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\title{Indestructible Strong Compactness and Level by
Level Inequivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
Mahlo cardinal, indestructibility,
level by level inequivalence between strong compactness
and supercompactness, Prikry forcing,
Prikry sequence, non-reflecting stationary set of ordinals,
lottery sum.}}
%\thanks{The authors wish to acknowledge helpful conversations on
% the subject matter of this paper with James Cummings and
% Joel Hamkins.}}
% The authors also wish to thank Grigor Sargsyan,
% to whom they owe a huge debt of gratitude.
% Without him, this paper would not be possible.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank the two
anonymous referees, whose perceptive comments
and suggestions have been incorporated
into the current version of the paper.}\\
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{August 7, 2012\\(revised May 29, 2013)}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
If $\gd < \gg$ are such that
$\gd$ is indestructibly supercompact and
$\gg$ is measurable, then it must be
the case that level by level
inequivalence between strong compactness
and supercompactness fails. We prove a
theorem which points to this result being
best possible. Specifically, we show that
relative to the existence of cardinals $\gk_1 < \gl$
such that $\gk_1$ is $\gl$ supercompact
and $\gl$ is inaccessible, there is a model for level by level
inequivalence between strong compactness and
supercompactness containing a supercompact
cardinal $\gk < \gk_1$ in which $\gk$'s strong
compactness, but not supercompactness, is indestructible under
$\gk$-directed closed forcing.
In this model, $\gk$ is the least
strongly compact cardinal, and
no cardinal is supercompact
up to an inaccessible cardinal.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Say that a model of ZFC containing at least
one supercompact cardinal $\gk$ satisfies
{\em level by level inequivalence between
strong compactness and supercompactness}
if for every non-supercompact measurable
cardinal $\gd$, there is some $\gg > \gd$
such that $\gd$ is $\gg$ strongly compact
yet $\gd$ is not $\gg$ supercompact.
By \cite[Theorem 2]{A07a}, this is incompatible
with $\gk$ being indestructibly supercompact,
assuming there is a measurable cardinal $\gl > \gk$.
Specifically, \cite[Theorem 2]{A07a} shows that
if $\gk$ is indestructibly supercompact and
$\gl > \gk$ is measurable, then
$\{\gd < \gk \mid \gd$ is measurable and
$\gd$ is not $\gd^+$ strongly compact$\} \subseteq
\{\gd < \gk \mid \gd$ is measurable and
level by level inequivalence between
strong compactness and supercompactness fails
at $\gd\}$
is unbounded in $\gk$. This raises the
following
\bigskip
\noindent Question: Is it possible to
obtain a model with a restricted large
cardinal structure containing an
indestructibly supercompact cardinal in
which level by level inequivalence between
strong compactness and supercompactness holds?
\bigskip
Unfortunately, an answer to this Question
remains unknown. The purpose of this
paper is to establish a result giving evidence that
a positive answer to this Question is
indeed plausible. Specifically, we
prove the following theorem.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH + $\gk_1 < \gl$
are such that $\gk_1$ is $\gl$ supercompact and
$\gl$ is inaccessible''. There is then a
partial ordering $\FP \in V$, a submodel
$\ov V \subseteq V^\FP$, and $\gk < \gk_1$ such that
$\ov V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
In $\ov V$, no cardinal is supercompact up to
an inaccessible cardinal, and
level by level inequivalence between strong
compactness and supercompactness holds. Further,
in $\ov V$, $\gk$ is the least strongly
compact cardinal, and $\gk$'s strong
compactness, but not supercompactness, is indestructible under
$\gk$-directed closed forcing.
\end{theorem}
Theorem \ref{t1} should be contrasted with
%the work of \cite{A07b}. In particular,
\cite[Theorem 1]{A07b}.
In particular, a model witnessing the
dual notion of {\em level by level equivalence
between strong compactness and
supercompactness}\footnote{This notion
was first introduced by the author and
Shelah in \cite{AS97a}. For the purposes
of this paper and \cite{A07b},
{\em level by level equivalence between
strong compactness and supercompactness}
means that for any two regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly compact
iff $\gk$ is $\gl$ supercompact.
