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\title{Indestructibility and Stationary Reflection
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal,
strong cardinal, weakly compact cardinal,
Mahlo cardinal, indestructibility, stationary reflection,
non-reflecting stationary set of ordinals.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{January 8, 2008\\
(revised March 21, 2008)}
\begin{document}
\maketitle
\begin{abstract}
If $\gk < \gl$
are such that $\gk$ is a strong cardinal
%is indestructibly strong
whose strongness is indestructible
under $\gk$-strategically closed forcing
and $\gl$ is weakly compact,
then we show that
$A = \{\gd < \gk \mid \gd$
is a non-weakly compact
Mahlo cardinal which reflects
stationary sets$\}$
must be unbounded in $\gk$.
This phenomenon, however, need not
occur in a universe with
%an indestructibly strong cardinal and
relatively few large cardinals.
In particular,
we show how to construct a model
where no cardinal is supercompact
up to a Mahlo cardinal
in which the least supercompact
cardinal $\gk$ is also the
least strongly compact cardinal,
$\gk$'s strongness is indestructible under
$\gk$-strategically closed forcing,
$\gk$'s supercompactness is indestructible
under $\gk$-directed closed forcing
not adding any new subsets of $\gk$,
and $\gd$ is Mahlo and reflects
stationary sets iff $\gd$ is weakly compact.
In this model, no strong cardinal
$\gd < \gk$ is indestructible
under $\gd$-strategically closed forcing.
It therefore follows that it is
relatively consistent for the least
strong cardinal $\gk$ whose
strongness is indestructible under
$\gk$-strategically closed forcing
to be the same as the least supercompact
cardinal, which also has its supercompactness
indestructible under $\gk$-directed closed
forcing not adding any new subsets of $\gk$.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
It is known that
the structure of the universe above either a
strong or supercompact
cardinal $\gk$ exhibiting suitable
indestructibility properties
can affect what happens
at large cardinals below $\gk$, assuming
the universe is sufficiently rich.
%has enough large cardinals.
On the other hand, these effects
can be mitigated if the universe
contains relatively few large cardinals.
For instance, as shown in \cite{AH4} and \cite{A07},
if $\gk$ is either strong or supercompact
and appropriately indestructible and
$\gl > \gk$ is $2^\gl$ supercompact, then
there must always be measurable
cardinals $\gd < \gk$ which are not
themselves limits of measurable cardinals
such that $\gd$ is $\gd^+$ strongly compact
but $\gd$ is not $\gd^+$ supercompact.
On the other hand, as shown in \cite{A07},
the existence of such measurable cardinals
below a suitably indestructible strong or
supercompact cardinal $\gk$ need not occur
if there are no measurable cardinals above $\gk$.
This and related phenomena have been examined
further in \cite{AH4}, \cite{A07}, and \cite{AA}.
%This phenomenon has been examined
%in, e.g., \cite{AH4} and \cite{A07}.
The purpose of this paper is
to continue the studies
mentioned in the preceding
paragraph, but in the context
of stationary reflection.
Specifically, we prove the following.
\begin{theorem}\label{t1}
If
$\gk < \gl$ are such that $\gk$
%is indestructibly strong and
is a strong cardinal whose
strongness is indestructible
under $\gk$-strategically closed forcing and
$\gl$ is weakly compact, then
$A = \{\gd < \gk \mid \gd$ is a
non-weakly compact Mahlo cardinal
which reflects stationary sets$\}$
is unbounded in $\gk$.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal is supercompact up to a
Mahlo cardinal''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact +
No cardinal is supercompact up to a
Mahlo cardinal''. In $V^\FP$, $\gk$ is both
the least strongly compact and least
supercompact cardinal. In addition,
in $V^\FP$, $\gk$'s strongness
is indestructible under $\gk$-strategically
closed forcing, and $\gk$'s supercompactness is
indestructible under $\gk$-directed closed
forcing not adding any new subsets of $\gk$.
Finally, in $V^\FP$, $\gd$ is Mahlo
and reflects stationary sets iff $\gd$
is weakly compact.
\end{theorem}
As a corollary to Theorem \ref{t2},
we have the following theorem.
\begin{theorem}\label{t3}
It is consistent, relative to the
existence of a supercompact cardinal,
for the least strong cardinal $\gk$
whose strongness is indestructible
under $\gk$-strategically closed forcing
to be the same as the least supercompact
cardinal, which also has its supercompactness
indestructible under $\gk$-directed
closed forcing not adding any new subsets
of $\gk$.
