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\title{Indestructibility, Strongness, and
Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, strong cardinal, level by level equivalence
between strong compactness and supercompactness}}
% non-reflecting stationary set of ordinals}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank Joel Hamkins
for helpful conversations on the
subject matter of this paper.
The author also wishes to thank the
referee for helpful comments and
suggestions that have been incorporated
into this version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{June 30, 2002\\
(revised February 5, 2003)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model in which
there is
%an indestructible strong cardinal $\gk$
a strong cardinal $\gk$ whose
strongness is indestructible under
$\gk$-strategically closed forcing
and in which level by level
equivalence between strong
compactness and supercompactness
holds non-trivially.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
A very surprising fact (see Theorems
5 - 7 of \cite{AH4}) is that if
there are large enough cardinals in
the universe, then indestructibility
for either a strong or supercompact
cardinal (in the sense of \cite{GS}
or \cite{L}) is incompatible with
level by level equivalence between
strong compactness and supercompactness.
Indeed, Theorem 6 of \cite{AH4}
states that if $\gk$ is a
strong cardinal such that forcing
with any
$\gk$-strategically closed partial
ordering preserves $\gk$'s strongness
(where for the rest of this paper,
we will refer to such a cardinal as an
{\it indestructible strong cardinal}) and
level by level equivalence between
strong compactness and supercompactness
holds below $\gk$
(where for the rest of this paper,
{\it level by level equivalence between
strong compactness and supercompactness}
means that for $\gd \le \gl$,
$\gd$ and $\gl$ both
regular, $\gd$ is $\gl$
strongly compact iff $\gd$
is $\gl$ supercompact), then
no cardinal $\gl > \gk$ is
$2^\gl$ supercompact.
However, a question left unanswered
in \cite{AH4} is whether
it is possible for there to be
level by level equivalence
non-trivially below an indestructible
strong cardinal $\gk$ in a universe
in which no cardinal $\gl > \gk$ is
$2^\gl$ supercompact.
%By Theorem 6 of \cite{AH4},
By the next to last sentence,
the existence of such a universe is the
best possible outcome for which
one could hope.
The purpose of this paper is to provide
an affirmative answer to
the aforementioned question.
Specifically, we prove the
following theorem.
\begin{theorem}\label{t1}
Suppose
$V_0 \models ``$ZFC + $\gk_0$ is
supercompact + There are no
cardinals $\gd < \gl$ such
that $\gd$ is $\gl$ supercompact
and $\gl$ is measurable''.
There is then a cardinal
$\gk < \gk_0$, a submodel
$V$ of a (possibly trivial)
forcing extension of $V_0$, and a
partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
No cardinal $\gl > \gk$ is
$2^\gl$ supercompact +
$\gk$ is an indestructible strong cardinal +
For any $\gg < \gk$,
there is a cardinal
$\gd \in [\gg, \gk)$ which is
$\gd^{+ \gg}$ supercompact +
%$\{\gd < \gk : \gd$ is $\gd^{+ \gd}$
%supercompact$\}$ is unbounded in $\gk$ +
%For any $\gg < \gk$, there are cardinals
%$\gd < \gl$, $\gd, \gl \in (\gg, \gk)$
%such that $\gd$ is $\gl$ supercompact +
Level by level equivalence between
strong compactness and supercompactness
holds''.
\end{theorem}
It will in fact be the case, roughly
speaking, that however much supercompactness
$\gk_0$ reflects will also occur
unboundedly often below $\gk$ in $V_0$,
$V$, and $V^\FP$.
Thus, e.g.,
$\{\gd < \gk : \gd$ is $\gd^{+ \gd}$
supercompact$\}$ is unbounded in $\gk$
in these models.
We will comment again on this both at
the beginning and at the conclusion
of the proof of Theorem \ref{t1}.
%and in our concluding remarks.
Before beginning the proof of
Theorem \ref{t1}, 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 $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
%is an Easton support iteration of length $\gk$
%so that at stage $\ga$, a nontrivial 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 {\it $\gk$-distributive}
if for every sequence
$\la D_\ga : \ga < \gk \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$
is dense open.
%$\FP$ is $\gk$-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
%every two elements
%$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
%there is an
%upper bound $p \in \FP$.
