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\title{Aspects of Strong Compactness,
Measurability, and Indestructibility
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Measurable cardinal,
strongly compact cardinal, supercompact
cardinal, indestructibility}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\}
\date{December 12, 1999\\
(revised September 17, 2000)}
\begin{document}
\maketitle
\begin{abstract}
We prove three theorems concerning Laver indestructibility,
strong compactness, and measurability.
We then state some related open questions.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
Since its introduction in the 1970s, Laver's notion
of forcing \cite{L} for making a supercompact
cardinal $\gk$ indestructible under
$\gk$-directed closed forcing has proven to be
one of the most important and useful techniques
in the theory of large cardinals and forcing.
Its applications are numerous, beginning with
Magidor's construction \cite{Ma} of a model
in which GCH holds below $\ha_\omega$ yet
$2^{\ha_\omega} = \ha_{\omega + 2}$, and
continuing with many more applications.
In spite of its importance as a tool when
dealing with supercompactness, until
recently, the connections between strong
compactness and indestructibility have not
been studied. Lately, however, there has
been research carried out on how much
indestructibility a non-supercompact
strongly compact cardinal can satisfy.
The work done has followed two lines of
inquiry, namely using supercompactness as
a tool, and seeing what can be proven for
an arbitrary strongly compact cardinal.
The results of \cite{A98}, \cite{AG},
and \cite{AH} follow the first of the
above paths. In \cite{A98}, it is
shown that any and every measurable
limit $\gk$ of supercompact cardinals
(which by a result of Menas \cite{Me}
must be strongly compact) can be forced
to have its strong
compactness indestructible under any
$\gk$-directed closed partial ordering
not destroying $\gk$'s measurability.
Since the work of \cite{Me} shows that if
$\ga < \gk$ and $\gk$ is the
$\ga^{\rm th}$ measurable limit of
strongly compact or supercompact cardinals, then
$\gk$ isn't $2^\gk$ supercompact,
this provides an example of a weak form of
indestructibility for certain
non-supercompact strongly compact cardinals.
Full indestructibility for a non-supercompact
strongly compact cardinal was provided in
\cite{AG} and \cite{AH}. In \cite{AG}, the
consistency of the least measurable cardinal
being both the least strongly compact cardinal
and fully indestructible, relative to a
supercompact cardinal, was shown. In \cite{AH},
Hamkins showed the consistency, relative to a
supercompact limit of supercompact cardinals,
of the least measurable limit of supercompact
cardinals having its strong
compactness be fully indestructible.
His techniques easily generalize to give a model,
relative to a sequence
$\la \gk_\gb : \gb \le \ga \ra$ in which
$\ga < \gk_0$ and each $\gk_\gb$ is a
supercompact limit of supercompact cardinals,
where for every $\gb \le \ga$,
$\gk_\gb$ is the $\gb^{\rm th}$ measurable
limit of supercompact cardinals and $\gk_\gb$'s
strong compactness
is fully indestructible. This improves the
result of the preceding paragraph,
albeit from stronger hypotheses.
Hamkins has done work following the second of
the above paths. Using a technique he calls
the lottery preparation, Hamkins has shown
that an arbitrary strongly compact cardinal
$\gk$ can be made simultaneously indestructible
under the partial orderings which add a Cohen
subset to $\gk$, which add a ``long Prikry
sequence'' to $\gk$, and which add certain
types of clubs to $\gk$. The relevant
details and definitions can be found in
\cite{H4}.
The purpose of this paper is to pursue
further what sorts of indestructibility
properties can be forced for sequences
of measurable cardinals and non-supercompact
strongly compact cardinals. Specifically,
we prove the following three theorems.
\begin{theorem}\label{t1}
Let $V \models ``$ZFC + GCH +
$\gk_1 < \gk_2 < \gl$ are so that
$\gl$ is the least inaccessible above
$\gk_2$ and $\gk_1$ and $\gk_2$ are
both $\gl$ supercompact''.
There is then a partial ordering
$\FP \in V_\gl$, $|\FP| = \gk_2$ so that
$V^\FP_\gl \models ``$ZFC + $\gk_1$ and
$\gk_2$ are both the first two strongly
compact and first two measurable cardinals +
$\gk_1$'s strong compactness is fully
indestructible under $\gk_1$-directed closed
forcing + $\gk_2$'s measurability is fully
indestructible under $\gk_2$-directed
closed forcing''.
\end{theorem}
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + GCH + There is a (proper)
class of supercompact cardinals''.
There is then a class partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC + There is a proper
class of measurable cardinals + Every
measurable cardinal $\gk$ has its
measurability fully indestructible
under $\gk$-directed closed forcing''.
\end{theorem}
\begin{theorem}\label{t3}
Let $V \models ``$ZFC + GCH + $\gk$ is
almost huge''. There is then a cardinal
$\gl > \gk$ and a partial ordering
$\FP \in V_\gl$, $|\FP| = \gk$ so that
$V^\FP_\gl \models ``$ZFC + $\gk$ is a supercompact
limit of supercompact cardinals +
The strongly compact and supercompact cardinals
coincide except at measurable limit points +
Every supercompact cardinal $\gd$ is
fully indestructible under $\gd$-directed
closed forcing + Every non-supercompact
strongly compact cardinal $\gd$ has both its strong
compactness and its degree of
supercompactness fully indestructible
under $\gd$-directed closed forcing''.
\end{theorem}
We take this opportunity to make a few
explanatory remarks about the above
three theorems. Although the forcing
of \cite{L} is easily iterable
(see, e.g., \cite{A98}, \cite{A83}, or
\cite{CFM}), the forcing of \cite{AG}
can't be iterated above any strongly
compact cardinal so as to preserve
strong compactness. Thus, the
techniques of \cite{AG} won't by
themselves yield a model in which the first two
strongly compact cardinals are both the
first two measurable cardinals and satisfy
some sort of indestructibility properties.
New ideas are necessary in order to prove
Theorem \ref{t1}. These new ideas do provide
partial orderings which are to a certain
extent iterable, i.e., can be iterated to
preserve indestructibility if not strong
compactness, thereby allowing Theorem \ref{t2}
to be proven. Finally, in \cite{A98},
relative to a supercompact limit of supercompact
cardinals, a model for most of the conclusions
of Theorem \ref{t3} was constructed, except
that one could only infer the aforementioned
weak form of indestructibility for non-supercompact
measurable limits of compact cardinals.
Theorem \ref{t3} shows that from a much stronger
hypothesis than in \cite{A98}, full
indestructibility can be obtained
both for strong compactness as well as
whatever degree of supercompactness such
measurable limits exhibit.
In fact, in each of Theorems \ref{t1},
\ref{t2}, and \ref{t3}, not only will we
have the relevant indestructibility properties
under forcing with set partial orderings
(which is what we will show in Lemmas
\ref{l2}, \ref{l5}, and \ref{l9}),
but as in \cite{L} and \cite{A98},
we will have these same indestructibility
properties under forcing with the appropriately
defined Easton support class iterations and products.
The structure of this paper is as follows.
Section \ref{s0} contains our introductory
comments and preliminary remarks concerning
notation, terminology, etc.
Sections \ref{s1} - \ref{s3} contain proofs
of Theorems \ref{t1} - \ref{t3} respectively.
Section \ref{s4} contains our concluding remarks.
Before giving the proofs of Theorems
\ref{t1} - \ref{t3}, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb], [\ga, \gb), (\ga, \gb]$, and $(\ga,
\gb)$ are as
in standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If we also have that
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an Easton support iteration of length $\gk$
so that at stage $\ga$, a non-trivial forcing is done
based on the ordinal $\gd_\ga$, then we will say that
$\gd_\ga$ is in the field of $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\gk$ is a regular cardinal and $\FP$ is
a partial ordering, $\FP$ is $\gk$-closed if for every sequence
$\langle p_\ga: \ga < \gk \rangle$ of elements of $\FP$ so that
$\beta < \gamma < \gk$ implies $p_\beta \le p_\gamma$ (an increasing
chain of length
$\gk$), there is some $p \in \FP$ (an upper bound to this chain)
so that $p_\ga \le p$ for all $\ga < \gk$.