\end{theorem}
Theorem \ref{t3} should be contrasted with
\cite[Lemma 2.1]{AC}, which says that
any supercompact cardinal must be
a limit of strong cardinals.
It is thus not possible for the
least supercompact cardinal to
be the same as the least strong cardinal.
If we add on the requirement of a
certain degree of indestructibility,
then Theorem \ref{t3} indicates that
not only can
the least strong cardinal with
this degree of indestructibility
be the same as the least supercompact
cardinal $\gk$, but the supercompactness
of $\gk$ can have a certain amount of
indestructibility as well.
%We conclude Section \ref{s1}
%with a discussion of
%some preliminary material.
%We presume a basic knowledge
%of large cardinals and forcing.
%A good reference in this
%regard is \cite{J}.
We now very briefly give some
preliminary information
concerning notation and terminology.
%For anything left unexplained,
%readers are urged to consult \cite{A03},
%\cite{A01a},
%\cite{AS97a}, or \cite{AS97b}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
%and $p \decides \varphi$ means that $p$ decides $\varphi$.
%For $\gk$ a regular cardinal and $\gl$ an ordinal,
%$\add(\gk, \gl)$ is the standard partial ordering for adding
%$\gl$ many Cohen subsets of $\gk$.
%For $\ga < \gb$ ordinals,
%$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
%$(\a, \b)$ are as in standard interval notation.
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 $\FP$ is a reverse Easton iteration
%such that at stage $\ga$, a nontrivial
%forcing is done adding a subset
%of $\gd$, then we will say that
%$\gd$ is in the field of $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
The partial ordering
$\FP$ is {\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\em ${\prec}\gk$-strategically closed}
if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$ \FP$ is {\em ${<}\gk$-strategically closed}
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
$\FP$ is {\em $(\gk, \infty)$-distributive} if
given a sequence $\la D_\ga \mid \ga < \gk \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$ is also a dense
open subset of $\FP$.
Note that if $\FP$ is either
$\gk$-strategically closed or
$(\gk, \infty)$-distributive and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
In addition, if $\FP$ is $\gk$-directed closed, then
$\FP$ is ${\prec}\gk$-strategically closed.
%An {\em indestructibly supercompact
%cardinal} is one as first given
%by Laver in \cite{L}, i.e.,
%$\gk$ is indestructibly supercompact
%if $\gk$'s supercompactness is
%preserved in any generic extension
%via a $\gk$-directed closed
%partial ordering (including
%of course trivial forcing).
The cardinal $\gk$ {\em reflects stationary sets}
if for every stationary $S \subseteq \gk$,
there is some $\gd < \gk$ such that
$S \cap \gd$ is stationary in $\gd$.
The cardinal $\gk$ is {\em supercompact
up to the cardinal $\gl$} if
$\gk$ is $\gd$ supercompact
for every $\gd < \gl$.
As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra \mid \FP \in {\cal A}$
and $p \in \FP\} \bigcup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. The terminology ``lottery sum''
is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
\section{The Proofs of Theorems
\ref{t1} -- \ref{t3}}\label{s2}
\begin{pf}
We turn now to the proof of Theorem \ref{t1},
which we begin with the following key lemma.
\begin{lemma}\label{l1t1}
Suppose
$V \models ``\gk < \gl$ are such that
$\gk$ is a regular cardinal and
$\gl$ is weakly compact''.
There is then a $\gk$-strategically
closed notion of forcing $\FQ \in V$
such that
$V^\FQ \models ``\gl$ is a non-weakly
compact Mahlo cardinal which reflects
stationary sets''.
\end{lemma}
\begin{proof}
Let $\FP$
be the reverse Easton iteration of
length $\gl + 1$ which adds a
Cohen subset to every inaccessible
cardinal in the half-open interval
$(\gk, \gl]$. We begin by showing that
$V^\FP \models ``\gl$ is weakly compact''.
To do this, we adapt an argument
from \cite[Theorem 1.4]{H4},
quoting liberally from Hamkins' presentation.
Write $\FP = \FP_\gl \ast \dot \FQ_\gl$.
By the definition of $\FP$, it is
clearly the case that $\gl$ remains
inaccessible in $V^\FP = V^{\FP_\gl \ast \dot \FQ_\gl}$.
It therefore is enough to show that
$\gl$ has the tree property in $V^{\FP_\gl \ast \dot \FQ_\gl}$.