The general case is treated in \cite{AS97a}.}
which otherwise satisfies the same conclusions
as Theorem \ref{t1} of this paper is constructed
in \cite[Theorem 1]{A07b},
starting from a model of ``ZFC + There exists
a supercompact cardinal''.
Also, note that the only known restrictions
on the large cardinal structure of any
possible model witnessing a positive answer
to our Question are provided by \cite[Theorem 2]{A07a}.
We will touch upon this again at the end of the paper.
Before beginning the proof of Theorem \ref{t1}, we briefly mention
some preliminary information and terminology.
%Before beginning the proofs of
%Theorems \ref{t1} and
%\ref{p1}, 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 the usual interval notation.
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 $\gk \ge \go$ is a regular cardinal, then
$\add(\gk, 1)$ is the standard partial ordering
for adding a single Cohen subset of $\gk$.
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.
The partial ordering $\FP$ used in
the proof of Theorem \ref{t1} will be a {\em Gitik iteration}.
By this we will mean an Easton support iteration $\FP$
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
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).
We do explicitly mention that if
$p, q \in \FP$, then $q \ge p$ roughly speaking
means that $q \ge p$ as in a usual reverse
Easton iteration, except that stems of Prikry
conditions in $p$ can only be extended nontrivially finitely often.
If $q \ge p$ but no stems of Prikry conditions in $p$
are extended, then $q$ is called an {\em Easton
extension of $p$}.
%Roughly speaking, this means that
%$p_\gb$ extends $p_\ga$ as in a usual
%reverse Easton iteration, except that
%at coordinates at which Prikry forcing occurs in $p_\ga$,
%measure 1 sets are shrunk and stems are not extended.
For a more precise definition,
readers are urged to consult \cite{G}.
%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 $\gd_0$.
Key to the proof of Lemma \ref{l5},
which shows that the supercompactness
of the cardinal $\gk$ of Theorem \ref{t1}
is not indestructible under $\gk$-directed
closed forcing, is the following theorem
due to Gitik.
\begin{theorem}{\bf (\cite[Proposition 1.1]{AGS})}\label{t2}
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{theorem}
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.''}
Finally, we mention that we are assuming
familiarity with the large cardinal
notions of measurability,
strong compactness, and supercompactness.
Interested readers may consult \cite{J}
or \cite{SRK} for further details.
We do note, however, that the cardinal
$\gk$ is said to be {\em supercompact (strongly compact)
up to the cardinal $\gl$} if
$\gk$ is $\gd$ supercompact ($\gd$ strongly compact) for
every $\gd < \gl$.
$\gk$ is said to be {\em indestructibly supercompact} if,
as in \cite{L}, $\gk$'s supercompactness is
indestructible under arbitrary $\gk$-directed closed forcing.
The measurable cardinal $\gk$ is said to have
{\em 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 note 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{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + GCH +
$\gk_1 < \gl$ are such that $\gk_1$
is $\gl$ supercompact and $\gl$ is inaccessible''.
Without loss of generality, assume that
$\gl = \gk_1'$, where for the duration of
this paper, for any ordinal $\gd$, $\gd'$ is
the least inaccessible cardinal above $\gd$ in $V$.
%is the least inaccessible above $\gk_1$.
%For any ordinal $\gd$, let
%$\gd'$ be the least $V$-inaccessible cardinal above $\gd$.
The partial ordering
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk_1 \ra$
to be used in the proof of
Theorem \ref{t1}
is the Gitik iteration of length $\gk_1$
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 other stages $\ga > 0$ at which
$\FP$ (possibly) does nontrivial forcing are
those ordinals $\gd$ which are, in $V$,
Mahlo cardinals which are not
supercompact up to $\gd'$.
%the next inaccessible cardinal.
(In particular, we emphasize that if
$\gd < \gk_1$ is such that $V \models ``\gd$
is supercompact up to $\gd'$'',
then $\gd$ is a trivial stage of forcing.)