$\FP$ is $<\gk$-closed if $\FP$ is $\delta$-closed for all cardinals
$\gd < \gk$.
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$. $\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$ \FP$ is $< \gk$-strategically closed if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
$\FP$ is $\prec \gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
Note that trivially, if $\FP$ is $<\gk$-closed, then $\FP$ is
$< \gk$-strategically
closed and $\prec \gk $-strategically closed. The converse of
both of these facts is false.
Suppose that
$\gk < \gl$ are regular cardinals.
A partial ordering $\FP$ that will be used
throughout the course of this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP$ is
defined as
$\{ p$ : For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\gb \in S_p$ implies
$\gb > \gk$ and cof$(\gb) = \gk \}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP$ (see
\cite{Bu} or \cite{KiMa}), in $V[G]$,
if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\cup\{S_p:p\in G \} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced, the
bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH
have been preserved.
It is also virtually immediate that $\FP$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu} or \cite{KiMa}) that
$\FP$
is $\prec \gl$-strategically closed.
We mention that we are assuming familiarity with the
large cardinal notions of measurability,
strong compactness, supercompactness,
and almost hugeness.
Interested readers may consult \cite{Ka}
or \cite{SRK} for further details.
We note only that when we say $\gk$ is
$< \gl$ supercompact, we mean $\gk$ is
$\gd$ supercompact for all cardinals
$\gd < \gl$, and when we say $\gk$ is
$< \gl$ strongly compact, we mean $\gk$ is
$\gd$ strongly compact for all cardinals
$\gd < \gl$.
When discussing indestructibility, we will
say that a measurable, strongly compact,
or supercompact cardinal $\gk$ is fully
indestructible if after forcing with an
arbitrary $\gk$-directed closed partial
ordering $\FP$, $\gk$ retains its
measurability, strong compactness, or
supercompactness.
If additional restrictions are placed on
$\FP$, then we will say that $\gk$ is
weakly indestructible.
If $\gk$ exhibits some degree of strong
compactness or supercompactness yet
isn't necessarily fully strongly
compact or supercompact, and if
forcing with an arbitrary $\gk$-directed
closed partial ordering $\FP$ preserves
the degree of strong compactness or
supercompactness $\gk$ exhibits, then
$\gk$'s degree of strong compactness or
supercompactness will be said to be
fully indestructible.
Finally, suppose $j : V \to M$ is
an elementary embedding with critical
point $\gk$ so that $M$ is (at least)
$\gk$ closed with respect to $V$.
Suppose also that $\FP =
\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$ is an Easton support
iteration of length $\gk$ defined so that
each $\dot \FQ_\ga$ is chosen to be
an element of $V_\gk$.
By assuming that $\le_{\cal W}$ is
a well-ordering of $V_\gk$ so that
$j(\le_{\cal W}) \rest V_\gk =
\le_{\cal W}$,
we will also be able to assume that
$j(\FP) = \FP \ast \dot \FQ$ for
some term $\dot \FQ \in M$.
This will characterize all of the
Easton support iterations used
throughout the course of this paper.
\section{The Proof of Theorem \ref{t1}}\label{s1}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk_1 < \gk_2 < \gl$
are so that $\gl$ is the least inaccessible above
$\gk_2$ and $\gk_1$ and $\gk_2$ are both
$\gl$ supercompact''. Working in
$V_\gl$, let $\FP^0$ be the partial ordering
from \cite{AG} so that
$V^{\FP^0}_\gl \models ``\gk_1$ is both the
least strongly compact and least measurable
cardinal + $\gk_1$'s strong compactness is
fully indestructible''. We then have that
$V^{\FP^0} \models ``\gk_1$ is $< \gl$
strongly compact + $\gk_1$ is the least
measurable cardinal + The $< \gl$
strong compactness of $\gk_1$ is indestructible
when forcing with $\gk_1$-directed closed partial
orderings of rank $< \gl$''. Since
$|\FP^0| = \gk_1$, the L\'evy-Solovay results
\cite{LS} also tell us that
$V^{\FP^0} \models ``\gk_2$ is $\gl$ supercompact +
$\gl$ is the least inaccessible above $\gk_2$''.
In addition, by the usual cardinality arguments,
$V^{\FP^0} \models ``$For every cardinal
$\gd \ge \gk_1$, $2^\gd = \gd^+$''.
Take $V_1 = V^{\FP^0}$. Let
$\la \gd_\ga : \ga < \gk_2 \ra$ enumerate the
measurable cardinals in the interval
$(\gk_1, \gk_2)$. We define
in $V_1$ an Easton support
iteration
$\FP^1 = \la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk_2 \ra$ of length $\gk_2$ as follows:
\begin{enumerate}
\item\label{c1} $\FP_0 = \{\emptyset\}$.
\item\label{c2} If $\forces_{\FP_\ga}
``$There is a $\gd_\ga$-directed closed
partial ordering of rank below the
least inaccessible above $\gd_\ga$ so
that forcing with it destroys $\gd_\ga$'s
measurability'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for such a
partial ordering.
\item\label{c3} If case \ref{c2} above
doesn't hold (which will mean that
$\forces_{\FP_\ga} ``$The measurability of
$\gd_\ga$ is indestructible under forcing
with $\gd_\ga$-directed closed partial
orderings of rank below the least
inaccessible above $\gd_\ga$''), then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ is a term for
the partial ordering which adds a non-reflecting
stationary set of ordinals of cofinality
$\gk_1$ to $\gd_\ga$''.
\end{enumerate}
\setlength{\parindent}{0in}
Note that by its definition,
$|\FP^1| = \gk_2$.
\setlength{\parindent}{1.5em}
\begin{lemma}\label{l1}
$V^{\FP^1}_1 \models ``$There are no
measurable cardinals in the interval
$(\gk_1, \gk_2)$ + $\gk_1$ is the
least measurable cardinal + $\gk_1$
is a $< \gl$ strongly compact cardinal
whose $< \gl$ strong compactness is
indestructible under forcing with
$\gk_1$-directed closed partial orderings
of rank $< \gl$''.
\end{lemma}
\begin{proof}
The definition of $\FP^1$ ensures that it is
a $\gk_1$-directed closed partial ordering of
rank $< \gl$. Thus,
$V^{\FP^1}_1 \models ``\gk_1$ is the least
measurable cardinal + $\gk_1$ is a $< \gl$
strongly compact cardinal whose $< \gl$
strong compactness is indestructible under
forcing with $\gk_1$-directed closed partial
orderings of rank $< \gl$''. Further, by
its definition, we know that for any
$\ga < \gk_2$,
$V^{\FP_{\ga + 1}}_1 \models ``\gd_\ga$
isn't measurable''. If we write
$\FP^1 = \FP_{\ga + 1} \ast \dot \FQ^{\ga + 1}$, then since
$\forces_{\FP_{\ga + 1}} ``\dot \FQ^{\ga + 1}$ is
$< \gd_{\ga + 1}$-strategically closed'',
$V^{\FP_{\ga + 1} \ast \dot \FQ^{\ga + 1}}_1 =
V^{\FP^1}_1 \models ``\gd_\ga$ isn't measurable''. Thus,
$V^{\FP^1}_1 \models ``$No $V_1$-measurable cardinal
$\gd \in (\gk_1, \gk_2)$ is measurable''.