Suppose as a consequence that
$\dot T$ is a name for a $\gl$-tree in
$V^{\FP_\gl \ast \dot \FQ_\gl}$. In $V$, let
$N$ be a transitive elementary
substructure of $H(\gl^+)$ of size
$\gl$ containing $\FP_\gl \ast \dot \FQ_\gl$ and
$\dot T$ which is closed under
${<}\gl$ sequences.
Since $\gl$ is weakly compact in $V$,
there is an elementary embedding
$j : N \to M$ having critical point $\gl$.
As in \cite[Theorem 1.4]{H4}, we
may also assume that $\card{M} = \gl$ and
$V \models ``M^{< \gl} \subseteq M$''.
Write $j(\FP_\gl) = \FP_\gl \ast \dot \FQ_\gl
\ast \dot \FR \ast j(\dot \FQ_\gl)$.
For any $V$-generic object $G \ast H$ over
$\FP_\gl \ast \dot \FQ_\gl$, the fact $\FP_\gl$ is
$\gl$-c.c$.$ and $\FQ_\gl$ is
$\gl$-directed closed allows us to infer that
$V[G][ H] \models ``M[G][ H]^{< \gl} \subseteq
M[G][ H]$''. Further,
it is the case that
$M[G][H] \models ``\FR \ast j(\dot \FQ_\gl)$ is
${\prec}\gl$-strategically closed'', since
$M[G][H] \models ``\FR \ast j(\dot \FQ_\gl)$ is
$\gl$-directed closed''.
From this, because
$V[G][ H] \models ``M[G][ H]^{< \gl} \subseteq
M[G][ H]$'',
$V[G][H] \models ``\FR \ast j(\dot \FQ_\gl)$ is
${\prec}\gl$-strategically closed'' as well.
In addition, it follows that
$\forces_{\FP_\gl \ast \dot \FQ_\gl}
``\bigcup \dot H$ is a condition in the
partial ordering $j(\dot \FQ_\gl) \in
N^{\FP_\gl \ast \dot \FQ_\gl
\ast \dot \FR}$''.
Therefore, since by the fact
$V[G][ H] \models ``\card{M[G][ H]} = \gl$'',
there are only $\gl$ many dense open
subsets of $\FR \ast j(\dot \FQ_\gl)$
present in $M[G][ H]$,
%and since $V[G][ H] \models ``M[G][ H]^{< \gl} \subseteq M[G][ H]$'',
we may use a standard diagonalization argument
%(see, e.g., the proof of Lemma \ref{l3t2})
to meet the $\gl$ many dense open
subsets of $\FR \ast j(\dot \FQ_\gl)$ and
construct in $V[G][ H]$ an
$M[G][ H]$-generic object $H' \ast H''$ for
$\FR \ast j(\dot \FQ_\gl)$ such that
$\bigcup H \in H''$.
Specifically, let
$\la D_\gs \mid \gs < \gl \ra$ enumerate in
$V[G][H]$ the dense open subsets of
$\FR \ast j(\dot \FQ_\gl)$ present in
$M[G][H]$ such that
every dense open subset of $\FR\ast j(\dot \FQ_\gl)$
occurring in $M[G][H]$ 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 $q_\tau$
%$\sup(\la q_\gb \mid \gb < \gs \ra)$
(initially, $q_{0} = \la \emptyset, \bigcup H \ra$),
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
$q \in \la q_\gb \mid \gb < \gs \ra$.
By the ${\prec} \gl$-strategic closure of
$\FR \ast j(\dot \FQ_\gl)$ in $V[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs \mid \gs < \gl \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gl$.
Let
$H' = \{p \in \FR \ast j(\dot \FQ_\gl) \mid \exists \gs <
\gl [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR \ast j(\dot \FQ_\gl)$.
As $j '' (G \ast H) \subseteq G \ast H \ast H' \ast H''$,
we then have that
$j$ lifts in $V[G][ H]$ to an elementary embedding
$j : N[G][H] \to M[G][H][H'][H'']$.
Because $\dot T \in N$,
$T \in N[G][H]$.
Since $T$ is a $\gl$-tree in both
$V[G][H]$ and $N[G][H]$, by elementarity,
$j(T)$ is a $j(\gl)$-tree in $M[G][H][H'][H'']$.
Any element on the $\gl^{\rm th}$ level of
of $j(T)$ gives a branch of length
$\gl$ through $T$.
This means that $\gl$ has the tree
property in $V[G][H]$, as desired.
We continue using results of Kunen found in
the proof of the Theorem of
\cite[Section 3, pages 68--71]{Ku}.
