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\title{Mixed Levels of Indestructibility
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal, indestructibility, Prikry forcing,
Prikry sequence, non-reflecting stationary set of ordinals,
lottery sum.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY 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/aapter\\
awapter@alum.mit.edu}
\date{September 27, 2014}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
Starting from a supercompact cardinal $\gk$,
we force and construct a model in which
$\gk$ is both the least strongly compact
and least supercompact cardinal and
$\gk$ exhibits mixed levels of indestructibility.
Specifically, $\gk$'s strong compactness,
but not its supercompactness,
is indestructible under any $\gk$-directed
closed forcing which also
adds a Cohen subset of $\gk$.
On the other hand, in this model, $\gk$'s supercompactness
is indestructible under any $\gk$-directed closed forcing
which doesn't add a Cohen subset of $\gk$.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{AGS}, the following theorem was proven.
\begin{theorem}\label{t1}
Let $V \models ``$ZFC +
$\gk$ is supercompact''.
There is then a partial ordering
$\FP \subseteq V$ %$\card{\FP} = \gk$
such that $V^\FP \models ``\gk$ is both
supercompact and the least strongly
compact cardinal''.
For any $\FQ \in V^\FP$ which is
$\gk$-directed closed,
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact''.
Further, there is $\FR \in V^\FP$ which is
$\gk$-directed closed and nontrivial such that
$V^{\FP \ast \dot \FR} \models ``\gk$
is not supercompact''.
Moreover, for this $\FR$,
$V^{\FP \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank''.
\end{theorem}
The partial ordering $\FR$ of Theorem \ref{t1}
turns out to be $(\add(\gk, 1))^{V^{\FP}}$
(where for any regular cardinal $\gd$, {\em $\add(\gd, 1)$}
is the standard partial ordering for adding a single
Cohen subset of $\gd$).
We use this to motivate the terminology that for
a model $\ov V$ of ZFC, partial ordering $\FQ \in \ov V$,
and regular cardinal $\gd$ of $\ov V$,
{\em $\FQ$ adds a Cohen subset of $\gd$} means that in
${\ov V}^{\FQ}$, there is a subset of $\gd$ which is
$\ov V$-generic for $((\add(\gd, 1))^{\ov V}$.
Theorem \ref{t1} may be thought of as being
complementary to Laver's celebrated result of \cite{L},
where it is shown that any supercompact cardinal
$\gk$ can have its supercompactness forced to be
indestructible under arbitrary $\gk$-directed
closed forcing. Theorem \ref{t1} and the work of
\cite{L}, however, together raise the following
\bigskip
\noindent Question: Is it possible to force a supercompact
cardinal $\gk$ to have its strong compactness, but not
its supercompactness, indestructible under
$\gk$-directed closed partial orderings in a certain
class ${\mathcal C}$, and also have its supercompactness
indestructible under $\gk$-directed closed partial orderings
lying in the complement of ${\mathcal C}$?
\bigskip
The purpose of this paper is to answer the
above question in the affirmative.
Specifically, we will prove the following theorem.
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + $\gk$ is supercompact''.
There is then a partial ordering $\FP \subseteq V$
such that $V^\FP \models ``\gk$ is both
supercompact and the least strongly
compact cardinal''.
For any $\FQ \in V^\FP$ which is
$\gk$-directed closed and adds a Cohen
subset of $\gk$,
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact but not supercompact''.
In fact,
$V^{\FP \ast \dot \FQ} \models ``\gk$
has trivial Mitchell rank''.
On the other hand, for any $\FQ \in V^\FP$
which is $\gk$-directed closed and doesn't
add a Cohen subset of $\gk$,
$V^{\FP \ast \dot \FQ} \models ``\gk$ is supercompact''.
\end{theorem}
\noindent Forcing to obtain a model in which
the least strongly compact cardinal is the
same as the least supercompact cardinal was
of course first done by Magidor in \cite{Ma}.
Before beginning the proof of our theorem, we briefly mention
some preliminary information and terminology.
