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\title{Indestructibility, Measurability,
and Degrees of Supercompactness
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal,
indestructibility,
nonreflecting stationary set of ordinals.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{December 13, 2010}
\date{December 24, 2010\\
(revised May 18, 2011)}
\begin{document}
\maketitle
\begin{abstract}
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, and $\gd$ is not $\gd^+$ supercompact$\}$
is unbounded in $\gk$.
If in addition
$\gl$ is $2^\gl$ supercompact, then
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is $\gd^+$ supercompact$\}$
is unbounded in $\gk$ as well.
%In fact, $A_1$ must be unbounded in $\gk$ if $\gl$
%is only measurable.
The large cardinal
hypotheses on $\gl$ are 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, there is an
indestructibly supercompact cardinal $\gk$,
and a restricted large cardinal structure above $\gk$.
If we weaken the indestructibility requirement on
$\gk$ to indestructibility under partial orderings
which are both $\gk$-directed closed and
$(\gk^+, \infty)$-distributive, then it is possible to
construct a model containing a supercompact cardinal
$\gk$ witnessing this degree of indestructibility
in which {\em every} measurable cardinal $\gd < \gk$
is (at least) $\gd^+$ supercompact.
\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 what happens at large cardinals below
$\gk$.
On the other hand, it is possible to mitigate
these effects
if the universe contains relatively few large cardinals.
These sorts of occurrences have been
previously investigated in \cite{A07, A08, A09, A11, A10, 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 the degree of supercompactness
certain measurable cardinals can manifest in
universes containing a supercompact
cardinal with various
indestructibility properties.
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.
\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, and $\gd$ is not $\gd^+$ supercompact$\}$
is unbounded in $\gk$.
\end{theorem}
In fact, if we assume additional hypotheses on $\gl$,
then it is possible to infer even more.
Specifically, we have:
\begin{theorem}\label{t2}
Suppose that
$\gk$ is indestructibly supercompact
and there is a cardinal $\gl > \gk$
which is $2^\gl$ supercompact. Then
besides $A_1$ being unbounded in $\gk$,
%$A_1 = \{\gd < \gk \mid \gd$ is
%measurable, $\gd$ is not a limit of
%measurable cardinals, and $\gd$ is not $\gd^+$ supercompact$\}$
%and
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is $\gd^+$ supercompact$\}$
is unbounded in $\gk$ as well.
\end{theorem}
With a limited large cardinal structure above $\gk$,
Theorems \ref{t1} and \ref{t2} need not be true.
Specifically, we have:
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is $2^\gd$ supercompact''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gd > \gk$ is $2^\gd$ supercompact +
$\gk$ is indestructibly supercompact +
If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then
$\gd$ is not $\gd^+$ supercompact''.
\end{theorem}
\begin{theorem}\label{t4}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gd > \gk$ is measurable +
$\gk$ is indestructibly supercompact +
If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ supercompact''.
\end{theorem}
If we are willing to weaken the amount of
indestructibility on our supercompact cardinal $\gk$,
then it is possible to obtain an improved version
of Theorem \ref{t4}. In particular:
\begin{theorem}\label{t5}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gd > \gk$ is measurable +
$\gk$ is a supercompact cardinal whose supercompactness
is indestructible under partial orderings which
are both $\gk$-directed closed and
$(\gk^+, \infty)$-distributive +
If $\gd < \gk$ is a measurable cardinal, then
$\gd$ is (at least) $\gd^+$ supercompact''.
\end{theorem}
Of course, reflection easily yields that
if $\gk$ is supercompact, then
$\{\gd < \gk \mid \gd$ is (at least)
$\gd^+$ supercompact$\}$ %(and much more)
must be unbounded in $\gk$.
It is therefore not possible to obtain an analogue
of Theorem \ref{t5} for Theorem \ref{t3}.
We take this opportunity to make a few additional
remarks concerning Theorems \ref{t1} -- \ref{t5}.
The limited amount of indestructibility forced
in Theorem \ref{t5} is due to
the necessity in our proofs
of preserving a nontrivial degree
of supercompactness.
However, if we weaken the requirement
in Theorem \ref{t5} of all measurable
cardinals $\gd < \gk$ being %(at least)
$\gd^+$ supercompact to only measurable cardinals
$\gd < \gk$ which are not themselves
limits of measurable cardinals being %(at least)
$\gd^+$ supercompact, then Theorem \ref{t4}
shows that it is possible for
$\gk$ to be a fully indestructible supercompact cardinal.
Also, as our proof will show,
the degrees of indestructibility mentioned
in the statement of Theorems \ref{t1} and \ref{t2}
can be weakened. In particular, $\gk$'s
supercompactness can be indestructible
under $\gk$-directed closed,
$(\gk^+, \infty)$-distributive forcing.
This provides a nice balance between
Theorems \ref{t1} and \ref{t5},
which complements the balance between
Theorems \ref{t2} and \ref{t3} and
Theorems \ref{t1} and \ref{t4}.
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 $\gk$ is a regular cardinal.
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 $\gk$-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.
$\FP$ 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
It follows that forcing with
any partial ordering $\FP$
which is $(\gk^+, \infty)$-distributive
preserves the
%either the $\gk^+$ strong compactness or
$\gk^+$ supercompactness of $\gk$,
since forcing with $\FP$ adds no new
subsets of $P_\gk(\gk^+)$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorems \ref{t2} -- \ref{t5}.
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 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{A Key Forcing Notion}\label{s2}
%The proofs of Theorems \ref{t1}, \ref{t4}, and \ref{t5}
The proof of Theorem \ref{t1}
will depend on the existence of a
certain partial ordering $\FP(\gd, \gk)$.
We isolate the existence of this key forcing
notion in the following theorem.
\begin{theorem}\label{t6}
Let $V \models ``\gd < \gk$ are such that $\gd$ is
a regular cardinal and $\gk$ is $2^\gk$ supercompact''.
%is the least $2^\gk$ supercompact cardinal greater than $\gd$.
There is then a $\gd$-directed closed partial ordering
$\FP(\gd, \gk) \in V$ such that
$V^{\FP(\gd, \gk)} \models ``\gk$ is the least
measurable cardinal greater than $\gd$ +
$2^\gk = 2^{\gk^+} = \gk^{++}$ +
$\gk$ is $\gk^+$ supercompact''.