At such a stage $\gd$,
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FL_\gd
\ast \dot \FR_\gd$, where $\dot \FL_\gd$
is a term for the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$.
\item If $V^{\FP_\gd \ast \dot \FL_\gd} \models
``\gd$ is not measurable'', then
$\dot \FR_\gd$ is a term for trivial forcing.
\item If $V^{\FP_\gd \ast \dot \FL_\gd} \models
``\gd$ is measurable'', then
$\dot \FR_\gd$ 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.
%For $\gk$ as in the conclusions of Theorem \ref{t1},
The fact that nothing is done at stage $\gd$
%when the lottery selects trivial forcing,
unless $\gd$ is a Mahlo cardinal in $V$
which is not supercompact up to $\gd'$,
i.e., that no Prikry sequence is added,
ensures that $\gk_1$ is $\gl$ supercompact in $V^\FP$.
By reflection, this means that there is
$\gk < \gk_1$ such that in $V^\FP$,
$\gk$ is supercompact up to $\gk'$.
Let $\ov V = (V_{\gk'})^{V^\FP}$.
In $\ov V$, $\gk$ is supercompact.
Since a Prikry sequence is added when
a nontrivial forcing at stage $\gd$
preserves the measurability of $\gd$,
there will be a partial ordering
$\FR \in \ov V$ such that
$\ov V^\FR \models ``\gk$ is not supercompact''.
%$\gk$'s supercompactness will be destroyed.
The lottery sum at stage $\gd$, in
conjunction with the Prikry forcing, will
allow us to show that in $\ov V$,
$\gk$'s strong compactness is preserved
by nontrivial forcing.
Because unboundedly many in $\gk$
Prikry sequences will have been added
by $\FP$, $\ov V \models ``$No cardinal
below $\gk$ is strongly compact'', i.e.,
$\ov V \models ``\gk$ is the least strongly
compact cardinal''.
The definition of $\ov V$
will ensure that in $\ov V$,
level by level inequivalence between strong
compactness and supercompactness holds.
The following lemmas show that
$\FP$ is as desired.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk_1$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
%quoting verbatim when appropriate.
Take $j : V \to M$ as an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk_1$.
%such that $M \models ``\gk$ is not $\gl$ supercompact''.
Since $M^\gl \subseteq M$, $M \models ``\gk_1$ is
supercompact up to $\gk_1' = \gl$''.
This means
by the definition of $\FP$ that only trivial
forcing occurs at stage $\gk_1$ in $M$
in the definition of $j(\FP)$.
Consequently, in $M$, above the appropriate condition,
$j({\FP})$ is forcing equivalent to ${\FP} \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gl$.
We now follow the proofs of \cite[Lemma 2.1]{A06a}
and \cite[Lemma 2.1]{AGS}
and apply the argument of \cite[Lemma 1.5]{G}.
Specifically, let $G$ be $V$-generic over ${\FP}$.
%By the definition of ${\FP}$,
%$j '' G = G$.
Since GCH in $V$ implies that
$V \models ``2^{\gl} = \gl^+$'', we may let
$\la \dot x_\ga \mid \ga < \gl^+ \ra$ be an
enumeration in $V$ of all of the
canonical ${\FP}$-names of subsets of
$P_{\gk_1}(\gl)$.
Because $\FP$ is
$\gk_1$-c.c$.$ and $M^{\gl} \subseteq M$,
$M[G]^\gl \subseteq M[G]$.
By \cite[Lemmas 1.4 and 1.2]{G},
we may therefore
define in $V[G]$ an increasing sequence
$\la p_\ga \mid \ga < \gl^+ \ra$
of elements of $j({\FP})/G$
such that if $\ga < \gb < \gl^+$,
$p_\gb$ is an
Easton extension of $p_\ga$,
every initial segment of
the sequence is in $M[G]$, and for every
$\ga < \gl^+$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x_\ga)$''.
(For $\gb < \gl^+$ a limit ordinal,
$p_\gb$ exists since $M[G]^\gl \subseteq M[G]$
and the first nontrivial stage of forcing
in $p_\ga$ for $\ga < \gb$ takes place well after $\gl$.
