forcing (which is of course
also a generalization of the result
of \cite{L}, this time for
non-supercompact strongly compact cardinals).
In \cite{AC2}, the author and Cummings
showed that it is consistent, relative
to a supercompact cardinal, for the
least strong cardinal $\gk$ to be the
least strongly compact cardinal.
As shown in Lemma 2.1 of \cite{AC2}
and the succeeding remarks, under these
circumstances, $\gk$ isn't $2^\gk$ supercompact.
The purpose of this paper is to provide
in some sense a synthesis of the
aforementioned results of \cite{GS}, \cite{AG},
and \cite{AC2}.
We will construct two
distinct models in which the
least strongly compact cardinal is
also the least strong cardinal and
satisfies indestructibility properties
for both its strong compactness and
its strongness.
Specifically, we prove the following two theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is Mahlo''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
No cardinal $\gd > \gk$ is Mahlo +
$\gk$ is
both the least strongly compact
and least strong cardinal + $\gk$'s
strongness is Gitik-Shelah indestructible +
$\gk$'s strong compactness is indestructible
under ${<} \gk$-directed closed forcing''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is
supercompact''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is
both the least strongly compact
and least strong cardinal +
Each of $\gk$'s strong compactness
and strongness is indestructible
under ${<} \gk$-directed closed
partial orderings which are also
$\gk$-strategically closed''.
\end{theorem}
We remark that in some
way, Theorem \ref{t1}
represents the best possible
combination of the results
of \cite{GS}, \cite{AG}, and
\cite{AC2} mentioned above,
in the sense that
the model constructed manifests
exactly all of the relevant
features of these earlier theorems.
We do have to pay the
price, however, of having to settle for
both a ground model and a generic
extension in which the
large cardinal structure is severely limited.
On the other hand,
Theorem \ref{t2} allows us to deal
with the situation in which there are
no restrictions placed on the large
cardinal structure of either our
ground model or generic extension.
Here, though, the amount of indestructibility
obtained is less than optimal.
Before beginning the proofs of
Theorems \ref{t1} and
\ref{t2}, 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.
For $x$ a set of ordinals,
$\ov x$ is the order type of $x$.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
%If we also have that $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
%is an Easton support iteration of length $\gk$
%so that at stage $\ga$, a nontrivial forcing is done
%based on the ordinal $\gd_\ga$, then we will say that
%$\gd_\ga$ is in the field of $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
Any term for trivial forcing will
always be taken as a term for the
partial ordering $\{\emptyset\}$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\FP$ is
an arbitrary partial ordering
and $\gk$ is a regular cardinal,
%$\FP$ is {\it $\gk$-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 {\it $\gk$-closed}
if every increasing chain of elements
$\la p_\ga : \ga < \gk \ra$ of $\FP$
has an upper bound $p \in \FP$.
$\FP$ is {\it ${<} \gk$-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 {\it 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 $p \in \FP$.
If in addition, any directed subset of
$\FP$ of size $\gk$ has an upper bound,
then $\FP$ is said to be
{\it ${\le} \gk$-directed closed}.
$\FP$ is {\it $\gk$-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
${\le} \gk$-directed closed, then $\FP$ is
$\gk$-strategically closed.
%Note that
%if $\FP$ is $\gk$-strategically closed,
%then $\FP$ is $\gk$-distributive.
Also, if $\FP$ is
$\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$\FP$ is {\it ${\prec}\gk$-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.
As in \cite{GS}, we will say that
the partial ordering $\FP$
is {\it $\gk^+$-weakly closed
and satisfies the Prikry condition} if
it meets the following criteria.
\begin{enumerate}
\item $\FP$ has two partial
orderings $\le$ and $\le^*$ with
$\le^* \ \subseteq \ \le$.
\item For every $p \in \FP$
and every statement $\varphi$
in the forcing language
with respect to $\FP$, there
is some $q \in \FP$ such that
$p \le^* q$ and $q \decides \varphi$
($q$ decides $\varphi$).
\item The partial ordering
$\le^*$ is $\gk$-closed.
\end{enumerate}
For $\gk$ an inaccessible
cardinal, we will say that
the partial ordering $\FP$ is
%will be said to be
{\it ${\prec}\gk$-weakly closed and
satisfies the Prikry condition} if
it meets the criteria just given,
except that $\le^*$ is
$\gd$-closed for every $\gd < \gk$.
%it is $\gd$-weakly closed and
%satisfies the Prikry condition
%for every $\gd < \gk$.
For more details on these definitions,
readers are urged to consult
\cite{GS}.\footnote{Readers will
note that Gitik and Shelah use
``$\gk$-closed'' to mean what we
would call
``$\gd$-closed for every $\gd < \gk$,''
which is different from our usage.
Our definition of a partial ordering
being $\gk^+$-weakly closed and
satisfying the Prikry condition,
however, has been presented so
as to coincide with theirs.}
Throughout the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik style iterations of
Prikry-like forcings.}
By this we will mean an iteration
as first given by Gitik in \cite{G}
(and elaborated upon further in
\cite{GS}),
where at any stage $\gd$ at which
a nontrivial forcing is done,
we assume the partial ordering
$\FQ_\gd$ with which we force is
${\prec}\gd$-weakly closed
and satisfies the Prikry condition.
By Lemmas 1.2 and 1.4 of \cite{G},
if $\gd_0$ is the first stage in the
definition of $\FP$ at which a nontrivial
forcing is done, then forcing with
$\FP$ adds no bounded subsets to $\gd_0$.
In this paper, we will use non-reflecting
stationary set forcing $\FP_{\eta,\gl}$.