\end{theorem}
\begin{proof}
Assume $\gd$ and $\gk$ are as in the hypotheses
of Theorem \ref{t6}. We define
$\FP(\gd, \gk)$ as $\FP^1 \ast \dot \FP^2
\ast \dot \FP^3$.
Let $\gr = (2^\gk)^V$.
$\FP^1$ is defined as the reverse Easton iteration
$\la \la \FP_n, \dot \FQ_n \ra \mid n < \go \ra$, where
$\FP_0 = \add(\gr^+, 1)$.
For each $n < \go$, if
$\forces_{\FP_n} ``$There is a cardinal greater than
$\gr$ violating GCH'', then
$\forces_{\FP_n} ``\dot \FQ_n = \dot \add(\gg^+, 1)$ where
$\gg$ is the least cardinal greater than $\gr$
violating GCH''.
If this is not the case, i.e., if
$\forces_{\FP_n} ``$All cardinals greater than
$\gr$ satisfy GCH'', then
$\forces_{\FP_n} ``\dot \FQ_n$ is trivial
forcing''.\footnote{We slightly abuse notation
here when we write $\forces_{\FP_n}$,
since we always assume we are forcing
above the relevant condition when necessary.}
Standard arguments (see \cite[Exercise 15.16]{J}
and \cite[Lemma 4, Case 2]{A00}) show that if $\gg$
is a cardinal, then after
forcing with $\add(\gg^+, 1)$,
all cardinals less than or equal to $\gg^+$ are preserved,
$2^\gg = \gg^+$, $2^\gg$ of the ground model is
collapsed to $\gg^+$, and all cardinals greater
than or equal to $(2^\gg)^+$ of the ground model are preserved.
This means that
$V^{\FP^1} \models ``$For every $n < \go$,
$2^{\gr^{+ n}} = \gr^{+ n + 1}$''.
Since $\FP^1$ is $\gr^+ = ((2^\gk)^+)^V$-directed closed,
$(2^\gk)^V = (2^\gk)^{V^{\FP^1}} = \gr$.
In addition, it follows that
$V^{\FP^1} \models ``\gk$ is $2^\gk$ supercompact''.
Work now in $V_* = V^{\FP^1}$.
$\FP^2$ is defined as $\FP_\gk \ast \dot \add(\gk^+, 1)$, where
$\FP_\gk$ is the reverse Easton iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
of length $\gk$ which does trivial
forcing except for those cardinals $\gl \in (\gd, \gk)$
which are inaccessible in $V_*$. In this case,
$\FP_{\gl + 1} = \FP_\gl \ast \dot \FQ_\gl$, where
$\forces_{\FP_\gl \ast \dot \FQ_\gl} ``$For every
$n < \go$, $2^{\gl^{+ n}} = \gl^{+ n + 1}$''.
Assuming $\dot \FQ_\gl$ is a term for the analogue
of the iteration given in the preceding paragraph,
$\dot \FQ_\gl$ may be written as
$\dot \add(\gl^+, 1) \ast \dot \FR_\gl$, where
$\forces_{\FP_\gl \ast \dot \add(\gl^+, 1)} ``\dot \FR_\gl$ is
$(2^\gl)^+$-directed closed'', i.e.,
$\forces_{\FP_\gl \ast \dot \add(\gl^+, 1)} ``\dot \FR_\gl$ is
$\gl^{++}$-directed closed''.
Standard arguments once again show that
$V_*^{\FP^2} \models ``2^\gg = \gg^+$ if
$\gl \in (\gd, \gk]$ is inaccessible and
$\gg \in [\gl, \gl^{+ \go})$''.
It is also the case that
$V_*^{\FP^2} \models ``\gk$ is
$2^\gk = \gk^+$ supercompact''.
To see this, %let $\gr = (2^\gk)^{V_*}$, and
let $j : V_* \to M$ be an elementary embedding
witnessing the $\gr$ supercompactness of $\gk$ in $V_*$
generated by a supercompact ultrafilter over
$P_\gk(\gr)$. In particular, $M^{\gr} \subseteq M$.
We use a standard lifting argument
(a form of which is given, e.g.,
in the proof of \cite[Lemma 2.2]{A10})
to show that $j$ lifts in $V_*^{\FP_\gk \ast \dot \add(\gk^+, 1)}$ to
$j : V_*^{\FP_\gk \ast \dot \add(\gk^+, 1)}
\to M^{j(\FP_\gk \ast \dot \add(\gk^+, 1))}$. Specifically, let
$G_0$ be $V_*$-generic over $\FP_\gk$, and let
$G_1$ be $V_*[G_0]$-generic over
$\add(\gk^+, 1)$.
Observe that $j(\FP_\gk \ast \dot \add(\gk^+, 1)) =
\FP_\gk \ast \dot \add(\gk^+, 1) \ast
\dot \FQ \ast \dot \add(j(\gk^+), 1)$.
Working in $V_*[G_0][G_1]$,
we first note that since $\FP_\gk \ast \dot \add(\gk^+, 1)$
is $\gr^+$-c.c.,
%and $\add(\gk^+, 1)$ is $\gk^+$-directed closed in $V_*[G_0]$,
$M[G_0][G_1]$ remains $\gr$ closed with respect to $V_*[G_0][G_1]$.
This means that $\FQ$ is $\gr^+$-directed closed
in both $M[G_0][G_1]$ and $V_*[G_0][G_1]$.
Since $M[G_0][G_1] \models ``\card{\FQ} = j(\gk)$'',
the number of dense open subsets of $\FQ$ present
in $M[G_0][G_1]$ is $(2^{j(\gk)})^M$.
In $V_*$,
since $M$
is given via an ultrapower by a supercompact
ultrafilter over $P_\gk(2^\gk)$,
this is calculated as
$\card{\{f \mid f : [2^\gk]^{< \gk} \to 2^\gk\}} =
\card{\{f \mid f : 2^\gk \to 2^\gk\}} = 2^{2^\gk} = 2^\gr$.
Since $V_* \models ``2^{2^\gk} = (2^\gk)^+ = \gr^+$''
and $\gr^+$ is preserved from $V_*$ to
$V_*[G_0][G_1]$, we may let
$\la D_\ga \mid \ga < \gr^+ \ra \in V_*[G_0][G_1]$
enumerate the dense open subsets of $\FQ$ present in $M[G_0][G_1]$.