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 the usual interval notation.
%If $\gk \ge \go$ is a regular cardinal, then
%$\add(\gk, 1)$ is the standard partial ordering
%for adding a single Cohen subset of $\gk$.
When forcing, $q \ge p$ will mean that
{\it $q$ 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 $x \in V[G]$, 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 $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
The abuse of notation mentioned above will be compounded
by writing $x \in V^\FP$ instead of $\dot x \in V^\FP$.
Any term for trivial forcing will
always be taken as a term for the
partial ordering $\{\emptyset\}$.
If $\varphi$ is a formula in the forcing language
with respect to $\FP$ and $p \in \FP$, then
$p \decides \varphi$ means that
{\it $p$ decides $\varphi$}.
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 \mid \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha \mid \alpha < \delta \rangle$
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
of length $\gk + 1$
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 stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\it ${<}\gk$-strategically closed} if
$\FP$ is $\gd$-strategically closed for every
$\gd < \gk$.
Note that if $\FP$ is
$\gk$-directed closed, then $\FP$ is
${<}\gk$-strategically closed.
%(so since $\add(\gk, 1)$ is $\gk$-directed
%closed, $\add(\gk, 1)$ is ${<}\gk$-strategically closed as well).
We adopt Hamkins' terminology of \cite{H3, H2, H03}
and say that {\it $x \subseteq \gk$
is a fresh subset of $\gk$
with respect to $\FP$} if
$\FP$ is nontrivial forcing,
$x \in V^\FP$, $x \not\in V$, yet
$x \cap \ga \in V$ for every $\ga < \gk$.
%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.
From time to time within the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik iterations}.
By this we will mean an Easton support iteration
as first given by Gitik in \cite{G},
to which we refer readers for a discussion
of the basic properties of
and terminology associated with such an iteration.
For the purposes of this paper,
at any stage $\gd$ at which
a nontrivial forcing is done in a Gitik iteration,
we assume the partial ordering
$\FQ_\gd$ with which we force
%is either $\gd$-directed closed or is
has the form $\FR_\gd \ast \dot \FR'_\gd$, where
$\FR_\gd$ is $\gd$-directed closed and $\dot \FR_\gd$
is a term for either trivial forcing or
Prikry forcing defined with respect to
a normal measure over $\gd$
(although other types of partial orderings
may be used in the general case --- see
\cite{G} for additional details).
%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 $\gd_0$.
We recall for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{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
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
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.''}
Key to the proof of Theorem \ref{t2}
(specifically the fact that $\gk$'s supercompactness
is not indestructible when forcing with any
$\gk$-directed closed partial ordering adding
a Cohen subset of $\gk$)
is the following
result due %theorem due
to Gitik \cite[Proposition 1.1]{AGS}.
\begin{proposition}\label{p1}
Suppose $\gk$ is a Mahlo cardinal and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga \le \gk \ra$
is an Easton support iteration of length $\gk + 1$ satisfying
the following properties.
\begin{enumerate}
\item $\FP_0 = \{\emptyset\}$.
\item For each $\ga < \gk$,
$\forces_{\FP_\ga} ``\card{\dot \FQ_\ga} < \gk$''.
\item $\forces_{\FP_\gk} ``\dot \FQ_\gk$ is
${<} \gk$-strategically closed''.
%\item For some $\ga < \gk$, $\forces_{\FP_\ga} ``\dot \FQ_\ga$
%is nontrivial''.
\item For some $\ga, \gd < \gk$,
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ adds a new subset of $\gd$''.
\item $\gk$ is Mahlo in $V^{\FP_{\gk + 1}} = V^\FP$.
\end{enumerate}
Then in $V^\FP$, %$V^{\FP_{\gk + 1}}$,
there are no fresh subsets of $\gk$.
\end{proposition}
We note that Proposition \ref{p1} is an %weak
analogue of results due to Hamkins
(see \cite{H3, H2, H03}).