This means the proof of Lemma \ref{l1}
will be complete once we have shown that forcing with
$\FP^1$ creates no new measurable cardinals in
the interval $(\gk_1, \gk_2)$.
To see this last fact, observe that as above, we can write
$\FP^1 = \FP_1 \ast \dot \FQ^1$. Let $\gg_0$ be the
least inaccessible above $\gd_0$.
It is then the case that
$|\FP_1| < \gg_0$ and
$\forces_{\FP_1} ``\dot \FQ^1$ is $\gg_0$-strategically
closed''. Thus, in Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP^1$ is a `` gap forcing admitting a gap at
$\gg_0$'', so the results of
\cite{H1}, \cite{H2}, and \cite{H3} tell us that
forcing with $\FP^1$ creates no new measurable
cardinals in the interval
$[\gg_0, \gk_2)$. Since $\FP^1$ is
$< \gd_0$-strategically closed, forcing with
$\FP^1$ creates no new measurable cardinals below
$\gd_0$. Hence, as $\gd_0$ is the least
$V_1$-measurable cardinal in the interval
$(\gk_1, \gk_2)$, the work of the preceding
paragraph combined with what was just done
shows that
$V^{\FP^1}_1 \models ``$No cardinal
$\gd \in (\gk_1, \gk_2)$ is measurable''.
This proves Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^{\FP^1}_1 \models ``\gk_2$ is a measurable
cardinal whose measurability is indestructible
under forcing with $\gk_2$-directed closed
partial orderings of rank $< \gl$''.
\end{lemma}
\begin{proof}
Let $k_1 : V_1 \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk_2$.
Suppose towards a contradiction that
$\forces_{\FP^1} ``\dot \FQ$ is a term for a
$\gk_2$-directed closed partial ordering of
rank $< \gl$ so that forcing with $\dot \FQ$
destroys $\gk_2$'s measurability''. Since
$M^\gl \subseteq M$, $\gl$ is the least
inaccessible above $\gk_2$ in $M$ as well as $V_1$,
and $M$ and $V_1$ agree up to rank $\gl$. This means
$\FP^1 \ast \dot \FQ \in M$ and
$M^{\FP^1 \ast \dot \FQ} \models
``\gk_2$ isn't measurable''. Thus, since
${k_1(\FP^1)}_{\gk_2} = \FP^1$,
in $M$ in the definition of $k_1(\FP^1)$,
$\dot \FQ_{\gk_2}$ is a term for a
$\gk_2$-directed closed partial ordering
of rank $< \gl$ so that
$M^{{k_1(\FP^1)}_{\gk_2} \ast \dot \FQ_{\gk_2}} =
M^{\FP^1 \ast \dot \FQ_{\gk_2}} \models
``\gk_2$ isn't measurable''. Once again, since
$M^\gl \subseteq M$,
$V^{\FP^1 \ast \dot \FQ_{\gk_2}}_1 \models
``\gk_2$ isn't measurable'' as well.
We are now in a position to run the
standard reverse Easton arguments with
$\dot \FQ_{\gk_2}$. Specifically,
$k_1(\FP^1 \ast \dot \FQ_{\gk_2}) =
\FP^1 \ast \dot \FQ_{\gk_2} \ast \dot \FR \ast
k_1(\dot \FQ_{\gk_2})$, where $\dot \FR$ is a
term for the portion of the forcing whose field
is in the interval $[\gl, k_1(\gk_2))$.
Since $\gl$ is the least inaccessible above
$\gk_2$ in both $V_1$ and $M$, the field of
$\dot \FR$ is actually composed of ordinals
in the interval $(\gl, k_1(\gk_2))$.
Let $G_0 \ast G_1 \ast G_2$ be $V_1$-generic for
$\FP^1 \ast \dot \FQ_{\gk_2} \ast \dot \FR$.
The usual arguments show that
$M[G_0][G_1]$ remains $\gl$ closed with respect to
$V_1[G_0][G_1]$,
$M[G_0][G_1][G_2]$ remains $\gl$ closed with respect to
$V_1[G_0][G_1][G_2]$, and
since ${k_1}''\FP^1 = \FP^1$,
$k_1 : V_1 \to M$ extends to
$k_1 : V_1[G_0] \to M[G_0][G_1][G_2]$.
Thus, in
$V_1[G_0][G_1][G_2]$, ${k_1}''G_1$ generates
a master condition $p$ for $k_1(\FQ_{\gk_2})$.
If $G_3$ is a $V_1[G_0][G_1][G_2]$-generic
object containing $p$, then in
$V_1[G_0][G_1][G_2][G_3]$, $k_1$ extends once more to
$k_1 : V_1[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$ and
thereby generates a normal measure $\mu$ over
$\gk_2$ present in
$V_1[G_0][G_1][G_2][G_3]$.
As $|\FP^1 \ast \dot \FQ_{\gk_2}| < \gl$ in both
$M$ and $V_1$,
$\gl$ remains the least inaccessible above
$\gk_2$ in both
$M[G_0][G_1]$ and $V_1[G_0][G_1]$.
Therefore, since
$\FR \ast k_1(\dot \FQ_{\gk_2})$ is
$\gl$-strategically closed
in both $M[G_0][G_1]$ and $V_1[G_0][G_1]$,
$\mu$ must be present in
$V_1[G_0][G_1]$ as well. Thus,
$V^{\FP^1 \ast \dot \FQ_{\gk_2}}_1 \models
``\gk_2$ is measurable'', a contradiction.
This proves Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^{\FP^1}_1 \models ``\gk_2$ is $\gl$
strongly compact''.
\end{lemma}
\begin{proof}
If $k_1 : V_1 \to M$ and $\dot \FQ_{\gk_2}$
are as in Lemma \ref{l2}, then we know by
the contradiction just derived, the
definition of $\FP^1$,
and the fact $k_1(\gk_1) = \gk_1$ that
$\dot \FQ_{\gk_2}$ must be a term for the
partial ordering which adds a
non-reflecting stationary set of ordinals of
cofinality $\gk_1$ to $\gk_2$.
This allows us to complete the proof of
Lemma \ref{l3}
using ideas of Magidor
unpublished by him
which are also exposited in
\cite{AC1} and \cite{AC2}. Specifically,
$\gl$ is large enough so that we
may assume by choosing a normal ultrafilter of
Mitchell order $0$ over ${\gk_2}$ that
$k_2 : M \to N$ is an embedding witnessing the
measurability of ${\gk_2}$ definable in $M$ so that
$N \models ``{\gk_2}$ isn't measurable''.
It is the case that if
$k : V_{1} \to N$ is an elementary embedding with
critical point ${\gk_2}$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
so that $x \subseteq y$ and
$N \models ``|y| < k({\gk_2})$'',
then $k$ witnesses the $\gl$
strong compactness of ${\gk_2}$.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of ${\gk_2}$.
We show that $j$ extends in
$V^{\FP_1}_1$ to
$j : V_{1}^{\FP^{1}} \to N^{j(\FP^1)}$.
Since this extended embedding witnesses
the $\gl$ strong compactness of ${\gk_2}$ in
$V_{1}^{\FP^1}$, this proves Lemma \ref{l3}.
To do this, write
$j(\FP^1)$ as
$\FP^1 \ast \dot \FQ^{\gk_2} \ast \dot \FR^{\gk_2}$, where
$\dot \FQ^{\gk_2}$ is a term for the portion of
$j(\FP^1)$ between ${\gk_2}$ and $k_2({\gk_2})$ and
$\dot \FR^{\gk_2}$ is a term for the rest of
$j(\FP^1)$, i.e., the part above $k_2({\gk_2})$.
Note that since
$N \models ``{\gk_2}$ isn't measurable'',
${\gk_2} \not\in {\rm field}(\dot \FQ^{\gk_2})$.