We may now use the fact that $\FQ$ is
$\gr^+$-directed closed in $V_*[G_0][G_1]$ to meet each $D_\ga$
and thereby construct in $V_*[G_0][G_1]$
an $M[G_0][G_1]$-generic
object $H_0$ over $\FQ$. Our construction guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast H_0$,
so $j$ lifts in $V_*[G_0][G_1]$ to
$j : V_*[G_0] \to M[G_0][G_1][H_0]$.
It remains to lift $j$ in $V_*[G_0][G_1]$ through
$\add(\gk^+, 1)$.
Because $V_*[G_0] \models ``\card{\add(\gk^+, 1)}
= 2^\gk = (2^\gk)^{V_*} = \gr$'',
$M[G_0][G_1][H_0] \models ``\card{\add(j(\gk^+), 1)}
= 2^{j(\gk)} = (2^{j(\gk)})^M$''.
Therefore, since
$M[G_0][G_1][H_0]$ remains $\gr$ closed
with respect to $V_*[G_0][G_1]$,
$M[G_0][G_1][H_0] \models ``\add(j(\gk^+), 1)$ is
$j(\gk^+)$-directed closed'', and
$j(\gk^+) > j(\gk) > \gr$, there is a master condition
$q \in \add(j(\gk^+), 1)$ for $j '' \{p \mid p \in G_1\}$.
Further, the number of dense open subsets of
$\add(j(\gk^+), 1)$ present in $M[G_0][G_1][H_0]$ is
$(2^{2^{j(\gk)}})^M$.
This is calculated in $V_*$ as
$\card{\{f \mid f : [2^\gk]^{< \gk} \to 2^{2^\gk}\}} =
\card{\{f \mid f : 2^\gk \to (2^\gk)^+\}} =
\card{\{f \mid f : \gr \to \gr^+\}} = 2^\gr =
(2^\gk)^+ = \gr^+$.
Working in $V_*[G_0][G_1]$,
since $\add(j(\gk^+), 1)$ is $\gr^+$-directed
closed in both $M[G_0][G_1][H_0]$ and $V_*[G_0][G_1]$,
we may consequently use
the arguments of the preceding paragraph to construct
an $M[G_0][G_1][H_0]$-generic object $H_1$
over $\add(j(\gk^+), 1)$ containing $q$.
Since by the definition of $H_1$,
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast H_0 \ast H_1$,
$j$ lifts in $V_*[G_0][G_1]$ to
$j : V_*[G_0][G_1] \to M[G_0][G_1][H_0][H_1]$.
As $V_*[G_0][G_1] \models ``\card{\gr} = \gk^+$'',
this means that
$V_*^{\FP^2} \models ``\gk$ is $2^\gk = \gk^+$ supercompact''.
Note that by its definition, $\FP^1 \ast \dot \FP^2$
is $\gd$-directed closed.
Work now in $V^{\FP^1 \ast \dot \FP^2} = \ov V$.
Fix $\gl \in (\gd, \gk]$ an inaccessible cardinal.
We define three notions of forcing.
In particular, we describe now
a specific form of
the partial orderings
of \cite[Section 4]{AS97b}.
Following the notation of \cite[Section 4]{AS97b},
we will denote these partial orderings by
$\FP^0_{\gl, \gl^{++}}$,
$\FP^1_{\gl, \gl^{++}}[S]$,
and
$\FP^2_{\gl, \gl^{++}}[S]$.
So that readers are not overly burdened, we
abbreviate our definitions and descriptions somewhat.
Full details may be found by consulting
\cite{AS97b}, along with
the relevant portions of \cite{AS97a}.
We do mention explicitly, however, that
(more than) the amount of GCH required for the definitions of
$\FP^0_{\gl, \gl^{++}}$,
$\FP^1_{\gl, \gl^{++}}[S]$,
and
$\FP^2_{\gl, \gl^{++}}[S]$
to be given and for these partial orderings to have
the properties described below has been forced by
$\FP^1 \ast \dot \FP^2$.
%Fix $\gd < \gl$, $\gl > \gd^+$ regular cardinals in our
%ground model $V$, with $\gd$ inaccessible and $\gl$ either
%inaccessible or the successor of a cardinal of cofinality
%greater than $\gd$.
%We assume GCH holds for all
%cardinals $\eta \ge \gd$.
The first notion of forcing $\FP^0_{\gl, \gl^{++}}$ is just
the standard notion of forcing
for adding a nonreflecting stationary
set of ordinals $S$ of cofinality
$\gd $ to $\gl^{++}$.
For further details on the definition of
this partial ordering, we refer readers to
\cite{AS97a} or \cite{AS97b}. We note only that
$\FP^0_{\gl, \gl^{++}}$ is $\gd$-directed closed.
Next, work in
$V_1 = {\ov V}^{\FP^0_{\gl, \gl^{++}}}$, letting $\dot S$
be a term always forced to denote $S$.
$\FP^2_{\gl, \gl^{++}}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ nonstationary).
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\ga \mid \ga \in S \ra$,
the existence of which is given by
\cite[Lemma 1]{AS97a} and \cite[Lemma 1]{AS97b}.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gl, \gl^{++}}
[S] $.
First, since each element of
$S$ has cofinality $\gd$, the proof of Lemma 1
of \cite{AS97a} and \cite{AS97b}
shows each $x \in X$ can be assumed to be
such that order-type$(x) = \gd$. Then,
$\FP^1_{\gl, \gl^{++}}[ S]$ is defined as the set of all
4-tuples $\la w, \ga, \bar r, Z \ra$ satisfying the
following properties.
\begin{enumerate}
\item $w \in {[\gl^{++}]}^{< \gl}$.
\item $\ga < \gl$.
\item $ \bar r = \la r_i \mid i \in w \ra$ is a
sequence of functions from $\ga$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\ga$.
\item $Z \subseteq \{x_\gb \mid \gb \in S\}$
is a set such that if $z \in Z$, then for some
$y \in {[w]}^\gd$, $y \subseteq z$ and $z - y$
is bounded in the $\gb$ such that $z = x_\gb$.