Adopting the terminology of these papers,
Hamkins shows that for a suitably large
cardinal $\gk$ (measurable, supercompact, etc.)
and an iteration $\FP$ {\it admitting
a gap below $\gk$}
(i.e., for some $\gd < \gk$,
$\FP$ can be written as
$\FQ \ast \dot \FR$, where $\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is $\gd$-strategically closed''),
after forcing with $\FP$,
there are no fresh subsets of $\gk$.
The iterations we consider will not be
gap forcings, yet they retain
this crucial property vital to the proof
of Theorem \ref{t2}.
%the crucial property of adding no fresh subsets of $\gk$.
%for what he calls {\it gap iterations},
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{J}
or \cite{SRK} for further details.
We do note, however, that $\gk$ is said to be
{\em supercompact up to the cardinal $\gl$} if
$\gk$ is $\gd$ supercompact for every $\gd < \gl$.
The measurable
cardinal $\gk$ is said to have {\it trivial Mitchell rank} if
there is no elementary embedding
$j : V \to M$ generated by a
normal measure ${\cal U}$ over $\gk$ such that
$M \models ``\gk$ is a measurable cardinal''.
We explicitly observe that if $\gk$ has trivial
Mitchell rank, then $\gk$ is not supercompact
(and in fact, if $\gk$ has trivial Mitchell
rank, then $\gk$ is not even $2^\gk$ supercompact).
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + $\gk$ is supercompact''.
Without loss of generality, we assume that
$V \models {\rm GCH}$ as well.
For any ordinal $\gd$, let
$\gd'$ be the least
$V$-strong cardinal above
$\gd$.
The partial ordering
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk \ra$
to be used in the proof of
Theorem \ref{t2} is a modification of the one used
in the proof of \cite[Theorem 1]{AGS}. Specifically, $\FP$
is the Gitik iteration of length $\gk$
which has the following properties.
%which is defined as follows.
\begin{enumerate}
\item $\FP$ begins by forcing with $\add(\go, 1)$, i.e.,
$\FP_0 = \{\emptyset\}$ and
$\forces_{\FP_0} ``\dot \FQ_0 = \dot \add(\go, 1)$''.
\item The only stages at which
$\FP$ (possibly) does nontrivial forcing are
those ordinals $\gd$ which are, in $V$,
Mahlo limits of strong cardinals.
At such a stage $\gd$,
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FL_\gd
\ast \dot \FS_\gd$, where $\dot \FL_\gd$
is a term for the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$.
\item If either $V^{\FP_\gd \ast \dot \FL_\gd} \models ``$There is no
subset of $\gd$ which is $V^{\FP_\gd}$-generic for
$(\add(\gd, 1))^{V^{\FP_\gd}}$'', or
$V^{\FP_\gd \ast \dot \FL_\gd} \models
``\gd$ is not measurable'', then
$\dot \FS_\gd$ is a term for trivial forcing.
\item If $V^{\FP_\gd \ast \dot \FL_\gd} \models ``$There is a
subset of $\gd$ which is $V^{\FP_\gd}$-generic for
$(\add(\gd, 1))^{V^{\FP_\gd}}$'' and
$V^{\FP_\gd \ast \dot \FL_\gd} \models ``\gd$ is measurable'', then
$\dot \FS_\gd$ is a term for Prikry forcing
defined with respect to some normal measure
over $\gd$.
\end{enumerate}
The intuition behind the
above definition of
$\FP$ is as follows.
$\FP$ begins by forcing with $\add(\go, 1)$ to ensure that
Proposition \ref{p1} is applicable.
The fact that no Prikry forcing is done when the forcing at
stage $\gd$ doesn't add a Cohen subset of $\gd$ ensures that in
$V^\FP$, $\gk$'s supercompactness is indestructible under any
$\gk$-directed closed partial ordering not adding
a Cohen subset of $\gk$.