$\FP$ is {\it $\gk$-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
%$\gk^+$-directed closed, then $\FP$ is
%$\gk$-strategically closed, and
Note that
if $\FP$ is $\gk$-strategically closed,
then $\FP$ is $\gk$-distributive.
Also, if $\FP$ is
$\gk$-distributive and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$\FP$ is {\it ${\prec}\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (again choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
We recall for the benefit of readers Hamkins'
definition from Section 3 of \cite{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\it lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings'', ``side-by-side forcing'', and
``choosing which partial ordering to force
with generically''.}
A result which will be key in the proof
of Theorem \ref{t1} is Hamkins'
Gap Forcing Theorem of \cite{H2} and \cite{H3}.
We therefore state this theorem now, along
with some associated terminology, quoting
freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| < \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ {\it admits a gap at $\gd$}.
Also, as in the terminology of
\cite{H2} and \cite{H3} (and elsewhere),
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The Gap Forcing Theorem is then
the following.
\begin{theorem}\label{t2}
(Hamkins' Gap Forcing Theorem)
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a gap at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ 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 $V$.
\end{theorem}
Finally, we
mention that we are assuming familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{K}
%or \cite{SRK}
for further details.
Also, unlike \cite{K}, we will say that
the cardinal $\gk$ is {\it $\gl$ strong}
for $\gl > \gk$ if there is
$j : V \to M$ an elementary embedding having
critical point $\gk$ so that
$j(\gk) > |V_\gl|$ and $V_\gl \subseteq M$.
As always, $\gk$ is {\it strong} if $\gk$ is $\gl$
strong for every $\gl > \gk$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V_0 \models ``$ZFC + $\gk_0$ is supercompact +
There are no cardinals
$\gd < \gl$ such that
$\gd$ is $\gl$ supercompact and
$\gl$ is measurable''.
Without loss of generality, by
first forcing GCH and then following
this by the forcing of \cite{AS97a}
if necessary,
%and finally truncating the universe
%at the appropriate cardinal if necessary,
we may also assume that
in addition to the preceding,
$V_0 \models ``$GCH +
%No cardinal
%$\gl > \gk_0$ is measurable + If
%$\gd < \gl < \gk_0$ are such that
%$\gd$ and $\gl$ are both measurable, then
%$\gd$ isn't $\gl$ supercompact +
Level by level equivalence between
strong compactness and supercompactness
holds''.
%For all cardinals $\gd \le \gl < \gk_0$,
%$\gd$ is $\gl$ strongly compact iff
%$\gd$ is $\gl$ supercompact''.
We then let $\gk < \gk_0$ be the
least strong cardinal in $V_0$
(we know $\gk < \gk_0$ by
Lemma 2.1 of \cite{AC2}), and take
$V$ as $V_0$ truncated at
$\gk^*$,
where for
the rest of this paper,
for $\gg$ an ordinal,
$\gg^*$ is the least cardinal above
$\gg$ which is $2^{\gg^*}$ supercompact.
%the least cardinal $\gl$ above $\gk$ which is $2^\gl$ supercompact.
Note that
$V \models ``\gk$ is the least strong cardinal'',
since otherwise, if
$V \models ``\gd < \gk$ is a strong cardinal'', then
$V_0 \models ``\gd$ is $\gg$ strong for every
$\gg < \gk$ and $\gk$ is strong'',
so by the proof of Lemma 2.1 of
\cite{AC2},
$V_0 \models ``\gd < \gk$ is a strong
cardinal'', a contradiction.
Note also that by reflection in $V_0$
and the supercompactness of $\gk_0$,
for any $\gg < \gk$,
$\{\gd < \gk_0 : \gd$ is $\gd^{+ \gg}$
supercompact$\}$ is unbounded in $\gk_0$,
meaning that a $\gd$ with this property
exists above $\gk$.
Therefore, by further reflection in $V_0$
via the strongness of $\gk$,
$\{\gd < \gk : \gd$ is $\gd^{+ \gg}$
supercompact$\}$ is unbounded in $\gk$.
In fact, this same argument shows that, e.g.,
$\{\gd < \gk : \gd$ is $\gd^{+ \gd}$
supercompact$\}$ is unbounded in $\gk$,
$\{\gd < \gk : \gd$ is $\gd^{+ \gd + 1}$
supercompact$\}$ is unbounded in $\gk$, etc.