This means that it is possible to let $p_\gb$ be
an Easton extension of $p_\ga$ for $\ga < \gb$.)
The remainder of the argument of
\cite[Lemma 1.5]{G} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_{\gk_1}(\gl))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_{\gk_1}(\gl))}^{V[G]}$ and
for some $\ga < \gl^+$ and some
${\FP}$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j({\FP})/G}
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gk(\gl)$.
Thus, $\forces_{{\FP}} ``\gk_1$ is
$\gl$ supercompact''.
%Since $\gl$ was arbitrary,
%$V^\FP \models ``\gk$ is supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
Suppose $V \models ``\gk \le
\gk_1$ is supercompact up to $\gk'$''. Let
$\FQ \in V^\FP$ be a
partial ordering which is $\gk$-directed
closed and has rank below $\gk'$.
%the least ($V$ or $V^\FP$) inaccessible cardinal above $\gk$.
Then $V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact up to $\gk'$''.
\end{lemma}
\begin{proof}
We follow the proofs of \cite[Lemma 2.2]{A06}
and \cite[Lemma 2.2]{AGS},
quoting verbatim when appropriate.
Let $\gk \le \gk_1$ be such that
$V \models ``\gk$ is supercompact up to $\gk'$''.
Write $\FP = \FP_\gk \ast \dot \FP^\gk$.
Since by the definition of $\FP$,
$\card{\FP_\gk} = \gk$ and
$\forces_{\FP_\gk} ``$Forcing with $\dot \FP^\gk$ adds no
new subsets of $(\gk')^V$'',
$\gk'$ has the same meaning in
$V$, $V^{\FP_\gk}$, and $V^{\FP_\gk \ast \dot \FP^\gk} = V^\FP$.
%the least inaccessible cardinal $\gk'$ above $\gk$'',
Writing now $\gk'$ without fear of
ambiguity, it suffices to show that if
$\FQ \in V^{\FP_\gk}$ is $\gk$-directed closed and
has rank below $\gk'$, then
%the least inaccessible cardinal above $\gk$, then
$V^{\FP_\gk \ast \dot \FQ} \models ``\gk$ is
strongly compact up to $\gk'$''.
%the next inaccessible cardinal''.
To do this,
let $\gz > \max(\gk, |{\rm TC}(\dot \FQ)|)$,
$\gz < \gk'$ be an
arbitrary regular cardinal.
By GCH in $V$ and the choice of $\gz$, it is the case that
$(2^{[\gz]^{< \gk}})^V =
(2^{[\gz]^{< \gk}})^{V^{{\FP_\gk} \ast \dot \FQ}} =
(2^\gz)^V = (2^\gz)^{V^{{\FP_\gk} \ast \dot \FQ}} =
(\gz^+)^V = (\gz^+)^{V^{{\FP_\gk} \ast \dot \FQ}}$.
Without ambiguity, we may write $\gr = \gz^+$ and
$\gs = \gr^+ = 2^\gr = 2^{\gz^+} = 2^{2^\gz}$.
%Let $\gs > \gl > \max(|{\rm TC}(\dot \FQ)|, \gk)$
%be sufficiently large regular cardinals, and let
%Let $\gs > \gl$ be a sufficiently large
%regular cardinal, and take
%be sufficiently large regular cardinals, and let
%be a sufficiently large regular cardinal.
By assuming $j(\gk)$ is minimal, we may
take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
Since $\gs < \gk'$,
by the choice of $\gs$ and the definition of $\FP$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j({\FP_\gk})$.
In addition, the next nontrivial
forcing in the definition of
$j({\FP_\gk})$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
$j({\FP_\gk} \ast \dot \FQ)$ is forcing equivalent to
${\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{{\FP_\gk} \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either Prikry forcing
or trivial forcing''.
The remainder of the proof of Lemma \ref{l2}
is as in the proof of \cite[Lemma 2]{AG}.