Also, since
$M \models ``{\gk_2}$ is measurable'',
by elementarity,
$N \models ``k_2({\gk_2})$ is measurable''.
Thus, the field of $\dot \FQ^{\gk_2}$
is composed of all $N$-measurable cardinals
in the interval
$({\gk_2}, k_2({\gk_2})]$ (so
$k_2({\gk_2}) \in {\rm field}(\dot \FQ^{\gk_2})$),
a non-reflecting stationary set of ordinals of
cofinality $\gk_1$ is added to
$k_2(\gk_2)$ when forcing over
$N^{\FP^1}$ with $\FQ^{\gk_2}$,
and the field of $\dot \FR^{\gk_2}$ is composed of all
$N$-measurable cardinals in the interval
$(k_2({\gk_2}), k_2(k_1({\gk_2})))$.
Let $G_0$ be $V_{1}$-generic over $\FP^1$.
We construct in $V_{1}[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ^{\gk_2}$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR^{\gk_2}$. Since $\FP^1$ is an
Easton support iteration of length ${\gk_2}$,
a direct limit is taken at stage ${\gk_2}$,
and no forcing is done at stage ${\gk_2}$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
This means that
$j : V_{1} \to N$ extends in $V_1[G_0]$ to
$j : V_{1}[G_0] \to N[G_0][G_1][G_2]$.
To build $G_1$, note that since $k_2$
is generated by an
ultrafilter ${\cal U}$ over ${\gk_2}$ and
since in both $V_{1}$ and $M$, $2^{\gk_2} = \gk^+_2$,
$|k_2(\gk^+_2)| = |k_2(2^{\gk_2})| =
|\{ f : f : {\gk_2} \to \gk^+_2$ is a function$\}| =
|{[\gk^+_2]}^{\gk_2}| = \gk^+_2$. Thus, as
$N[G_0] \models ``|\wp(\FQ^{\gk_2})| = k_2(2^{\gk_2})$'', we can let
$\la D_\ga : \ga < \gk^+_2 \ra$ enumerate in
$V_{1}[G_0]$ the dense open subsets of
$\FQ^{\gk_2}$ present in $N[G_0]$.
Since the ${\gk_2}$ closure of $N$ with respect to either
$M$ or $V_{1}$ implies the least element of the field of
$\FQ^{\gk_2}$ is $> \gk^+_2$, the definitions of $\FP^1$ and
$\FQ^{\gk_2}$
imply that
$N[G_0] \models ``\FQ^{\gk_2}$ is
$\prec \gk^+_2$-strategically closed''.
By the fact the standard arguments show that
forcing with the ${\gk_2}$-c.c$.$ partial ordering
$\FP^1$ preserves that $N[G_0]$ remains
${\gk_2}$-closed with respect to either
$M[G_0]$ or $V_{1}[G_0]$,
$\FQ^{\gk_2}$ is $\prec \gk^+_2$-strategically closed
in both $M[G_0]$ and $V_{1}[G_0]$ as well.
We can now construct $G_1$ in either
$M[G_0]$ or $V_{1}[G_0]$ as follows.
Player I picks
$p_\ga \in D_\ga$ extending
$\sup(\la q_\gb : \gb < \ga \ra)$
(initially, $q_{-1}$ is the empty condition)
and player II responds by picking
$q_\ga \ge p_\ga$ (so $q_\ga \in D_\ga$).
By the $\prec \gk^+_2$-strategic closure of
$\FQ^{\gk_2}$ in both $M[G_0]$ and $V_{1}[G_0]$,
player II has a winning strategy for this
game, so
$\la q_\ga : \ga < \gk^+_2 \ra$ can be taken
as an increasing sequence of conditions with
$q_\ga \in D_\ga$ for $\ga < \gk^+_2$. Clearly,
$G_1 = \{p \in \FQ^{\gk_2} : \exists \ga <
\gk^+_2 [q_\ga \ge p]\}$ is an
$N[G_0]$-generic object over $\FQ^{\gk_2}$.
It remains to construct in $V_{1}[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR^{\gk_2}$.
To do this, we first write
$k_1(\FP^1)$ as
$\FP^1 \ast \dot \FQ_{\gk_2} \ast \dot \FT^{\gk_2}$, where
$\dot \FT^{\gk_2}$ is a term for the portion of
$k_1(\FP^1)$ above $\gk_2$.
Since as in Lemma \ref{l2},
$\gl$ is the least inaccessible cardinal
above $\gk_2$ in both $V_1$ and $M$,
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is measurable''.
Thus, the field of
$\dot \FT^{\gk_2}$ is composed of all
$M$-measurable cardinals in the interval
$(\gl, k_1({\gk_2}))$, which implies that in
$M$,
$\forces_{\FP^1 \ast \dot \FQ_{\gk_2}}
``\dot \FT^{\gk_2}$ is $\prec \gl^+$-strategically
closed''. Further,
$V_{1} \models 2^\gl = \gl^+$, and as
$\gl$ is inaccessible,
$|{[\gl]}^{< {\gk_2}}| = \gl$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter ${\cal U}$ over
$P_{\gk_2}(\gl)$,
$|k_1(\gk_2^+)| = |k_1(2^{\gk_2})| =
|2^{k_1(\gk_2)}| =
|\{ f : f : P_{\gk_2}(\gl) \to \gk_2^+$ is a function$\}| =
|{[\gk_2^+]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{C}, Section 1.2.5, page 8) associated with
$\dot \FT^{\gk_2}$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^1 \ast \dot \FQ_{\gk_2}$ and
$\forces_{\FP^1 \ast \dot \FQ_{\gk_2}} ``\tau \in
\dot \FT^{\gk_2}$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP^1 \ast \dot \FQ_{\gk_2}} ``\tau \ge \sigma$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP^1 \ast \dot \FQ_{\gk_2}} ``\dot \FT^{\gk_2}$ is
$\prec \gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
$\prec \gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V_{1}$ as well. Therefore, as
$\forces_{\FP^1 \ast \dot \FQ_{\gk_2}} ``|\dot \FT^{\gk_2}| =
k_1(\gk_2)$ and
$2^{k_1(\gk_2)} = {(k_1(\gk_2))}^+ = k_1(\gk_2^+)$'',
we can assume without loss of generality that in $M$,
$|\FT^*| = k_1(\gk_2)$. This means we can let
$\la D_\ga : \ga < \gl^+ \ra$
enumerate in $V_{1}$ the dense open subsets of $\FT^*$
present in $M$ and argue as before to construct in
$V_{1}$ an $M$-generic object $H_2$ over $\FT^*$.
Note now that since $N$ is given
by an ultrapower of $M$ via a normal ultrafilter
${\cal U} \in M$ over ${\gk_2}$,
Fact 2 of Section 1.2.2 of \cite{C} tells us that
$k_2 '' H_2$ generates an $N$-generic object
$G^*_2$ over $k_2(\FT^*)$. By elementariness,
$k_2(\FT^*)$ is the term forcing in $N$
defined with respect to
$k_2(k_1(\FP^1_{\gk_2})_{{\gk_2} + 1}) =
\FP^1 \ast \dot \FQ^{\gk_2}$.
Therefore, since
$j(\FP^1) = k_2(k_1(\FP^1)) =
\FP^1 \ast \dot \FQ^{\gk_2} \ast
\dot \FR^{\gk_2}$,
$G^*_2$ is $N$-generic over
$k_2(\FT^*)$, and $G_0 \ast G_1$ is
$k_2(\FP^1 \ast \dot \FQ_{\gk_2})$-generic over
$N$, Fact 1 of Section 1.2.5 of \cite{C}
tells us that for
$G_2 = \{i_{G_0 \ast G_1}(\tau) : \tau \in
G^*_2\}$, $G_2$ is $N[G_0][G_1]$-generic over
$\FR^{\gk_2}$.