In any case,
for any $\gg < \gk$, there is a cardinal
$\gd \in [\gg, \gk)$
such that $\gd$ is $\gd^{+ \gg}$ supercompact.
Also, since level by level equivalence
between strong compactness and
supercompactness holds
%below $\gk_0$ and $\gk < \gk_0$,
in $V_0$ and $V \subseteq V_0$,
level by level equivalence between
strong compactness and supercompactness
holds in $V$ as well.
Working in $V$ (which is clearly a
model for GCH), we define the
partial ordering $\FP$ used
in the proof of Theorem \ref{t1}.
$\FP$ is the Easton support iteration
of length $\gk$ which
begins by adding a Cohen subset of
$\go$ and then is
trivial forcing except at those
stages $\gd < \gk$ which are not
themselves $2^\gd = \gd^+$ supercompact
in $V$ but which are in
$V$ measurable limits of cardinals
$\gg$ which are $2^\gg = \gg^+$ supercompact.
At such a stage $\gd$, the forcing
done is the lottery sum of all
$\gd$-strategically closed
partial orderings having rank below
${(\gd^*)}^V$.
%the least cardinal $\gl$ above $\gd$
%which is $2^\gl$ supercompact.
%= \gl^+$ supercompact.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is an
indestructible strong cardinal''.
\end{lemma}
\begin{proof}
Let
$\FQ \in V^\FP$ be such that
$V^\FP \models ``\FQ$ is
$\gk$-strategically closed''.
%Since we are assuming GCH,
By the definition of $V$, let
$\gl > |{\rm TC}(\dot \FQ)|$ be such that
$\gl$ is inaccessible.
%$\gl = \ha_\gl = \beth_\gl$ and $\gl$ has cofinality at least $\gk^+$.
Fix $j : V \to M$ an elementary embedding
witnessing the $\gl$ strongness of $\gk$. Since
$V \models ``$No cardinal $\gg > \gk$ is
$2^\gg$ supercompact'',
%= \gg^+$ supercompact'',
$M \models ``$No cardinal $\gg \in
(\gk, \gl)$
is $2^\gg$ supercompact''.
%= \gg^+$ supercompact''.
Further, since
$V \models ``\gk$ is the least
strong cardinal'', by Lemma 2.1 of
\cite{AC2} and the choice of $\gl$,
$M \models ``\gk$ isn't
$2^\gk$ supercompact''.
%= \gk^+$ supercompact''.
Thus, in $M$, $\gk$ is a stage at
which a forcing given by a lottery sum
is done in $j(\FP)$,
$\dot \FQ$ is a term for a partial
ordering allowed at the stage $\gk$ lottery
in $j(\FP)$, and the first ordinal above
$\gk$ which is a non-trivial stage of
forcing for $j(\FP)$ is above $\gl$ as well.
We show that in $V^{\FP \ast \dot \FQ}$,
$j$ lifts to a $\gl$ strong embedding
$j : V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
This will complete the proof of
Lemma \ref{l1}, since $\gl$ may be chosen
arbitrarily large in the universe.
The argument that the embedding $j$
lifts is quite similar to the
argument given in the proof of
Theorem 4.10 of \cite{H4},
or the argument given in the
proof of Lemma 4.2 of \cite{A03}.
For the benefit of readers, we give
the argument here as well, again taking the
liberty to quote freely from it.
We may assume that
$M = \{j(f)(a) : a \in {[\gl]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq V\}$.
Since we may write
$j(\FP)$ as $\FP \ast \dot \FQ \ast \dot \FR$,
by our remarks above,
we know that the first ordinal at which
$\dot \FR$ is forced to do a
lottery sum forcing is above $\gl$.
%Since $\gl$ has been chosen to have cofinality at least $\gk^+$,
Since $\gl$ is inaccessible,
we have that $M^\gk \subseteq M$.
This means that if
$G$ is $V$-generic over $\FP$ and
$H$ is $V[G]$-generic over $\FQ$,
$\FR$ is ${\prec} \gk^+$-strategically closed in both
$V[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$.