Specifically, if $\eta<\lambda$ are both regular
cardinals, then conditions in $\FP_{\eta,\gl}$
are bounded subsets $s\subseteq\gl$ consisting of
ordinals of cofinality $\eta$ such that for every
$\alpha<\lambda$, the initial segment
$s\intersect\alpha$ is non-stationary in
$\alpha$, ordered by end-extension. It is
well-known that if $G$ is $V$-generic over
$\FP_{\eta, \gl}$ (see \cite{Bu} or \cite{A01})
and GCH holds in $V$, then in
$V[G]$, the set $S=S[G]=\bigcup G \subseteq \gl$
is a non-reflecting stationary set of ordinals of
cofinality $\eta$, the bounded subsets of $\gl$
are the same as those in $V$, and cardinals,
cofinalities and GCH have been preserved. It
is virtually immediate that $\FP_{\eta, \gl}$ is
${<} \eta$-directed closed, and it can
be shown (see \cite{Bu} or
\cite{A01}) that $\FP_{\eta, \gl}$
is $\gd$-strategically closed
for every $\gd < \gl$. It follows from
work of Solovay
(Theorem 4.8 of \cite{SRK} and the
succeeding remarks) that the existence of a
non-reflecting stationary subset of $\lambda$,
consisting of ordinals of confinality $\eta$,
implies that no cardinal
$\delta\in(\eta,\lambda]$ is $\lambda$ strongly
compact. Thus, iterations of this forcing provide
a way to destroy all strongly compact cardinals
in an interval.
We recall for the benefit of readers the
definition given by Hamkins in
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 {\it 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.''}
A result which will be used in the proof
of Theorem \ref{t2} is a corollary
of Theorem 3 of
Hamkins' paper \cite{H5}.
These two theorems are generalizations of
Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}
(and we refer readers to \cite{H2},
\cite{H3}, and \cite{H5} for
further details).
We therefore state the theorem
we will be using now, along
with some associated terminology.
%quoting freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| \le \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{AH2},
$\FP$ {\it admits a closure point at $\gd$}.
Also, as in the terminology of
\cite{H2}, \cite{H3}, \cite{H5},
and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific generalization of the
Gap Forcing Theorem we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
(Hamkins)
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a closure point
at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
\end{theorem}
Finally, 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.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of
Theorem \ref{t1}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is Mahlo''.
%Without loss of generality, by first
%doing a preliminary forcing if necessary,
%we assume in addition that $V \models {\rm GCH}$.
For the rest of the
proof of Theorem \ref{t1}, for any
ordinal $\gd$, we let $\gd^*$ be
the least $V$-Mahlo cardinal above $\gd$.
We are now in a position to define the
partial ordering $\FP$ to be used
in the proof of Theorem \ref{t1}.
$\FP$ is a Gitik style iteration
%(as first given in \cite{G})
of Prikry-like forcings
$\la \la \FP_\gd, \dot \FQ_\gd \ra :
\gd < \gk \ra$
of length $\gk$
which does a nontrivial forcing
only at those ordinals $\gd < \gk$
which are Mahlo cardinals in $V$.
At such a stage $\gd$, we first
force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank below
$\gd^*$ which are in $V^{\FP_\gd}$
either ${<} \gd$-directed closed or
$\gd^+$-weakly closed and satisfy
the Prikry condition. If in
$V^{\FP_\gd \ast \dot \FQ^*_\gd}$,
there is some Gitik style iteration
$\FR$ of Prikry-like forcings having length
$\gk$ which does a nontrivial forcing
only at $V$-Mahlo cardinals above $\gd$
such that
$V^{\FP_\gd \ast \dot \FQ^*_\gd \ast \dot \FR}
\models ``\gd$ is a strong cardinal'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If this is not the case, then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
We note that this definition makes sense, i.e.,
that under the appropriate
circumstances, it is possible to
choose $\dot \FS_\gd$ as a
term for Prikry forcing.
This is since if
$V^{\FP_\gd \ast \dot \FQ^*_\gd \ast \dot \FR}
\models ``\gd$ is a strong cardinal'' for
$\FQ^*_\gd$ and $\FR$ as in
the above paragraph, then
in particular,
$V^{\FP_\gd \ast \dot \FQ^*_\gd \ast \dot \FR}
\models ``\gd$ is a measurable cardinal''.
Because $\FR$ is a Gitik style iteration
of Prikry-like forcings which has its
first nontrivial stage at $\gd^*$, it must
therefore be the case that
$\forces_{\FP_\gd \ast \dot \FQ^*_\gd}
``$Forcing with $\dot \FR$ adds no bounded subsets to
$\gd^*$''. As
$\FP_\gd \ast \dot \FQ^*_\gd$ is forcing
equivalent to a partial ordering having
size below $\gd^*$, we know that
$\forces_{\FP_\gd \ast \dot \FQ^*_\gd}
``$Forcing with $\dot \FR$ adds no bounded subsets to
the least Mahlo cardinal above $\gd$'',
i.e., that $\gd^*$ retains its
identity as the least Mahlo cardinal
above $\gd$ in $V^{\FP_\gd \ast
\dot \FQ^*_\gd}$.
From this, we may further infer that
$\forces_{\FP_\gd \ast \dot \FQ^*_\gd}
``\gd$ is a measurable cardinal'', which
means that $\dot \FS_\gd$ may be chosen as a
term for Prikry forcing when appropriate.
\begin{lemma}\label{l1}
$V^\FP \models ``$No cardinal $\gd < \gk$
is a strong cardinal''.
\end{lemma}
\begin{proof}
Suppose to the contrary that
$\gd < \gk$ is such that
$V^\FP \models ``\gd$ is a strong cardinal''.
Since no partial ordering can create
a new Mahlo cardinal, and since a
strong cardinal is clearly also
a Mahlo cardinal, it must be the case that
$V \models ``\gd$ is a Mahlo cardinal''.
Write
$\FP = \FP_{\gd + 1} \ast
\dot \FP^{\gd + 1} =
\FP_\gd \ast \dot \FQ^*_\gd
\ast \dot \FS_\gd \ast
\dot \FP^{\gd + 1}$.
%where $\dot \FP^{\gd + 1}$ is a term
%for the portion of $\FP$ defined at
%ordinals starting from $\gd^*$.
As $V^\FP \models ``\gd$ is a strong cardinal'',
it must consequently be the case that
$\dot \FS_\gd$ is a term for trivial forcing.
This is since otherwise,
$\dot \FS_\gd$ is a term for
Prikry forcing. This means that
$V^\FP \models ``{\rm cof}(\gd) = \go$'',
which contradicts that
$V^\FP \models ``\gd$ is a strong cardinal''.
