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\title{Indestructibility and Destructible Measurable Cardinals
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
indestructibility, superdestructibility, lottery sum,
nonreflecting stationary set of ordinals.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{This paper is dedicated to the memory
of Rich Laver, a friend and inspiration.
It is truly a privilege to be able to contribute
a paper to this volume in his honor.}
\thanks{The author wishes to thank the referee
for helpful comments and suggestions which
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 17, 2013}
\date{December 25, 2013\\
(revised February 22, 2014)}
\begin{document}
\maketitle
\begin{abstract}
Say that $\gk$'s measurability is {\em destructible}
if there exists a ${<}\gk$-closed forcing
adding a new subset of $\gk$ which
destroys $\gk$'s measurability.
For any $\gd$, let
$\gl_\gd =_{\rm df}$ The least beth fixed point above $\gd$.
Suppose that $\gk$ is indestructibly supercompact
and there is a measurable cardinal $\gl > \gk$.
It then follows that
$A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability
%is superdestructible$\}$
is destructible when forcing with
partial orderings having rank below $\gl_\gd\}$
%either $\add(\gd, 1)$ or $\add(\gd, \gd^{+})\}$
is unbounded in $\gk$.
On the other hand, under the same hypotheses,
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability is
indestructible when forcing with either
$\add(\gd, 1)$ or $\add(\gd, \gd^{+})\}$
is unbounded in $\gk$ as well.
%In fact, $A_1$ must be unbounded in $\gk$ if $\gl$
%is only measurable.
The large cardinal
hypothesis on $\gl$ is necessary,
as we further demonstrate by
constructing via forcing two distinct models in which
either $A_1 = \emptyset$ or $A_2 = \emptyset$.
In each of these models,
both of which have restricted large cardinal structures above $\gk$,
every measurable cardinal $\gd$ which is not a limit of
measurable cardinals is $\gd^+$ strongly compact, and
there is an
indestructibly supercompact cardinal $\gk$.
%and a restricted large cardinal structure above $\gk$.
In the model in which $A_1 = \emptyset$,
%{\em every} measurable cardinal $\gd$ has its
%measurability indestructible when forcing with
%either $\add(\gd, 1)$ or $\add(\gd, \gd^{+})$. In addition,
every measurable cardinal $\gd$ which is
not a limit of measurable cardinals
is ${<} \gl_\gd$ strongly compact and has its
${<} \gl_\gd$ strong compactness (and hence also its
measurability) indestructible when forcing with $\gd$-directed
closed partial orderings having rank below $\gl_\gd$.
%$\gl_\gd =_{\rm df}$ The least beth fixed point above $\gd$.
The choice of the least beth fixed point above
$\gd$ is arbitrary, and other
values of $\gl_\gd$ are also possible.
%variants of this result are also possible.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
%We begin with some key definitions.
%The cardinal $\gk$ is {\em indestructibly supercompact}
%if $\gk$'s supercompactness is preserved after forcing
%with a $\gk$-directed closed partial ordering.
It is a very interesting fact that the large cardinal
structure of the universe above a
supercompact cardinal $\gk$
with suitable indestructibility properties
can affect the large cardinal structure
below $\gk$ in ways which are not immediately apparent.
%influence what happens at large cardinals below $\gk$.
On the other hand, these effects may not be present
%this control can vanish
%it is possible to mitigate these effects
if the universe contains relatively few large cardinals.
These sorts of occurrences have previously
been investigated in \cite{A07, A08, A09, A11, A10, A12, AH4}.
%\cite{A07}, \cite{A08}, and \cite{A09}.
The purpose of this paper is to continue
studying this phenomenon, but in the context of
investigating destructibility and indestructibility properties
certain measurable cardinals can manifest in
universes containing an indestructibly supercompact cardinal.
We begin with the following theorem,
where as in \cite{L}, $\gk$ is
{\em indestructibly supercompact} if
$\gk$'s supercompactness is preserved by
arbitrary $\gk$-directed closed forcing.
In analogy to \cite{H5}, $\gk$'s measurability is {\em destructible}
if there exists a ${<}\gk$-closed forcing
adding a new subset of $\gk$ which
destroys $\gk$'s measurability.
As in \cite{H5}, $\gk$'s measurability is {\em superdestructible} if
{\em every} ${<}\gk$-closed forcing adding a new subset of $\gk$
destroys $\gk$'s measurability.
$\gk$'s measurability is {\em indestructible when forcing
with a partial ordering $\FP$} if after forcing with
$\FP$, $\gk$ remains measurable.
For any $\gd$, we take
$\gl_\gd =_{\rm df}$ The least beth fixed point above $\gd$.
We also say that {\em $\gk$ is ${<} \gl$ supercompact
(${<} \gl$ strongly compact) for
$\gl > \gk$} if $\gk$ is $\gd$ supercompact
($\gd$ strongly compact) for every $\gd < \gl$.
\begin{theorem}\label{t1}
Suppose that
$\gk$ is indestructibly supercompact and
there is a measurable cardinal $\gl > \gk$.
Then $A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
%and $\gd$ is a
%superdestructible measurable cardinal$\}$
and $\gd$'s measurability
is destructible when forcing with
partial orderings having rank below $\gl_\gd\}$
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$
and
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability
is indestructible when forcing with
either $\add(\gd, 1)$ or
$\add(\gd, \gd^{+})\}$
are both unbounded in $\gk$.
\end{theorem}
With a limited large cardinal structure above $\gk$,
Theorem \ref{t1} need not be true.
Specifically, we have:
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is measurable +
$\gk$ is indestructibly supercompact +
If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ strongly compact and
$\gd$'s measurability is destructible when forcing with
partial orderings having rank below $\gl_\gd$''.
%either $\add(\gd, 1)$ or $\add(\gd, \gd^{+})$''.
\end{theorem}
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is inaccessible''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is inaccessible +
$\gk$ is indestructibly supercompact and is
also the least strongly compact cardinal +
%Any measurable cardinal $\gd < \gk$ is
%${<} \gl_\gd$ strongly compact and has its
%${<} \gl_\gd$ strong compactness (and hence
%also its measurability) indestructible when
%when forcing with $\gd$-directed closed
%partial orderings having rank below $\gl_\gd$''.
Any measurable cardinal $\gd < \gk$ which is
not a limit of measurable cardinals is
${<} \gl_\gd$ strongly compact and has its
${<} \gl_\gd$ strong compactness (and hence
also its measurability) indestructible
when forcing with $\gd$-directed closed
partial orderings having rank below $\gl_\gd$''.
%+ Every measurable cardinal $\gd$ has its measurability
%indestructible when forcing with either
%$\add(\gd, 1)$ or $\add(\gd, \gd^{+})$''.
\end{theorem}
We take this opportunity to make a few %additional
remarks concerning Theorems \ref{t1} -- \ref{t3}.
As our proofs will show, each relevant measurable
cardinal $\gd$ in both Theorems \ref{t1} and
\ref{t2} can have its measurability destructible when
forcing with many more partial orderings than just
those having rank below $\gl_\gd$.
%$\add(\gd, 1)$ and $\add(\gd, \gd^{+})$.
Also, in Theorem \ref{t2}, it is possible
for each measurable cardinal which is not
a limit of measurable cardinals to be
$\gl$ strongly compact for many different
regular cardinals $\gl > \gd^+$.
Further, in Theorem \ref{t3},
$\gl_\gd$ can take on values different from
the least beth fixed point above $\gd$.
%each measurable
%cardinal $\gd$ will have its measurability indestructible when
%forcing with many more partial orderings than just
%those having rank below $\gl_\gd$.
We will comment on these issues later in the paper.
%following the proof of Theorem \ref{t2}.
In addition, with just the hypothesis of
the existence of a measurable cardinal $\gl$ above
an indestructibly supercompact cardinal $\gk$, it
does not seem possible to be able to infer, e.g., that
$A_3 = \{\gd < \gk \mid \gd$ is measurable, $\gd$
is not a limit of measurable cardinals, and $\gd$'s
measurability is indestructible when forcing with
either $\add(\gd, 1)$, $\add(\gd, \gd^+)$, or
$\add(\gd, \gd^{++})\}$ is unbounded in $\gk$.
Gitik's work of \cite{G89, G93} seems to suggest
that stronger hypotheses on $\gl$ are required.
Finally, Theorem \ref{t3} should be contrasted with
\cite[Theorem 1.1]{A05}.
For its statement, we take as our notation that
%Here,
$\gr_\gd =_{\rm df}$ The least inaccessible cardinal
above $\gd$.
In this theorem, relative to the existence
of a model for GCH and %the existence of
a cardinal $\gk$ which is $\gr_\gk$ supercompact,
%$\gl$ strongly compact for $\gl$ the least inaccessible
%cardinal above $\gk$,
a model $V^*$ is constructed
in which $\gk$ is indestructibly supercompact
and is also the least strongly compact cardinal,
no cardinal $\gd > \gk$ is inaccessible,
and for every measurable cardinal $\gd < \gk$,
$\gd$ is ${<} \gr_\gd$ strongly compact and has its
${<} \gr_\gd$ strong compactness indestructible when forcing
with $\gd$-directed closed partial orderings having rank
below $\gr_\gd$.
Although this is {\em prima facie} a stronger result
than Theorem \ref{t3} of this paper,
$V^*$ is constructed by truncating a forcing
extension of $V$ at $\gr_\gk$.
It is therefore only a set model, and not a proper
class model as is the model witnessing the
conclusions of the current Theorem \ref{t3}.
In addition, ZFC + GCH + There exists $\gk$ which is
$\gr_\gk$ supercompact $\vdash$ Con(ZFC + GCH + There exists
a supercompact cardinal with no inaccessible
cardinals above it), which of course easily follows
by considering $V_{\gr_\gk}$.
Thus, Theorem \ref{t3} has two advantages over
\cite[Theorem 1.1]{A05}, in that Theorem \ref{t3}'s
witnessing model is a proper class, and Theorem
\ref{t3} is established using provably weaker hypotheses.
We conclude Section \ref{s1}
with a very brief discussion of
some preliminary material.
We presume a basic knowledge
of large cardinals and forcing.