Working now in $V^{\FP_\gl}$,
we may factor $\FQ_\gl$ as
$\FR_\gl \ast \dot \FT_\gl$, where
$\FR_\gl$ is a notion of forcing adding a
$\gl$-Souslin tree ${\cal T}$, and $\FT_\gl$
is the partial ordering adding a generic
path through ${\cal T}$.
It is the case (see \cite[pages 69--70]{Ku})
that $\FR_\gl$ is
${<} \gl$-strategically closed and
$\FT_\gl$ is $\gl$-c.c. In addition,
$\FR_\gl$ is $\ha_1$-directed closed but
not necessarily $\ha_2$-directed closed.
We claim that $\FQ = \FP_\gl \ast \dot \FR_\gl$
is our desired partial ordering.
To see this, by its definition, $\FQ$ is clearly
$\gk$-strategically closed. Since
$V^{\FP_\gl \ast \dot \FQ_\gl} \models ``\gl$
is weakly compact'',
$V^{\FP_\gl \ast \dot \FQ_\gl} \models ``\gl$
is a Mahlo cardinal''. Thus, since
$\FQ_\gl = \FR_\gl \ast \dot \FT_\gl$,
forcing can't create a Mahlo cardinal, and
$V^{\FP_\gl \ast \dot \FR_\gl} \models ``$There
is a $\gl$-Souslin tree'',
$V^{\FP_\gl \ast \dot \FR_\gl} \models ``\gl$ is a
non-weakly compact Mahlo cardinal''.
It therefore remains to show that
$V^{\FP_\gl \ast \dot \FR_\gl} \models ``\gl$
reflects stationary sets''. To do this, let
$S \in V^{\FP_\gl \ast \dot \FR_\gl}$ be stationary.
Since $\FT_\gl$ is $\gl$-c.c., by
\cite[Exercise H2, page 247]{Ku2},
$V^{\FP_\gl \ast \dot \FR_\gl \ast \dot \FT_\gl} =
V^{\FP_\gl \ast \dot \FQ_\gl} \models
``S$ is stationary''. Another appeal to the
weak compactness of $\gl$ in
$V^{\FP_\gl \ast \dot \FQ_\gl}$ then yields
the existence of a fixed $\gd < \gl$ such that
$V^{\FP_\gl \ast \dot \FQ_\gl} \models ``S
\cap \gd$ is a stationary
subset of $\gd$''.
Because $V^{\FP_\gl} \models ``\FQ_\gl$ is
$\gl$-directed closed'',
$S \cap \gd \in V^{\FP_\gl}$, and
$V^{\FP_\gl} \models ``S
\cap \gd$ is a stationary
subset of $\gd$''.
Since $\FR_\gl$ is
${<} \gl$-strategically closed,
%and $S \in V^{\FP_\gl \ast \dot \FR_\gl}$,
%this last fact is true in
$V^{\FP_\gl \ast \dot \FR_\gl} = V^\FQ \models
``S \cap \gd$ is a stationary subset of $\gd$''.
This completes the proof of Lemma \ref{l1t1}.
\end{proof}
We note that the proof given in
Lemma \ref{l1t1} of the fact that
$V^{\FP_\gl \ast \dot \FQ_\gl}
\models ``\gl$ is weakly compact''
remains valid if we only force with $\FP_\gl$.
In this case,
$N$ may be chosen so as to
contain $\FP_\gl$ as a member, with $j : N \to M$
as before an elementary embedding having
critical point $\gl$. We then have that
$j(\FP_\gl) = \FP_\gl \ast
\dot \FR^*$, and there is neither
a forcing nor a generic object $H$ at stage
$\gl$ in $V[G]$ (although in $M[G]$,
$\FR^*$ contains a forcing at stage $\gl$).
Since $M[G] \models ``\FR^*$ is
${\prec} \gl$-strategically closed'',
an $M[G]$-generic object $H^*$ for $\FR^*$
may be constructed in $V[G]$ as above, without
having to worry about a master condition for
a forcing done at stage $\gl$ in $V[G]$.
(The game starts with $q_0$ being the empty condition.)
As $j '' G \subseteq G \ast H^*$,
the elementary embedding $j$ will then lift to
$j : N[G] \to M[G][H^*]$, and the remainder of
the argument that $V[G] \models ``\gl$ is
weakly compact'' is as before.
This observation will be critical in the
proof of Lemma \ref{l4at2} to be given in
the proof of Theorem \ref{t2}.