However, since $\FP^{\gd + 1}$ is a Gitik
style iteration of Prikry-like
forcings which has its first
nontrivial stage at $\gd^*$,
it must be the case by the
definition of $\FP$ that
$\dot \FS_\gd$ is a term for Prikry forcing
defined over $\gd$.
%This means that
%$V^{\FP_\gd \ast \dot \FQ^*_\gd
%\ast \dot \FS_\gd \ast \dot \FP^{\gd + 1}} =
%V^{\FP_{\gd + 1} \ast \dot \FP^{\gd + 1}} =
%V^\FP \models ``\gd$ has cofinality $\go$'',
%which contradicts that
%$V^\FP \models ``\gd$ is a strong cardinal''.
%Thus, $V^\FP \models ``{\rm cof}(\gk) = \go$''.
This contradiction
completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``\gk$ is a
strongly compact cardinal
whose strong compactness
is indestructible under ${<} \gk$-directed
closed forcing''.
\end{lemma}
\begin{proof}
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is ${<}\gk$-directed
closed''.
Let $\gl > \max(\gk, \card{{\rm TC}(\dot \FQ)})$
be an arbitrary regular cardinal large
enough so that
${(2^{[\gl]^{< \gk}})}^V = \rho =
{(2^{[\gl]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \rho^+$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gk$.
%such that $M \models ``\gk$ isn't $\gs$ supercompact''.
Since $V \models ``$No cardinal $\gd > \gk$
is Mahlo'', by the closure properties of $M$,
$M \models ``\card{{\rm TC}(\dot \FQ)}$ is
below the least Mahlo cardinal above $\gk$''.
%This has as an immediate consequence that
Thus, as $M \models ``\gk$ is Mahlo'',
$\dot \FQ$ is a term for an allowable choice
in the stage $\gk$ lottery sum performed in
$M$ in the definition of $j(\FP)$.
We hence assume without loss of generality
for the remainder of this proof that in $M$,
we are forcing above a condition that opts
for $\FQ$ in $M^\FP$, i.e., that
$j(\FP)$ is forcing equivalent to
$\FP_\gk \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR =
\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR$.
The remainder of the proof of Lemma \ref{l3}
is as in the proof of Lemma 2 of \cite{AG}.
For concreteness, we provide a sketch
of the proof, and refer readers to the
proof of Lemma 2 of \cite{AG} for any
missing details.
By the last sentence of the
preceding paragraph, in $M$,
$j(\FP \ast \dot \FQ)$ is
forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either Prikry forcing
or trivial forcing''. Further, since
$M \models ``$There are no Mahlo
cardinals in the interval
$(\gk, \gs]$'', the next nontrivial
stage in the definition of
$j(\FP)$ after $\gk$ takes place
well above $\gs$.
Consequently, as in Lemma 2 of \cite{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga : \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of
${(P_\gk(\gl))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
$\FP \ast \dot \FQ \ast \dot \FS_\gk$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) : \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb : \gb < \eta \ra$''.
If we then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gl]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gl)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FS_\gk$
either the trivial condition
(if $\FS_\gk$ is trivial forcing)
or of the
form $\la \emptyset, B \ra$
(if $\FS_\gk$ is Prikry forcing), in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) : \gb < \gl \ra \in
\dot X$'' for some name $\dot X$ of $X$,
then as in Lemma 2 of \cite{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gl$ strongly compact''.
Since $\gl$ was arbitrary,
and since trivial forcing is
${<}\gk$-directed closed,
this completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``\gk$ is a strong cardinal
whose strongness is
Gitik-Shelah indestrucible''.
\end{lemma}
\begin{proof}
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is $\gk^+$-weakly
closed and satisfies the
Prikry condition''.
Let $\gl > \max(\gk, \card{{\rm TC}(\dot \FQ)})$
be any beth fixed point, with
$\gg$ the least beth fixed point of
cofinality $\gk^+$ above $\gl$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gg$ strongness
of $\gk$
%such that $M \models ``\gk$ isn't $\gg$ strong''.
which is generated by a
$(\gk, \gg)$-extender
${\cal E} = \la E_a : a \in
{[\gg]}^{< \go} \ra$.
Note that since ${\rm cof}(\gg) > \gk$,
as mentioned in the paragraph
immediately prior to
the statement of the Main Theorem
in Hamkins and Woodin's paper \cite{HW},
we may assume without loss of
generality that $M^\gk \subseteq M$.
In addition, since
$V \models ``\gg$ isn't a limit of
inaccessible cardinals'' and
$V_\gg \subseteq M$,
$M \models ``\gg$ isn't a limit
of inaccessible cardinals''.
In particular, since
$V \models ``$No cardinal
$\gd \in (\gk, \gg]$ is Mahlo'',
$M \models ``$No cardinal
$\gd \in (\gk, \gg]$ is Mahlo''.
By essentially the same reasoning
as given in the proof of Lemma \ref{l3},
$\dot \FQ$ is a term for an allowable
choice in the stage $\gk$ lottery
sum performed in $M$ in the definition
of $j(\FP)$. Again by essentially the
same reasoning as given in the
proof of Lemma \ref{l3}, we may assume
that $j(\FP)$ is forcing equivalent in $M$ to
a partial ordering of the form
$\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR$.
We consider now two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: $\dot \FS_\gk$ is a
term for Prikry forcing.
In this situation, by the
definition of $\FP$, it must
be the case that in
$M^{\FP \ast \dot \FQ}$,
there is some Gitik style
iteration $\FR'$
of Prikry-like forcings
having length $j(\gk)$ which
does a nontrivial forcing only
at $M$-Mahlo cardinals above $\gk$
such that
$M^{\FP \ast \dot \FQ \ast \dot \FR'}
\models ``\gk$ is a strong cardinal''.
Further, since
%$V \models ``$No cardinal $\gd > \gk$
%is Mahlo'' and $V_\gg \subseteq M$,
$M \models ``$No cardinal $\gd \in
(\gk, \gg]$ is Mahlo'',
the first nontrivial stage in the
definition of $\dot \FR'$ takes
place well above $\gg$.