A good reference in this
regard is \cite{J}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
When $G$ is $V$-generic over $\FP$,
we abuse notation slightly and
take both $V[G]$ and
$V^\FP$ as being the generic
extension of $V$ by $\FP$.
We also abuse notation slightly by
occasionally confusing terms with the
sets they denote, especially for
ground model sets and variants of the generic object.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$(\ga, \gb]$, $[\ga, \gb)$, and
$(\ga, \gb)$ are as in standard interval notation.
%For $\gk$ a measurable cardinal, the
%normal measure ${\cal U}$ over $\gk$ has
%{\em trivial Mitchell rank} if for
%$j : V \to M$ the elementary embedding
%generated by ${\cal U}$,
%$M \models ``\gk$ is not measurable''.
Suppose $\gk < \gl$ are regular cardinals.
For $\ga$ an arbitrary ordinal, the
partial ordering $\add(\gk, \ga)$ is
the standard Cohen partial ordering
for adding $\ga$ many Cohen subsets of $\gk$.
The partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D \subseteq \FP$
of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
$\FP$ is {\em $\gk$-closed} if every increasing
chain of members of $\FP$ of length $\gk$ has an upper bound.
$\FP$ is {\em ${<}\gk$-closed} if $\FP$ is
$\gd$-closed for every $\gd < \gk$.
$\FP$ is {\em $\gk$-strategically closed}
if in the two person game in which the
players construct an increasing sequence
$\la p_\ga \mid \ga \le \gk \ra$,
where player I plays odd stages and
player II plays even stages,
player II has a strategy ensuring 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
$\la p_\ga \mid \ga < \gk \ra$,
where player I plays odd stages and
player II plays even stages,
player II has a strategy ensuring the game
can always be continued.
It therefore follows that
any partial ordering $\FP$ which is
$\gk$-directed closed is also
${\prec }\gk$-strategically closed
and consequently adds no new subsets of
any cardinal $\gd < \gk$.
%$\FP$ is {\em ${<}\gk$-strategically closed}
%if $\FP$ is $\gd$-strategically closed
%for every $\gd < \gk$.
%$\FP$ is {\em $(\gk, \infty)$-distributive}
%if the intersection of $\gk$ many
%dense open subsets of $\FP$ is dense open.
%It therefore follows that
%any partial ordering $\FP$ which is
%$\gk$-directed closed is also
%${<}\gk$-strategically closed, and any
%partial ordering which is $\gk$-strategically
%closed is $(\gk, \infty)$-distributive.
%It further
In the proof of Theorem \ref{t1},
we will use the partial ordering
$\FP(\gk, \gl)$, the standard notion of
forcing for adding a nonreflecting stationary
set of ordinals of cofinality $\gk$ to $\gl$.
For further details on the definition of this
partial ordering, we refer readers to either
\cite{AC1} or \cite{A01}.
We note only that $\FP(\gk, \gl)$ is
$\gk$-directed closed and ${\prec} \gl$-strategically closed.
In the proof of Theorem \ref{t3},
we will refer to our
partial ordering $\FP$ as being a
{\it Gitik style iteration of
Prikry-like forcings.}
By this we will mean an Easton
support iteration
as first given by Gitik in \cite{G}.
%(and elaborated upon further in
%\cite{G2}),
The ordering,
roughly speaking, is the
usual one associated with
reverse Easton iterations, except that
when extending Prikry conditions,
we take larger stems only finitely often.
For a more precise definition, we urge
readers to consult either \cite{G}
%\cite{G2},
or \cite{AG}.
%By Lemmas 1.2 and 1.3 of
%\cite{G} and \cite{G2}
%respectively and Lemma 1.4 of \cite{G},
%if $\gd_0$ is the first stage in the
%definition of $\FP$ at which a non-trivial
%forcing is done, then forcing with
%$\FP$ adds no bounded subsets to $\gd_0$,
%assuming the forcing done at
%any non-trivial stage $\gd$
%is $\gd$-directed closed forcing
%followed by either Prikry forcing over $\gd$
%or trivial forcing.
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.''}
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorems \ref{t1} -- \ref{t3}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
%$\gd$-strategically closed''.
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ is {\em mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$
that admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an elementary embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proofs of Theorems \ref{t1} -- \ref{t3}}\label{s2}
\begin{pf}
We turn now to the proof of Theorem \ref{t1},
which will be established via a sequence of lemmas.
\begin{lemma}\label{l1}
If $\gk < \gl$ are such that $\gk$ is indestructibly
supercompact and $\gl$ is measurable, then
$A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability
is destructible when forcing with
partial orderings having rank below $\gl_\gd\}$
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$
is unbounded in $\gk$.
\end{lemma}
\begin{proof}
We follow the proofs of \cite[Theorem 2]{A07}
and \cite[Theorem 1]{A10}.
Suppose that
$\gk$ is indestructibly supercompact and
that $\gl$ is the least measurable cardinal greater than $\gk$.
%there is a measurable cardinal $\gl > \gk$.
We show that $A_1$ is unbounded in $\gk$.
%= \{\gd < \gk \mid \gd$ is measurable, $\gd$ is not a limit of
%measurable cardinals, and $\gd$'s measurability
%is destructible when forcing with either
%$\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$ is unbounded in $\gk$.
%Let $\eta > \gk$ be the least measurable cardinal.
First force with $\FP(\gk, \gl^+)$,
the partial ordering which adds a nonreflecting stationary
set of ordinals of cofinality $\gk$ to $\gl^+$.
By \cite[Theorem 4.8]{SRK} and the fact that
$\FP(\gk, \gl^+)$ is ${\prec} \gl^+$-strategically closed,
after this forcing,
$\gl$ is not $\gl^+$ strongly compact
and is the least measurable cardinal above $\gk$.
Next, force with $\add(\gk, 1)$.
%to add a Cohen subset of $\eta^+$.
After this forcing, which %is $\gk$-directed closed and
has cardinality $\gk < \gl$,
by the results of \cite{LS},
$\gl$ is not $\gl^+$ strongly compact
and $\gl$ remains the least measurable
cardinal above $\gk$.
In particular, after the forcing,
$\gl$ is a measurable cardinal which
is not a limit of measurable cardinals.
In addition, by \cite[Theorem I]{H5},
after the forcing, $\gl$ has become a superdestructible
measurable cardinal. In particular, $\gk$'s measurability is
destructible when forcing with
partial orderings having rank below $\gl_\gk$.
%either $\add(\gl, 1)$ or $\add (\gl, \gl^+)$.
Since $\gk$'s supercompactness is indestructible
when forcing with $\gk$-directed closed partial orderings,
$\FP(\gk, \gl^+) \ast \dot \add(\gk, 1)$
is $\gk$-directed closed,
and the destructibility of $\lambda$ when forcing with
partial orderings having rank below $\gl_\gd$
%either $\add(\gl, 1)$ or $\add (\gl, \gl^+)$
is detectible in
a large enough $V_\eta$, by reflection,
$A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability is
destructible when forcing with
partial orderings having rank below $\gl_\gd\}$
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$
is unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact $\FP(\gk, \gl^+) \ast \dot \add(\gk, 1)$
is $\gk$-directed closed that
$A_1$ is unbounded in $\gk$ in the ground model.
This completes the proof of Lemma \ref{l1}.
\end{proof}
%We remark that the proof of Lemma \ref{l1} does
%not allow us to infer that
We remark that in general, it is not possible to infer that
if $\gd \in A_1$, then $\gd$ is a superdestructible
measurable cardinal. To see this, suppose
$\ov V \models ``\gk$ is indestructibly supercompact'',
$\ov V = V^\FP$ for $\FP$ the reverse Easton iteration
of length $\gk$ of
\cite{L} which forces indestructibility for $\gk$, and
$j : \ov V^{\FP(\gk, \gl^+) \ast \dot \add(\gk, 1)} \to \ov M$
is an elementary embedding witnessing a fixed but arbitrary degree of
supercompactness of $\gk$ in
$\ov V^{\FP(\gk, \gl^+) \ast \dot \add(\gk, 1)}$. By elementarity,
$\ov M$ will be a generic extension of a ground model $M$ via
the partial ordering
$j(\FP) \ast \dot \FP(j(\gk), j(\gl^+)) \ast \dot \add(j(\gk), 1) =
\FP \ast \dot \FP(\gk, \gl^+) \ast \dot \add(\gk, 1) \ast
\dot \FQ \ast \dot \FP(j(\gk), j(\gl^+)) \ast \dot \add(j(\gk), 1)$, where
$\dot \FQ$ is a term for the forcing done between stages $\gk + 1$ and
$j(\gk)$ in $M^{\FP \ast \dot \FP(\gk, \gl^+) \ast \dot
\add(\gk, 1)}$. The nontrivial forcing
$\FQ \ast \dot \FP(j(\gk), j(\gl^+)) \ast \dot \add(j(\gk), 1)$
could have caused $\gl$ not to be a superdestructible measurable
cardinal in $\ov M$, the model over
which reflection is done.\footnote{It should be noted
that at the moment, it is unknown if the nontrivial forcing
$\FQ \ast \dot \FP(j(\gk), j(\gl^+)) \ast \dot \add(j(\gk), 1)$
actually destroys $\gl$'s superdestructibility.}
%affected the superdestructibility of $\gl$'s measurability in $\ov M$.
%This means that there is nontrivial forcing
%done above $\gl$ in $M^{\FP \ast \dot \FP(
Returning now to the proof of Theorem \ref{t1},
in order to complete it,
%In order to complete the proof of Theorem \ref{t1},
we first need to define a certain partial ordering $\FP$.
As in the proof of Lemma \ref{l1}, assume that
$\gk < \gl$ are such that $\gk$ is indestructibly
supercompact and $\gl$ is the least
measurable cardinal above $\gk$ in our ground
model $V$. We then let $\FP = \FP^0 \ast
\dot \FP(\gk, \gl^+) \ast \dot \FP^1$, where
$\FP^0 = \add(\gl^+, 1)$.
$\dot\FP^1$ will be a term for the reverse Easton iteration of
length $\gl$ which begins by forcing with
$\add(\gk, 1)$ and then performs nontrivial forcing
only at those stages $\gd \in (\gk, \gl)$ which are
inaccessible cardinals. At such a $\gd$, we force with
the lottery sum of trivial forcing $\{\emptyset\}$,
$\add(\gd, 1)$, and $\add(\gd, \gd^+)$.