Since Prikry forcing is performed when %a Prikry sequence is added when
a nontrivial forcing at stage $\gd$
both adds a Cohen subset of $\gd$ and
preserves the measurability of $\gd$, we ensure that
$\gk$'s strong compactness, but not its
supercompactness, is indestructible in $V^\FP$ under any
$\gk$-directed closed partial ordering adding a Cohen subset of $\gk$.
Because unboundedly many in $\gk$
Prikry sequences will have been added
by $\FP$, $V^\FP \models ``$No cardinal
below $\gk$ is strongly compact'', i.e.,
$V^\FP \models ``\gk$ is the least strongly
compact cardinal''.
The following lemmas show that
$\FP$ is as desired.
\begin{lemma}\label{l1}
Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk$-directed
closed and adds a Cohen subset of $\gk$. Then
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact''.
\end{lemma}
\begin{proof}
We follow the proof of %\cite[Lemma 2.2]{A06} and
\cite[Lemma 2.2]{AGS}, quoting verbatim when appropriate.
Suppose $\FQ \in V^\FP$ is $\gk$-directed closed
and adds a Cohen subset of $\gk$.
Let $\gl > \max(2^\gk, |{\rm TC}(\dot \FQ)|)$ be an
arbitrary regular cardinal large enough so that
$(2^{[\gl]^{< \gk}})^V = \gr =
(2^{[\gl]^{< \gk}})^{V^{\FP \ast \dot \FQ}}$ and
$\gr$ is regular in both $V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \gr^+ = 2^\gr$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
By \cite[Lemma 2.1]{AC2},
$\gk$ is a Mahlo limit of strong cardinals in $M$.
Consequently, by the choice of $\gs$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j(\FP)$.
Further, $M \models ``$No cardinal $\gd$ in the half-open interval
$(\gk, \gs]$ is strong''.
This is since otherwise, in $M$, $\gk$ is supercompact up to a
strong cardinal, so by the proof of \cite[Lemma 2.4]{AC2},
$\gk$ is supercompact in $M$.
Therefore, the next nontrivial
forcing in the definition of
$j(\FP)$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
%by the definition of $\FP$,
because forcing with $\FQ$ adds a Cohen subset of $\gk$,
$j(\FP \ast \dot \FQ)$ 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 Prikry forcing''.
The remainder of the proof of Lemma \ref{l1}
is as in the proof of \cite[Lemma 2]{AG}.
We outline the argument,
and refer readers to the
proof of \cite[Lemma 2]{AG} for any
missing details.
By the last two sentences of the preceding paragraph,
as in \cite[Lemma 2]{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
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
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 $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga \mid \ga \le \rho \ra$,
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 $\dot p_\ga$\footnote{Roughly speaking,
{\em $p_\gb$ is an Easton extension of $p_\ga$}
means that $p_\gb$ extends $p_\ga$ as in a usual
Easton support iteration, except that no stems
of any components of $p_\ga$
which are conditions in Prikry forcing
are extended. For a more precise
definition, readers are urged to consult either
\cite{G} or \cite{AG}.}
deciding $`\la j(\gb) \mid \gb < \gl \ra \in
j(\dot A_\ga)$' '', 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
$\la \dot p_\gb \mid \gb < \eta \ra$''.
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 $q' \in \FS_\gk$
of the form $\la \emptyset, B \ra$,
in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces_{j(\FP \ast \dot \FQ)} ``\la j(\gb) \mid \gb < \gl \ra \in
\dot X$'' for a name $\dot X$ of $X$.
As in \cite[Lemma 2]{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{l1}.
\end{proof}
\begin{lemma}\label{l1a}
Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk$-directed
closed and doesn't add a Cohen subset of $\gk$. Then
$V^{\FP \ast \dot \FQ} \models ``\gk$
is supercompact''.
\end{lemma}
\begin{proof}
Let $\FQ \in V^\FP$ be such that $\FQ$ is
$\gk$-directed closed and in $V^{\FP \ast \dot \FQ}$,
there is no subset of $\gk$ which is $V^\FP$-generic for
$(\add(\gk, 1))^{V^\FP}$.