%As mentioned in Section \ref{s1},
By the fact $\FR'$ is a Gitik style
iteration of Prikry-like forcings and
Lemmas 1.2 and 1.4 of \cite{G},
it must be true that
$\forces_{\FP \ast \dot \FQ}
``$Forcing with $\dot \FR'$ adds
no subsets to $\gg$''.
Therefore, since
$M^{\FP \ast \dot \FQ \ast \dot \FR'} \models
``\gk$ is $\gl$ strong'', we may immediately
infer that
$M^{\FP \ast \dot \FQ} \models ``\gk$ is
$\gl$ strong'', a fact which is witnessed in
${(V_\gg)}^{M^{\FP \ast \dot \FQ}}$.
Since $V_\gg \subseteq M$,
by the choice of $\gg$,
${(V_\gg)}^{V^{\FP \ast \dot \FQ}}
\subseteq M^{\FP \ast \dot \FQ}$, i.e.,
${(V_\gg)}^{V^{\FP \ast \dot \FQ}} =
{(V_\gg)}^{M^{\FP \ast \dot \FQ}}$.
%from which we again immediately conclude
%by the choice of $\gg$ that
We hence conclude that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
\bigskip
Case 2: $\dot \FS_\gk$ is a
term for trivial forcing.
In this situation, we can
mimic the proof of
Gitik-Shelah indestructibility
given in \cite{GS}.
For concreteness, we give
a sketch of the argument,
assuming without loss
of generality that we have
ignored the term $\dot \FS_\gk$
and rewritten all other terms
if necessary with respect to
the appropriate forcing language.
As in Case 1 for $\dot \FR'$,
the first
nontrivial stage in the
definition of $\dot \FR$
takes place well above $\gg$.
Because
$V \models ``2^\gk = \gk^+$'',
as in the proof of Lemma
2.2 of \cite{GS},
we can let
$\la \dot D_\ga : \ga < \gk^+ \ra$
be an enumeration in $V$ of the set
$X = \{\dot D \in M$ : In $M$,
$\forces_{\FP \ast \dot \FQ}
%\ast \dot \FS_\gk}
``\dot D$ is a
$\le^*$-dense subset of $\dot \FR$''
and there is a function
$f : \gk \to \wp(V_\gk)$ such that
$\dot D = j(f)(\gk)\}$.
Therefore, again as in the proof
of Lemma 2.2 of \cite{GS}, since in $M$,
$\forces_{\FP \ast \dot \FQ}
``\dot \FR$ is $\gg^+$-weakly closed'',
for every $\dot D \in M$ such that
$\forces_{\FP \ast \dot \FQ} ``\dot D$
is a $\le^*$-dense subset of $\dot \FR$'',
there is some $\ga < \gk^+$ such that
$\forces_{\FP \ast \dot \FQ} ``\dot D
\supseteq \dot D_\ga$''.
If we now let $G \ast H$ be
$V$-generic over $\FP \ast \dot \FQ$,
then as in \cite{GS},
since $M^\gk \subseteq M$, there is in
$V[G \ast H]$ an increasing sequence
$\la r_\ga : \ga < \gk^+ \ra$ of
elements of $\FR$ such that
$r_\ga \in i_{G \ast H}(\dot D_\ga)$.
For each $a \in {[\gg]}^{< \go}$,
if we define in $V[G \ast H]$ an ultrafilter
$F_a$ over ${[\gk]}^{\ov a}$ by
$x \in F_a$ iff for some
$\la p, q \ra \in G \ast H$ and some
$\ga < \gk^+$,
$\la p, q, r_\ga \ra \forces_{\FP \ast
\dot \FQ \ast \dot \FR} ``a \in j(\dot x)$'',
then the arguments of Lemmas 2.3 - 2.5 of
\cite{GS} show that
${\cal F} = \la F_a : a \in {[\gg]}^{< \go} \ra$
is in $V[G \ast H]$ a
$(\gk, \gg)$-extender witnessing the
$\gg$ strongness of $\gk$.
Since $\gg > \gl$,
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
\bigskip
\setlength{\parindent}{1.5em}
Thus, for an arbitrarily large
$\gl$,
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
Since trivial forcing is
$\gk^+$-weakly closed and
satisfies the Prikry condition,
this completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
By Lemma \ref{l4} and the
fact that
$\forces_{\FP} ``\dot \FP_{\go, \gk^+}$
is $\gk$-strategically closed'',
$V^{\FP \ast \dot \FP_{\go, \gk^+}} \models
``\gk$ is a strong cardinal''.
In addition,
$V^{\FP \ast \dot \FP_{\go, \gk^+}} \models
``\gk^+$ contains
a non-reflecting stationary set of
ordinals of cofinality $\go$''.
By reflection,
the preceding two sentences therefore
imply that in $V^{\FP \ast \dot \FP_{\go, \gk^+}}$,
unboundedly in $\gk$ many successor
cardinals $\gd < \gk$ contain
non-reflecting stationary sets
of ordinals of cofinality $\go$. Further,
again since
$\forces_{\FP} ``\dot \FP_{\go, \gk^+}$ is
$\gk$-strategically closed'',
this same fact is true in
$V^\FP$ as well.
%Again by the definition of $\FP$,
%we then immediately have that
%$V^\FP \models ``$Unboundedly
%in $\gk$ many cardinals below $\gk$
%$\gd < \gk$
%contain non-reflecting stationary
%sets of ordinals of cofinality $\go$''.
As mentioned in Section \ref{s1},
Solovay's Theorem 4.8 of \cite{SRK}
and the succeeding remarks then
immediately imply that
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
Since no partial ordering can create
a new Mahlo cardinal,
$V^\FP \models ``$No cardinal
$\gd > \gk$ is Mahlo''.
Lemmas \ref{l1} - \ref{l2} therefore complete
the proof of Theorem \ref{t1}.
\end{proof}
We remark that in the absence of
any measurable cardinals above
$\gk$ in $V^\FP$,
it is clear that it is impossible
to define in this model any sort of iterations
of Prikry or Radin forcing based
on cardinals above $\gk$.
One may therefore wonder
if in $V^\FP$ there are any partial orderings
under which $\gk$'s strongness
is indestructible other than
those which are either $\gk$-strategically
closed, $\gk$-closed, or ${\le} \gk$-directed
closed.