\end{enumerate}
%\noindent As in Section 4 of \cite{AS97b}
%and Section 1 of \cite{AS97a}, the
%definition of $Z$ implies
%$|Z| < \gd$.
\noindent
The ordering on $\FP^1_{\gl, \gl^{++}}[S]$ is given by
$\la w^1, \ga^1, \bar r^1, Z^1 \ra \le
\la w^2, \ga^2, \bar r^2, Z^2 \ra$ iff the following hold.
\begin{enumerate}
\item $w^1 \subseteq w^2$.
\item $\ga^1 \le \ga^2$.
\item If $i \in w^1$, then $r^1_i
\subseteq r^2_i$.
\item $Z^1 \subseteq Z^2$.
\item If $z \in Z^1 \cap {[w^1]}^\gd$ and
$\ga^1 \le \ga < \ga^2$, then $|\{i \in z \mid
r^2_i(\ga) = 0\}| = |\{i \in z \mid r^2_i(\ga) = 1\}| = \gd$.
\end{enumerate}
The proof of \cite[Lemma 4]{AS97a} shows that
$\FP^0_{\gl, \gl^{++}} \ast (\FP^1_{\gl, \gl^{++}}[\dot S] \times
\FP^2_{\gl, \gl^{++}}[\dot S])$ is equivalent to
${\hbox{\rm Add}}(\gl^{++}, 1) \ast \dot {\hbox{\rm Add}}(\gl, \gl^{++})$.
The proofs of \cite[Lemmas 3 and 5]{AS97a}
and \cite[Lemma 6]{AS97b} show that
$\FP^0_{\gl, \gl^{++}} \ast \FP^1_{\gl, \gl^{++}}[\dot S]$
preserves cardinals
and cofinalities, is $\gl^{+++}$-c.c.,
is ${<}\gl$-strategically closed, and is such that \break
$V^{\FP^0_{\gl, \gl^{++}} \ast \FP^1_{\gl, \gl^{++}}[\dot S]} \models
``2^\gl = \gl^{++}$, $2^{\gl^+} = \gl^{++}$,
and $\gl$ is nonmeasurable''.
By the remarks in \cite[middle of page 108]{AS97a},
$\FP^0_{\gl, \gl^{++}} \ast \FP^1_{\gl, \gl^{++}}[\dot S]$
is $\gd$-directed closed.
Let $\FP^3$ be the reverse Easton iteration of length
$\gk + 1$ which forces with
$\FP^0_{\gl, \gl^{++}} \ast \FP^1_{\gl, \gl^{++}}[\dot S]$
whenever $\gl \in (\gd, \gk)$ is inaccessible,
forces with
$\FP^0_{\gk, \gk^{++}} \ast (\FP^1_{\gk, \gk^{++}}[\dot S] \times
\FP^2_{\gk, \gk^{++}}[\dot S])$ %is equivalent to
at stage $\gk$, and does trivial forcing otherwise.
By the facts mentioned in the preceding paragraph,
$\FP^3$ is $\gd$-directed closed.
The proof of \cite[Lemma 9]{AS97b} in conjunction with
the facts mentioned in the preceding paragraph show that
$\ov V^{\FP^3} \models ``\gk$ is the least measurable
cardinal greater than $\gd$ +
$2^\gk = 2^{\gk^+} = \gk^{++}$ +
$\gk$ is $\gk^+$
supercompact''. If we now define
$\FP(\gd, \gk) = \FP^1 \ast \dot \FP^2 \ast \dot \FP^3$,
then $\FP(\gd, \gk)$, which is
$\gd$-directed closed, is our desired partial ordering.
This completes the proof of Theorem \ref{t6}.
\end{proof}
We conclude Section \ref{s2} by observing that
the definitions of $\FP^1$ and $\FP^2$ given above may be
changed. All that is necessary is that enough
GCH is forced to allow the arguments of
\cite{AS97a} and \cite{AS97b} to be used to
establish that after forcing with $\FP^3$,
$\gk$ has become the least measurable cardinal
greater than $\gd$ and $\gk$ remains $\gk^+$ supercompact.
\section{The Proofs of Theorems \ref{t1} -- \ref{t5}}\label{s3}
We begin with the proof of Theorem \ref{t1},
after which the proof of Theorem \ref{t2} follows immediately.
\begin{proof}
We follow the proofs of \cite[Theorem 2]{A07}
and \cite[Theorem 1]{A10}.
Suppose that
%$\gl > \gk$ is $2^\gl$ supercompact and
$\gk$ is indestructibly supercompact and there
is a measurable cardinal $\gl > \gk$.
We show that
$A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is not $\gd^+$ supercompact$\}$
is unbounded in $\gk$.
Let $\eta > \gk$ be the least measurable cardinal.
Force with $\add(\eta^+, 1)$.
%to add a Cohen subset of $\eta^+$.
After this forcing,
%which is $\gk$-directed closed,
which is both $\gk$-directed closed
and $(\gk^+, \infty)$-distributive,
$2^\eta = \eta^+$ and
$\eta$ remains the least measurable
cardinal above $\gk$.
In particular, after the forcing,
$\eta$ is a measurable cardinal which
is not a limit of measurable cardinals,
so automatically, $\eta$ is not $2^\eta = \eta^+$
supercompact.
Since $\gk$'s supercompactness is suitably indestructible,
by reflection,
$A_1 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is not $\gd^+$ supercompact$\}$
is unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact $\add(\eta^+, 1)$ %adding a Cohen subset of $\eta^+$
is $\gk$-directed closed that
$A_1$ is unbounded in $\gk$ in the ground model.
This completes the proof of Theorem \ref{t1}.
\end{proof}
%Having completed the proof of Theorem \ref{t1},
%we turn now to the proof of Theorem \ref{t2}.
\begin{proof}
We argue in analogy to the proof of Theorem \ref{t1}.
Suppose that $\gk$ is indestructibly supercompact
and there is a cardinal $\gl > \gk$ which is $2^\gl$ supercompact.
To show that
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is $\gd^+$ supercompact$\}$
is unbounded in $\gk$,
force with $\FP(\gk^{++}, \gl)$.
By Theorem \ref{t6}, after this forcing,
which is both $\gk$-directed closed and
$(\gk^+, \infty)$-distributive,
$\gl$ has become the least measurable cardinal greater than
both $\gk$ and $\gk^{++}$, and $\gl$ is $\gl^+$ supercompact.