For the proofs of Lemmas \ref{l2} -- \ref{l4}, we use
well known ideas.
%ideas that have appeared extensively in the literature.
Our reference is \cite[Theorem 2.1]{A12}
for Lemmas \ref{l2} and \ref{l3} and \cite[Lemma 2.2]{A12b}
for Lemma \ref{l4},
from which we feel free to quote verbatim if necessary.
\begin{lemma}\label{l1a}
$V^\FP \models ``\gl$ is not $\gl^+$ strongly compact''.
\end{lemma}
\begin{proof}
Consider $V^{\FP^0}$. Standard arguments
(see, e.g., \cite[Exercise 15.16]{J}) show that in $V^{\FP^0}$,
$2^\gl = \gl^+$, all cardinals less than or equal to
$\gl^+$ are preserved, $(2^\gl)^V$ is collapsed to $\gl^+$,
and all cardinals greater than or equal to $((2^\gl)^+)^V$
are preserved. Since $\FP(\gk, \gl^+)$ is
${\prec} \gl^+$-strategically closed and
$V^{\FP^0} \models ``\card{\FP(\gk, \gl^+)}
= \gl^+$'', these facts
are preserved to $\ov V = V^{\FP^0 \ast \dot \FP(\gk, \gl^+)}$ as well.
Therefore, as in the proof of Lemma \ref{l1}, by
\cite[Theorem 4.8]{SRK},
%$V^{\FP^0 \ast \dot \FP(\gk, \gl^+)} \models
$\ov V \models ``\gl$ is not $\gl^+$ strongly compact''.
To complete the proof of Lemma \ref{l1a}, we must now show that
$\ov V^{\FP^1} = V^\FP \models ``\gl$ is not $\gl^+$ strongly compact''.
To do this,
write $\FP^1 = \add(\gk, 1) \ast \dot \FR$.
Since $\card{\add(\gk, 1)} = \gk$,
$\add(\gk, 1)$ is nontrivial, and
$\forces_{\add(\gk, 1)} ``\dot \FR$ is $\gk^{++}$-directed closed'', it
follows that $\FP^1$ admits a gap at $\gk^+$.
Further, by its definition, $\FP^1$ is mild with respect to $\gl$.
Therefore, by Theorem \ref{tgf}, if $\gl$ were $\gl^+$ strongly
compact in $\ov V^{\FP^1} = V^\FP$, it would have had to have been
$\gl^+$ strongly compact in $\ov V$.
As we have just proven in the preceding paragraph,
this is false.
This completes the proof of Lemma \ref{l1a}.
\end{proof}
For the remainder of the proof of Theorem \ref{t1},
we adopt the notation used in the proof of Lemma \ref{l1a}.
\begin{lemma}\label{l2}
$V^\FP \models ``\gl$ is the least measurable cardinal
above $\gk$''.
\end{lemma}
\begin{proof}
%Let $\ov V = V^{\FP^0}$. Standard arguments
%(see, e.g., \cite[Exercise 15.16]{J}) show that in $\ov V$,
%$2^\gl = \gl^+$, all cardinals less than or equal to
%$\gl^+$ are preserved, $(2^\gl)^V$ is collapsed to $\gl^+$,
%and all cardinals greater than or equal to $((2^\gl)^+)^V$
%are preserved.
By its definition, $\add(\gl^+, 1) \ast \dot \FP(\gk, \gl^+)$
is ${\prec} \gl^+$-strategically closed and therefore
adds no new subsets of $\gl$.
Since $V \models ``\gl$ is the least
measurable cardinal above $\gk$'' and
$\ov V = V^{\add(\gl^+, 1) \ast \dot \FP(\gk, \gl^+)}$
contains no new subsets of $\gl$,
$\ov V \models ``\gl$ is the least
measurable cardinal above $\gk$'' as well.
We can therefore let $j : \ov V \to M$ be an elementary
embedding generated by a normal measure over
$\gl$. %such that $M \models ``\gl$ is not measurable''.
We can also let $G$ be $\ov V$-generic over $\FP^1$.
Because $\FP^1$ is a reverse Easton iteration having
length $\gl$, $\FP^1$ is $\gl$-c.c. Consequently, as
$M^\gl \subseteq M$, $M[G]^\gl \subseteq M[G]$, and
the cardinal structure in $\ov V[G]$ at and above $\gl$
is the same as in $\ov V$.
By opting for a condition in $M$ which ensures that
trivial forcing is chosen at stage $\gl$ in the
definition of $j(\FP^1)$, $j(\FP^1)$ can be taken to be
forcing equivalent to $\FP^1 \ast \dot \FQ$, where
the first nontrivial stage in $\dot\FQ$ occurs well after $\gl$.
Further, since $j$ is generated by a normal measure over $\gl$,
$2^\gl = \gl^+$ in $\ov V$, and
$M[G] \models ``\card{\FQ} = j(\gl)$'', the number of dense open
subsets of $\FQ$ present in $M[G]$ is
$(2^{j(\gl)})^M = (2^{j(\gl)})^{M[G]} =
j(\gl^+)$.
This is calculated in either $\ov V$ or $\ov V[G]$ as
$\card{\{f \mid f : \gl \to 2^\gl\}} =
\card{\{f \mid f : \gl \to \gl^+\}} = [\gl^+]^\gl = \gl^+$.
We may consequently let
$\la D_\ga \mid \ga < \gl^+ \ra \in \ov V[G]$ enumerate the
dense open subsets of $\FQ$ present in $M[G]$.
Because $M[G]^\gl \subseteq M[G]$,
by the definition of $\FP^1$,
$\FQ$ is $\gl^+$-directed closed in both
$M[G]$ and $\ov V[G]$.
We may hence in $\ov V[G]$ meet each $D_\ga$ and thereby
construct in $\ov V[G]$ an $M[G]$-generic object $H$ over $\FQ$.
Since by construction, $j '' G \subseteq G \ast H$,
$j$ lifts in $\ov V[G]$ to $j : \ov V[G] \to M[G][H]$.
This means that $\ov V[G] \models ``\gl$ is a measurable
cardinal'', i.e., $V^\FP \models ``\gl$ is a measurable cardinal''.
Finally, since $\FP^1$ admits a gap at $\gk^+$,
by Theorem \ref{tgf}, any cardinal greater than $\gk^+$
which is measurable
in $\ov V[G] = V^\FP$ had to have been measurable in $\ov V$.
Since $\ov V \models ``\gl$ is the least measurable cardinal
above $\gk$'', this means that
$V^\FP \models ``\gl$ is the least measurable cardinal above $\gk$''
as well.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^{\FP \ast \dot \add(\gl, 1)} \models ``\gl$ is a measurable cardinal''.
\end{lemma}
\begin{proof}
%We adopt the notation of Lemma \ref{l2}.
Consider as before $j : \ov V \to M$.
By opting for a condition in $M$ which ensures that
$\add(\gl, 1)$ is chosen at stage $\gl$ in the
definition of $j(\FP^1)$, $j(\FP^1)$ can be taken
to be forcing equivalent to $\FP^1 \ast \dot \add(\gl, 1)
\ast \dot \FQ \ast \dot \add(j(\gl), 1)$, where once again,
the first nontrivial stage in $\dot \FQ$ takes place well after $\gl$.
Let $G_0 \ast G_1$ be $\ov V$-generic over
$\FP^1 \ast \dot \add(\gl, 1)$. Since $\FP^1 \ast \dot \add(\gl, 1)$
is $\gl^+$-c.c. and $M^\gl \subseteq M$,
$M[G_0][G_1]^\gl \subseteq M[G_0][G_1]$ as well.
This means that the same analysis as given in the proof of Lemma \ref{l2}
allows us to build in $\ov V[G_0][G_1]$ an
$M[G_0][G_1]$-generic object $G_2$ over $\FQ$ and lift $j$ in
$\ov V[G_0][G_1]$ to $j : \ov V[G_0] \to M[G_0][G_1][G_2]$.
Therefore, since
$M[G_0][G_1][G_2]^\gl \subseteq M[G_0][G_1][G_2]$ in $\ov V[G_0][G_1]$,
$\ov V[G_0] \models ``\card{\add(\gl, 1)} = \gl$'',
$M[G_0][G_1][G_2] \models ``\add(j(\gl), 1)$ is
$j(\gl)$-directed closed'', and
$j(\gl) > \gl^+$, there is a master condition
$q \in \add(j(\gl), 1)$ for $j '' G_1$.
Further, the number of dense open subsets of
$\add(j(\gl), 1)$ present in $M[G_0][G_1][G_2]$ is
$(2^{j(\gl)})^M$. As in the proof of Lemma \ref{l2},
since $(2^{j(\gl)})^M = (2^{j(\gl)})^{M[G_0][G_1][G_2]}$,
this is calculated in either $\ov V$ or $\ov V[G_0][G_1]$ as $\gl^+$.
Consequently, we can once again use the same argument as
given in the proof of Lemma \ref{l2} and build in
$\ov V[G_0][G_1]$ an $M[G_0][G_1][G_2]$-generic object $G_3$
over $\add(j(\gl), 1)$ containing $q$. Since by construction,
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3$,
$j$ now fully lifts to
$j : \ov V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
Hence, $\ov V[G_0][G_1] \models ``\gl$ is a
measurable cardinal'', i.e., $V^{\FP \ast \dot \add(\gl^+, 1)}
\models ``\gl$ is a measurable cardinal''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^{\FP \ast \dot \add(\gl, \gl^+)} \models
``\gl$ is a measurable cardinal''.
\end{lemma}
\begin{proof}
%We once more adopt the notation of the previous two lemmas.
Consider again $j : \ov V \to M$.
By opting for a condition in $M$ which ensures that
$\add(\gl, \gl^+)$ is chosen at stage $\gl$ in the
definition of $j(\FP^1)$, $j(\FP^1)$ can be taken
to be forcing equivalent to $\FP^1 \ast \dot \add(\gl, \gl^+)
\ast \dot \FQ \ast \dot \add(j(\gl), j(\gl^+))$,
where as before,
the first nontrivial stage in $\dot \FQ$ takes place well after $\gl$.