Thus, in $V_{1}[G_0]$,
$j : V_{1} \to N$ extends to
$j : V_{1}[G_0] \to N[G_0][G_1][G_2]$.
This proves Lemma \ref{l3}.
\end{proof}
If we now let $\FP = \FP^0 \ast \dot \FP^1$,
then Lemmas \ref{l1} - \ref{l3} show that
$V^\FP_\gl$ is our desired model.
This completes the proof of Theorem \ref{t1}.
\end{proof}
We conclude Section \ref{s1} by remarking that
we don't know if $\gk_2$'s strong compactness,
and not just its measurability, is fully
indestructible in $V^\FP_\gl$.
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\la \gk_\ga : \ga
\in {\rm Ord} \ra$ is a proper class of
supercompact cardinals''.
Without loss of generality, by ``cutting off''
the universe if necessary at the least
inaccessible limit of supercompact cardinals,
we may assume that there are no inaccessible
limits of supercompact cardinals.
Let $\la \gd_\ga : \ga \in {\rm Ord} \ra$ enumerate
$\{\gd : \gd$ is measurable but $\gd \neq \gk_\ga$
for any $\ga \in {\rm Ord}\}$. In analogy to
Theorem \ref{t1}, we define an Easton support
class iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga \in {\rm Ord} \ra$ as follows:
\begin{enumerate}
\item\label{d1} $\FP_0$ is the partial ordering
for adding a Cohen subset to $\omega$.
\item\label{d2} If $\forces_{\FP_\ga} ``\gd_\ga$
isn't measurable'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the trivial
partial ordering $\{\emptyset\}$.
\item\label{d3} If $\forces_{\FP_\ga} ``\gd_\ga$
is measurable and there is a $\gd_\ga$-directed
closed partial ordering so that forcing with it
destroys $\gd_\ga$'s measurability'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for a
partial ordering of least possible rank
destroying $\gd_\ga$'s measurability.
\item\label{d4} If none of the above cases holds, then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
the trivial partial ordering $\{\emptyset\}$.
\end{enumerate}
\begin{lemma}\label{l4}
For any $\ga$, $\dot \FQ_\ga$ has rank below
the least $V$-supercompact cardinal above
$\gd_\ga$.
\end{lemma}
\begin{proof}
Since this is obviously true if cases
\ref{d2} and \ref{d4}
in the definition of $\FP$ hold, we assume
instead that
$\ga$ is the least ordinal so that at stage
$\ga$ in the definition of $\FP$,
case \ref{d3} holds. Thus,
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ is a term
for a $\gd_\ga$-directed closed partial ordering
destroying $\gd_\ga$'s measurability''.
Let $\gb$ be so that $\gk_\gb$ is the least
supercompact cardinal above $\gd_\ga$. If
$\dot \FQ_\ga$ has rank $\gd \ge \gk_\gb$,
then let $j : V \to M$ be an elementary
embedding witnessing enough supercompactness
of $\gk_\gb$ so that
$M \models ``\forces_{\FP_\ga} `$There is a
$\gd_\ga$-directed closed partial ordering
of rank $\gd$ so that forcing with it
destroys $\gd_\ga$'s measurability' ''.
It is the case that $\ga, \gd_\ga < \gk_\gb$,
$\FP_\ga$ inductively has rank $< \gk_\gb$,
$j(\ga) = \ga$, and
$j(\gd_\ga) = \gd_\ga$. Hence, by reflection,
$\{\gg < \gk_\gb :$ $\forces_{\FP_\ga}
``$There is a $\gd_\ga$-directed closed
partial ordering of rank $\gg$ so that
forcing with it destroys $\gd_\ga$'s
measurability''$\}$ is unbounded in $\gk_\gb$.
This proves Lemma \ref{l4}.
\end{proof}
We observe that the proof of Lemma \ref{l4}
actually shows that in the above iteration,
if $\dot \FQ_\ga$ is chosen to be a term for a
$\gd_\ga$-directed closed partial ordering of
least rank so that
$\forces_{\FP_\ga} ``$Forcing with $\dot \FQ_\ga$
makes $\gd_\ga$ non-$\zeta$ supercompact for some
$\zeta$'', then $\dot \FQ_\ga$ has rank below
the least $V$-supercompact cardinal $\gk_\gb$
above $\gd_\ga$
as well. To see this, as in the above, let
$j : V \to M$ witness enough supercompactness of
$\gk_\gb$ so that
$M \models ``\forces_{\FP_\ga} `$There is a
$\gd_\ga$-directed closed partial ordering of
rank $\gd$ so that forcing with it makes
$\gd_\ga$ non-$\zeta$ supercompact for some
$\zeta$' ''.
Reflection once again yields the desired
result and tells us that the minimal $\zeta$
so that
$V^{\FP_\ga \ast \dot \FQ_\ga} \models
``\gd_\ga$ isn't $\zeta$ supercompact''
is below $\gk_\gb$ as well.
Note now that by Lemma \ref{l4} and the definition of
$\FP$, for any $\gk_\ga$, we can write
$\FP = \FP^{0, \ga} \ast \dot \FP^{1, \ga}$, where
$|\FP^{0, \ga}| = \gk_\ga$,
$\FP^{0, \ga}$ is $\gk_\ga$-c.c., and
$\forces_{\FP^{0, \ga}} ``\dot \FP^{1, \ga}$ is
$\gk_\ga$-directed closed''.
Hence, the standard Easton arguments
(see also \cite{A98}) yield that
$V \models {\rm ZFC}$.
\begin{lemma}\label{l5}
$V^\FP \models ``\gk_\ga$ is a measurable
cardinal whose measurability is
indestructible under forcing with
$\gk_\ga$-directed closed partial orderings''.
\end{lemma}
\begin{proof}
By the factorization of $\FP$ just given,
it suffices to show
$\forces_{\FP^{0, \ga}} ``\gk_\ga$ is a
measurable cardinal whose measurability
is indestructible under forcing with
$\gk_\ga$-directed closed partial orderings''.
By GCH and Lemma 1.5 of \cite{G} and Lemma 5 of
\cite{A98},
$\forces_{\FP^{0, \ga}} ``\gk_\ga$ is
measurable''. We show that
$\forces_{\FP^{0, \ga}} ``\gk_\ga$'s measurability
is indestructible under forcing with
$\gk_\ga$-directed closed partial orderings''.
To see this, assume to the contrary that
$\forces_{\FP^{0, \ga}} ``\dot \FQ$ is a term for a
$\gk_\ga$-directed closed partial ordering so that
forcing with $\dot \FQ$ destroys
$\gk_\ga$'s measurability''. Let
$\gl > \max({\rm rank}(\dot \FQ), \gk_\ga)$
be an inaccessible cardinal. If
$j : V \to M$ is an elementary embedding
witnessing the $\gl$ supercompactness of $\gk_\ga$
so that $M \models ``\gk_\ga$
isn't supercompact'',
$\FP^{0, \ga} \ast \dot \FQ \in M$, and in $M$,
$\forces_{\FP^{0, \ga}} ``\gk_\ga$ is measurable
and $\dot \FQ$ is a term for a
$\gk_\ga$-directed closed partial ordering so that
forcing with it destroys $\gk_\ga$'s measurability''.
Thus, by the definition of $\FP^{0, \ga}$,
${j(\FP^{0, \ga})}_{\gk_\ga} = \FP^{0, \ga}$ and
$\dot \FQ_{\gk_\ga}$ is a term
for a partial ordering
%of rank $\le {\rm rank}(\dot \FQ)$
so that forcing with
$\FP^{0, \ga} \ast \dot \FQ_{\gk_\ga}$ over
$M$ destroys $\gk_\ga$'s measurability.