As in the proof of Lemma \ref{l1},
we outline the argument,
%For concreteness, we provide a sketch
%of the proof,
and refer readers to the
proof of \cite[Lemma 2]{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,
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
$j({\FP_\gk})$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over ${\FP_\gk} \ast \dot \FQ$,
$\forces_{j({\FP_\gk})} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $j(\dot \FQ)$.
Thus, if
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
${\FP_\gk} \ast \dot \FQ$ names of subsets of
${(P_\gk(\gz))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
${\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk$
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 $\FR$),
$\forces_{{\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gz \ra \in
j(\dot A_\ga)$'\nobreak$\ $'', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{{\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
We can then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gz]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gz)$ 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 $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) \mid \gb < \gz \ra \in
\dot X$'' for some name $\dot X$ of $X$.
As in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gz))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gz$ strongly compact''.
Since $\gz < \gk'$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l2}.
%Since $\gl$ was arbitrary,
%$V^{\FP \ast \dot \FQ} \models ``\gk$ is
%strongly compact''.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Any measurable cardinal
$\gk \le \gk_1$ is strongly compact
up to $\gk'$''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gk \le \gk_1$
is measurable''.
By Lemma \ref{l1}, we assume without loss
of generality that $\gk < \gk_1$.
Since $V^\FP \models ``\gk$
is a Mahlo cardinal'' and forcing can't create
a new Mahlo cardinal,
$V \models ``\gk$ is a Mahlo cardinal'' as well.
Write $\FP = \FP_{\gk + 1} \ast \dot \FP^{\gk + 1}$.
We will show that
$V \models ``\gk$ is supercompact up to $\gk'$''.
Otherwise, by the definition of $\FP$, if
$V \models ``\gk$ is not supercompact up to $\gk'$'',
$V^{\FP_{\gk + 1}} \models ``\gk$ is not measurable''.
Hence, since $\forces_{\FP_{\gk + 1}} ``$Forcing with
$\dot \FP^{\gk + 1}$ adds no new subsets of $\gk'$'',
$V^{\FP_{\gk + 1} \ast \dot \FP^{\gk + 1}} = V^\FP \models
``\gk$ is not measurable'' as well.
This is contrary to our assumptions, so
$V \models ``\gk$ is supercompact up to $\gk'$''.
Therefore, since trivial forcing is $\gk$-directed closed
and is taken to have rank below $\gk'$,
by Lemma \ref{l2},
$V^\FP \models ``\gk$ is strongly compact up to $\gk'$''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
As one of the referees has pointed out,
as will be seen in the final construction
of the model witnessing the conclusions of Theorem \ref{t1},
Lemmas \ref{l1} -- \ref{l3} provide the existence of a model for
level by level inequivalence between strong compactness and supercompactness
containing a supercompact, indestructibly strongly compact cardinal.
Lemma \ref{l4} will show that $\gk$ is the least strongly
compact cardinal.
Lemma \ref{l5} will show that there is a serious technical obstacle
to using the techniques of this paper in answering our
Question, i.e., obtaining a model for
level by level inequivalence between strong compactness and supercompactness
containing an indestructibly supercompact cardinal.
We will return to this issue at the end of the paper.
\begin{lemma}\label{l4}
Suppose $V^\FP \models ``\gk \le \gk_1$
is a measurable cardinal''. Then
$V^\FP \models ``$No cardinal
$\gd < \gk$ is strongly compact up to $\gk$''.
%In fact, $V^\FP \models ``$No cardinal
\end{lemma}
\begin{proof}
We modify the proof of \cite[Lemma 2.3]{AGS},
once more quoting verbatim when appropriate.
Let $\gg = \gk^{+ \go}$.
By the proof of Lemma \ref{l3},
$V \models ``\gk$ is supercompact up to $\gk'$''. Consequently,
by assuming $j(\gk)$ is minimal, we may once again
take $j : V \to M$ as an elementary embedding witnessing
the $\gg$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gg$ supercompact''.
Suppose $\FQ \in V^{\FP_\gk}$ is
trivial forcing $\{\emptyset\}$.
%Add$(\gk, 1)$, i.e., the partial ordering for adding
%one Cohen subset of $\gk$.