As in Lemma \ref{l1},
suppose $\gl > \max(2^\gk, |{\rm TC}(\dot \FQ)|)$ is an
arbitrary regular cardinal large enough so that
$(2^{[\gl]^{< \gk}})^V = \gr =
(2^{[\gl]^{< \gk}})^{V^{\FP \ast \dot \FQ}}$ and
$\gr$ is regular in both $V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \gr^+ = 2^\gr$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
Again as in Lemma \ref{l1}, by \cite[Lemma 2.1]{AC2},
$\gk$ is a Mahlo limit of strong cardinals in $M$.
Consequently, by the choice of $\gs$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j(\FP)$.
Further, once more as in Lemma \ref{l1}, since
$M \models ``$No cardinal
$\gd \in (\gk, \gs]$ is strong'',
the next nontrivial
forcing in the definition of
$j(\FP)$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
because forcing with $\FQ$ doesn't add a Cohen subset of $\gk$,
$j(\FP \ast \dot \FQ)$ 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 trivial forcing''.
With a slight abuse of notation, we will henceforth say that in $M$,
above the appropriate condition,
$j(\FP \ast \dot \FQ)$ is forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$.
As in the proof of Lemma \ref{l1},
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 once again a term for a
``master condition'' for $\dot \FQ$.
Thus, if as before,
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
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
$\FP \ast \dot \FQ$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga \mid \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$ once more
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
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
$\la r, q \ra \in G \ast H$,
in $M$,
$\la r, \dot q, \dot p_\rho \ra
\forces_{j(\FP \ast \dot \FQ)} ``\la j(\gb) \mid \gb < \gl \ra \in
\dot X$'' for some name $\dot X$ of $X$.
As in \cite[Lemma 2]{AG}, %and Lemma \ref{l1},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G \ast H]}$.
We show that ${\cal U}$ is normal as well.
To do this, suppose $\la r, q \ra \in G \ast H$ is such that
$\la r, \dot q \ra \forces %\forces_{\FP \ast \dot \FQ}
``\dot f : (P_\gk(\gl))^{V[G \ast H]} \to
\gl$ is such that $\{s \mid \dot f(s) \in s\} \in \dot {\cal U}$''.
By the definition of ${\cal U}$ just given, we may assume that in $M$,
$\la r, \dot q, \dot p_\gr \ra \forces %\forces_{j(\FP \ast \dot \FQ)}
``\la j(\ga) \mid \ga < \gl \ra \in
\{s \mid j(\dot f(s)) \in s\}$''.
Let $\la \varphi_\ga \mid \ga < \gl \ra$ be such that
$\varphi_\ga$ is the statement in the forcing language
with respect to $j(\FP \ast \dot \FQ)$ given by
``$j(\dot f)(\la j(\gb) \mid \gb < \gl \ra) = j(\ga)$''.
Since $\gs > \gl$ and $M^\gs \subseteq M$,
$\la \varphi_\ga \mid \ga < \gl \ra \in M$.
Therefore, by forcing above $\la r, \dot q \ra$ and arguing
as in the definition of $\dot p_\gr$, we may assume that
$\dot p'_\gr$ is a term for
an Easton extension of $p_\gr$ such that for every
$\ga < \gl$,
$\la r, \dot q, \dot p'_\gr \ra \decides \varphi_\ga$
(so in $M[G \ast H]$, $p'_\gr \decides \varphi_\ga$,
where we assume that $\varphi_\ga$ has been rewritten
in the appropriate forcing language).
Because $\la r, \dot q, \dot p_\gr \ra \forces
%\forces_{j(\FP \ast \dot \FQ)}
``\la j(\ga) \mid \ga < \gl \ra \in \{s \mid j(\dot f(s)) \in s\}$'',
there must be some fixed $\ga_0 < \gl$ such that
$p'_\gr \forces \varphi_{\ga_0}$.