By the closure properties of $M$, forcing with
$\FP^{0, \ga} \ast \dot \FQ_{\gk_\ga}$ over $V$
destroys $\gk_\ga$'s measurability as well.
We can now use the same argument found in the
last paragraph of the proof of Lemma \ref{l2}
to show that
$V^{\FP^{0, \ga} \ast \dot \FQ_{\gk_\ga}} \models
``\gk_\ga$ is measurable''.
This contradiction proves Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$If $\gk$ is measurable, then
$\gk = \gk_\ga$ for some ordinal $\ga$''.
\end{lemma}
\begin{proof}
For any $\ga$, write
$\FP = \FP_{\ga + 1} \ast \dot \FR$.
By the definition of $\FP$,
$\forces_{\FP_{\ga + 1}} ``\dot \FR$ is
$\gd$-directed closed for $\gd$ the
least measurable cardinal above $\gd_\ga$''. Thus,
$\forces_{\FP} ``\gd_\ga$ isn't measurable'' iff
$\forces_{\FP_{\ga + 1}} ``\gd_\ga$ isn't measurable'',
so it suffices to show that
$\forces_{\FP_{\ga + 1}} ``\gd_\ga$ isn't measurable''.
To show this last fact, we observe that by its
definition, we can assume
that at stage $\ga$
we are not in case \ref{d2}
of the definition of $\FP$.
We may thus assume that
$\forces_{\FP_\ga} ``\gd_\ga$ is measurable''.
Note, however, that we can write
$\FP_\ga = \FP_0 \ast \dot \FR^*$, where
$|\FP_0| = \omega$ and
$\forces_{\FP_0} ``\dot \FR^*$ is $\ha_1$-strategically
closed''. Thus, once again using Hamkins' terminology,
$\FP_\ga$ is a ``gap forcing admitting a gap at
$\ha_1$.''
Therefore, by Corollary 5.4 of \cite{H4},
which says that a gap forcing admitting a gap at
$\ha_1$ can only make the measurability of a
measurable cardinal $\gg$ indestructible under
arbitrary $\gg$-directed closed forcing if
$\gg$ is supercompact in the ground model,
as $\gd_\ga$ isn't supercompact in $V$,
we must at stage $\ga$ be
in case \ref{d3} of the definition of
$\FP$. This means that
$\forces_{\FP_{\ga + 1}} ``\gd_\ga$ isn't measurable''.
We now know that
$V^\FP \models ``$The only $V$-measurable cardinals
remaining measurable were supercompact in $V$''.
Thus, the proof of Lemma \ref{l6} will be complete
once we have shown that any $V^\FP$-measurable
cardinal had to have been measurable in $V$.
Note, though, that as in the last paragraph,
we can write
$\FP = \FP_0 \ast \dot \FR^*$, where
$|\FP_0| = \omega$ and
$\forces_{\FP_0} ``\dot \FR^*$ is
$\ha_1$-strategically closed''.
Hence, once again by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
since $\FP$ is a
``gap forcing admitting a gap at $\ha_1$'',
forcing with $\FP$ can't create any new
measurable cardinals, i.e.,
any $V^\FP$-measurable cardinal was
measurable in $V$.
This proves Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l5} and \ref{l6}
complete the proof of Theorem \ref{t2}.
\end{proof}
We conclude Section \ref{s2} by observing
that by Corollary 5.4 of \cite{H4},
since we use a partial ordering which is a
``gap forcing admitting a gap at $\ha_1$''
to produce a proper class of fully
indestructible measurable cardinals,
we had to have started with a proper class
of supercompact cardinals. We could not
have started with anything weaker.
\section{The Proof of Theorem \ref{t3}}\label{s3}
We turn now to the proof of Theorem \ref{t3}.
\begin{proof}
The proof of Theorem \ref{t3} combines ideas of
\cite{A98} and \cite{AS} with ideas from \cite{AH}.
Let $V \models ``$ZFC + GCH + $\gk$ is almost huge'',
and let $j : V \to M$ witness $\gk$'s almost
hugeness. If $j(\gk) = \gl'$, then $\gl'$ is
inaccessible in $V$, and
$V_{\gl'} \models ``\gk$ is a supercompact
limit of supercompact cardinals''.
If $\gl$ is the least inaccessible above $\gk$,
then $\gl < \gl'$ and
$V_\gl \models ``\gk$ is a supercompact limit
of supercompact cardinals'' as well.
Work for the time being in $\ov V = V_{\gl'}$.
Let $\gg < \gd < \gk$ be so that $\gg$ is
regular and $\gd$ is supercompact.
By Lemma 13, pages 2028 - 2029 of \cite{AS}
(see also the proof of the Theorem of \cite{A98}),
there is a $\gg$-directed closed partial ordering
$\FP_{\gg, \gd} \in \ov V$
of rank $\gd + 1$ with
$|\FP_{\gg, \gd}| = \gd$ so that
${\ov V}^{\FP_{\gg, \gd}} \models
``$There are no strongly compact cardinals in
the interval $(\gg, \gd)$
since unboundedly many cardinals in
$(\gg, \gd)$ contain non-reflecting
stationary sets of ordinals of
cofinality $\gg$ + $\gd$ is a
fully indestructible supercompact cardinal''.
This has as a consequence that
${\ov V}^{\FP_{\gg, \gd}} \models
``$Any partial ordering not adding bounded
subsets to $\gd$ preserves that there are
no strongly compact cardinals in the
interval $(\gg, \gd)$''. Further,
$\FP_{\gg, \gd}$ is defined as a modification
of Laver's indestructibility partial ordering
of \cite{L}, i.e., as an Easton support
iteration of length $\gd$
defined in the style of \cite{L} so that
every stage at which a non-trivial forcing
is done is a ground model measurable cardinal,
the least stage at which a non-trivial forcing
is done can be chosen to be an arbitrarily large
measurable cardinal in $(\gg, \gd)$,
and at a stage
$\ga$ when a non-trivial
forcing $\FQ$ is done,
$\FQ = \FQ^0 \ast \dot \FQ^1$
where $\FQ^0$ is $\ga$-directed closed
and $\dot \FQ^1$ is a term for the
forcing adding
a non-reflecting stationary set of
ordinals of cofinality $\gg$
to some cardinal $\gb > \ga$.
Let $\la \gd_\ga : \ga < \gk \ra$
enumerate the supercompact cardinals
below $\gk$ together with their
measurable limits.
We define now an Easton support iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
of length $\gk$ as follows:
\begin{enumerate}
\item\label{e1} $\FP_1 = \FP_0 \ast \dot \FQ_0$,
where $\FP_0$ is the partial ordering for
adding a Cohen subset to $\omega$, and
$\dot \FQ_0$ is a term for
$\FP_{\ha_2, \gd_0}$.
\item\label{e2} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and
$\forces_{\FP_\ga} ``$There is a $\gd_\ga$-directed
closed partial ordering so that after forcing with it,
$\gd_\ga$ isn't $\zeta$ supercompact
for some $\zeta$'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for such
a partial ordering of minimal rank which
destroys the $\zeta$ supercompactness of
$\gd_\ga$ for the minimal possible $\zeta$.
\item\label{e3} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and case
\ref{e2} above doesn't hold
(which will mean that
$\forces_{\FP_\ga} ``\gd_\ga$ is a measurable
limit of supercompact cardinals whose
degree of supercompactness is fully
indestructible and whose strong
compactness is fully indestructible''), then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the
trivial partial ordering
$\{\emptyset\}$.
\item\label{e4} If $\gd_\ga$ is not a
measurable limit of supercompact cardinals,
$\ga = \gb + 1$, $\gd_\gb$ is a measurable
limit of supercompact cardinals,
and case \ref{e2} above holds for $\gd_\gb$,
then inductively, since a direct limit must
be taken at stage $\gb$,
$|\FP_\gb| = \gd_\gb < \gd_{\gb + 1} = \gd_\ga$.