We have been informed by Gitik \cite{G2}
that via forcing, it is possible
to construct a universe in which
some cardinal $\gl$ is supercompact,
there are no inaccessible cardinals
above $\gl$, and there is a partial
ordering $\FP^*$ which is $\gl^+$-weakly closed,
satisfies the Prikry condition,
but isn't either $\gl$-strategically
closed, $\gl$-closed, or ${\le} \gl$-directed
closed. Unfortunately, however, Gitik's
construction requires the use of a
L\'evy collapse which is $\gl$-closed
in order to build $\FP^*$,
and there is no reason to believe that
$\FP^*$ retains its desired properties
after the forcing used in the proof
of Theorem \ref{t1}.
We thus conclude this section by asking
if in the model witnessing the
conclusions of Theorem \ref{t1}, there
are any partial orderings under which
$\gk$'s strongness is indestructible
other than those mentioned in the
preceding paragraph.
\section{The Proof of Theorem \ref{t2}}\label{s3}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact''.
%As in the proof of
%Theorem \ref{t1}, without loss of generality,
%we assume in addition that $V \models {\rm GCH}$.
For the rest of the
proof of Theorem \ref{t2}, for any
ordinal $\gd$, we redefine $\gd^*$ as
the least $V$-strong cardinal above $\gd$.
We are now in a position to define the
partial ordering $\FP$ to be used
in the proof of Theorem \ref{t2}.
$\FP$ is once again a Gitik style iteration
of Prikry-like forcings
$\la \la \FP_\gd, \dot \FQ_\gd \ra :
\gd < \gk \ra$
of length $\gk$.
This time, however, $\FP$
does a nontrivial forcing
only at those ordinals $\gd < \gk$
which are either strong
cardinals or regular
limits of strong cardinals in $V$.
If $\gd$ is a $V$-strong cardinal
but not a $V$-(regular) limit of
strong cardinals, $\dot \FQ_\gd$
is a term for Prikry forcing over
$\gd$ defined using the appropriate
normal measure.
If, on the other hand, $\gd$ is a $V$-regular
limit of strong cardinals,
we first
force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank below
$\gd^*$ which are in $V^{\FP_\gd}$
both ${<}\gd$-directed closed and
$\gd$-strategically closed.
If in
$V^{\FP_\gd \ast \dot \FQ^*_\gd}$,
there is some Gitik style iteration
$\FR$ of Prikry-like forcings having length
$\gk$ which does a nontrivial forcing
only at either $V$-strong cardinals above $\gd$
or $V$-regular limits of strong cardinals
above $\gd$ such that
$V^{\FP_\gd \ast \dot \FQ^*_\gd \ast \dot \FR}
\models ``\gd$ is a strong cardinal'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If this is not the case, then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
We note that once again,
this definition makes sense, i.e.,
that it is possible to choose
either $\dot \FQ_\gd$ or
$\dot \FS_\gd$ as a term
for Prikry forcing when appropriate.
For $\dot \FQ_\gd$, this follows
%Note that this makes sense,
since it is inductively the case that
$\card{\FP_\gd} < \gd$, so by
the L\'evy-Solovay results \cite{LS},
$\forces_{\FP_\gd} ``\gd$ is
a measurable cardinal''.
For $\dot \FS_\gd$, this follows via
essentially the same reasoning
as given in the paragraph
immediately prior to the
statement of Lemma \ref{l1},
using the Main Theorem of
\cite{HW} to infer that
anything forcing
equivalent to a partial
ordering having size below
$\gd^*$ preserves the fact that
$\gd^*$ is the least strong
cardinal above $\gd$.
\begin{lemma}\label{l5}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
By Lemma 2.1 of \cite{AC2},
since
$V \models ``\gk$ is supercompact'',
unboundedly in $\gk$ many
$\gd < \gk$ are strong cardinals.
Therefore, by the definition of $\FP$,
in $V^\FP$, unboundedly in $\gk$
many $\gd < \gk$ contain Prikry sequences.
However, by Theorem 11.1 of
\cite{CFM}, the presence of a Prikry sequence
implies the presence of a
non-reflecting stationary set
of ordinals of cofinality $\go$.
Since as was mentioned in
Section \ref{s1}, Theorem 4.8
of \cite{SRK} and the
succeeding remarks imply such a set
can't exist above a
strongly compact cardinal,
we may
now immediately infer that no
cardinal $\gd < \gk$ is strongly
compact. This completes the
proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$ is
a strongly compact cardinal
whose strong compactness is
indestructible under
${<}\gk$-directed closed forcing
which is also $\gk$-strategically closed''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6} follows closely
the proof of Lemma \ref{l3}. Specifically,
suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is both ${<}\gk$-directed
closed and $\gk$-strategically closed''.
Let $\gl > \max(\gk, \card{{\rm TC}(\dot \FQ)})$
be an arbitrary regular cardinal large
enough so that
${(2^{[\gl]^{< \gk}})}^V = \rho =
{(2^{[\gl]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \rho^+$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gk$
such that $M \models ``\gk$ isn't $\gs$ supercompact''.
As in Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal $\gd \in (\gk, \gs]$
is strong'' (since otherwise, $\gk$ would
be supercompact up to a strong cardinal and
hence be fully supercompact). Therefore,
%by the closure properties of $M$,
$M \models ``\card{{\rm TC}(\dot \FQ)}$ is
below the least strong cardinal above $\gk$''.
%This has as an immediate consequence that
Since by Lemma 2.1 of \cite{AC2} and the
succeeding remarks,
$M \models ``\gk$ is a strong cardinal
which is a limit of strong cardinals'',
we may now infer that
$\dot \FQ$ is a term for an allowable choice
in the stage $\gk$ lottery sum performed in
$M$ in the definition of $j(\FP)$.
We hence assume without loss of generality
for the remainder of this proof that in $M$,
we are forcing above a condition that opts
for $\FQ$ in $M^\FP$, i.e., that
$j(\FP)$ is forcing equivalent to
$\FP_\gk \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR =
\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either Prikry forcing
or trivial forcing''. Further, since
$M \models ``$There are no strong
cardinals in the interval
$(\gk, \gs]$'', the next nontrivial
stage in the definition of
$j(\FP)$ after $\gk$ takes place
well above $\gs$.