In other words, by extending $\la r, \dot q \ra$ if necessary
(and abusing notation by denoting the extended condition by
$\la r, \dot q \ra$ as well),
we may assume that
$\la r, \dot q \ra \forces ``\{s \mid \dot f(s) = \ga_0\} =
\dot A_\gg$'' and
$\la r, \dot q, \dot p'_\gr \ra \forces
%\forces_{j(\FP \ast \dot \FQ)}
``\la j(\gb) \mid \gb < \gl \ra \in
\{s \mid j(\dot f(s)) = j(\ga_0)\}$''
%and $\{s \mid j(\dot f(s)) = j(\ga_0)\} = j(\dot A_\gg)$'' for
for some fixed $\ga_0 < \gl$ and fixed $\gg < \gr$.
It must consequently be the case that
$\la r, \dot q, \dot p_\gr \ra \forces ``\la j(\gb) \mid \gb < \gl \ra \in
j(\dot A_\gg)$''. This is since otherwise,
by the definition of $\dot p_\gr$,
$\la r, \dot q, \dot p_\gr \ra \forces ``\la j(\gb) \mid \gb < \gl \ra
\not\in j(\dot A_\gg)$''.
However, $\la r, \dot q, \dot p'_\gr \ra \ge
\la r, \dot q, \dot p_\gr \ra$ and
$\la r, \dot q, \dot p'_\gr \ra \forces
``\la j(\gb) \mid \gb < \gl \ra \in j(\dot A_\gg)$''. Thus,
$\la r, \dot q, \dot p_\gr \ra \forces ``\{s \mid \dot f(s) = \ga_0\}
\in \dot {\cal U}$''.
This completes the proof of Lemma \ref{l1a}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.3]{AGS},
again quoting verbatim when appropriate.
Let $\gl = \gk^{+ \go}$.
Take $j : V \to M$ as an elementary embedding witnessing
the $\gl$ supercompactness of $\gk$.
Suppose $\FQ \in V^\FP$ is Add$(\gk, 1)$.
%i.e., the partial ordering for adding
%one Cohen subset of $\gk$.
By Lemma \ref{l1}, $V^{\FP \ast \dot \FQ} \models
``\gk$ is measurable'' (since
$V^{\FP \ast \dot \FQ} \models ``\gk$ is strongly compact'').
Because $\gl$ has been chosen large enough,
it therefore follows that
$M^{\FP \ast \dot \FQ} \models ``\gk$ is measurable''.
In addition, as in Lemma \ref{l1}, it is possible to opt
for $\FQ$ in the stage $\gk$ lottery held in $M$ in
the definition of $j(\FP)$. Therefore, by the
definition of $\FP$,
since $\FQ = \add(\gk, 1)$ and so of course
adds a Cohen subset of $\gk$,
above the appropriate condition,
$(j(\FP \ast \dot \FQ))_{\gk + 1} =
\FP_\gk \ast \dot \FQ_\gk = \FP_{\gk + 1}$ is forcing
equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS_\gk$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$ is
Prikry forcing defined over $\gk$''.
%This means that above the appropriate condition,
%$\forces_{\FP_\gk \ast \dot \FQ_\gk} ``\gk$ contains
%a Prikry sequence''.
This means that in $M$,
$\forces_{\FP_\gk} ``$By forcing above a condition
$\dot p^*_\gk$ ensuring that $\dot \add(\gk, 1)$
is chosen in the stage $\gk$ lottery held in the definition
of $j(\FP)$, $\dot \FQ_\gk$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gk$''.
Consequently, by reflection, for unboundedly
many $\gd < \gk$,
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that $\dot \add(\gd, 1)$
is chosen in the stage $\gd$ lottery held in the definition
of $\FP$, $\dot \FQ_\gd$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gd$''.
%the forcing has taken place
%above a condition yielding that
%$\forces_{\FP_\gd \ast \dot \FQ_\gd} ``\gd$ contains
%a Prikry sequence''.
It now follows that
$\forces_{\FP} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''. To see this,
let $\gg < \gk$ be fixed but arbitrary.
Choose $p = \la \dot p_\ga \mid \ga < \gk \ra \in \FP$.