This means inductively $\FP_\gb$
has been defined so as
to have rank $< \gd_\ga$, so by Lemma \ref{l4}
and the succeeding remark, $\dot \FQ_\gb$ can be
chosen to have rank $< \gd_\ga$.
Also, by Lemma \ref{l4} and the succeeding remark,
$\zeta < \gd_\ga$ for $\zeta$ the least so that
${\ov V}^{\FP_\gb \ast \dot \FQ_{\gb}} =
{\ov V}^{\FP_\ga} \models ``\gd_\gb$
isn't $\zeta$ supercompact''. Let
$\dot \gg_\ga$ be so that
$\forces_{\FP_\ga} ``\dot \gg_\ga = \gd^+_\gb$'',
and let
$\sigma \in (\gd_\gb, \gd_\ga)$ be the least
measurable cardinal (in either $V$ or
$V^{\FP_\ga}$) so that
$\forces_{\FP_{\ga}} ``\sigma >
\max(\dot \gg_\ga, \dot \zeta,
{\rm rank}(\dot \FQ_\gb))$''. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$ defined so that
$\sigma$ is below the least stage at which,
in the definition of
$\FP_{\gg_\ga, \gd_\ga}$, a non-trivial forcing is done.
\item\label{e5} If $\gd_\ga$ is not a measurable limit
of supercompact cardinals and case \ref{e4} doesn't hold,
then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where for
$\gg_\ga = {(\cup_{\gb < \ga} \gd_\gb)}^+$,
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$.
\end{enumerate}
\begin{lemma}\label{l7}
${\ov V}^\FP \models ``$If $\gd < \gk$ is a measurable
limit of supercompact cardinals, then
$\gd$'s degree of supercompactness is fully
indestructible''.
\end{lemma}
\begin{proof}
Write
$\FP = \FP_0 \ast \dot \FR$, where
$|\FP_0| = \omega$ and
$\forces_{\FP_0} ``\dot \FR$ is
$\ha_1$-strategically closed''. Thus,
$\FP$ is a ``gap forcing admitting a
gap at $\ha_1$'', so by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
any supercompact cardinal in ${\ov V}^\FP$
had to have been supercompact in $\ov V$, and
$\gd$ must be in $\ov V$ a measurable limit
of supercompact cardinals.
This means $\gd = \gd_\ga$ for some
$\ga < \gk$.
If
${\ov V}^\FP \models ``\gd_\ga$'s degree of
supercompactness isn't fully indestructible'', then let
$\FQ \in {\ov V}^\FP$ of minimal rank
and $\zeta$ smallest be
so that
${\ov V}^\FP \models ``\FQ$ is $\gd_\ga$-directed closed and
$\gd_\ga$ is $\zeta$ supercompact'' and
${\ov V}^{\FP \ast \dot \FQ} \models
``\gd_\ga$ isn't $\zeta$ supercompact''.
Write $\FP = \FP_\ga \ast \dot \FQ_\ga \ast
\dot \FQ_{\ga + 1} \ast \dot \FR^* =
\FP_{\ga + 1} \ast \dot \FQ_{\ga + 1} \ast
\dot \FR^* = \FP_{\ga + 1} \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FR^* = \FP_{\ga + 2} \ast \dot \FR^*$.
Since the definition of $\FP$ ensures that
$\forces_{\FP_{\ga + 2}} ``\dot \FR^*$ is
$\gd_{\ga + 1}$-directed closed'',
${\ov V}^{\FP_{\ga + 1} \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FR^*} =
{\ov V}^\FP \models ``\gd_{\ga + 1}$ is a fully
indestructible supercompact cardinal''.
Hence, by the proofs of Lemma \ref{l4}
and the succeeding remark,
$\FQ, \zeta \in {\ov V}^{\FP}_{\gd_{\ga + 1}}$.
Therefore, the preceding tells us
$\FQ \in {\ov V}^{\FP_{\ga + 2}} =
{\ov V}^{\FP_\ga \ast \dot \FQ_\ga \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}}}$ and
${\ov V}^{\FP_\ga \ast \dot \FQ_\ga \ast
\dot \FP_{\gg_{\ga + 1}, \gd_{\ga + 1}} \ast
\dot \FQ} =
{\ov V}^{\FP_\ga \ast \dot \FQ^*} \models
``\gd_\ga$ isn't $\zeta$ supercompact''.
Since $\forces_{\FP_\ga}
``\dot \FQ^*$ is $\gd_\ga$-directed
closed'',
we must be in case \ref{e4} at stage
$\ga + 2$ of the definition of $\FP$.
This means for some $\zeta' \le \zeta$,
${\ov V}^{\FP_\ga \ast \dot \FQ_\ga} =
{\ov V}^{\FP_{\ga + 1}} \models
``\gd_\ga$ isn't $\zeta'$ supercompact'',
and consequently, $\gd_\ga$ isn't
$\zeta'$ supercompact in
${\ov V}^\FP$ as well. Since
${\ov V}^\FP \models ``\gd_\ga$ is
$\zeta$ supercompact'', this is a
contradiction.
This proves Lemma \ref{l7}.
\end{proof}
We remark that Lemma \ref{l7} and its proof
show that if $\gd_\ga$ is supercompact in
$\ov V$, then by writing
$\FP = \FP_\ga \ast
\dot \FP_{\gg_\ga, \gd_\ga} \ast
\dot \FR^{**}$
if $\gd_\ga$ isn't a limit of
supercompact cardinals in
$\ov V$, and
$\FP = \FP_\ga \ast \dot \FQ_\ga
\ast \dot \FR^{**}$ where
$\dot \FQ_\ga$ is a term for
the trivial partial ordering
$\{\emptyset\}$ if $\gd_\ga$
is a limit of supercompact
cardinals in $\ov V$, we have that
${\ov V}^{\FP_\ga \ast
\dot \FP_{\gg_\ga, \gd_\ga} \ast
\dot \FR^{**}} =
{\ov V}^\FP \models ``\gd_\ga$ is a
fully indestructible supercompact cardinal''.
And, if $\gd = \gd_\gb$ is a measurable
limit of supercompact cardinals
in ${\ov V}^\FP$, by the
first paragraph of the proof of Lemma \ref{l7},
$\gd_\gb = \cup_{\gg < \gb} \gd_\gg$, where each
$\gd_\gg$ is supercompact in $\ov V$.
This means Lemma \ref{l7} implies that if
$\FQ \in {\ov V}^\FP$ is so that
${\ov V}^\FP \models ``\FQ$ is
$\gd_\gb$-directed closed'',
${\ov V}^{\FP \ast \dot \FQ} \models
``\gd_\gb$ is a measurable limit of
supercompact cardinals'', i.e.,
${\ov V}^{\FP \ast \dot \FQ} \models
``\gd_\gb$ is strongly compact''. Hence,
${\ov V}^\FP \models ``$If $\gd < \gk$ is a
measurable limit of supercompact cardinals,
then $\gd$ is a fully indestructible
strongly compact cardinal''.
\begin{lemma}\label{l8}
${\ov V}^\FP \models ``$The supercompact and
strongly compact cardinals coincide below
$\gk$ except at measurable limit points''.
\end{lemma}
\begin{proof}
We first show that if
$\gd < \gk$ is strongly compact in
${\ov V}^\FP$, then
$\gd = \gd_\ga$ for some
$\ga < \gk$.