%We note also that Theorem \ref{t1}
%does not require
Having completed the proof of
Lemma \ref{l1t1}, the proof of
Theorem \ref{t1} now follows
fairly easily. Suppose
$\gk < \gl$ are such that
$\gk$ is a strong cardinal whose
strongness is indestructible under
$\gk$-strategically closed forcing and
$\gl$ is weakly compact. Force with the
$\gk$-strategically closed partial
ordering $\FQ$ of Lemma \ref{l1t1}.
After this forcing, $\gl$ has become a
non-weakly compact Mahlo cardinal which
reflects stationary sets, and $\gk$
remains a strong cardinal. In addition,
by reflection, $A = \{\gd < \gk \mid \gd$
is a non-weakly compact Mahlo cardinal which
reflects stationary sets$\}$ is unbounded
in $\gk$. Since $\FQ$ is $\gk$-strategically
closed, $A$ must be unbounded in $\gk$ in
the ground model as well.
This completes the proof of Theorem \ref{t1}.
\end{pf}
With the proof of Theorem \ref{t1} finished,
we turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal is supercompact up to a Mahlo cardinal''.
Our partial ordering $\FP$ is the reverse
Easton iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk \ra$
of length $\gk$ which begins by
adding a Cohen real and whose
only later nontrivial stages occur at ordinals
$\gd < \gk$ which are $V$-Mahlo cardinals.
If $\gd$ is not weakly compact in $V$, then
$\dot \FQ_\gd$ is a term for the partial
ordering adding a non-reflecting stationary
set of ordinals of cofinality
$\go$ to $\gd$.\footnote{The precise
definition of this partial ordering may
be found in \cite{AC}. We do wish to
note here, however, that it is
${\prec} \gd$-strategically closed.
In particular, the partial ordering for
adding a non-reflecting stationary set
of ordinals of cofinality $\go$ to
$\gd^+$ is $\gd$-strategically closed.}
If $\gd$ is weakly compact in $V$, then
$\dot \FQ_\gd$ is a term for the lottery
sum of all partial orderings having
rank below the least $V$-Mahlo cardinal
above $\gd$ which are either $\gd$-strategically
closed or $\gd$-directed closed and do not
add a new subset of $\gd$.
\begin{lemma}\label{l1t2}
$V^\FP \models ``$No cardinal is
supercompact up to a Mahlo cardinal''.
\end{lemma}
\begin{proof}
Write $\FP = \FP' \ast \dot \FP''$, where
$\FP'$ is nontrivial, $\card{\FP'} = \go$, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically
closed''. Because $\FP$ admits
this factorization, by Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}, if
$V^\FP \models ``\gd$ is $\gl$ supercompact'', then
$V \models ``\gd$ is $\gl$ supercompact'' as well.
Thus, since $V \models ``$No cardinal is supercompact
up to a Mahlo cardinal'', $V^\FP \models ``$No
cardinal is supercompact up to a Mahlo cardinal''.
This completes the proof of Lemma \ref{l1t2}.
\end{proof}
\begin{lemma}\label{l2t2}
$V^\FP \models ``\gk$'s supercompactness is
indestructible under $\gk$-directed
closed forcing not adding any new
subsets of $\gk$''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 1.1]{A07}.
Let $\FQ \in V^\FP$ be such that
$V^\FP \models ``\FQ$ is
$\gk$-directed closed and does not
add a new subset of $\gk$''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{\FP} ``\dot \FQ$ is
$\gk$-directed closed and does not
add a new subset of $\gk$''.
Suppose $\gl \ge
\max(\gk^+, \card{{\rm TC}(\dot \FQ)})$
is an arbitrary cardinal, and let
$\gg = 2^{\card{[\gl]^{< \gk}}}$. Take
$j : V \to M$ as an elementary
embedding witnessing the $\gg$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gg)$.
%such that $j(f)(\gk) = \dot \FQ$.
Since
$V \models ``$No cardinal above
$\gk$ is Mahlo'', $\gg \ge \gk$,
and $M^\gg \subseteq M$,
$M \models ``\gk$ is Mahlo and
no cardinal in the half-open interval
$(\gk, \gg]$ is Mahlo''.
In addition, by our assumptions,
$\FQ$ is an allowable choice
in the stage $\gk$ lottery
held in $M^\FP$ in the definition of $j(\FP)$.
Hence,
by forcing above a condition opting
for $\FQ$,
%the definition of $\FP$ implies that
$j(\FP \ast \dot \FQ)$ is forcing
equivalent to $\FP \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first nontrivial stage of
forcing in $\dot \FR$ takes
place well above $\gg$.