As in \cite{H4} and \cite{A03},
by using a suitable coding that allows us
to identify finite subsets of $\gl$
with elements of $\gl$,
by the definition of $M$,
there must be some
$\ga < \gl$ and function $g$ such that
$\dot \FQ = j(g)(\ga)$.
%(assuming that $\dot \FQ$ has been chosen reasonably).
Let $N = \{i_{G \ast H}(\dot z) : \dot z =
j(f)(\gk, \ga, \gl)$ for some function $f \in V\}$.
It is easy to verify that
$N \prec M[G][H]$, that $N$ is closed under
$\gk$ sequences in $V[G][H]$, and that
$\gk$, $\ga$, $\gl$, $\FQ$, and $\FR$ are all
elements of $N$.
Further, since
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : \gk \to V_\gk$ in $V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G][H]$ and $V[G][H]$, we can
%use the argument for the construction of the
%generic object $G_1$ to be given in Lemma \ref{l5} to
build $H' \subseteq \FR$
in $V[G][H]$ as follows. Let
%an $N$-generic object $H'$ over $\FR$ as follows. Let
$\la D_\gs : \gs < \gk^+ \ra$ enumerate in
$V[G][H]$ the dense open subsets of
$\FR$ present in $N$ so that
every dense open subset of $\FR$
occurring in $N$ appears at an
odd stage at least once in the
enumeration.
If $\gs$ is an odd ordinal,
$\gs = \tau + 1$ for some $\tau$.
Player I picks
$p_\gs \in D_\gs$ extending $q_\tau$
%$\sup(\la q_\gb : \gb < \gs \ra)$
(initially, $q_{0}$ is the empty condition),
and player II responds by picking
$q_\gs \ge p_\gs$
according to a fixed strategy ${\cal S}$
(so $q_\gs \in D_\gs$).
If $\gs$ is a limit ordinal, player II
uses ${\cal S}$ to pick
$q_\gs$ extending each
$q \in \la q_\gb : \gb < \gs \ra$.
%$\sup(\la q_\gb : \gb < \gs \ra)$.
By the ${\prec} \gk^+$-strategic closure of
$\FR$ in $V[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs : \gs < \gk^+ \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gk^+$.
Let
$H' = \{p \in \FR : \exists \gs <
\gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $H'$ is actually
$M[G][H]$-generic over $\FR$.
If $D$ is a dense open subset of
$\FR$ in $M[G][H]$, then
$D = i_{G \ast H}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$,
where $\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that
$j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of
$\FR$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$ and therefore $\gl$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is
definable without the parameters
$\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N$.
Thus, since $H'$ meets every dense open
subset of $\FR$ present in $N$,
$H' \cap i_{G \ast H}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$H' \cap i_{G \ast H}(\dot D) \neq \emptyset$.
This means $H'$ is
$M[G][H]$-generic over $\FR$, so in
$V[G][H]$, as
$j''G \subseteq G \ast H \ast H'$,
$j$ lifts to
$j : V[G] \to M[G][H][H']$ via the definition
$j(i_G(\tau)) = i_{G \ast H \ast H'}(j(\tau))$.
It remains to lift $j$ through the forcing $\FQ$
while working in $V[G][H]$.
To do this, it suffices to show that
$j''H \subseteq j(\FQ)$ generates an
$M[G][H][H']$-generic object $H''$ over $j(\FQ)$.
Given a dense open subset
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = i_{G \ast H \ast H'}(\dot D)$ for some name
$\dot D = j(\vec D)(a)$
for some $a \in {[\gl]}^{< \go}$ and
some function
$\vec D = \la D_\gs : \gs \in {[\gk]}^{|a|} \ra$.
%{[\gk]}^{< \go} \ra$.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $\gk$-distributive, it follows that
%$D' = \bigcap_{\gs \in {[\gk]}^{< \go}} D_\gs$ is
$D' = \bigcap_{\gs \in {[\gk]}^{|a|}} D_\gs$ is
also a dense open subset of $\FQ$. As
$j(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$j''H \cap D \neq \emptyset$. Thus,
$H'' = \{p \in j(\FQ) : \exists q \in j''H
[q \ge p]\}$ is our desired generic object, and
$j$ lifts to
$j : V[G][H] \to M[G][H][H'][H'']$.
This final lifted version of $j$ is
$\gl$ strong since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$.