Let $G_0 \ast G_1$ be $\ov V$-generic over
$\FP^1 \ast \dot \add(\gl, \gl^+)$.
Since $\FP^1 \ast \dot \add(\gl, \gl^+)$
is $\gl^+$-c.c. and $M^\gl \subseteq M$, as in Lemma \ref{l3},
$M[G_0][G_1]^\gl \subseteq M[G_0][G_1]$ as well.
This means that the same analysis as given previously
allows us to build in $\ov V[G_0][G_1]$ an
$M[G_0][G_1]$-generic object $G_2$ over $\FQ$ and lift $j$ in
$\ov V[G_0][G_1]$ to $j : \ov V[G_0] \to M[G_0][G_1][G_2]$.
As in Lemma \ref{l3}, $M[G_0][G_1][G_2]^\gl \subseteq M[G_0][G_1][G_2]$
in $\ov V[G_0][G_1]$.
%remains $\gl$ closed with respect to $\ov V[G_0][G_1]$.
To lift $j$ fully to $\ov V[G_0][G_1]$,
we now use the idea found in \cite[Lemma 2.2]{A12b}
(which has appeared elsewhere in the literature as well ---
readers may consult \cite[Lemma 2.2]{A12b} for
additional references).
We again feel free to quote verbatim as needed.
We will construct in $\ov V[G_0][G_1]$ an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gl), j(\gl^+))$.
For $\ga \in (\gl, \gl^+)$ and
$p \in \add(\gl, \gl^+)$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p \mid
\gs < \ga\}$ and
$G_1 \rest \ga = \{p \rest \ga \mid p \in G_1\}$. Clearly,
$\ov V[G_0][G_1] \models ``|G_1 \rest \ga| \le \gl$
for all $\ga \in (\gl, \gl^+)$''. Thus, since
${(\add(j(\gl), j(\gl^+))}^{M[G_0][G_1][G_2]}$ is
$j(\gl)$-directed closed and $j(\gl) > \gl^+$,
$q_\ga = \bigcup\{j(p) \mid p \in G_1 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gl), j(\gl^+))}^{M[G_0][G_1][G_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \ga$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2] \models
``2^{j(\gl)} = j(\gl^+)$'',
%``$GCH holds for all cardinals at or above $j(\gl)$'',
$M[G_0][G_1][G_2] \models ``\add(j(\gl),
j(\gl^+))$ is
$j(\gl^+)$-c.c$.$ and has
$j(\gl^+)$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2]$ is a
maximal antichain of $\add(j(\gl), j(\gl^+))$,
${\cal A} \subseteq \add(j(\gl), \gb)$ for some
$\gb \in (j(\gl), j(\gl^+))$.
Thus, since $\ov V \models ``2^\gl = \gl^+$''
and the fact $j$ is generated by a normal measure over $\gl$ imply that
$\ov V \models ``|j(\gl^+)| = \gl^+$'', we can let
$\la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra \in
\ov V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gl), j(\gl^+))$
present in $M[G_0][G_1][G_2]$.
Working in $\ov V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga \mid \ga \in (\gl, \gl^+) \ra$ of
elements of $\add(j(\gl), j(\gl^+))$ such that
$\forall \ga \in (\gl, \gl^+) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gl), j(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra
\exists \gb \in (\gl, \gl^+)
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_3 = \{p \in \add(j(\gl), j(\gl^+)) \mid
\exists r \in \la r_\ga \mid \ga \in (\gl, \gl^+) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gl), j(\gl^+))$. To define
$\la r_\ga \mid \ga \in (\gl, \gl^+) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gl, \ga)} r_\gb$.
By the facts
$\la r_\gb \mid \gb \in (\gl, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2]^\gl \subseteq M[G_0][G_1][G_2]$
in $\ov V[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb \mid \gb < \eta < \gl^+ \ra$
be the subsequence of
$\la {\cal A}_\gb \mid \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gl), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gl), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gl), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2]^\gl \subseteq M[G_0][G_1][G_2]$
in $\ov V[G_0][G_1]$, define by induction an increasing sequence
$\la s_\gb \mid \gb < \eta \ra$ such that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_3$ is
$M[G_0][G_1][G_2]$-generic over
$\add(j(\gl), j(\gl^+))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra
\exists \gb \in (\gl, \gl^+)
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) \mid \ga < \gl^+ \ra$ is
unbounded in $j(\gl^+)$. To see this, if
$\gb < j(\gl^+)$ is an ordinal, then for some
$f : \gl \to M$ representing $\gb$,
we can assume that for $\ga < \gl$,
$f(\ga) < \gl^+$. Thus, by the regularity of
$\gl^+$ in $\ov V$,
$\gb_0 = \bigcup_{\ga < \gl} f(\ga) <
\gl^+$, and $j(\gb_0) \ge \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga \mid \ga <
\gl^+ \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gl, \gl^+)$ be such that
${\cal A} \subseteq \add(j(\gl), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gl, \gl^+)$ is such that for some
$\ga \in (\gl, \gl^+)$, $p = p \rest \ga$,
$G_3$ is such that if
$p \in G_1$, $j(p) \in G_3$.
Thus, working in $\ov V[G_0][G_1]$,
we have shown that $j$ lifts to
$j : \ov V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
This means that $\ov V[G_0][G_1] \models ``\gl$
is a measurable cardinal'', i.e.,
$V^{\FP \ast \dot \add(\gl, \gl^+)} \models ``\gl$
is a measurable cardinal''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
We may now complete the proof of Theorem \ref{t1}
by mimicking the proof of Lemma \ref{l1}.
We show that $A_2
= \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals,
$\gd$ is not $\gd^+$ strongly compact,
and $\gd$'s measurability is
indestructible when forcing with either
$\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$
is unbounded in $\gk$.
Force with $\FP$.
By Lemma \ref{l1a}, after this forcing,
$\gl$ is not $\gl^+$ strongly compact.
In addition, after this forcing, which is $\gk$-directed closed,
by Lemma \ref{l2},
$\gl$ remains the least measurable
cardinal above $\gk$.
In particular, after the forcing,
$\gl$ is a measurable cardinal which
is not a limit of measurable cardinals.
In addition, by Lemmas \ref{l3} and \ref{l4},
after the forcing, $\gl$ has become a
measurable cardinal whose measurability is
indestructible when forcing with either
$\add(\gl, 1)$ or $\add (\gl, \gl^+)$.
Since $\gk$'s supercompactness is indestructible
when forcing with $\gk$-directed closed partial orderings
%$\add(\gl^+, 1) \ast \dot \FP(\gk, \gl^+)$ is $\gk$-directed closed,
and the indestructibility of $\lambda$ when forcing with either
$\add(\gl, 1)$ or $\add (\gl, \gl^+)$ is detectible in
a large enough $V_\eta$, by reflection,
$A_2$ is unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact $\FP$ is $\gk$-directed closed that
$A_2$ is unbounded in $\gk$ in the ground model.
This completes the proof of Theorem \ref{t1}.
\end{pf}
Having finished with the proof of Theorem \ref{t1},
we turn now to the proof of Theorem \ref{t2}.
Recall that this theorem states that if
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable'', then
there is a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is measurable +
$\gk$ is indestructibly supercompact +
If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ strongly compact and
$\gd$'s measurability is destructible when forcing with
partial orderings having rank below $\gl_\gd$''.
\begin{proof}
Suppose $V$ is as in the hypotheses for Theorem \ref{t2}.
%$V \models ``$ZFC + $\gk$ is supercompact +
%No cardinal $\gz > \gk$ is measurable''.
Without loss of generality, by
%first doing a preliminary forcing if necessary,
\cite[Theorem 2]{A01} and the remarks at the end of \cite{A01},
we assume in addition that $V \models ``{\rm GCH}$ +
Every measurable cardinal $\gd$ is
$\gd^+$ strongly compact''.\footnote{Forcing GCH
may require the use of a proper class partial ordering.
This in turn implies that the forcing conditions which are
defined and used to prove Theorem \ref{t1} are in actuality
a proper class as well.}
Let $f$ be a Laver function
\cite{L} for $\gk$, i.e.,
$f : \gk \to V_\gk$ is such that
for every $x \in V$ and every
$\gl \ge \card{{\rm TC}(x)}$, there is
an elementary embedding
$j : V \to M$ generated by a
supercompact ultrafilter over
$P_\gk(\gl)$ such that
$j(f)(\gk) = x$.
Our partial ordering $\FP$ is the
reverse Easton iteration of
length $\gk$ which begins by
forcing with $\add(\go, 1)$
and then (possibly) does nontrivial
forcing only at
cardinals $\gd < \gk$ which are
measurable limits of measurable cardinals in $V$.
At such a stage $\gd$, if
$f(\gd) = \dot \FQ$ and
$\forces_{\FP_\gd} ``\dot \FQ$
is a $\gd$-directed closed
%closed, $(\gd^+, \infty)$-distributive partial
partial ordering having rank below
the least measurable cardinal above $\gd$ in $V$'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$.
If this is not the case, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$,
where $\dot \FQ$ is a term for trivial forcing.
%we perform trivial forcing.
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is
indestructibly supercompact''.
\end{lemma}
\begin{proof}
We follow the proofs of \cite[Lemma 2.1]{A07},
\cite[Lemma 2.1]{A10}, and \cite[Lemma 3.1]{A12},
quoting verbatim when appropriate.
Let $\FQ \in V^{\FP}$ be such that
$V^{\FP} \models ``\FQ$ is
$\gk$-directed closed''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{{\FP}} ``\dot \FQ$ is
$\gk$-directed closed''.
Suppose $\gl \ge
\max(2^\gk, \card{{\rm TC}(\dot \FQ)})$
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 $\gd > \gk$ is measurable'',
$\gg \ge 2^\gk$, %2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
$M \models ``\gk$ is a measurable limit of
measurable cardinals,
and no cardinal $\gd$ in the %half-open
interval $(\gk, \gg]$ is measurable''. Hence,
the definition of ${\FP}$ implies that
$j({\FP} \ast \dot \FQ) = {\FP} \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first stage at which
$\dot \FR$ is forced to do nontrivial forcing
is 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{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$If $\gd < \gk$ is a measurable
cardinal which is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ strongly compact and
$\gd$'s measurability is destructible when forcing with
partial orderings having rank below $\gl_\gd$''.