%$2^{[\gg]^{< \gk}}$.
Laver's original argument from \cite{L} now applies
and shows
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be $V$-generic over
$\FP \ast \dot \FQ \ast \dot \FR$,
lift $j$ in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
$V[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ)$ containing $p$, lift $j$
again in $V[G_0][G_1][G_2][G_3]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gg^+$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gl))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0][G_1]$.)
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of
Lemma \ref{l2t2}.
\end{proof}
Since trivial forcing is $\gk$-directed
closed and does not add any new subsets
of $\gk$, Lemma \ref{l2t2} immediately
allows us to infer that
$V^\FP \models ``\gk$ is supercompact''.
\begin{lemma}\label{l3t2}
$V^\FP \models ``\gk$'s strongness is
indestructible under $\gk$-strategically
closed forcing''.
\end{lemma}
\begin{proof}
Let $\FQ \in V^\FP$ be such that
$V^\FP \models ``\FQ$ is
$\gk$-strategically closed''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{\FP} ``\dot \FQ$ is
$\gk$-strategically closed''.
Suppose $\gl \ge
\max(\gk^+, \card{{\rm TC}(\dot \FQ)})$
is an arbitrary cardinal, and let
%$\gg = {(2^{\card{[\gl]^{< \gk}}})}^+$.
$\gg$ be the least singular strong
limit cardinal of cofinality $\gk^+$
above $\gl$.
Take
$j : V \to M$ as an elementary
embedding witnessing the $\gg$
strongness of $\gk$
generated by a $(\gk, \gg)$-extender
such that
$M = \{j(f)(a) \mid a \in {[\gg]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq V\}$.
Since
$V \models ``$No cardinal above
$\gk$ is Mahlo''
and $V_\gg \subseteq M$,
by the choice of $\gg$,
$M \models ``\gk$ is Mahlo and
no cardinal in the half-open interval
$(\gk, \gg]$ is Mahlo''.
In addition, as in Lemma \ref{l2t2},
by our assumptions, $\FQ$ is an allowable choice
in the stage $\gk$ lottery
held in $M^\FP$ in the definition of $j(\FP)$.
Hence,
by forcing above a condition $p$ opting
for $\FQ$, we may assume that
$j(\FP \ast \dot \FQ)$ is forcing
equivalent to $\FP \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first nontrivial stage of
forcing in $\dot \FR$ takes
place well above $\gg$.
Consequently, to complete the proof
of Lemma \ref{l3t2}, we
show that in $V^{\FP \ast \dot \FQ}$,
$j$ lifts to a $\gg$ strongness embedding
$j : V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
%This will complete the proof of Lemma \ref{l3t2}.
To do this,
we use ideas found in the proof of
\cite[Theorem 4.10]{H4}.
(See also \cite[Lemma 2.2]{GS}.)
%to show that the embedding $j$ lifts.
For the benefit of readers, we give
the argument here as well, once again
taking the
liberty to quote freely from Hamkins' presentation.
%Since $j(\FP)$ is forcing equivalent to
%$\FP \ast \dot \FQ \ast \dot \FR$,
%we know that the first ordinal in the
%field of $\dot \FR$ is above $\gg$.
Since by choice of $\gg$,
we may also 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 $\gg$-strategically closed in
$M[G][H]$.
As in \cite{H4}, by using a suitable coding
which allows us to identify finite
subsets of $\gg$ with elements of $\gg$,
the definition of $M$ enables us to find some
$\ga < \gg$ and function $g$ such that
$\dot \FQ = j(g)(\ga)$. Let
%(assuming that $\dot \FQ$ has been chosen reasonably). Let
$N = \{{\rm den}_{G \ast H}(\dot z) \mid \dot z =
j(f)(\gk, \ga, \gg)$ 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
$\gk$, $\ga$, $\gg$, $p$, $\FQ$, and $\FR$ are all
elements of $N$.
Further, since
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : \gk \to V_\gk$ in $V$, 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 an $N$-generic object $H'$
over $\FR$ in $V[G][H]$ using
the diagonalization argument given
in the proof of Lemma \ref{l1t1}
(where once again the game starts with $q_0$
being the empty condition).
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
$D = {\rm den}_{G \ast H}(\dot D)$ 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
$\gk < \gk_1 < \cdots < \gk_n < \gg$. Let
$\ov D$ be a name for the intersection of all
${\rm den}_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$, where
$\gk < \ga_1 < \cdots < \ga_n < \gg$ 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 $\gg$-strategically closed in
$M[G][H]$ and therefore $(\gg, \infty)$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is definable without
the parameters $\gk_1, \ldots, \gk_n$.