%Therefore, $V[G][H] \models ``\gk$ is $\gl$ strong''.
%Since $\gl$ may be chosen arbitrarily large,
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l1a}
If $\gl > \gd$ is a cardinal, then
$V \models ``\gd$ is
$\gl$ supercompact''
iff
$V^\FP \models ``\gd$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
Suppose $\gl > \gd$.
Assume first that
$V^\FP \models ``\gd$ is
$\gl$ supercompact''.
Write
$\FP = \FP_0 \ast \dot \FP_1$, where
$|\FP_0| = \go$ and
$\forces_{\FP_0} ``\dot \FP_1$ is
$\ha_1$-strategically closed''.
Since $\FP$ admits a
gap at $\ha_1$, by the
Gap Forcing Theorem of
\cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$
supercompact'' as well.
In fact, the Gap Forcing Theorem of
\cite{H2} and \cite{H3} implies that
for any $\gg \ge \gd$
(cardinal or ordinal) such that
$V^\FP \models ``\gd$ is $\gg$ supercompact'',
$V \models ``\gd$ is $\gg$ supercompact''.
Assume now that
$V \models ``\gd$ is $\gl$ supercompact''
and $\gl$ is regular.
By our assumptions on $V$ and $V_0$,
$\gd < \gk$ and
$\gl < \eta$, where for the
remainder of the proof of
Lemma \ref{l1a},
$\eta$ is the least $V$-measurable
cardinal above $\gd$.
Let $A$ be the set of non-trivial
stages of forcing below $\gd$, and let
$\FP_A$ be the portion of $\FP$
defined using the elements of
$A$. Rewrite
$\FP = \FP_A \ast \dot \FQ$.
We begin by showing that
$V^{\FP_A} \models ``\gd$ is
$\gl$ supercompact''.
If $A$ is bounded below $\gd$,
then if $A$ contains a largest
element $\gg$, since $\gl > \gd$
(meaning that $\gd$ is at least
$2^\gd = \gd^+$ supercompact),
$\gg^* \le \gd$.
By the definition of
the lottery sum done at stage $\gg$,
the forcing selected must have
cardinality below $\gg^*$.
This means that $\FP_A$ is
forcing equivalent to a
partial ordering having size
below $\gd$, so by the
L\'evy-Solovay results \cite{LS},
$V^{\FP_A} \models ``\gd$ is
$\gl$ supercompact''.
If, however, $A$ is bounded below
$\gd$ and doesn't contain a
largest element, then by its
definition, $|\FP_A| < \gd$.
Hence, once again, the results
of \cite{LS} tell us that
$V^{\FP_A} \models ``\gd$ is
$\gl$ supercompact''.
We may therefore assume that
$A$ is unbounded in $\gl$.
Let $j : V \to M$ be an elementary
embedding withnessing the $\gl$
supercompactness of $\gd$ generated
by a normal ultrafilter over
$P_\gd(\gl)$ such that
$M \models ``\gd$ isn't $\gl$ supercompact''.
Write
$j(\FP_A) = \FP_A \ast \dot \FR$, and let
$G$ be $V$-generic over $\FP_A$.
Since $\FP_A$ is an Easton support
iteration and therefore satisfies
$\gd$-c.c.,
standard arguments show that
$M[G]$ remains $\gl$ closed with respect to
$V[G]$.
Further, by opting for trivial forcing at
stage $\gd$ in $M$ if necessary,
we may infer that
$\forces_{\FP_A} ``$The first ordinal
which is a non-trivial stage of forcing
for $\dot \FR$ above $\gd$ is above $\gl$''.
%the definition of $\FP_A$ tells us as
This is since
$\gl < \eta$,
%is below the least
%$V$-measurable cardinal above $\gd$,
$M^\gl \subseteq M$,
$M \models ``\gd$ isn't $\gl$ supercompact'',
and every non-trivial stage of forcing
in $M$ must occur at an $M$-measurable
cardinal.
We may therefore assume without
loss of generality that
$\forces_{\FP_A} ``\dot \FR$ is
$\gl$-strategically closed''.