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)$''.
\end{lemma}
\begin{proof}
Suppose $\gd < \gk$ is such that
$V \models ``\gd$ is a measurable cardinal
which is not a limit of measurable cardinals''.
Write $\FP = \FP_\gd \ast \dot \FP^\gd$.
Since $\gd$ is not a limit of measurable cardinals
in $V$, by the definition of $\FP$,
%$\card{\FP_\gd} < \gd$
$\FP_\gd$ is forcing equivalent to a partial ordering
having cardinality less than $\gd$.
In addition, $\FP_\gd$ is nontrivial.
Consequently, by the results of \cite{LS} and \cite[Theorem I]{H5},
$V^{\FP_\gd} \models ``\gd$ is a measurable
cardinal which is not a limit of
measurable cardinals,
$\gd$ is $\gd^+$ strongly compact,
and $\gd$'s measurability is
destructible when forcing with
partial orderings having rank below $\gl_\gd$''.
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)$''.
Thus, since $\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$\eta$-directed closed for $\eta$ the least
measurable limit of measurable cardinals above $\gd$''
(which inductively is the same in both $V$ and $V^{\FP_\gd}$),
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models ``\gd$
is a measurable cardinal which is not a limit
of measurable cardinals,
$\gd$ is $\gd^+$ strongly compact,
and $\gd$'s measurability is destructible when forcing with
partial orderings having rank below $\gl_\gd$''.
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)$''.
To complete the proof of Lemma \ref{l6}, it
therefore suffices to show that
any $V^\FP$-measurable cardinal which is not
a limit of measurable cardinals is a
$V$-measurable cardinal which is not a limit
of measurable cardinals as well.
To do this, suppose now that $\gd < \gk$ is such that
$V^\FP \models ``\gd$ is a measurable cardinal
which is not a limit of measurable cardinals''.
Write $\FP = \add(\go, 1) \ast \dot \FQ$.
Since $\card{\add(\go, 1)} = \go$, $\add(\go, 1)$
is nontrivial, and $\forces_{\add(\go, 1)} ``\dot \FQ$ is
$\ha_2$-directed closed'', $\FP$ admits a gap at $\ha_1$.
By Theorem \ref{tgf}, this means that
$V \models ``\gd$ is a measurable cardinal''.
If $V \models ``\gd$ is a measurable cardinal which is a
limit of measurable cardinals'', then
$V \models ``\gd$ is a measurable cardinal which is a limit of
measurable cardinals which themselves are not limits of
measurable cardinals''. By the preceding paragraph,
since such cardinals are preserved to $V^\FP$, this
means that $V^\FP \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals'', a contradiction.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Since $\FP$ may be defined so that $\card{\FP} = \gk$,
the results of \cite{LS} show that
$V^\FP \models ``$No cardinal $\gd > \gk$ is measurable''.
Lemmas \ref{l5} and \ref{l6} therefore complete
the proof of Theorem \ref{t2}.
\end{proof}
As we mentioned earlier, our methods of proof for both
Theorems \ref{t1} and \ref{t2} show that the
relevant measurable cardinals $\gd$ can have their
measurability destructible
%allow us to infer that
%the relevant measurable cardinals $\gd$ are destructible
when forcing with partial orderings other than just
those having rank below the least beth fixed point above $\gd$.
%$\add(\gd, 1)$ or $\add(\gd, \gd^+)$.
For instance, the definition of $\gl_\gd$
could be changed to the second beth fixed point above $\gd$, etc.
%anything of the form $\add(\gd, \eta)$ where
%$\eta$ is below the least inaccessible cardinal above
%$\gd$, etc., provides additional examples of such partial orderings.
Further, the proof of Lemma \ref{l6} actually shows that
any measurable cardinal which is not a limit of measurable cardinals
has its measurability destructible when forcing with
partial orderings having rank below the least $V$-measurable
limit of measurable cardinals above it.
In addition, the methods of \cite{A01} allow us
to assume that in the ground model $V$ for Theorem \ref{t2},
each measurable cardinal $\gd$ is $\gl$ strongly compact,
where $\gl$ is any fixed regular cardinal below the least
measurable cardinal above $\gd$ (such as, e.g.,
$\gd^{++}$, $\gd^{+ 17}$, the least inaccessible or
Ramsey cardinal above $\gd$, etc.).
Thus, in the model $V^\FP$ witnessing the conclusions of
Theorem \ref{t2}, each measurable cardinal $\gd$
which is not a limit of measurable cardinals
may also be $\gl$ strongly compact for the aforementioned
values of $\gl$.
Finally, it is not possible to extend Theorem \ref{t2}
and obtain a model in which {\em every} measurable cardinal
$\gd < \gk$ has its measurability destructible
when forcing with
partial orderings having rank below $\gl_\gd$.
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)$.
This follows since if
$\gk$ is indestructibly supercompact and
$j : V \to M$ is an elementary embedding witnessing
the $\gl_\gk$ supercompactness of $\gk$, then
$M \models ``\gk$ is a measurable cardinal which
is a limit of measurable cardinals and $\gk$'s
measurability is indestructible when forcing with
partial orderings having rank below $\gl_\gk$''.
%either $\add(\gk, 1)$ or $\add(\gk, \gk^+)$''.
Hence, by reflection,
$\{\gd < \gk \mid \gd$ is a measurable cardinal which is a
limit of measurable cardinals and $\gd$'s measurability
is indestructible when forcing with
partial orderings having rank below $\gl_\gd\}$
%either $\add(\gd, 1)$ or $\add(\gd, \gd^+)\}$
must be unbounded in $\gk$ in $V$.
%Having completed
We turn now to the proof of Theorem \ref{t3}.
Recall that this theorem states that if
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is inaccessible'', then
there is a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is inaccessible +
$\gk$ is indestructibly supercompact and is
also the least strongly compact cardinal +
Any measurable cardinal $\gd < \gk$ which is
not a limit of measurable cardinals is
${<} \gl_\gd$ strongly compact and has its
${<} \gl_\gd$ strong compactness (and hence
also its measurability) indestructible
when forcing with $\gd$-directed closed
partial orderings having rank below $\gl_\gd$''.
\begin{proof}
Suppose $V$ is as in the hypotheses for Theorem \ref{t3}.
%Let $V \models ``$ZFC + $\gk$ is supercompact +
%No cardinal $\gz > \gk$ is inaccessible''.
Without loss of generality, by doing a preliminary
forcing if necessary, we assume that
$V \models {\rm GCH}$ as well.
The proof of Theorem \ref{t3} now proceeds
as a modification of
the proof of \cite[Theorem 1.1]{A05}.
%with ideas used in the proofs of Lemmas \ref{l3} and \ref{l4}.
We will quote verbatim from the relevant portions of \cite{A05}
in our presentation when appropriate.
%in our definition of the forcing conditions
%and from the relevant arguments
%in the corresponding lemmas when appropriate.
In particular, whereas the partial ordering found in the
proof of \cite[Theorem 1.1]{A05} has a uniform definition
for every measurable cardinal $\gd$ (due to the
strong hypotheses used, which as we have already
indicated prove the consistency of the
hypotheses found in the proof of Theorem \ref{t3}),
%the partial ordering
%depending on whether a measurable cardinal $\gd$ is
%${<} \gl_\gd$ supercompact in our
%ground model $V$,
the definition of the partial ordering
$\FP$ employed in the proof of Theorem \ref{t3} splits
into two cases.
These will depend on whether a $V$-measurable cardinal $\gd$
is such that
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact
but is a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact'' or
the negation of this,
i.e., if either
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact
and is not a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact'' or
$V \models ``\gd$ is measurable but is not
${<} \gl_\gd$ supercompact''.
%$V \models ``\gd$ is ${<} \gl_\gd$ supercompact
%but is not a limit of cardinals $\gg$
%which are ${<} \gl_\gg$ supercompact''.
The difference between these two cases will be
in how the lottery sum is formed.
Specifically, the partial ordering $\FP$
used in the proof of Theorem \ref{t3}
is a Gitik style iteration of
Prikry-like forcings of length $\gk$,
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga < \gk \ra$,
which possibly does a nontrivial forcing
only at those ordinals $\gd < \gk$ which are
$V$-measurable cardinals.
%If $\gd$ is inaccessible but not measurable,
%we force with the lottery sum of trivial
%forcing $\{\emptyset\}$, $\add(\gd, 1)$,
%and $\add(\gd, \gd^+)$.
If $V \models ``\gd$ is ${<} \gl_\gd$ supercompact
and is a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact'',
%if $V \models ``\gd$ is not
%${<} \gl_\gd$ supercompact'',
we first force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank less than
the least inaccessible cardinal above $\gd$
which are $\gd$-directed closed in $V^{\FP_\gd}$.
If this is not the case, i.e., if either
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact
and is not a limit of cardinals $\gg$
which are ${<} \gl_\gg$ supercompact'' or
$V \models ``\gd$ is measurable but is not
${<} \gl_\gd$ supercompact'',
%if $V \models ``\gd$ is not
%${<} \gl_\gd$ supercompact'',
we first force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank less than
$\gl_\gd$ which are $\gd$-directed closed in $V^{\FP_\gd}$.
If $V \models ``\gd$ is not ${<} \gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If the preceding conjunction of
conditions does not hold,
i.e., if either $V \models ``\gd$
is ${<} \gl_\gd$ supercompact''
or $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is not measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
\begin{lemma}\label{l7}
$V^\FP \models ``\gk$ is an
indestructibly supercompact cardinal''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\FQ$ is a $\gk$-directed
closed partial ordering''.
We show that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is supercompact''.
To do this, we argue as in \cite[Lemma 2.1]{A05}
(which combines arguments found in the proofs of \cite[Lemmas 2 and 3]{AG}).
For any missing details, readers
are urged to consult
%the aforementioned lemmas of
\cite{AG}.