If not, then let
$\gd < \gk$ be so that $\gd$ is strongly compact, but
$\gd \neq \gd_\ga$ for any $\ga < \gk$. For
$\ga$ least so that
$\gd_\ga > \gd$, $\gd_\ga$ must be supercompact
in ${\ov V}$, and
$\gd$ must be so that
$\gd \in (\gg_\ga, \gd_\ga)$. By the definition of
$\FP_{\gg_\ga, \gd_\ga}$ and $\FP$, it is then the
case that in both
${\ov V}^{\FP_\ga \ast \dot \FP_{\gg_\ga, \gd_\ga}}$
and ${\ov V}^\FP$,
unboundedly many cardinals in the interval
$(\gg_\ga, \gd_\ga)$ contain non-reflecting
stationary sets of ordinals of cofinality
$\gg_\ga$. Thus, ${\ov V}^\FP \models ``
\gd$ isn't strongly compact''.
Since Lemma \ref{l7} and the
succeeding remark show that any supercompact
$\gd_\ga$ remains supercompact in ${\ov V}^\FP$,
and since by the first paragraph of the
proof of Lemma \ref{l7},
any $\gd < \gk$ which is a
measurable limit of supercompacts in
${\ov V}^\FP$ must be a $\gd_\ga$ for
some $\ga$,
Lemma \ref{l8} is proved.
\end{proof}
\begin{lemma}\label{l9}
${\ov V}^\FP \models ``\gk$ is a fully
indestructible supercompact cardinal''.
\end{lemma}
\begin{proof}
If not, then let
$\FQ \in {\ov V}^\FP$ of minimal rank and
$\zeta$ least be so that
${\ov V}^\FP \models ``\FQ$ is
$\gk$-directed closed'' and
${\ov V}^{\FP \ast \dot \FQ} \models
``\gk$ isn't $\zeta$ supercompact''.
Since $\FQ$ has minimal rank and
$\zeta$ is smallest, it must
be the case as before that $\FQ$ has rank
$< \gk^*$ and $\zeta < \gk^*$,
where by the agreement of
$V$ and $M$ up to rank $\gl'$, $\gk^*$
is the least $< \gl'$ supercompact
cardinal above $\gk$ in both $V$ and $M$.
Note that as earlier, $\FP$ is an
initial segment of $j(\FP)$.
This means by $V$'s agreement with $M$
through rank $\gl'$ and the definition of
$\FP$ that in $M$, $\dot \FQ_\gk$ has rank
$< \gk^*$ and is so that
$M^{\FP \ast \dot \FQ_\gk} \models
``\zeta$ is the smallest so that
$\gk$ isn't $\zeta$ supercompact'', i.e., again by
the agreement of $V$ and $M$ through rank $\gl'$,
$V^{\FP \ast \dot \FQ_\gk} \models
``\gk$ isn't $\zeta$ supercompact'' as well.
Thus, since
$\FP \ast \dot \FQ_\gk$ is
an initial segment of
$j(\FP \ast \dot \FQ_\gk)$,
the definition of $\FP$
allows us to apply the
arguments of Lemma \ref{l2} to show that
$j : V \to M$ extends to
$j : V^{\FP \ast \dot \FQ_\gk} \to
M^{j(\FP \ast \dot \FQ_\gk)}$ and can
be used to define an ultrafilter
${\cal U} \in V^{\FP \ast \dot \FQ_\gk}$
witnessing the $\zeta$ supercompactness of
$\gk$ in $V^{\FP \ast \dot \FQ_\gk}$.
This contradiction proves Lemma \ref{l9}.
\end{proof}
Since $\gl$ is the least inaccessible cardinal
above $\gk$, there are no measurable
cardinals between $\gk$
and $\gl$ in either $V$ or $V^\FP$.
Further, by the definition of $\FP$,
$\FP \in V_\gl$ and $|\FP| = \gk$.
Thus, Lemmas \ref{l7} - \ref{l9}
and the remark after Lemma \ref{l7} show
that $V^\FP_\gl$ is our desired model.
This completes the proof of Theorem \ref{t3}.
\end{proof}
Let $\varphi(\ga, \gb)$ be the statement
``$\ga$ is a $< \gb$
supercompact cardinal whose degree of
supercompactness is fully indestructible
by partial orderings of rank below $\gb$,
$\ga$ is a limit of
$< \gb$ supercompact cardinals,
the $< \gb$ supercompact and $< \gb$
strongly compact cardinals coincide below
$\ga$ except at measurable limit points,
below $\ga$, each
$< \gb$ supercompact cardinal
has its
degree of supercompactness fully indestructible
by partial orderings of rank below $\gb$,
and below
$\ga$, each measurable limit of
$< \gb$ supercompact cardinals has both its
degree of supercompactness and its degree of
strong compactness fully indestructible
by partial orderings of rank below $\gb$''.
We conclude Section \ref{s3} by observing that
since in either $V$ or $M$, by the agreement
of $V$ and $M$ up to rank $\gl'$,
$\forces_{\FP} \varphi(\gk, \gl')$,
$A = \{\gd < \gk :$ $\forces_{\FP_\gd}
\varphi(\gd, \gk)\}$
is unbounded in $\gk$.
Fix $\gd \in A$, and let $\ga$ be so that
$\gd = \gd_\ga$. Write
$\FP = \FP_\ga \ast \dot \FQ$.
Since the first paragraph of the proof of
Lemma \ref{l7} shows that both $\FP$
and $\FP_\ga$ are ``gap forcings admitting
a gap at $\ha_1$'', the results of
\cite{H1}, \cite{H2}, and \cite{H3} show
that no cardinal in $V$ can have either
its degree of supercompactness or its
degree of strong compactness increased in
$V^{\FP_\ga}$ or $V^\FP$. Therefore,
since $\forces_{\FP_\ga} ``\dot \FQ$ is
$\gd_\ga$-directed closed'',
$\forces_{\FP_\ga \ast \dot \FQ}
\varphi(\gd_\ga, \gk)$, i.e.,
$\forces_\FP \varphi(\gd_\ga, \gk)$.
Thus, $V^\FP_\gk$ is a model of ZFC in which the
supercompact and strongly compact cardinals
coincide except at measurable limit points,
each non-supercompact
measurable limit of supercompact cardinals
has both its degree of supercompactness
and its strong compactness fully
indestructible,
each supercompact cardinal is fully indestructible,
and there are many
supercompact limits of supercompact cardinals.
This allows us to produce a universe essentially
as rich in supercompact limits of supercompact
cardinals as the one produced in \cite{A98}.
\section{Concluding Remarks}\label{s4}
We conclude this paper by giving some open problems
related to the material contained herein.
These are as follows.
\begin{enumerate}
\item\label{p1} In Theorem \ref{t1}, is
$\gk_2$'s strong compactness, and not just
its measurability, fully indestructible?
\item\label{p2} In general, is it possible
for the first $\ga$ strongly compact cardinals
to be non-supercompact and to exhibit some
form of indestructibility?
\item\label{p3} How much indestructibility can
be forced for an arbitrary non-supercompact
strongly compact cardinal?
\item\label{p4} What are the consistency strengths
for various forms of indestructibility exhibited
by a non-supercompact strongly compact cardinal?
In general, is the existence of a fully indestructible
non-supercompact strongly compact cardinal
equiconsistent with the existence of a supercompact
cardinal?
\end{enumerate}
As the remarks at the end of Section \ref{s2} indicate,
Hamkins' work of \cite{H1}, \cite{H2}, \cite{H3},
and \cite{H4} shows that
if a non-supercompact strongly compact cardinal is
to be made fully indestructible, the forcing used
can't in any way naively resemble the
partial orderings of \cite{A98} or \cite{L}, i.e.,
it can't be an Easton support iteration of partial
orderings which have the appropriate degree of
strategic closure.
We conjecture, however, that no such partial ordering
can exist, i.e., that the existence of a fully indestructible
non-supercompact strongly compact cardinal is equiconsistent
with the existence of a supercompact cardinal.
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\end{document}