In addition, the definition of
$\FP_A$ tells us that
$\forces_{\FP_A} ``|\dot \FR| = j(\gd)$'',
meaning by GCH in both $V$ and $M$
that the number of dense open
subsets of $\FR$ in
$M[G]$ is at most $2^{j(\gd)} =
{j(\gd)}^+ = j(\gd^+)$.
Since $M$ is generated via an
ultrapower and $\gl$ is regular, by GCH in $V$,
$|{[\gl]}^{{<}\gd}| = \gl$ and
$|j(\gd^+)| =
|\{ f : f : P_\gd(\gl) \to \gd^+$ is a
function$\}| =
|\{ f : f : \gl \to \gd^+$ is a function$\}| =
2^\gl = \gl^+$.
%regardless of whether $\gl = \gd$ or $\gl > \gd$.
This means we can use the argument given
in Lemma \ref{l1} for the construction of
the generic object
$H'$ to construct in $V[G]$ an
$M[G]$-generic object $H$ over $\FR$ such that
$j''G \subseteq G \ast H$ and lift
in the usual way $j$ to
$j : V[G] \to M[G][H]$
via the definition
$j(i_G(\tau)) = i_{G \ast H}(j(\tau))$.
Hence, it is once again true that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
Observe that $\gd$ can't be a stage
at which a lottery sum forcing is done,
since for such $\gd$,
$\gd$ isn't $2^\gd = \gd^+$ supercompact in $V$,
which is impossible because $\gl > \gd$ and
$V \models ``\gd$ is $\gl$ supercompact''.
Therefore, as $\gd$ is a stage at which
no lottery sum forcing is done
(meaning that only trivial forcing
is done at stage $\gd$), then by the
definition of $\FP$,
$\forces_{\FP_A} ``\dot \FQ$ is
$\eta$-strategically closed''.
%for $\eta$ the least $V$-measurable cardinal
%above $\gd$''.
Since we have already observed that
$\gl < \eta$,
%by our assumptions on $V$,
%$V \models ``\gd$ isn't $\gg$ supercompact'',
%this means that $\gl < \gg$.
we are now able to infer that
$V^{\FP_A \ast \dot \FQ} = V^\FP \models
``\gd$ is $\gl$ supercompact''.
We complete the proof of Lemma \ref{l1a}
by noting what happens when
$V \models ``\gd$ is $\gl$ supercompact''
and $\gl$ is singular.
If this is the case, then if
$\gl$ has cofinality at least $\gd$,
since GCH in $V$ tells us that
$|{[\gl]}^{< \gd}| = \gl$ and
$|j(\gd^+)| = \gl^+$,
the same proof as just given
immediately yields that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
If the
cofinality of $\gl$ is below $\gd$, then by GCH in $V$,
$V \models ``\gd$ is $\gl^+$ supercompact''
(see the introductory section of \cite{AS97a} for a
discussion of this fact). We therefore once again have that
$V^{\FP_A} \models ``\gd$ is both
$\gl^+$ and $\gl$ supercompact''.
Since
$\forces_{\FP_A} ``\dot \FQ$ is
$\eta$-strategically closed'', this means that
$V^{\FP_A \ast \dot \FQ} = V^\FP \models
``\gd$ is $\gl$ supercompact'',
regardless of our cofinality
assumptions on $\gl$.
This completes the proof of Lemma \ref{l1a}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models
``$Level by level equivalence between
strong compactness and supercompactness
holds below $\gk$''.
\end{lemma}
\begin{proof}
We begin by noting that the
case $\gd = \gl$ is immediate,
since for any cardinal
$\sigma$ and any ordinal $\rho$
having cardinality $\sigma$
(in particular, $\rho = \sigma$),
$\sigma$ is measurable iff
$\sigma$ is $\rho$ strongly compact iff
$\sigma$ is $\rho$ supercompact.
%since $\gd$ is $\gd$ strongly
%compact iff $\gd$ is $\gd$
%supercompact iff $\gd$ is measurable.
We may therefore
fix $\gd < \gl < \gk$ such that
$\gd$ and $\gl$ are both regular
cardinals and
$V^\FP \models ``\gd$ is $\gl$
strongly compact''.
It is then automatic that
$V^\FP \models ``\gd$ is measurable'',
and by the argument given in the
first paragraph of the proof of
Lemma \ref{l1a},
$V \models ``\gd$ is measurable''
as well.