Let $\gd > \max(2^{[\gl_\gk]^{< \gk}}, \card{{\rm TC}(\dot \FQ)})$
be an arbitrary $V$-cardinal large
enough such that
${(2^{[\gd]^{< \gk}})}^V = \rho =
{(2^{[\gd]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$.
%and $\rho$ is regular in both
%$V$ and $V^{\FP \ast \dot \FQ}$.
%and let $\gs = \rho^+$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gr$ supercompactness
of $\gk$.
%such that $M \models ``\gk$ is not $\gs$ supercompact''.
Since $V \models ``$No cardinal
$\gz > \gk$ is inaccessible'',
$M \models ``$There are no inaccessible
cardinals in the interval
$(\gk, \gr]$''.
Further, because $\gr > 2^{[\gl_\gk]^{< \gk}}$ and
$V \models ``\gk$ is supercompact'',
$M \models ``\gk$ is ${<} \gl_\gk$ supercompact and is
a limit of cardinals $\gg$ which are ${<} \gl_\gg$ supercompact''.
Hence, by the definition of $\FP$,
$\FQ$ is an allowable choice in the
stage $\gk$ lottery held in
$M^{\FP_\gk} = M^\FP$
in the definition of $j(\FP)$.
In addition, as $M \models ``\gk$ is
${<} \gl_\gk$ supercompact'',
$j(\FP \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS$ is a term for trivial forcing.
Because $M \models ``$There are no inaccessible
cardinals in the interval
$(\gk, \gr]$'', the next nontrivial
stage in the definition of
$j(\FP)$ after $\gk$ takes place
well above $\gr$.
Hence, 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 $j '' \dot H$.
Let $K = K_0 \ast K_1 \ast K_2$ be $V[G][H]$-generic over
$\FS \ast \dot \FR \ast j(\dot \FQ)$.
Define an embedding
$j^* : V[G][H] \to M[G][H][K]$ by
$j^*(i_{G \ast H}(\ov \tau)) =
i_{G \ast H \ast K}(j(\ov \tau))$ for any term
$\ov \tau$ denoting a set in $V[G][H]$.
Since the closure properties of $M$ imply any term for a
condition in $K_2$ can be assumed to extend the
``master condition'' $\tau$ above,
as in \cite[Lemma 3]{AG}, $j^*$ is a
well-defined elementary embedding lifting $j$ which can
be used to define a supercompact ultrafilter
${\cal U} \in V[G][H][K]$ over
${(P_\gk(\gd))}^{V[G][H]}$ by
$X \in {\cal U}$ iff $\la j(\ga) \mid \ga < \gd \ra
\in j^*(X)$.
Since $\FP \ast \dot \FQ$ is $\gr$-c.c$.$,
the usual arguments show that
$M[G][H]$ remains $\gr$ closed
with respect to $V[G][H]$. Hence,
since forcing with
$\FS \ast \dot \FR \ast j(\dot \FQ)$
over either $M[G][H]$ or $V[G][H]$
adds no subsets of $\gr$,
${\cal U} \in V[G][H]$, i.e.,
$V[G][H] \models ``\gk$ is $\gd$ supercompact''.
Since $\gd > \gk$ was an arbitrarily
large enough $V$-cardinal,
and since trivial forcing is
$\gk$-directed closed,
this completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ was ${<} \gl_\gd$ supercompact in $V$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gk$ is measurable''.
Write
$\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$.
Since by the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ adds
%no subsets of $\gl_\gd$'', it must be the case that
no bounded subsets of the least inaccessible
cardinal above $\gd$'', it must be the case that
$\forces_{\FP_{\gd + 1}} ``\gd$ is measurable''.
Note now that
$V \models ``\gd$ is measurable''.
For, if this were not the case, then
again by the definition of $\FP$,
since $\dot \FQ_\gd$ is a term
for trivial forcing, it must
be true that
$\forces_{\FP_\gd} ``\gd$ is measurable''.
In addition, observe that
as any measurable cardinal is also Mahlo,
$V^\FP \models ``\gd$ is a Mahlo cardinal''.
Because forcing cannot create a new Mahlo cardinal,
it must also be true that
$V \models ``\gd$ is a Mahlo cardinal'' as well.
Therefore, since when a lottery sum is performed
at a nontrivial stage of forcing $\gg$, it is of
partial orderings having rank below the least
$V$-inaccessible cardinal above $\gg$, the definition
of $\FP$ allows us to infer that $\FP_\gd \subseteq V_\gd$.
Hence, as $\FP$ is a Gitik style iteration
of Prikry-like forcings, $\FP_\gd$
is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$.
This means that since $\FP_\gd$
satisfies $\gd$-c.c$.$ in
$V^{\FP_\gd}$ (this follows because
$\gd$ is measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$ is
%isomorphic to
a subordering of the Easton support product of
$\la \FQ_\ga \mid \ga < \gd \ra$ as
calculated in $V^{\FP_\gd}$),
the proof of \cite[Lemma 3]{AC1}
tells us that the restriction of
every $\gd$-additive
ultrafilter over $\gd$ present in
$V^{\FP_\gd}$
%which measures only the subsets of $\gd$ in $V$ to $V$ must in fact be
to $V$ is a $\gd$-additive ultrafilter over $\gd$
which is a member of $V$. This contradiction
to our supposition that
$V \models ``\gd$ is not measurable''
consequently yields that
$V \models ``\gd$ is measurable''.
However, if it is not the case that
$\gd$ is ${<} \gl_\gd$
supercompact in $V$, then by the
definition of $\FP$,
$V^{\FP_{\gd + 1}} \models ``\gd$
%contains a Prikry sequence and hence
is not measurable''.
This completes the proof of Lemma \ref{l8}.
\end{proof}
\begin{lemma}\label{l9}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ not a limit of
cardinals $\gg$ which are
${<} \gl_\gg$ supercompact in $V$ is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$''.
\end{lemma}
\begin{proof}
By Lemma \ref{l8}, we know that any
measurable cardinal $\gd < \gl_\gd$
had to have been ${<} \gl_\gd$
supercompact in $V$.
As in Lemma \ref{l8},
by the definition of $\FP$,
write
$\FP = \FP_{\gd + 1} \ast
\dot \FP^{\gd + 1}$, where
$\forces_{\FP_{\gd + 1}} ``$Forcing
with $\dot \FP^{\gd + 1}$
does not add any subsets of $\gl_\gd$''.
Thus, to prove Lemma \ref{l9}, it
suffices to show that
its conclusions hold in $V^{\FP_{\gd + 1}}$.
Towards this end, let
$\FQ \in V^{\FP_{\gd + 1}}$ be such that
$V^{\FP_{\gd + 1}} \models
``\FQ$ is $\gd$-directed closed and
has rank less than $\gl_\gd$''.
By the definition of $\FP$, since
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact
and is not a limit of cardinals $\gg$ which are
${<} \gl_\gg$ supercompact'',
$\FP_{\gd + 1} \ast \dot \FQ$ is forcing
equivalent to $\FP_\gd \ast \dot \FQ'
\ast \dot \FS_\gd \ast \dot \FQ$, where $\dot \FQ'$ is
a term for the partial ordering of rank
less than $\gl_\gd$ selected in the
stage $\gd$ lottery held in the
definition of $\FP$, and
$\dot \FS_\gd$ is a term for trivial forcing.
Since $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$
can be taken to be a term in the forcing language
with respect to $\FP_\gd$ for a
$\gd$-directed closed partial
ordering having rank less than $\gl_\gd$,
we abuse notation in what follows and
assume without loss of generality that
what we will write as $\dot \FQ$ is
actually $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$.
We proceed now in analogy to the
argument given in the second and third
paragraphs of the proof of Lemma \ref{l7}.
Specifically, the fact that $V \models {\rm GCH}$
and the definition of $\FP$ allow us to choose
$\gg > \max(\gd, \card{{\rm TC}(\dot \FQ)})$,
$\gg < \gl_\gd$
as an arbitrary $V$-regular cardinal large
enough such that
${(2^{[\gg]^{< \gd}})}^V = \rho =
{(2^{[\gg]^{< \gd}})}^{V^{\FP_\gd \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP_\gd \ast \dot \FQ}$.
Let $\gs = \rho^+$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gd$
such that $M \models ``\gd$ is not $\gs$ supercompact''.
By the definition of $\FP_\gd$,
because in both $V$ and $M$, $\gs < \gl_\gd$,
$\FQ$ is an allowable choice in the
stage $\gd$ lottery held in $M^{\FP_\gd}$
in the definition of $j(\FP_\gd)$.
Consequently, $j(\FP_\gd \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd ' \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS_\gd '$ is either
a term for trivial forcing or for
Prikry forcing over $\gd$ defined
using the appropriate normal measure.
Further, since
$M \models ``$There are no inaccessible
cardinals in the interval
$(\gd, \gs]$'', as before, the next nontrivial
stage in the definition of
$j(\FP_\gd)$ after $\gd$ takes place
well above $\gs$.
Hence, as in Lemma \ref{l7} and
\cite[Lemma 2]{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP_\gd)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP_\gd \ast \dot \FQ$,
$\forces_{j(\FP_\gd)} ``\gt$ 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 $j '' \dot H$.
Thus, if
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP_\gd \ast \dot \FQ$ names of subsets of
${(P_\gd(\gg))}^{V[G][H]}$,
we can as
%we did earlier in the proof of Lemma \ref{l1}
is done in the proof of \cite[Lemma 2]{AG}
define in $M$ a sequence of
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '$
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 $\FR$),
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gg \ra \in
j(\dot A_\ga)$'$\ $'',\footnote{Roughly
speaking, Easton extension means that $p_\gb \ge p_\ga$
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}.} and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
If we then in $V[G][H]$ define a set
${\cal U} \subseteq 2^{[\gg]^{< \gd}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gd(\gg)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FS_\gd '$ either
the trivial condition (if $\FS_\gd '$
is trivial forcing) or of the form
$\la \emptyset, B \ra$ (if $\FS_\gd '$ is
Prikry forcing),
in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) \mid \gb < \gg \ra \in
j(\dot X)$'' for some name $\dot X$ of $X$,
then as in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gd$-additive, fine
ultrafilter over
${(P_\gd(\gg))}^{V[G][H]}$, i.e.,
$V[G][H] \models ``\gd$ is
$\gg$ strongly compact''.