Hence, by its definition,
${\rm den}_{G \ast H}(\ov D) \in N$.
Thus, since $H'$
meets every dense open subset
of $\FR$ present in $N$,
%is $N$-generic over $\FR$,
$H' \cap {\rm den}_{G \ast H}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$H' \cap {\rm den}_{G \ast H}(\dot D) \neq \emptyset$.
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
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = {\rm den}_{G \ast H \ast H'}(\dot D)$
for some name $\dot D = j(\vec D)(a)$,
where $a \in {[\gg]}^{< \go}$ and
$\vec D = \la D_\gs \mid \gs \in {[\gk]}^{|a|} \ra$
is a function.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $(\gk, \infty)$-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,
$H'' = \{p \in j(\FQ) \mid \exists q \in j''H
[q \ge p]\}$ 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
$\gg$ strong since $V_\gg \subseteq M$, meaning
${(V_\gg)}^{V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$. Therefore,
$V[G][H] \models ``\gk$ is $\gg$ strong''.
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of Lemma \ref{l3t2}.
\end{proof}
\begin{lemma}\label{l4t2}
$V^\FP \models ``\gk$ is both the least
strongly compact and least supercompact cardinal''.
\end{lemma}
\begin{proof}
Let $\FQ \in V^\FP$ be the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality $\go$
to $\gk^+$. Since
$V^\FP \models ``\FQ$ is $\gk$-strategically closed'',
by Lemma \ref{l3t2},
$V^{\FP \ast \dot \FQ} \models ``\gk$ is a
strong cardinal''.
Since $V^{\FP \ast \dot \FQ} \models ``\gk^+$
contains a non-reflecting stationary set of
ordinals of cofinality $\go$'', by reflection,
$V^{\FP \ast \dot \FQ} \models
``$There are unboundedly in $\gk$ many cardinals
$\gd < \gk$ which contain non-reflecting
stationary sets of ordinals of cofinality $\go$''.
Therefore, since
$V^\FP \models ``\FQ$ is $\gk$-strategically closed'',
this last fact must be true in $V^\FP$ as well.
By a theorem of Solovay (see
\cite[Theorem 4.8]{SRK} and the succeeding remarks),
$V^\FP \models ``$No cardinal $\gd < \gk$ is
strongly compact''.
As $V^\FP \models ``\gk$ is supercompact'',
$V^\FP \models ``\gk$ is both the least
strongly compact and least supercompact cardinal''.
This completes the proof of Lemma \ref{l4t2}.
\end{proof}
\begin{lemma}\label{l4at2}
Suppose that $\gd < \gk$ is weakly compact. Then
$V^{\FP_\gd} \models ``\gd$ is weakly compact''.
\end{lemma}
\begin{proof}
In analogy to the proof of
Lemma \ref{l1t1}, let
$\dot T$ be the name of a $\gd$-tree in
$V^{\FP_\gd}$, with $N$ a transitive
elementary substructure of $H(\gd^+)$ of
size $\gd$ containing $\FP_\gd$ and
$\dot T$ which is closed under
${<} \gd$ sequences and
$j : N \to M$ an elementary embedding
having critical point $\gd$.
As before, we also assume that
$\card{M} = \gd$ and
$V \models ``M^{< \gd} \subseteq M$''.
Write $j(\FP_\gd) = \FP_\gd \ast \dot \FQ$.
Regardless if
$M \models ``\gd$ is weakly compact'', it will
be the case that in $M$,
$\forces_{\FP_\gd} ``\dot \FQ$ is
${\prec} \gd$-strategically closed''.
This means that the argument outlined in
the paragraph immediately following the
proof of Lemma \ref{l1t1} may now be used
to show that
$V^{\FP_\gd} \models ``\gd$ is weakly compact''.
This completes the proof of Lemma \ref{l4at2}.
\end{proof}
\begin{lemma}\label{l5t2}
In $V^\FP$, $\gd$ is Mahlo and reflects
stationary sets iff $\gd$ is weakly compact.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd$ is Mahlo and
reflects stationary sets''. Since by
Lemmas \ref{l1t2} and \ref{l2t2},
$V^\FP \models ``\gk$ is supercompact and no
cardinal is supercompact up to a
Mahlo cardinal'', it is clearly true that
$\gd \le \gk$. Consequently, as
$V^\FP \models ``\gk$ is
weakly compact'', we may assume that
$\gd < \gk$. In addition, since forcing does not
create new Mahlo cardinals, we have that
$V \models ``\gd$ is Mahlo''.