By the proof of Lemma \ref{l2}, $V^{{\FP_\gk} \ast \dot \FQ} \models
``\gk$ is measurable'' (since
$V^{{\FP_\gk} \ast \dot \FQ} \models ``\gk$ is
strongly compact up to $\gk'$'').
Because $\gg$ has been chosen large enough,
it therefore follows that
$M^{{\FP_\gk} \ast \dot \FQ} \models ``\gk$ is measurable''.
In addition, as in Lemma \ref{l2}, it is possible to opt
for $\FQ$ in the stage $\gk$ lottery held in $M$ in
the definition of $j({\FP_\gk})$. Therefore, by the
definition of $\FP$, above the appropriate condition,
$(j({\FP_\gk} \ast \dot \FQ))_{\gk + 1} =
(\FP_\gk \ast \dot \FQ_\gk)^M = (\FP_{\gk + 1})^M$ is forcing
equivalent in $M$ to
${\FP_\gk} \ast \dot \FS_\gk$, where
$\forces_{{\FP_\gk}} ``\dot \FS_\gk$ 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 %$\add(\gk, 1)$
trivial forcing $\{\emptyset\}$ is chosen in the
stage $\gk$ lottery held in the definition
of $j({\FP_\gk})$, $\dot \FQ_\gk$ is forcing equivalent to
Prikry forcing defined using some normal measure over
$\gk$''.
Consequently, by reflection, for unboundedly
many $\gd < \gk$,
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that %$\add(\gd, 1)$
trivial forcing $\{\emptyset\}$
is chosen in the stage $\gd$ lottery held in the definition
of $\FP_\gk$, $\dot \FQ_\gd$ is forcing equivalent to
Prikry forcing defined using some normal measure over $\gd$''.
%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
$\forces_{{\FP_\gk}} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''. To see this,
let $\gg < \gk$ be fixed but arbitrary.
Choose $p = \la \dot p_\ga \mid \ga < \gk \ra \in {\FP_\gk}$.
Since ${\FP_\gk}$ 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 $\gd > \gr > \gg$, $\gd < \gk$ such that
$\forces_{\FP_\gd} ``$By forcing above a condition
$\dot p^*_\gd$ ensuring that %$\add(\gd, 1)$
trivial forcing $\{\emptyset\}$ is chosen in the
stage $\gd$ lottery held in the definition
of $\FP_\gk$, $\dot \FQ_\gd$ is forcing equivalent to
Prikry forcing defined using some normal measure over $\gd$''.
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, $\forces_{{\FP_\gk}} ``$Unboundedly many $\gd < \gk$
contain Prikry sequences''.
Hence, by \cite[Theorem 11.1]{CFM},
%$\forces_{\FP} ``$Unboundedly many
$V^{\FP_\gk} \models ``$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
$V^{\FP_\gk} \models ``$No
cardinal $\gd < \gk$ is strongly compact
up to $\gk$''.
Because $V \models ``\gk$ is supercompact
up to $\gk'$'', only trivial forcing occurs
at stage $\gk$ of the definition of $\FP$.
If we now write $\FP = \FP_\gk \ast \dot \FP^\gk$,
we may consequently infer that $\forces_{\FP_\gk}
``$Forcing with
$\dot \FP^\gk$ adds no new subsets of $\gk'$''.
From this, it immediately follows that
$V^{\FP} \models ``$No
cardinal $\gd < \gk$ is strongly compact
up to $\gk$''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
Suppose $V^\FP \models ``\gk \le \gk_1$
is a measurable cardinal''.
Then for $\FR = ({\rm Add}(\gk, 1))^{V^\FP}$,
$V^{\FP \ast \dot \FR} \models ``\gk$
is not supercompact''.
In fact, in $V^{\FP \ast \dot \FR}$, $\gk$
has trivial Mitchell rank.