Since $\gg$ was arbitrary,
and since trivial forcing is
${<} \gl_\gd$-directed closed
and can be defined so as to
have rank less than $\gl_\gd$, this
completes the proof of Lemma \ref{l9}.
\end{proof}
\begin{lemma}\label{l10}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ which is not a limit of measurable cardinals
was in $V$ a ${<} \gl_\gd$ supercompact cardinal which
is not a limit of cardinals $\gg$ which are
${<} \gl_\gg$ supercompact''.
\end{lemma}
\begin{proof}
By Lemma \ref{l8}, we know that
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact''.
%If $V^\FP \models ``\gd$ is a limit of measurable cardinals'', then
%again by Lemma \ref{l8},
If $V \models ``\gd$ is a limit of cardinals $\gg$ which
are ${<} \gl_\gg$ supercompact'', then
$V \models ``\gd$ is a limit of cardinals $\gz$ which
are ${<} \gl_\gz$ supercompact and each such $\gz$
is not a limit of cardinals $\eta$ which are
${<} \gl_\eta$ supercompact''.
By the proof of Lemma \ref{l9}, each such $\gz$ is ${<} \gl_\gz$
strongly compact and hence measurable in $V^\FP$.
This means that $V^\FP \models ``\gd$ is a limit
of measurable cardinals'', a contradiction.
This completes the proof of Lemma \ref{l10}.
\end{proof}
\begin{lemma}\label{l11}
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
We argue in analogy to the proof
of \cite[Lemma 4]{AG} and \cite[Lemma 2.4]{A05}.
Let $\gd > \gk^{++}$ be any
sufficiently large cardinal below $\gl_\gk$, e.g.,
the least strong limit cardinal
above $\gk$.
Take $j : V \to M$ as an elementary
embedding witnessing the $\gd$
supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gd$ supercompact''.
Since $\gd < \gl_\gk$, this means that
$M \models ``\gk$ is not
${<} \gl_\gk$ supercompact''.
By the choice of $\gd$, it is possible to
opt for $\add(\gk, \gk^{++})$ at stage
$\gk$ in $M^\FP =
M^{\FP_\gk}$ in the definition of $j(\FP)$.
Further, by Lemma \ref{l7} and the fact
$M^\gd \subseteq M$,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}
\models ``\gk$ is measurable''.
By the definition of $\FP$, this
therefore means that
above the appropriate condition,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR$,
where $\dot \FS_\gk$ is a term for
Prikry forcing over $\gk$
defined with respect to the
appropriate normal measure, and
$\forces_{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk} ``$Forcing with $\dot \FR$ does not
add any subsets to $\gk$''.
Consequently,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR} =
M^{j(\FP)} \models ``\gk$ is a singular
strong limit cardinal violating GCH''.
By reflection, this just means that
$V^\FP \models ``$There are unboundedly
in $\gk$ many singular strong limit
cardinals below $\gk$ violating GCH''.
%By \cite[Theorem 4.8]{SRK} and the
%succeeding remarks,
%we may infer that
By Solovay's theorem \cite{So}
that GCH must hold at any singular
strong limit cardinal above a
strongly compact cardinal,
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l11}.
\end{proof}
Lemmas \ref{l7} -- \ref{l11} complete the
proof of Theorem \ref{t3}.
\end{proof}
As we mentioned earlier, our proof methods
for Theorem \ref{t3} remain valid for
different values of $\gl_\gd$, e.g.,
$\gl_\gd =_{\rm df}$ The second beth fixed
point above $\gd$, etc.
The arguments for Lemmas \ref{l7} -- \ref{l11}
will literally go through unchanged.
In addition, the proof methods for Theorem \ref{t3}
also remain valid if we assume that
$V \models ``$ZFC + $\gk$ is supercompact + No cardinal
$\gz > \gk$ is Mahlo''.
Here, we assume that in the definition of
$\FP$, the word ``inaccessible'' is replaced with the word ``Mahlo'',
and $\gl_\gd$ is, e.g., as it was originally (or in fact is
replaced by any suitable value).
The key point is that the proof of
Lemma \ref{l8} is still sound.
In particular, if $V^\FP \models ``\gd < \gk$ is measurable''
and $V \models ``\gd$ is not measurable'',
it will still be the case that $\FP_\gd \subseteq V_\gd$.
This allows us to infer that in fact,
$V \models ``\gd$ is measurable''.
However, if $V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is Ramsey'',
the definition of $\FP$ is modified changing
the word ``inaccessible'' to ``Ramsey'',
$\gl_\gd$ is as before or is anything suitable,
$V^\FP \models ``\gd < \gk$ is measurable'', and
$V \models ``\gd$ is not measurable'',
we do not necessarily know that $\FP_\gd \subseteq V_\gd$.
It will of course still be true that
$V \models ``\gd$ is a Mahlo cardinal''.
However, there could be some $V$-measurable cardinal
$\gg < \gd$ at which in the stage $\gg$ lottery,
a partial ordering having size above $\gd$ were chosen.
If this were indeed the case, then
$\FP_\gd \not\subseteq V_\gd$, and we cannot
use our original methods to infer that
$V \models ``\gd$ is measurable''.
Because $\FP$ is a Gitik style iteration of
Prikry-like forcings, we cannot use the results
of \cite{H2, H3, H03} either to infer that
$V \models ``\gd$ is measurable''.
Thus, the proof of Lemma \ref{l8}, and hence also
the proof of Theorem \ref{t3}, break down.
We therefore conclude by asking if there is some
alternate method which will allow the appropriate
analogue of Theorem \ref{t3} to be proven if our ground model
$V$ is such that
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is Ramsey''
(or even if $V$ is such that
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable'').
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\begin{proof}
Let
$V \models ``$ZFC + GCH + $\gk$ is
$\gl = \gl_\gk$ supercompact''.
%+ There is some supercompact ultrafilter
%${\cal U}$ over $P_\gk(\gl)$ such that
%for $j : V \to M$ the associated
%elementary embedding,
%$M \models ``\gk$ is $\gl$ supercompact''.
The partial ordering $\FP$
used in the proof of Theorem \ref{t3}
is a Gitik style iteration of
Prikry-like forcings of length $\gk$,
$\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$,
which does a nontrivial forcing
only at those ordinals $\gd < \gk$ which are
$V$-measurable cardinals.
At such a stage $\gd$,
%if $V \models ``\gd$ is not
%${<} \gl_\gd$ supercompact'',
we first force with $\FQ^*_\gd$,
the lottery sum of all
partial orderings having rank less than
$\gl_\gd$ which are
$\gd$-directed closed
in $V^{\FP_\gd}$.
If $V \models ``\gd$ is not ${<} \gl_\gd$ supercompact'' and
$V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for Prikry forcing
over $\gd$ defined using the appropriate
normal measure.
If the preceding conjunction of
conditions does not hold,
i.e., if either $V \models ``\gd$
is ${<} \gl_\gd$ supercompact''
or $V^{\FP_\gd \ast \dot \FQ^*_\gd} \models
``\gd$ is not measurable'', then
$\FQ_\gd = \FQ^*_\gd \ast \dot \FS_\gd$, where
$\dot \FS_\gd$ is a term for
trivial forcing.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is a
${<} \gl$ supercompact
cardinal whose ${<} \gl$
supercompactness is
indestructible under forcing with
partial orderings having rank less than $\gl$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\FQ$ is a $\gk$-directed
closed partial ordering having rank
less than $\gl$''.
We show that
$V^{\FP \ast \dot \FQ} \models
``\gk$ is ${<} \gl$ supercompact''.
To do this, we combine arguments
found in the proofs of Lemmas 2
and 3 of \cite{AG}.
For any missing details, readers
are urged to consult
%the aforementioned lemmas of
\cite{AG}.
Let $\gd > \max(\gk, \card{{\rm TC}(\dot \FQ)})$,
$\gd < \gl$
be an arbitrary $V$-cardinal large
enough so that
${(2^{[\gd]^{< \gk}})}^V = \rho =
{(2^{[\gd]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$.
%and $\rho$ is regular in both
%$V$ and $V^{\FP \ast \dot \FQ}$.
%and let $\gs = \rho^+$.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness
of $\gk$.
%such that $M \models ``\gk$ is not $\gs$ supercompact''.
By the definition of $\FP$,
$\FQ$ is an allowable choice in the
stage $\gk$ lottery held in
$M^{\FP_\gk} = M^\FP$
in the definition of $j(\FP)$.
In addition, $M \models ``\gk$ is
${<} \gl$ supercompact''.
Consequently, $j(\FP \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FS \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS$ is a term for trivial forcing.
Further, since
$M \models ``$There are no measurable
cardinals in the interval
$(\gk, \gl]$'', the next nontrivial
stage in the definition of
$j(\FP)$ after $\gk$ takes place
well above $\gl$.
Hence, as in Lemma 2 of \cite{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Let $K$ be $V[G][H]$-generic over
$\FS \ast \dot \FR \ast j(\dot \FQ)$.
Define an embedding
$j^* : V[G][H] \to M[G][H][K]$ by
$j^*(i_{G \ast H}(\ov \tau)) =
i_{G \ast H \ast K}(j(\ov \tau))$ for any term
$\ov \tau$ denoting a set in $V[G][H]$.
Since the closure properties of $M$ imply any term for a
condition in $j(\dot \FQ)$ can be assumed to extend the
``master condition'' $\tau$ above,
as in Lemma 3 of \cite{AG}, $j^*$ is a
well-defined elementary embedding lifting $j$ which can
be used to define a supercompact ultrafilter
${\cal U} \in V[G][H][K]$ over
${(P_\gk(\gd))}^{V[G][H]}$ by
$X \in {\cal U}$ iff $\la j(\ga) : \ga < \gd \ra
\in j^*(X)$.
Since $\FP$ is $\gk$-c.c$.$,
the usual arguments show that
$M[G][H]$ remains $\gr$ closed
with respect to $V[G][H]$. Hence,
since forcing with
$\FS \ast \dot \FR \ast j(\dot \FQ)$
over either $M[G][H]$ or $V[G][H]$
adds no subsets of $\gr$,
${\cal U} \in V[G][H]$, i.e.,
$V[G][H] \models ``\gk$ is $\gd$ supercompact''.