In particular, after this forcing,
$\gl$ is a measurable cardinal which is not a
limit of measurable cardinals.
We now argue as in the proof of Theorem \ref{t1}.
Since $\gk$'s supercompactness is suitably indestructible,
by reflection,
$A_2 = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and $\gd$ is $\gd^+$ supercompact$\}$
is unbounded in $\gk$ after the
forcing has been performed.
As before, we infer by the
fact $\FP(\gk^{++}, \gl)$
is $\gk$-directed closed that
$A_2$ is unbounded in $\gk$ in the ground model.
This completes the proof of Theorem \ref{t2}.
\end{proof}
%\begin{pf}
%The argument given in the first paragraph of
%the proof of Theorem \ref{t1} easily yields
%Theorem \ref{t2} as a corollary of Theorem \ref{t1}.
%\end{pf}
Having completed the proofs of Theorems \ref{t1}
and \ref{t2},
we turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is $2^\gd$ supercompact''.
Without loss of generality, by first
doing a preliminary forcing if necessary,
we assume in addition that $V \models {\rm GCH}$.
This is accomplished using a standard argument.
In particular, it
is possible to force GCH via the class reverse
Easton iteration $\la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga \in {\rm Ord} \ra$, where $\FP_0 = \add(\go, 1)$.
For any ordinal $\ga$, if
$\forces_{\FP_\ga} ``$There is a cardinal violating GCH'',
then
$\forces_{\FP_\ga} ``\dot \FQ_\ga = \dot \add(\gg^+, 1)$ where
$\gg$ is the least cardinal violating GCH''.
If this is not the case, i.e., if
$\forces_{\FP_\ga} ``$All cardinals satisfy GCH'', then
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ is trivial forcing''.
Since this is a closure point forcing in Hamkins'
sense of \cite{AH01}, by Hamkins' results of
\cite{H03}, no new $\gd$ which is $2^\gd$
supercompact (or indeed, no new cardinal which is
measurable) is created.
Hence, if
$V \models ``\gd$ is $\gd^+$ supercompact'', then
$V \models ``\gd$ is $2^\gd$ supercompact''.
This has as a consequence that if
$V \models ``\gd$ is $\gd^+$ supercompact'', then
$V \models ``\gd$ is a limit of measurable cardinals''.
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
%which is used to establish Theorem \ref{t2} is the
reverse Easton iteration of
length $\gk$ which begins by
forcing with $\add(\go, 1)$
%adding a Cohen subset of $\go$
and then (possibly) does nontrivial
forcing only at
cardinals $\gd < \gk$ which are both limits
of cardinals $\eta$ which are $2^\eta$ supercompact in $V$
and are at least $2^\gd$ supercompact in $V$.
%\footnote{Note
%that since $V \models {\rm GCH}$, any cardinal
%$\gd$ which is $\gd^+$ supercompact is automatically
%a limit of measurable cardinals. In the proofs of
%Theorems \ref{t4} and \ref{t5}, this won't necessarily
%be the case, i.e., there will be cardinals
%$\gd$ which are $\gd^+$ supercompact but which
%are not limits of measurable cardinals.
%This is why we explicitly state here that any
%nontrivial stage of forcing must occur at a limit of
%measurable cardinals.}
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 $\eta > \gd$ such that
$\eta$ is $2^\eta$ supercompact 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{l1}
$V^\FP \models ``\gk$ is
indestructibly supercompact''.
\end{lemma}
\begin{proof}
We follow the proofs of \cite[Lemma 2.1]{A07}
and \cite[Lemma 2.1]{A10}.
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 $\gl \ge
\max(\gk^{+}, \card{{\rm TC}(\dot \FQ)})$
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 $\gd$ above
$\gk$ is $2^\gd$ supercompact'', $\gg \ge
2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
$M \models ``\gk$ is both $2^\gk = \gk^+$ supercompact
and a limit of cardinals $\eta$ which are $2^\eta$ supercompact,
and no cardinal $\gd$ in the %half-open
interval $(\gk, \gg]$ is $2^\gd$ supercompact''. 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{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $V \models ``\gd < \gk$ is a
$2^\gd$ supercompact cardinal
which is not a limit of
cardinals $\eta$ which are $2^\eta$ supercompact'', then
%$V^\FP \models ``2^\gd = \gd^+$ + $\gd$
$V^\FP \models ``\gd$ is $2^\gd$ supercompact''.
%is a $2^\gd$ supercompact cardinal''.
%which is not a
%limit of cardinals $\eta$ which are $2^\eta$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$V \models ``\gd < \gk$ is a
$2^\gd$ supercompact cardinal which is not a limit of
cardinals $\eta$ which are $2^\eta$ supercompact''. Write
$\FP = \FP_\gd \ast \dot \FP^\gd$.
By the definition of $\FP$,
$\card{\FP_\gd} < \gd$
and $\forces_{\FP_\gd} ``\dot \FP^\gd$ is (at least)
%$(2^\gd)^+$
$\beth_\go(\gd)$-directed closed''.
%\footnote{More precisely,
%$\FP_\gd$ is forcing equivalent to a partial
%ordering having size less than $\gd$.}
Therefore, %standard arguments in conjunction with
the L\'evy-Solovay results \cite{LS} show that
%$V^{\FP_\gd} \models ``2^\gd = \gd^+$ +
$V^{\FP_\gd} \models ``\gd$ is $2^\gd$ supercompact'', so
%is a $2^\gd$ supercompact cardinal'', so
%which is not a limit of
%cardinals $\eta$ which are $2^\eta$ supercompact'', so
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models
``\gd$ is $2^\gd$ supercompact''.
%``2^\gd = \gd^+$ +
%``\gd$ is a $2^\gd$ supercompact cardinal''.
%which is not a limit of
%cardinals $\eta$ which are $2^\eta$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$If $\gd < \gk$ is a measurable cardinal
which is not a limit of measurable cardinals, then
$\gd$ is not $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
We prove the contrapositive.
Suppose that $V^\FP \models ``\gd < \gk$ is $\gd^+$ supercompact''.
Write $\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_2$-directed closed''.