We may now argue as in the
proof of Lemma \ref{l3} to infer that
$V^{\FP \ast \dot \FQ} \models ``\gk$
is $\gl$ strongly compact''.
Since $\gl$ was arbitrary, and since
trivial forcing is both ${<}\gk$-directed closed
and $\gk$-strategically closed,
this completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``\gk$ is a strong cardinal
whose strongness is indestructible under
${<}\gk$-directed closed forcing which is
also $\gk$-strategically closed''.
\end{lemma}
\begin{proof}
In analogy to the proof of Lemma \ref{l6},
the proof of Lemma \ref{l7}
follows closely the proof of Lemma \ref{l4}.
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ$ is both
${<}\gk$-directed closed and $\gk$-strategically
closed''.
Let $\gl > \max(\gk, \card{{\rm TC}(\dot \FQ)})$
be any beth fixed point, with
$\gg$ the least beth fixed point
of cofinality $\gk^+$ above $\gl$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gg$ strongness
of $\gk$
such that $M \models ``\gk$ isn't $\gg$ strong''
which is generated by a
$(\gk, \gg)$-extender
${\cal E} = \la E_a : a \in
{[\gg]}^{< \go} \ra$.
As in the proof of Lemma \ref{l4},
we may assume without loss of generality that
$M^\gk \subseteq M$.
Also, by Lemma 2.5 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in [\gk, \gg]$ is strong''
(since otherwise, $\gk$ would be
strong up to a strong cardinal
and hence be fully strong).
%By essentially the same reasoning
%as given in the proof of Lemma \ref{l6},
Therefore, since $M \models ``\gk$
is a regular limit of strong cardinals'',
we may now infer that
$\dot \FQ$ is a term for an allowable
choice in the stage $\gk$ lottery
sum performed in $M$ in the definition
of $j(\FP)$.
%Again by essentially the
%same reasoning as given in the
%proof of Lemma \ref{l6},
Hence, as above, we may assume
that $j(\FP)$ is forcing equivalent in $M$ to
a partial ordering of the form
$\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR$.
We consider once again two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: $\dot \FS_\gk$ is a
term for Prikry forcing.
In this situation, by the
definition of $\FP$, it must
be the case that in
$M^{\FP \ast \dot \FQ}$,
there is some Gitik style
iteration $\FR'$
of Prikry-like forcings
having length $j(\gk)$ which
does a nontrivial forcing only
at either $M$-strong cardinals above $\gk$
or $M$-regular limits of strong
cardinals above $\gk$ such that
$M^{\FP \ast \dot \FQ \ast \dot \FR'}
\models ``\gk$ is a strong cardinal''.
Further, since
%as in Lemma 2.5 of \cite{AC2},
$M \models ``$No cardinal $\gd \in [\gk, \gg]$
is strong'',
%(since otherwise, $\gk$ would be
%strong up to a strong cardinal and
%hence be fully strong),
the first nontrivial stage in the
definition of $\dot \FR'$ takes
place well above $\gg$.
We may now argue as in the proof
of Case 1 of Lemma \ref{l4} to
infer that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
\bigskip
Case 2: $\dot \FS_\gk$ is a
term for trivial forcing.
As in Case 1 of the
proof of this lemma for $\dot \FR'$,
the first
nontrivial stage in the
definition of $\dot \FR$
takes place well above $\gg$.
We may now argue as in the proof
of Case 2 of Lemma \ref{l4} to infer that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
\bigskip
\setlength{\parindent}{1.5em}
Thus, once again, for an arbitrarily
large $\gl$,
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ strong''.
Since trivial forcing is
both ${<}\gk$-directed closed
and $\gk$-strategically closed,
this completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is a strong cardinal''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gk$
is a strong cardinal''.
By the definition of $\FP$
and the proof of Lemma \ref{l5},
$\gd$ cannot be in $V$ a strong
cardinal which isn't a (regular)
limit of strong cardinals,
since in $V^\FP$, all such
cardinals have cofinality $\go$.
Further, by the argument given
in the second paragraph of
the proof of Lemma \ref{l1},
$\gd$ cannot be in $V$ a
regular limit of strong cardinals,
since if it were, by the
definition of $\FP$, $\gd$ would
have to have cofinality $\go$ in
$V^\FP$, a contradiction to the
fact $\gd$ is strong in $V^\FP$.
These two facts immediately
imply that $\gd$ cannot be
a stage in the definition of
$\FP$ at which a nontrivial forcing occurs.
In particular, in $V$,
$\gd$ cannot be a strong cardinal.
Define now
$\gg = \sup(\{\ga < \gd : \ga$ is
a nontrivial stage of forcing in
the definition of $\FP\})$. Write
$\FP = \FP_{\gg + 1} \ast \dot
\FP^{\gg + 1}$.
It must be the case that
$V^{\FP_{\gg + 1}} \models
``\gd$ is a strong cardinal''.
To see this, note that
since $\gd$
is a trivial stage of forcing
in the definition of $\FP$,
$\gg < \gd$. Further,
$\gd < \gg^*$, since if
$\gg^* \le \gd$, then because
$\gg^*$ is a nontrivial
stage of forcing in the
definition of $\FP$,
$\gg$ is not the supremum
of the nontrivial stages
of forcing below $\gd$.
Thus, the first nontrivial stage of
forcing in $\FP^{\gg + 1}$, namely $\gg^*$,
must occur above $\gd$.
Because $\FP^{\gg + 1}$ is a
Gitik style iteration of Prikry-like
forcings that does a nontrivial
forcing only at ordinals at
and above $\gg^*$,
forcing with $\FP^{\gg + 1}$ adds
no bounded subsets to $\gg^*$.
We may therefore infer that
$V^{\FP_{\gg + 1}} \models
``\gd$ is $\ga$ strong for every
$\ga < \gg^*$''. Since
$\FP_{\gg + 1}$ is forcing equivalent
to a partial ordering having size
below $\gg^*$, the
Main Theorem of \cite{HW} tells us that
$V^{\FP_{\gg + 1}} \models
``\gg^*$ is a strong cardinal''.