Since $\gd < \gl$ was an arbitrarily
large enough $V$-cardinal,
and since trivial forcing is
$\gk$-directed closed,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l8}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ was ${<} \gl_\gd$ supercompact in $V$''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gd < \gk$ is measurable''.
Write
$\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$.
Since by the definition of $\FP$,
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ adds
no subsets of $\gl_\gd$'', it must be the case that
$\forces_{\FP_{\gd + 1}} ``\gd$ is measurable''.
Note now that
$V \models ``\gd$ is measurable''.
For, if this were not the case, then
again by the definition of $\FP$,
since $\dot \FQ_\gd$ is a term
for trivial forcing, it must
be true that
$\forces_{\FP_\gd} ``\gd$ is measurable''.
As $\FP$ is a Gitik style iteration
of Prikry-like forcings, $\FP_\gd$
is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
This means that since $\FP_\gd$
satisfies $\gd$-c.c$.$ in
$V^{\FP_\gd}$ (this follows because
$\gd$ is measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$ is
%isomorphic to
a subordering of the Easton support product of
$\la \FQ_\ga : \ga < \gd \ra$ as
calculated in $V^{\FP_\gd}$),
the proof of Lemma 3 of \cite{AC1}
tells us that every $\gd$-additive
ultrafilter over $\gd$ present in
$V^{\FP_\gd}$ must in fact be
present in $V$. This contradiction
to our supposition that
$V \models ``\gd$ is not measurable''
consequently yields that
$V \models ``\gd$ is measurable''.
However, if it is not the case that
$\gd$ is ${<} \gl_\gd$
supercompact in $V$, then by the
definition of $\FP$,
$V^{\FP_{\gd + 1}} \models ``\gd$
%contains a Prikry sequence and hence
is not measurable''.
This completes the proof of Lemma \ref{l8}.
\end{proof}
\begin{lemma}\label{l9}
$V^\FP \models ``$Any measurable cardinal
$\gd < \gk$ is ${<} \gl_\gd$ strongly
compact and has its ${<} \gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$''.
\end{lemma}
\begin{proof}
By Lemma \ref{l8}, we know that any
measurable cardinal $\gd < \gl_\gd$
had to have been ${<} \gl_\gd$
supercompact in $V$.
As in Lemma \ref{l8},
by the definition of $\FP$,
write
$\FP = \FP_{\gd + 1} \ast
\dot \FP^{\gd + 1}$, where
$\forces_{\FP_{\gd + 1}} ``$Forcing
with $\dot \FP^{\gd + 1}$
does not add any subsets of $\gl_\gd$''.
Thus, to prove Lemma \ref{l9}, it
suffices to show that
its conclusions hold in $V^{\FP_{\gd + 1}}$.
Towards this end, let
$\FQ \in V^{\FP_{\gd + 1}}$ be such that
$V^{\FP_{\gd + 1}} \models
``\FQ$ is $\gd$-directed closed and
has rank less than $\gl_\gd$''.
By the definition of $\FP$, since
$V \models ``\gd$ is ${<} \gl_\gd$ supercompact'',
$\FP_{\gd + 1} \ast \dot \FQ$ is forcing
equivalent to $\FP_\gd \ast \dot \FQ'
\ast \dot \FS_\gd \ast \dot \FQ$, where $\dot \FQ'$ is
a term for the partial ordering of rank
less than $\gl_\gd$ selected in the
stage $\gd$ lottery held in the
definition of $\FP$, and
$\dot \FS_\gd$ is a term for trivial forcing.
Since $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$
can be taken to be a term in the forcing language
with respect to $\FP_\gd$ for a
$\gd$-directed closed partial
ordering having rank less than $\gl_\gd$,
we abuse notation in what follows and
assume without loss of generality that
what we will write as $\dot \FQ$ is
actually $\dot \FQ' \ast \dot \FS_\gd \ast \dot \FQ$.
We proceed now in analogy to the
argument given in the second
paragraph of the proof of Lemma \ref{l1}.
Specifically,
let $\gg > \max(\gd, \card{{\rm TC}(\dot \FQ)})$,
$\gg < \gl_\gd$
be an arbitrary $V$-regular cardinal large
enough so that
${(2^{[\gg]^{< \gd}})}^V = \rho =
{(2^{[\gg]^{< \gd}})}^{V^{\FP_\gd \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP_\gd \ast \dot \FQ}$,
and let $\gs = \rho^+$. Take
$j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness
of $\gd$.
%such that $M \models ``\gd$ is not $\gs$ supercompact''.
By the definition of $\FP_\gd$,
$\FQ$ is an allowable choice in the
stage $\gd$ lottery held in $M^{\FP_\gd}$
in the definition of $j(\FP_\gd)$.
Consequently, $j(\FP_\gd \ast \dot \FQ)$
is forcing equivalent in $M$ to
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd ' \ast
\dot \FR \ast j(\dot \FQ)$, where
$\dot \FS_\gd '$ is either
a term for trivial forcing or for
Prikry forcing over $\gd$ defined
using the appropriate normal measure.
Further, since
$M \models ``$There are no inaccessible
cardinals in the interval
$(\gd, \gs]$'', as before, the next nontrivial
stage in the definition of
$j(\FP_\gd)$ after $\gd$ takes place
well above $\gs$.
Hence, as in Lemma \ref{l1} and
Lemma 2 of \cite{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP_\gd)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP_\gd \ast \dot \FQ$,
$\forces_{j(\FP_\gd)} ``\gt$ 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
$\la \dot A_\ga : \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP_\gd \ast \dot \FQ$ names of subsets of
${(P_\gd(\gg))}^{V[G \ast H]}$,
we can as
%we did earlier in the proof of Lemma \ref{l1}
is done in the proof of Lemma 2 of \cite{AG}
define in $M$ a sequence of
$\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) : \gb < \gg \ra \in
j(\dot A_\ga)$'$\ $'', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP_\gd \ast \dot \FQ \ast \dot \FS_\gd '}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb : \gb < \eta \ra$''.
If we then in $V[G \ast H]$ define a set
${\cal U} \subseteq 2^{[\gg]^{< \gd}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gd(\gg)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FS_\gd '$ either
the trivial condition (if $\FS_\gd '$
is trivial forcing) or of the form
$\la \emptyset, B \ra$ (if $\FS_\gd '$ is
Prikry forcing),
in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) : \gb < \gg \ra \in
j(\dot X)$'' for some name $\dot X$ of $X$,
then as in Lemma 2 of \cite{AG},
${\cal U}$ is a $\gd$-additive, fine
ultrafilter over
${(P_\gd(\gg))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gd$ is
$\gg$ strongly compact''.
Since $\gg$ was arbitrary,
and since trivial forcing is
${<} \gl_\gd$-directed closed
and can be defined so as to
have rank less than $\gl_\gd$, this
completes the proof of Lemma \ref{l9}.
\end{proof}
We explicitly note that Lemma \ref{l9}
actually tells us that in $V^\FP$, any cardinal
$\gd < \gk$ which was ${<}\gl_\gd$ strongly
compact in $V$ is ${<}\gl_\gd$
strongly compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$ (so in particular
is measurable in $V^\FP$).
This observation will be used in the
proof of Theorem \ref{t2}.
\begin{lemma}\label{l11}
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
We argue in analogy to the proof
of Lemma 4 of \cite{AG} and Lemma ??? of ???.
Let $\gd > \gk^{++}$ be any
sufficiently large cardinal, e.g.,
the least strong limit cardinal
above $\gk$.
Take $j : V \to M$ as an elementary
embedding witnessing the $\gd$
supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gd$ supercompact'',
which means since $\gd < \gl$ that
$M \models ``\gk$ is not
${<} \gl_\gk$ supercompact''.
By the choice of $\gd$, it is possible to
opt for $\add(\gk, \gk^{++})$ at stage
$\gk$ in $M^\FP =
M^{\FP_\gk}$ in the definition of $j(\FP)$.
Further, by Lemma \ref{l1} and the fact
$M^\gd \subseteq M$,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})}
\models ``\gk$ is measurable''.
By the definition of $\FP$, this
therefore means that
above the appropriate condition,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR$,
where $\dot \FS_\gk$ is a term for
Prikry forcing over $\gk$
defined with respect to the
appropriate normal measure, and
$\forces_{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk} ``$Forcing with $\dot \FR$ does not
add any subsets to $\gk$''.
Consequently,
$M^{\FP_\gk \ast \dot \add(\gk, \gk^{++})
\ast \dot \FS_\gk \ast \dot \FR} =
M^{j(\FP)} \models ``\gk$ is a singular
strong limit cardinal violating GCH''.
By reflection, this just means that
$V^\FP \models ``$There are unboundedly
in $\gk$ many singular strong limit
cardinals below $\gk$ violating GCH''.
By a theorem of Solovay \cite{So},
we may infer that
$V^\FP \models ``$No cardinal
$\gg < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l11}.
\end{proof}
Let now $V^*$ be
the model $V^\FP$ truncated at $\gl$.
By Lemmas \ref{l1} - \ref{l11},
their proofs, and the fact that
$\FP$ may be defined so as
to have cardinality $\gk$,
$V^* \models ``$ZFC + No cardinal
$\gd > \gk$ is inaccessible''. In
$V^*$,
$\gk$ is both the
least supercompact and least strongly
compact cardinal, and
$\gk$'s supercompactness
is indestructible under $\gk$-directed
closed forcing. In addition,
in $V^*$, for every measurable cardinal
$\gd < \gk$, $\gd$ is ${<}\gl_\gd$ strongly
compact and has its ${<}\gl_\gd$ strong
compactness indestructible under
$\gd$-directed closed forcing having
rank less than $\gl_\gd$.
This completes the proof of
Theorem \ref{t1}.
\end{proof}
We remark that in the model $V^\FP$ witnessing
the conclusions of Theorem \ref{t2}, $\gk$
can be the least strongly compact cardinal.
To see this, suppose that in addition to the
hypotheses for Theorem \ref{t2}, $V \models ``\gk$ is the
least strongly compact cardinal'' as well (something which
is possible by Magidor's theorem of \cite{Ma}).