%By Theorem \ref{tgf},
%$V \models ``\gd$ is $2^\gd$ supercompact''.
%By the factorization of $\FP$ given in Lemma \ref{l3} and
By Theorem \ref{tgf}, $V \models ``\gd$ is $(\gd^+)^{V^\FP}$
supercompact''. Consequently, since
$(\gd^+)^{V^\FP} \ge (\gd^+)^V$,
$V \models ``\gd$ is $\gd^+ = 2^\gd$ supercompact''.
This allows us now to consider the following two cases.
\bigskip\noindent Case 1: $V \models ``\gd$ is not
a limit of cardinals $\eta$ which are $2^\eta$ supercompact''.
In this case, by Lemma \ref{l2}, $V^\FP \models ``\gd$
is $2^\gd$ supercompact''. From this, we immediately
infer that $V^\FP \models ``\gd$ is a limit of
measurable cardinals''.
\bigskip\noindent Case 2: $V \models ``\gd$ is
a limit of cardinals $\eta$ which are $2^\eta$ supercompact''.
In particular,
$V \models ``\gd$ is a
%$2^\gd$ supercompact cardinal which is a
limit of cardinals $\eta$ which are $2^\eta$ supercompact
such that each $\eta$ is not a limit of cardinals
$\gg$ which are $2^\gg$ supercompact''.
%which are themselves not limits of measurable cardinals''.
By Lemma \ref{l2},
such $\eta$ are preserved to $V^{\FP}$, i.e.,
$V^\FP \models ``\gd$ is
%a $2^\gd$ supercompact cardinal which is
a limit of
cardinals $\eta$ which are $2^\eta$ supercompact''.
In other words,
$V^\FP \models ``\gd$ is a limit of measurable cardinals''.
\bigskip Cases 1 and 2 complete the proof of Lemma \ref{l3}.
\end{proof}
Since trivial forcing is $\gk$-directed closed,
Lemma \ref{l1} implies that
$V^\FP \models ``\gk$ is supercompact''.
Also, because $\FP$ may be defined so that
$\card{\FP} = \gk$, the arguments of \cite{LS} show that
%standard arguments in tandem with the results of \cite{LS} show that
$V^\FP \models
``$No cardinal $\gd > \gk$ is $2^\gd$ supercompact''.
These remarks, together with Lemmas \ref{l1} -- \ref{l3},
complete the proof of Theorem \ref{t3}.
\end{proof}
Having completed the proof of Theorem \ref{t3},
we turn now to the proof of Theorem \ref{t4}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
Without loss of generality, by first forcing
GCH and then doing the forcing of
\cite[Theorem 1]{A01}, we may assume in addition that
$V \models ``$If $\gd \le \gk$ is measurable, then
$2^\gd = 2^{\gd^+} = \gd^{++}$ and $\gd$ is $\gd^+$
supercompact''.
(The aforementioned property of $V$ may be
assumed to hold
because as we have already
observed, forcing GCH will not create any new
measurable cardinals. Since the forcing of
\cite[Theorem 1]{A01} may be defined so as
to have size $\gk$, by the results of
\cite{LS}, it will not create any new measurable
cardinals greater than $\gk$.)
%, and there is a $\gd^+$ supercompactness
%embedding $j : V \to M$ generated by a
%supercompact ultrafilter over
%$P_\gd(\gd^+)$ such that
%$M \models ``\gd$ is nonmeasurable''.
We then define $\FP$ as in the
proof of Theorem \ref{t3},
except that at each nontrivial stage of forcing
$\gd < \gk$
(so in particular,
$\FP$ (possibly) does nontrivial forcing only at
cardinals $\gd < \gk$ which are both $2^\gd$
supercompact in $V$ and are limits of cardinals
$\eta$ which are $2^\eta$ supercompact in $V$),
we require that for our Laver function $f$,
$f(\gd) = \dot \FQ$ and
$\forces_{\FP_\gd} ``\dot \FQ$ is a $\gd$-directed closed
partial ordering
having rank below the least $V$-measurable cardinal
greater than $\gd$''.
The same arguments as used in the proof of
Theorem \ref{t3}, replacing both instances
in the proof of Lemma \ref{l1} of
$\gd$ not being $2^\gd$ supercompact with
$\gd$ not being measurable, will now show that
$V^\FP \models ``\gk$ is indestructibly supercompact +
No cardinal $\gd > \gk$ is measurable''.
The proof of Theorem \ref{t4} will therefore
be complete once we have established the following lemma.
\begin{lemma}\label{l4}
$V^\FP \models ``$If $\gd < \gk$ is a measurable
cardinal which is not a limit of measurable cardinals, then
$\gd$ is $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that $V^\FP \models ``\gd < \gk$ is a
measurable cardinal which is not a limit of
measurable cardinals''.
By the factorization of $\FP$ given in the
proof of Lemma \ref{l3} and Theorem \ref{tgf},
$V \models ``\gd$ is a measurable cardinal''.
If $V \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals'', then
in particular,
$V \models ``\gd$ is a measurable cardinal which
is a limit of measurable cardinals which are
not themselves limits of measurable cardinals''.
Observe now that essentially the same argument as
given in the proof of Lemma \ref{l2} remains valid
and shows that if
$V \models ``\eta < \gk$ is a measurable cardinal
which is not a limit of measurable cardinals'', then
$V^\FP \models ``\eta$ is $\eta^+$ supercompact''.
Thus, $V^\FP \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals'',
a contradiction.
Hence, if
$V^\FP \models ``\gd < \gk$ is a measurable
cardinal which is not a limit of measurable cardinals'', then
$V \models ``\gd$ is a measurable cardinal which is
not a limit of measurable cardinals''.
From this,
we infer as earlier in the proof of this lemma that
$V^\FP \models ``\gd$ is $\gd^+$ supercompact''.
This completes the proof of both Lemma \ref{l4}
and Theorem \ref{t4}.
\end{proof}
\end{proof}
We finish our proofs with
the proof of Theorem \ref{t5}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is measurable''.
As in the proof of Theorem \ref{t4},
%by first forcing
%GCH and then doing the forcing of
%\cite[Theorem 1]{A01}, we may assume in addition that
we assume in addition that
$V \models ``$If $\gd \le \gk$ is measurable, then
$2^\gd = 2^{\gd^+} = \gd^{++}$ and $\gd$ is $\gd^+$
supercompact''
%, and there is a $\gd^+$ supercompactness
%embedding $j : V \to M$ generated by a
%supercompact ultrafilter over $P_\gd(\gd^+)$ such that
%$M \models ``\gd$ is nonmeasurable''.