As mentioned in the first paragraph
of the
proof of Lemma \ref{l7}, it now
immediately follows that
$V^{\FP_{\gg + 1}} \models
``\gd$ is a strong cardinal''.
Write $\FP_{\gg + 1} =
\FP_\gg \ast \dot \FQ_\gg$.
%By the definition of $\FP$,
%$\card{\FP_\gg} \le \gg$. If
%$\card{\FP_\gg} < \gg$, then
%once again, the definition of
%$\FP$ tells us that $\dot \FQ_\gg$
%is a term for trivial forcing.
If $\FP_{\gg + 1}$ is forcing
equivalent to a partial ordering
having size below $\gd$,
%in analogy to the preceding paragraph,
the Main Theorem of \cite{HW}
allows us once again to infer that
$V \models ``\gd$ is a strong cardinal'',
a contradiction.
Suppose consequently that this is not the case.
By the definition of $\FP$,
we hence have that $\gg$ is a
regular limit of strong cardinals.
Therefore, $\card{\FP_\gg} = \gg$,
and $\forces_{\FP_{\gg}} ``\dot \FQ_\gg =
\dot \FQ^*_\gg \ast \dot \FS_\gg$'',
where $\dot \FQ^*_\gg$ is a term for the
lottery sum of ${<}\gg$-directed closed
partial orderings which are also $\gg$-strategically
closed, and $\dot \FS_\gg$ is a term
for either trivial forcing or
Prikry forcing defined with respect
to the appropriate normal measure over $\gg$.
Further, since in either case
$\forces_{\FP_\gg \ast \dot \FQ^*_\gg}
``\card{\dot \FS_\gg} < \gd$'', the
Main Theorem of \cite{HW}
may be used over
$V^{\FP_\gg \ast \dot \FQ^*_\gg}$ to infer that
$\forces_{\FP_\gg \ast \dot \FQ^*_\gg}
``\gd$ is a strong cardinal''.
However,
since $\FP_\gg \ast \dot \FQ^*_\gg$
admits a closure point at $\gg$
(as witnessed by
$\card{\FP_\gg} = \gg$ and
$\forces_{\FP_\gg} ``\dot \FQ^*_\gg$ is
$\gg$-strategically closed'') and $\gg < \gd$,
we may now infer by Theorem \ref{t3} that
$V \models ``\gd$ is a strong cardinal''.
%This once again contradicts the what
%was established in the first
%paragraph of the proof of this lemma.
As before, this is a contradiction.
This completes the proof of Lemma \ref{l8}.
\end{proof}
Lemmas \ref{l5} - \ref{l8} complete
the proof of Theorem \ref{t2}.
\end{proof}
\section{Concluding Remarks}\label{s4}
In conclusion to this paper, we
discuss some possible generalizations
of Theorems \ref{t1} and \ref{t2}.
First, we note that any sort of
exact generalization of Theorem \ref{t1}
for two or more cardinals is impossible.
It is not even possible to obtain
a model $\ov V$ in which the first two
strongly compact cardinals $\gk_1$
and $\gk_2$ are the first two
strong cardinals and $\gk_2$
has its strongness Gitik-Shelah
indestructible.
%This is seen by using the
%same reasoning as given in
%the proof of Lemma \ref{l2}.
This is since the
argument given in the
proof of Lemma \ref{l2}
implies that if
the situation hypothesized
in the preceding sentence
were true, then there would be
by the Gitik-Shelah indestructibility
of $\gk_2$
a successor cardinal above
$\gk_1$ in $\ov V$ containing a
non-reflecting stationary
set of ordinals of cofinality $\go$.
As we have already
mentioned, this contradicts the strong
compactness of $\gk_1$.
This does not, of course, forestall
other possible generalizations of
either Theorem \ref{t1} or
Theorem \ref{t2}.
We can, for instance, ask if it is
possible to prove a version of
Theorem \ref{t1} in which there
are no restrictions on the large
cardinal structure of either our
ground model or generic extension.
The main obstacle to this
at the moment is any sort of
reasonable extension of
Hamkins' work of \cite{H2}, \cite{H3},
and \cite{H5} to forcing iterations
in which Prikry or Radin sequences
of ordinals are added.
Elaborating on this further,
we note that because of the
restrictions placed on the large
cardinal structure of our ground model
$V$ in the proof of Theorem \ref{t1},
we do not in the proof of Lemma \ref{l1}
worry about our forcing iteration
$\FP$ creating new strong cardinals,
as we do in Lemma \ref{l8} in
the proof of Theorem \ref{t2} when
there are no restrictions placed
on the large cardinal structure of
our ground model $V$.
If there were a version of
Theorem \ref{t3} for
a modification of the iteration
defined in the proof of Theorem \ref{t2}
in which the components of each
lottery sum were allowed to be as
in the proof of Theorem \ref{t1},
then it would be possible to prove
Theorem \ref{t1} without any
restrictions placed on the large
cardinal structures of either our
ground model or generic extension.
Indeed, the proof of Theorem \ref{t2}
witnesses the interesting
occurrence that even though the
partial ordering $\FP$ used is a Gitik style
iteration of Prikry-like forcings,
it has closure points ``locally''
in the sense that when necessary,
$\FP_\gg$ for the appropriate $\gg$
admits a closure point at $\gg$.
We can also ask if there is any
sort of generalization of
Theorem \ref{t2} possible for
two or more cardinals.
In fact, the methods of this
paper and \cite{Alot} allow
us to prove the following.
\begin{theorem}\label{t4}
Suppose
$V \models ``$ZFC + GCH + $\gk_1 <
\gk_2 < \gl$ are such that
$\gk_1$ is supercompact and
$\gk_2$ and $\gl$ are the
smallest cardinals above $\gk_1$
such that $\gk_2$ is $\gl$
supercompact and $\gl$ is
inaccessible''.