%, i.e., there is no normal measure $\mu$
%over $\gk$ in $V^{\FP \ast \dot \FR}$
%such that for
%$j : V^{\FP \ast \dot \FR} \to
%M^{j(\FP \ast \dot \FR)}$ the elementary
%embedding generated by the
%ultrapower via $\mu$,
%$M^{j(\FP \ast \dot \FR)} \models ``\gk$ is measurable''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l4}, write
$\FP = \FP_\gk \ast \dot \FP^\gk$. Since
$\forces_{\FP_\gk} ``$Forcing with $\dot \FP^\gk$
adds no new subsets of $\gk'$'',
$\FR \in V^{\FP_\gk}$, and
$V^{\FP \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank'' iff
$V^{\FP_\gk \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank''.
We therefore proceed by showing that
$V^{\FP_\gk \ast \dot \FR} \models ``\gk$
has trivial Mitchell rank''.
The proof of this fact is a modified version of the proof of
%essentially as in
\cite[Lemma 2.4]{AGS}.
In our argument, we again quote
verbatim from \cite{AGS} when appropriate.
%For the convenience of readers, we present the argument,
%again quoting verbatim from \cite{AGS} when appropriate.
Let $G \ast H$ be $V$-generic over
${\FP_\gk} \ast \dot \FR$. 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 Theorem \ref{t2}, 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 ${\FP_\gk} \ast \dot \FR$.
By the proof of Lemma \ref{l3},
$V \models ``\gk$ is supercompact up to $\gk'$''.
Since by the proof of Lemma \ref{l2},
$\gk$ must therefore be strongly compact up to $\gk'$
and hence both measurable and Mahlo in
$V[G \ast H]$, this contradicts Theorem \ref{t2}.
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 Mahlo cardinal'',
then $M \models ``j(\gd) = \gd$ is a Mahlo cardinal''.
Also, $M \models ``\gk$ is a Mahlo cardinal'', 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
$j({\FP_\gk}) \rest \gk = \FP_\gk$ and
$I_\gk = G$.
Further, as
$V[G \ast H] \models
``M[I]^\gk \subseteq M[I]$'',
$H \in M[I]$.
%It cannot be the case that
%$H \in M[G_\gd]$ for
%any $\gd < \gk$, since $H$ codes the generic
%added at stage $\gd$ for unboundedly many $\gd < \gk$.
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 $j({\FP_\gk})$ can take place.
%a nontrivial stage of forcing at stage
%$\gk$ in the definition of $j(\FP)$.
Consequently, $H \in M[I_{\gk + 1}] = M[G \ast I(\gk)]$, and
$M[I_{\gk + 1}] \models ``\gk$ is measurable''.
Note that since ${\FP_\gk}$ is defined by taking Easton supports,
${\FP_\gk}$ is $\gk$-c.c$.$ in both $V$ and $M$.
Because ${\FP_\gk}$ 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
${\FP_\gk}$ 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., there must be a stage
$\gk$ lottery held in $M[I_\gk]$
opting for some nontrivial forcing.
As we have already observed, because by hypothesis
$V[G \ast H] \models ``\gk$ does not have trivial Mitchell rank'',
in both $M[I] = M[j(G \ast H)]$ and
$M[G \ast I(\gk)] = M[I_{\gk + 1}]$,
$\gk$ is measurable. Consequently,
by the definition of ${\FP_\gk}$ and
$j({\FP_\gk})$, 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''
shows that
$V^{\FP_\gk \ast \dot \FR} \models
``\gk$ has trivial Mitchell rank''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
To complete the proof of Theorem \ref{t1},
by Lemma \ref{l1} and reflection, let
$\gk < \gk_1$ be the least cardinal such that
$V^\FP \models ``\gk$ is supercompact up to $\gk'$''.
Let $\ov V = (V_{\gk'})^{V^\FP}$.
%Lemmas \ref{l2} -- \ref{l5} then show that
It is then the case that
$\ov V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
% + No
%cardinal is supercompact up to an inaccessible cardinal''.
By the leastness of $\gk$ and Lemma \ref{l3},
in $\ov V$, no cardinal is supercompact up to an
inaccessible cardinal, and level by level inequivalence
between strong compactness and supercompactness holds.