Since $\FP$ is an Easton support iteration,
let $\gr > \gg$ be such that for every
$\ga \ge \gr$,
$\forces_{\FP_\ga} ``\dot p_\ga$ is a term
for the trivial condition''.
We may now find $\gd > \gr > \gg$ such that
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that $\dot \add(\gd, 1)$
is chosen in the stage $\gd$ lottery held in the definition
of $\FP$, $\dot \FQ_\gd$ is forcing equivalent to
a partial ordering adding a Prikry sequence to $\gd$''.
This means that we may find $q \ge p$ such that
$q \forces ``\gd$ contains a Prikry sequence''.
%It then immediately follows that
%$V^\FP \models ``$Unboundedly many $\gd < \gk$
%contain Prikry sequences''.
Thus, $\forces_{\FP} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''.
Hence, by \cite[Theorem 11.1]{CFM},
%$\forces_{\FP} ``$Unboundedly many
$V^\FP \models ``$Unboundedly many
$\gd < \gk$ (i.e., the successors of those
cardinals having Prikry sequences) contain
non-reflecting stationary sets of ordinals
of cofinality $\go$''.
By \cite[Theorem 4.8]{SRK} and the succeeding remarks,
it thus follows that
%$\forces_{\FP} ``$No
$V^\FP \models ``$No
cardinal $\gd < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l4}
Suppose $\FQ \in V^\FP$
is $\gk$-directed closed and
adds a Cohen subset of $\gk$.
%is such that $V^{\FP \ast \dot \FQ} \models ``$There is
%a subset of $\gk$ which is $((\add(\gk, 1))^{V^\FP}$-generic over $V^\FP$''.
Then $V^{\FP \ast \dot \FQ} \models ``\gk$
is not supercompact''.
In fact, in $V^{\FP \ast \dot \FQ}$, $\gk$
has trivial Mitchell rank.
\end{lemma}
\begin{proof}
We slightly modify the proof of \cite[Lemma 2.4]{AGS},
still quoting verbatim when appropriate.
Let $G \ast H$ be $V$-generic over
$\FP \ast \dot \FQ$.
Let $H' \subseteq \gk$, $H' \in V[G \ast H]$ be such that
$H'$ is $V[G]$-generic over $(\add(\gk, 1))^{V[G]}$.
If $V[G \ast H]
\models ``{\gk}$ does not have trivial Mitchell
rank'', then let $j : V[G \ast H] \to M[j(G \ast H)]$ be an
elementary embedding generated by a normal measure
${\cal U} \in V[G \ast H]$ over $\gk$ such that
$M[j(G \ast H)] \models ``\gk$ is measurable''.
Note that since $M = \bigcup_{{\ga \in {\rm Ord}}} j(V_\ga)$,
$j \rest V : V \to M$ is elementary.
Therefore, because $j \rest \gk = {\rm id}$, we may infer
that $(V_\gk)^V = (V_\gk)^M$.
%Without fear of ambiguity, we will thus write $V_\gk$.
However, by Proposition \ref{p1}, %by \cite[Proposition 1.1]{AGS},
we may further infer
that $(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$.
To see this, let $x \subseteq \gk$, $x \in M$.
Since $M \subseteq M[j(G \ast H)] \subseteq V[G \ast H]$,
$x \in V[G \ast H]$.
In addition, because $(V_\gk)^V = (V_\gk)^M$,
we know that $x \cap \ga \in V$ for every $\ga < \gk$.
This means that if $x \not\in V$, then $x$ is a
fresh subset of $\gk$ with respect to $\FP \ast \dot \FQ$.
Since by Lemma \ref{l1},
$\gk$ is strongly compact and hence both measurable and Mahlo in
$V[G \ast H]$, this contradicts Proposition \ref{p1}.
%\cite[Proposition 1.1]{AGS}.
Thus, $x \in V$, so
$(\wp(\gk))^M \subseteq (\wp(\gk))^V$.
From this, it of course immediately follows that
$(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$.