We then define $\FP$ as in the proof of Theorem \ref{t4},
except that at each nontrivial stage of forcing
$\gd < \gk$, we require that for our Laver function $f$,
$f(\gd) = \dot \FQ$ and
$\forces_{\FP_\gd} ``\dot \FQ$ is a $\gd$-directed closed,
$(\gd^+, \infty)$-distributive partial ordering
having rank below the least $V$-measurable cardinal
greater than $\gd$''.
The same arguments as used in the proofs of
Theorems \ref{t3} and \ref{t4} will now show that
$V^\FP \models ``\gk$ is a supercompact cardinal whose
supercompactness is indestructible under partial
orderings which are both $\gk$-directed closed and
$(\gk^+, \infty)$-distributive +
No cardinal $\gd > \gk$ is measurable''.
The proof of Theorem \ref{t5} will therefore be
complete once we have established the following lemma.
\begin{lemma}\label{l5}
$V^\FP \models ``$If $\gd < \gk$ is a measurable cardinal,
then $\gd$ is (at least) $\gd^+$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$V^\FP \models ``\gd < \gk$ is a measurable
cardinal''. As in the proof of Lemma \ref{l4},
%write $\FP = \FP' \ast \dot \FP''$, where
%$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
%$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
%Once again, by Theorem \ref{tgf},
$V \models ``\gd$ is a measurable cardinal''.
This allows us to write
$\FP = \FP_\gd \ast \dot \FP^\gd$ and consider
the following two cases.
\bigskip\noindent Case 1: $\card{\FP_\gd} < \gd$. In this case,
by the results of \cite{LS}, because
$V \models ``\gd$ is (at least) $\gd^+$ supercompact'',
$V^{\FP_\gd} \models ``\gd$ is (at least)
$\gd^+$ supercompact''. Since
by the definition of $\FP$,
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is both $\gd$-directed closed and
$(\gd^+, \infty)$-distributive'',
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models ``\gd$
is (at least) $\gd^+$ supercompact''.
\bigskip\noindent Case 2: $\card{\FP_\gd} = \gd$. In this case,
by our assumptions on $V$, let
$j : V \to M$ be an elementary embedding witnessing the
$\gd^+$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gd^+)$ such that
$M^{\gd^+} \subseteq M$
and $M \models ``\gd$ is not $\gd^+$ supercompact''.
We then have that
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$, where the first
ordinal at which $\dot \FQ$ is forced to do nontrivial
forcing is well beyond $\gd^+$.
We may then use a simplified version
of the standard lifting argument
given in the proof of Theorem \ref{t6}
%(as given, e.g., in the proof of \cite[Lemma 2.2]{A10})
to show that $j$ lifts in $V^{\FP_\gd}$ to
$j : V^{\FP_\gd} \to M^{j(\FP_\gd)}$.
For completeness, we give the details.
Let $G$ be $V$-generic over $\FP_\gd$. Working in $V[G]$,
we first note that since $\FP_\gd$ is $\gd$-c.c.,
$M[G]$ remains $\gd^+$ closed with respect to $V[G]$.
This means that $\FQ$ is $\gd^{++}$-directed closed
in both $M[G]$ and $V[G]$.
As before, because $M[G] \models ``\card{\FQ} = j(\gd)$'',
the number of dense open subsets of $\FQ$ present in
$M[G]$ is $(2^{j(\gd)})^M$.
Since $V \models ``2^\gd = 2^{\gd^+} = \gd^{++}$'' and $M$
is given via an ultrapower by a supercompact
ultrafilter over $P_\gd(\gd^+)$, this is calculated as
$\card{\{f \mid f : [\gd^+]^{< \gd} \to 2^\gd\}} =
\card{\{f \mid f : \gd^+ \to \gd^{++}\}} = \gd^{++}$.
We may therefore let
$\la D_\ga \mid \ga < \gd^{++} \ra \in V[G]$
enumerate the dense open subsets of $\FQ$ present in $M[G]$.
We may now use the fact that $\FQ$ is
$\gd^{++}$-directed closed in $V[G]$ to meet each $D_\ga$
and thereby construct in $V[G]$ an $M[G]$-generic
object $H$ over $\FQ$. Our construction guarantees that
$j '' G \subseteq G \ast H$, so $j$ lifts in $V[G]$ to
$j : V[G] \to M[G][H]$.
Hence, $V^{\FP_\gd} \models ``\gd$ is (at least)
$\gd^+$ supercompact''. As in Case 1 above, since
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is both $\gd$-directed closed and
$(\gd^+, \infty)$-distributive'',
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models ``\gd$
is (at least) $\gd^+$ supercompact''.
This completes the proof of both Lemma \ref{l5} and Theorem \ref{t5}.
\end{proof}
\end{proof}
%\section{Concluding Remarks}\label{s4}
%We finish with two remarks. First, as we have
%already indicated, although there exists a
%balance between Theorems \ref{t2} and \ref{t4} and
%Theorems \ref{t2} and \ref{t5}, this same
%degree of harmony between Theorems \ref{t1} and \ref{t3}
%is not present.
We conclude by remarking that
other than the fact that the proof of Lemma \ref{l5}
requires $(\gd^+, \infty)$-distributivity
at each nontrivial stage of forcing $\gd$
in the definition of $\FP$, there is no
reason prima facie to believe that this
restriction must be present.
We therefore end by asking if
the proof of Theorem \ref{t5} can be reworked so that
$V^\FP \models ``\gk$ is indestructibly supercompact''.
\begin{thebibliography}{99}
\bibitem{A07} A.~Apter, ``Indestructibility and
Level by Level Equivalence and Inequivalence'',
{\it Mathematical Logic Quarterly 53}, 2007, 78--85.
\bibitem{A08} A.~Apter, ``Indestructibility and
Measurable Cardinals with Few and Many Measures'',
{\it Archive for Mathematical Logic 47}, 2008, 101--110.