By Lemmas \ref{l2}, \ref{l4}, and \ref{l5},
in $\ov V$, $\gk$ is the least strongly compact cardinal,
and $\gk$'s strong compactness, but not supercompactness,
is indestructible under $\gk$-directed closed forcing.
This completes the proof of Theorem \ref{t1}.
\end{proof}
In conclusion to this paper, we pose two questions.
First, we ask if it is possible to produce a model
witnessing the conclusions of Theorem \ref{t1}
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}.
Finally, we note that if we assume stronger hypotheses
on our ground model $V$, then it is possible to
obtain models analogous to the one for Theorem \ref{t1}
in which there are large cardinals above $\gk$.
As an example, if we start with a model in which
$\gk_1 < \gl$ are such that $\gk_1$ is $\gl$ supercompact
and $\gl$ is Mahlo, then it is possible to force and
construct a model in which $\gk$ is supercompact,
no cardinal is supercompact up to a Mahlo cardinal, and
the additional conclusions of Theorem \ref{t1} hold.
(We simply change the definition of $\gd'$ to be the
least $V$-Mahlo cardinal above $\gd$ and proceed
as we did originally.) Such a model, of course, can
contain inaccessible cardinals above $\gk$.
However, because of the appropriate analogue of
Lemma \ref{l5}, $\gk$ will not be
indestructibly supercompact.
We therefore conclude by reiterating our original Question,
and ask whether it is possible to construct a model
with a restricted large cardinal structure containing
an indestructibly supercompact cardinal in which
level by level inequivalence between strong compactness
and supercompactness holds.
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\end{document}
To do this,
let $\gz > \max(\gk, |{\rm TC}(\dot \FQ)|)$,
$\gz < \gk'$ be an
arbitrary regular cardinal large enough so that
$(2^{[\gz]^{< \gk}})^V = \gr =
(2^{[\gz]^{< \gk}})^{V^{{\FP_\gk} \ast \dot \FQ}}$ and
$\gr$ is regular in both $V$ and $V^{{\FP_\gk} \ast \dot \FQ}$,
and let $\gs = \gr^+ = 2^\gr$.
%Let $\gs > \gl > \max(|{\rm TC}(\dot \FQ)|, \gk)$
%be sufficiently large regular cardinals, and let
%Let $\gs > \gl$ be a sufficiently large
%regular cardinal, and take
%be sufficiently large regular cardinals, and let
%be a sufficiently large regular cardinal.
By assuming $j(\gk)$ is minimal, we may
take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
Since $\gs < \gk'$,
by the choice of $\gs$ and the definition of $\FP$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j({\FP_\gk})$.
In addition, the next nontrivial
forcing in the definition of
$j({\FP_\gk})$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
$j({\FP_\gk} \ast \dot \FQ)$ is forcing equivalent to
${\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{{\FP_\gk} \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either Prikry forcing
or trivial forcing''.
The remainder of the proof of Lemma \ref{l2}
is as in the proof of \cite[Lemma 2]{AG}.
As in the proof of Lemma \ref{l1},
we outline the argument,
%For concreteness, we provide a sketch
%of the proof,
and refer readers to the
proof of \cite[Lemma 2]{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,
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
$j({\FP_\gk})$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over ${\FP_\gk} \ast \dot \FQ$,
$\forces_{j({\FP_\gk})} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $j(\dot \FQ)$.
Thus, if
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
${\FP_\gk} \ast \dot \FQ$ names of subsets of
${(P_\gk(\gz))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
${\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk$
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 $\FR$),
$\forces_{{\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gz \ra \in
j(\dot A_\ga)$'\nobreak$\ $'', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{{\FP_\gk} \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
We can then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gz]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gz)$ 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 $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) \mid \gb < \gz \ra \in
\dot X$'' for some name $\dot X$ of $X$.
As in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gz))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gz$ strongly compact''.
Since $\gz < \gk'$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l2}.
%Since $\gl$ was arbitrary,
%$V^{\FP \ast \dot \FQ} \models ``\gk$ is
%strongly compact''.