There is then a partial ordering
$\FP \in V$ such that for
$V' = {(V_\gl)}^{V^\FP}$,
$V' \models ``\gk_1$ and $\gk_2$
are both the first two strongly
compact and strong cardinals +
Each of $\gk_1$'s strong compactness
and strongness is indestructible
under ${<}\gk_1$-directed closed
partial orderings which are also
$\gk_1$-strategically closed +
$\gk_2$'s strongness, but not
its strong compactness, is
indestructible under $\gk_2$-strategically
closed forcing which is also
${<}\gk_1$-directed closed''.
\end{theorem}
\begin{sketch}
Suppose $V$ is as in the hypotheses
of Theorem \ref{t4}.
%As in the proofs of Theorems \ref{t1}
%and \ref{t2}, without loss of
%generality, we assume that in addition,
%$V \models {\rm GCH}$.
Let $\FP'$ be the partial ordering
used in the proof of Theorem \ref{t2}
defined with respect to $\gk_1$.
Since the definition of $\FP'$
may be given so that $\card{\FP'} = \gk_1$,
by the proof of Theorem \ref{t2} and
the results of \cite{LS},
$\ov V = V^{\FP'} \models ``\gk_1$
is both the least strongly compact
and least strong cardinal + Each of
$\gk_1$'s strong compactness and strongness
is indestructible under ${<}\gk_1$-directed
closed forcing which is also $\gk_1$-strategically
closed + $\gk_2$ and $\gl$ are the smallest
cardinals above $\gk_1$ such that
$\gk_1$ is $\gl$ supercompact and
$\gl$ is inaccessible''.
Work in $\ov V$. Let
$\la \gd_\ga : \ga < \gk_2 \ra$
enumerate in increasing order
$\{\gd \in (\gk_1, \gk_2) : \gd$ is $\gg$ strong for
all $\gg < \gl_\gd\}$, where
$\gl_\gd$ is the least inaccessible
cardinal above $\gd$.
Define $\FP''$
as the Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk_2 \ra$
of length $\gk_2$ which at stage $0$ adds
a Cohen subset to $\gk^+_1$.
At any stage $\ga > 0$, if
$\forces_{\FP_\ga} ``$After forcing with any
$\gd_\ga$-strategically closed partial ordering
of rank below $\gl_{\gd_\ga}$
which is also ${<}\gk_1$-directed
closed, $\gd_\ga$ remains
$\gg$ strong for all $\gg < \gl_{\gd_\ga}$'', then
$\dot \FQ_\ga$ is a term for $\FP_{\gk_1, \gd_\ga}$.
If this is not the case, then
$\dot \FQ_\ga$ is a term for a $\gd_\ga$-strategically
closed partial ordering of rank below
$\gl_{\gd_\ga}$ which is also
${<}\gk_1$-directed closed such that
$\forces_{\FP_\ga} ``$After forcing with
$\dot \FQ_\ga$, there is some $\gg < \gl_{\gd_\ga}$
such that $\gd_\ga$ is no longer $\gg$ strong''.
Note that $\FP''$ has been defined so
as to be both ${<}\gk_1$-directed
closed and $\gk_1$-strategically
closed.
Consequently, since $\FP''$
therefore cannot increase the
degree of strong compactness
or strongness of any cardinal
$\gd < \gk_1$, by the
indestructibility properties
of $\gk_1$ in $\ov V$,
$\ov V^{\FP''} \models ``\gk_1$
is both the least strongly compact
and least strong cardinal and
has each of its strong compactness
and strongness indestructible under
${<}\gk_1$-directed closed forcing
which is also $\gk_1$-strategically closed''.
In addition, $\FP''$ is a version of
the partial ordering used in the
proof of Theorem 5 of \cite{Alot},
so by the proof of Theorem 5 of \cite{Alot},
for $V' = {(V_\gl)}^{\ov V^{\FP''}}$,
$V' \models ``\gk_2$ is both
the first strongly compact and first
strong cardinal above $\gk_1$ and has
its strongness, but not its strong
compactness, indestructible under
$\gk_2$-strategically closed forcing
which is also ${<}\gk_1$-directed closed''.
Further, $\gk_1$ retains its strong
compactness, strongness, and
indestructibility properties in $V'$,
and by Lemma \ref{l5}, no cardinal
$\gd < \gk_1$ is strongly compact in $V'$.
Also, if there were a $\gd < \gk_1$
such that
$V' \models ``\gd$ is a strong cardinal'',
then $\ov V^{\FP''} \models ``\gd$ is
$\ga$ strong for every $\ga < \gk_1$ and
$\gk_1$ is a strong cardinal'',
which as we have already seen implies
$\ov V^{\FP''} \models ``\gd$ is a strong
cardinal''. This, of course, is
a contradiction to the fact that
$\gk_1$ is the least strong cardinal in
$\ov V^{\FP''}$.
If we now set $\FP = \FP' \ast \dot \FP''$,
the sketch of the proof of
Theorem \ref{t4} is complete.
\end{sketch}
We remark that the reason
$\gk_2$'s strong compactness
is not indestructible in $V^\FP$ under
$\gk_2$-strategically closed forcing
which is also ${<}\gk_1$-directed closed
is that if this were the case,
we could then force over
$V^\FP$ with $\FP_{\gk_1, \gl}$
for $\gl > \gk_2$ any
regular cardinal and preserve
the strong compactness of $\gk_2$.
This, however, contradicts
Theorem 4.8 of \cite{SRK} and
the succeeding remarks.
There is no prima facie reason, though,
that one could not construct
a model in which there is
more than one strong or
strongly compact cardinal, the strong
and strongly compact cardinals
coincide precisely, and each
cardinal $\gk$ which is either
strongly compact or strong has
both its strong compactness and
strongness indestructible under
${<}\gk$-directed closed forcing
which is also $\gk$-strategically
closed.
Unfortunately, this will not
be accomplished by iterating
any version of the partial ordering
used in the proof of Theorem \ref{t2}.
This is since this partial ordering
adds Prikry sequences, and as we
observed in the proof of Lemma \ref{l5},
a Prikry sequence can't exist above
a strongly compact cardinal.
We therefore conclude
this paper by asking if the
construction of such a model
is indeed possible.
\medskip
\noindent Added in revision: Sargsyan has
announced a positive solution to
the question posed above.
The details will appear in \cite{Sa}.
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\end{document}