\bibitem{A09} A.~Apter, ``Indestructibility and
Stationary Reflection'',
{\it Mathematical Logic Quarterly 55}, 2009, 228--236.
\bibitem{A11} A.~Apter, ``Indestructibility, HOD,
and the Ground Axiom'',
{\it Mathematical Logic Quarterly 57}, 2011, 261--265.
\bibitem{A10} A.~Apter, ``Indestructibility,
Instances of Strong Compactness, and
Level by Level Inequivalence'',
{\it Archive for Mathematical Logic 49}, 2010, 725--741.
\bibitem{A00} A.~Apter, ``On a Problem of Woodin'',
{\it Archive for Mathematical Logic 39}, 2000, 253--259.
%\bibitem{A02} A.~Apter, ``On Level by Level Equivalence
%and Inequivalence between Strong Compactness and
%Supercompactness'', {\it Fundamenta Mathematicae 171},
%2002, 77--92.
\bibitem{A01} A.~Apter, ``Some Structural Results
Concerning Supercompact Cardinals'',
{\it Journal of Symbolic Logic 66}, 2001, 1919--1927.
%\bibitem{A01} A.~Apter,
%``Strong Compactness, Measurability,
%and the Class of Supercompact Cardinals'',
%{\it Fundamenta Mathematicae 167}, 2001, 65--78.
%\bibitem{A97} A.~Apter, ``Patterns of
%Compact Cardinals'', {\it Annals of
%Pure and Applied Logic 89}, 1997, 101--115.
%\bibitem{A07} A.~Apter, ``Supercompactness
%and Level by Level Equivalence are
%Compatible with Indestructibility
%for Strong Compactness'', to appear
%in the {\it Archive for Mathematical Logic}.
%\bibitem{A01} A.~Apter, ``Supercompactness
%and Measurable Limits of Strong Cardinals'',
%{\it Journal of Symbolic Logic 66}, 2001,
%629--639.
%\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness'',
%{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
%\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness II: Strong Cardinals'',
%{\it Archive for Mathematical Logic 40}, 2001, 25--38.
%\bibitem{AG} A.~Apter, M.~Gitik,
%``The Least Measurable can be Strongly
%Compact and Indestructible'',
%{\it Journal of Symbolic Logic 63},
%1998, 1404--1412.
\bibitem{AH4} A.~Apter, J.~D.~Hamkins,
``Indestructibility and the Level-by-Level Agreement
between Strong Compactness and Supercompactness'',
{\it Journal of Symbolic Logic 67}, 2002, 820--840.
\bibitem{AH01} A.~Apter, J.~D.~Hamkins,
``Indestructible Weakly Compact Cardinals and the
Necessity of Supercompactness for Certain
Proof Schemata'', {\it Mathematical Logic Quarterly 47},
2001, 563--571.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong
Equality between Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349},
1997, 103--128.
\bibitem{AS97b} A.~Apter, S.~Shelah, ``Menas' Result is
Best Possible'',
{\it Transactions of the American Mathematical Society 349},
1997, 2007--2034.
%\bibitem{C} J.~Cummings, ``A Model in which GCH Holds
%at Successors but Fails at Limits'', {\it Transactions
%of the American Mathematical Society 329}, 1992, 1--39.
%\bibitem{F} M.~Foreman, ``More Saturated Ideals'', in:
%{\it Cabal Seminar 79-81}, {\bf Lecture Notes in
%Mathematics 1019}, Springer-Verlag, Berlin and
%New York, 1983, 1--27.
%\bibitem{G} M.~Gitik, ``Changing Cofinalities
%and the Nonstationary Ideal'',
%{\it Israel Journal of Mathematics 56},
%1986, 280--314.
%\bibitem{GS} M.~Gitik, S.~Shelah,
%``On Certain Indestructibility of
%Strong Cardinals and a Question of
%Hajnal'', {\it Archive for Mathematical
%Logic 28}, 1989, 35--42.
%\bibitem{H1} J.~D.~Hamkins,
%``Destruction or Preservation As You
%Like It'',
%{\it Annals of Pure and Applied Logic 91},
%1998, 191--229.
\bibitem{H03} J.~D.~Hamkins, ``Extensions with the
Approximation and Cover Properties have No New
Large Cardinals'', {\it Fundamenta Mathematicae 180},
2003, 257--277.
\bibitem{H2} J.~D.~Hamkins, ``Gap Forcing'',
{\it Israel Journal of Mathematics 125}, 2001, 237--252.
\bibitem{H3} J.~D.~Hamkins, ``Gap Forcing:
Generalizing the L\'evy-Solovay Theorem'',
{\it Bulletin of Symbolic Logic 5}, 1999, 264--272.
%\bibitem{H} J.~D.~Hamkins, {\it Lifting and
%Extending Measures; Fragile Measurability},
%Doctoral Dissertation, University of California,
%Berkeley, 1994.
%\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
%{\it Annals of Pure and Applied Logic 101},
%2000, 103--146.
%\bibitem{H5} J.~D.~Hamkins, ``Small Forcing Makes
%Any Cardinal Superdestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 51--58.
\bibitem{J} T.~Jech, {\it Set Theory:
The Third Millennium Edition,
Revised and Expanded}, Springer-Verlag,
Berlin and New York, 2003.
%\bibitem{K} A.~Kanamori, {\it The
%Higher Infinite}, Springer-Verlag,
%Berlin and New York, 1994.
%\bibitem{Ke} J.~Ketonen, ``Strong Compactness and
%Other Cardinal Sins'', {\it Annals of Mathematical
%Logic 5}, 1972, 47--76.
%\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
%between the Concepts of Compactness and Supercompactness'',
%circulated manuscript.
\bibitem{L} R.~Laver, ``Making the
Supercompactness of $\gk$ Indestructible
under $\gk$-Directed Closed Forcing'',
{\it Israel Journal of Mathematics 29},
1978, 385--388.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
%\bibitem{Ma} M.~Magidor, ``How Large is the First
%Strongly Compact Cardinal?'', {\it Annals of
%Mathematical Logic 10}, 1976, 33--57.
%\bibitem{Me} T.~Menas, ``On Strong Compactness and
%Supercompactness'', {\it Annals of Mathematical Logic 7},
%1974, 327--359.
%\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
%``Strong Axioms of Infinity and Elementary Embeddings'',
%{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}