We begin by assuming that
for no non-trivial stage of
forcing $\gg$ do we have
$\gd \in (\gg, \gg^*)$.
Note that under these circumstances,
$\FP$ is ``mild'' with respect to $\gd$,
where as in \cite{H2} and \cite{H3},
a partial ordering $\FQ$ is {\it mild}
with respect to a cardinal $\gg$
iff every set of ordinals $x$ in
$V^\FQ$ of size below $\gg$ has
a ``nice'' name $\tau$
in $V$ of size below $\gg$,
i.e., there is a set $y$ in $V$,
$|y| <\gg$, such that any ordinal
forced by a condition in $\FQ$
to be in $\tau$ is an element of $y$.
To see this,
as in the proof of Lemma \ref{l1a},
let $A$ be the set of non-trivial
stages of forcing below $\gd$, and
let $\FP_A$ be the portion of
$\FP$ defined using the elements of
$A$.
Rewrite
$\FP = \FP_A \ast \dot \FQ$. Since
$\gd$ is measurable in $V$,
by the definition of $\FP$,
$\forces_{\FP_A} ``\dot \FQ$ is (at least)
$\gd$-strategically closed''.
Hence, to show that $\FP$ is mild
with respect to $\gd$, it suffices
to show that
$\FP_A$ is mild with respect to $\gd$.
If $A$ contains a largest element $\gg$,
then by the definition of the
lottery sum done at stage $\gg$,
the forcing selected must have
cardinality below $\gg^*$.
Since $\gd \not\in (\gg, \gg^*)$,
$\gg^* \le \gd$.
This immediately yields that
$\FP_A$ is mild with respect to $\gd$.
If $A$ doesn't contain a largest
element, then if $A$ is unbounded in
$\gd$, the definition of $\FP_A$ as
an Easton support iteration
of strategically closed forcing
immediately yields that
$\FP_A$ is mild with respect to $\gd$.
If $A$ doesn't contain a largest
element, then if $A$ is bounded in
$\gd$, the definition of $\FP_A$ is
such that $|\FP_A| < \gd$.
Thus, regardless of the exact
nature of $A$,
$\FP_A$ is mild with respect to $\gd$,
so $\FP$ is mild with respect to $\gd$.
Therefore, by Corollary 7 of
\cite{H2} and Corollary 16 of \cite{H3},
which tell us that a partial ordering
admitting a gap at $\ha_1$
which is mild with respect to a
cardinal $\gr$ creates no new
instances of strong compactness for $\gr$,
it must be the case that
$V \models ``\gd$ is $\gl$ strongly compact''.
By our assumptions on $V_0$ and $V$,
we immediately have that
$V \models ``\gd$ is $\gl$ supercompact''
and that $\gl$ is below the least
$V$-measurable cardinal above $\gd$.
By Lemma \ref{l1a},
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
We next handle what happens when
%complete the proof of Lemma \ref{l2}
%by handling the situation where
there is a non-trivial stage of forcing
$\gg$ for which $\gd \in (\gg, \gg^*)$.
We show that when this holds,
$V^\FP \models ``\gd$ isn't $\gd^+$
strongly compact''.
By the remarks given in the
first paragraph of the proof
of this lemma,
this completes the proof of Lemma \ref{l2}.
Note that under these circumstances, since
$\gg$ isn't $2^\gg$ supercompact in $V$ and
$\gg^*$ is the least cardinal above $\gg$
which is $2^{\gg^*}$ supercompact in $V$,
$\gd$ isn't $2^\gd = \gd^+$ supercompact in $V$.
This means that in $V$, using level by level
equivalence, $\gd$ isn't $\gd^+$ strongly compact.
%exhibits no non-trivial degree of strong compactness.
We now assume
$V^\FP \models ``\gd$ is $\gd^+$
strongly compact'' and
consider the following two cases.
\bigskip
\setlength{\parindent}{0pt}
Case 1: ${(\gd^+)}^V < {(\gd^+)}^{V^\FP}$.
If this is the situation, then as
$\gd$ is measurable and hence a cardinal in $V^\FP$,
$V^\FP \models ``|{(\gd^+)}^V| = \gd$''.