We consider now two cases.
\bigskip\noindent Case 1: $V \models ``\gd$
is not weakly compact''. Write
$\FP = \FP_{\gd + 1} \ast \dot \FQ$.
By the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\gd$ contains a non-reflecting
stationary set of ordinals of
cofinality $\go$ and forcing with
$\dot \FQ$ adds no new subsets of $\gd$''. Thus,
$V^{\FP_{\gd + 1} \ast \dot \FQ} = V^\FP \models
``\gd$ does not reflect stationary sets''.
\bigskip\noindent Case 2: $V \models ``\gd$
is weakly compact''. Write
$\FP = \FP_\gd \ast \dot \FQ^*$. By Lemma \ref{l4at2},
%the same argument as given in the proof of Theorem \ref{t1},
%our remarks in the paragraph immediately
%following the proof of Lemma \ref{l1t1},
$V^{\FP_\gd} \models ``\gd$ is weakly compact''.
Further, by the definition of $\FP$,
$\forces_{\FP_\gd} ``$Forcing with
$\dot \FQ^*$ adds no new subsets of $\gd$''. Thus,
$V^{\FP_{\gd} \ast \dot \FQ^*} = V^\FP \models
``\gd$ is weakly compact''.
\bigskip Since
$V^\FP \models ``\gd$ reflects stationary
sets'', $\gd$ cannot fall into the rubric of Case 1.
This means that $V \models ``\gd$ is weakly
compact'', so by the proof given in Case 2,
$V^\FP \models ``\gd$ is weakly compact'' as well.
Since weakly compact cardinals are both
Mahlo and reflect stationary sets,
this completes the proof of Lemma \ref{l5t2}.
\end{proof}
Lemmas \ref{l1t2} -- \ref{l5t2} and
the intervening remarks complete
the proof of Theorem \ref{t2}.
\end{proof}
\begin{pf}
Theorem \ref{t3} now follows as an immediate
corollary to Theorem \ref{t2}, with the model $V^\FP$
constructed for Theorem \ref{t2} as its
witness. To see this, it suffices to show that
no strong cardinal $\gd < \gk$ has its
strongness indestructible under $\gd$-strategically
closed forcing. However, if this were the case, then
by Theorem \ref{t1}, there would be non-weakly
compact Mahlo cardinals $\gg < \gd$ which
reflect stationary sets. By the construction of
$V^\FP$, this is impossible.
This completes the proof of Theorem \ref{t3}.
\end{pf}
There are some interesting questions left
open by the above construction, with
which we conclude this paper.
%We conclude by giving them.
They are as follows:
\begin{enumerate}
\item\label{q1} Is it possible to improve the levels
of indestructibility for $\gk$ to the levels
given in \cite{GS} and \cite{L}, i.e., so that
$\gk$'s supercompactness is indestructible
under arbitrary $\gk$-directed closed forcing and
$\gk$'s strongness is indestructible under what
Gitik and Shelah in \cite{GS} call
``$\gk^+$-weakly closed forcings
satisfying the Prikry condition''?
\item\label{q2} Is it possible to prove an
analogue of Theorem \ref{t2} in which
there are Mahlo cardinals above $\gk$?
\item\label{q3} Is it possible to prove an analogue of
Theorem \ref{t3} for a universe with no
restrictions on its large cardinal structure?
\item\label{q4} Is it possible to prove an analogue of
Theorem \ref{t1} in which $\gk$ is supercompact
and exhibits some form of indestructibility for
its supercompactness?
\end{enumerate}
Observe that a positive answer to Question \ref{q3}
would require as its witnessing model something
quite different from the model for Theorem \ref{t2}.
Also, one way of obtaing
a positive answer to Question \ref{q4} would
be to construct a partial ordering $\FQ^*$
analogous to the partial ordering $\FQ$ of
Lemma \ref{l1t1} (which may not
even be $\ha_2$-directed closed), but with a sufficient
degree of directed closure.
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\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
``Strong Axioms of Infinity and Elementary Embeddings'',
{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}
\footnote{Let
$\mu$ be the measure generated by
$j$ and $\la \gk, \ga, \gg \ra$.
Since $\mu \in M$, let
$j_\mu : M \to N'$ be the
ultrapower embedding generated
by $\mu$. As the
referee has pointed out, these facts
also follow since $N$ is the image under
the lift of the embedding of
$M$ via the measure generated by
into $M[G][H]$
of}