Let $I = j(G \ast H)$.
Note that if $V \models ``\gd < \gk$ is a strong cardinal'',
then $M \models ``j(\gd) = \gd$ is a strong cardinal''.
Also, $M \models ``\gk$ is a Mahlo limit of strong
cardinals'', since
$M[j(G \ast H)] \models ``\gk$ is a Mahlo cardinal'', and
forcing can't create a new Mahlo cardinal.
Hence, by the results of the preceding paragraph,
it follows as well that
%and hence that
$j(\FP) \rest \gk = \FP_\gk = \FP$ and
$I_\gk = G$.
Further, as
$V[G \ast H] \models
``M[I]^\gk \subseteq M[I]$'',
$H' \in M[I]$.
We know in addition
that in $M$, $\forces_{\FP_\gk \ast \dot \FQ_\gk}
``$The forcing beyond stage $\gk$
adds no new subsets of $2^\gk$'' and $\gk$ is
a stage at which nontrivial forcing
in $j(\FP)$ can take place.
%a nontrivial stage of forcing at stage
%$\gk$ in the definition of $j(\FP)$.
Consequently, $H' \in M[I_{\gk + 1}] = M[G][I(\gk)]$, and
$M[I_{\gk + 1}] \models ``\gk$ is measurable''.
Note that since $\FP$ is defined by taking Easton supports,
$\FP$ is $\gk$-c.c$.$ in both $V$ and $M$.
Because $\FP$ is a Gitik iteration of
suitably directed closed partial orderings
together with Prikry forcing
and $(V_\gk)^V = (V_\gk)^M$,
$(V_\gk)^{V[G]} = (V_\gk)^{M[G]}$.
%$V_\gk[G]$ is the same when calculated in either
%$V[G]$ or $M[G]$.
It must therefore be the case that
$(\add(\gk, 1))^{V[G]} = (\add(\gk, 1))^{M[G]}$.
In addition, since $(V_{\gk + 1})^M \subseteq (V_{\gk + 1})^V$,
the fact
$\FP$ is $\gk$-c.c$.$ in $M$ yields that %(as well as in $V$),
$(V_{\gk + 1})^{M[G]} \subseteq (V_{\gk + 1})^{V[G]}$.
This means that $H'$ is $M[G]$-generic over $(\add(\gk, 1))^{M[G]}$,
since $H'$ is $V[G]$-generic over
$(\add(\gk, 1))^{V[G]} = (\add(\gk, 1))^{M[G]}$,
and a dense open subset of
$(\add(\gk, 1))^{M[G]}$ in $M[G]$ is a member of
$(V_{\gk + 1})^{M[G]}$.
Hence, $H'$ must be added by the stage $\gk$
forcing done in $M[G] = M[I_\gk]$, i.e., the stage
$\gk$ lottery held in $M[I_\gk]$
must opt for some forcing adding a Cohen subset of $\gk$.
%It also follows that
%$M[I_{\gk + 1}] \models ``\gk$ is measurable''.
By the definition of $\FP$ and
$j(\FP)$, we must then have that
$M[I_{\gk + 1}] \models ``\gk$ contains a Prikry
sequence''. %and hence has cofinality $\go$''.
This contradiction to the fact that
$M[I_{\gk + 1}] \models ``\gk$ is measurable''
completes the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} -- \ref{l4} complete the proof of Theorem \ref{t2}.
\end{proof}
We conclude this paper with two questions.
First, we ask what other classes of
$\gk$-directed closed partial orderings
${\mathcal C}$ will provide additional answers
to our Question posed in Section \ref{s1}.
Finally, as in \cite{AGS}, we finish by
asking if it is possible to get a model
witnessing the conclusions of Theorem \ref{t2}
in which $\gk$ is not the least strongly
compact cardinal.
Since Prikry forcing above a strongly
compact cardinal destroys strong compactness,
an answer to this question would require a different sort of
iteration from the one used in the
proof of Theorem \ref{t1}.
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\end{document}