Therefore, by our remarks above,
%since for any ordinal $\rho$
%having cardinality $\gd$,
%$\gd$ is measurable iff $\gd$ is $\rho$
%strongly compact iff $\gd$ is $\rho$ supercompact,
$V^\FP \models ``\gd$ is ${(\gd^+)}^V$
supercompact''.
By the Gap Forcing Theorem of \cite{H2} and
\cite{H3} (see also Theorem 5.3 of \cite{H4}),
$V \models ``\gd$ is ${(\gd^+)}^V = \gd^+$
supercompact'', an immediate contradiction.
\bigskip
Case 2: ${(\gd^+)}^V = {(\gd^+)}^{V^\FP}$.
To handle when this occurs, we use a new
idea due to Hamkins, which will appear in
\cite{H5} in a more general context.
%and which shows that
%$V \models ``\gd$ is $\gd^+$ strongly compact''.
Hamkins' argument is as follows. Let
$G$ be $V$-generic over $\FP$, and let
$j : V[G] \to M[j(G)]$ be
an elementary embedding witnessing
the $\gd^+$ strong compactness of
$\gd$ generated by a $\gd$-additive,
fine ultrafilter over $P_\gd(\gd^+)$
present in $V[G]$.
As $\FP$ admits a gap at $\ha_1$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$,
by the Gap Forcing Theorem of
\cite{H2} and \cite{H3}, the embedding
$j^* = j \rest V : V \to M$ is definable in $V$.
Note that $j$ and $j^*$ agree on the ordinals.
Since $j$ is a $\gd^+$ strong compactness
embedding in $V[G]$, there is some
$X \subseteq j(\gd^+)$, $X \in M[j(G)]$ with
$j '' \gd^+ \subseteq X$ and
$M[j(G)] \models ``|X| < j(\gd^+)$''.
Therefore, since $\gd^+$ is regular in
$V[G]$, $j(\gd^+)$ is regular in $M[j(G)]$,
%and $\gd^+$ is a regular cardinal,
so we can find an
$\ga < j(\gd^+)$ with $\ga > \sup(X)
\ge \sup(j''\gd^+)$.
This means that if $x \subseteq \gd^+$
is such that
$x \subseteq \gb < \gd^+$,
$j(\ga) \not\in j(x) \subseteq j(\gb)$.
But then,
${\cal U} = \{x \subseteq \gd^+ :
\ga \in j^*(x)\}$ defines in $V$
a $\gd$-additive,
uniform ultrafilter over $\gd^+$
which gives measure 1 to sets having
size $\gd^+$.
By a theorem of Ketonen \cite{Ke},
$\gd$ is $\gd^+$ strongly compact in $V$,
a contradiction.
\setlength{\parindent}{1.5em}
\bigskip
Thus, assuming that
$V^\FP \models ``\gd$ is $\gd^+$
strongly compact'' leads to absurdities.
Therefore,
Cases 1 and 2 complete the proof of Lemma \ref{l2}.
\end{proof}
We observe that in the
construction of $V^\FP$,
there will be non-trivial
stages of forcing $\gd$
at which measurable cardinals
between $\gd$ and ${(\gd^*)}^V$
are destroyed. These will occur, e.g.,
whenever the lottery at stage
$\gd$ opts for the appropriate
L\'evy collapse.
We thus can't infer in
general that
$V \models ``\gd$ is $\gl$ supercompact'' iff
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
However, Lemma \ref{l1a} shows that this
is indeed the situation if $\gl > \gd$.
%However, if $\gl > \gd$, then
%this is indeed the situation.
%In fact, we have the following lemma.
\begin{lemma}\label{l3}
$V^\FP \models
``$For any $\gg < \gk$, there is a cardinal
$\gd \in [\gg, \gk)$
such that $\gd$ is $\gd^{+ \gg}$ supercompact''.
\end{lemma}
\begin{proof}
By our discussion at the beginning of
the proof of Theorem \ref{t1},
for any $\gg < \gk$, we can find in $V$
a cardinal $\gd \in [\gg, \gk)$
such that $\gd$ is $\gd^{+ \gg}$ supercompact.
By Lemma \ref{l1a},
$V^\FP \models ``\gd$
is $\gd^{+ \gg}$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Since $\FP$ can be defined so that
$|\FP| = \gk$, by the results of \cite{LS},
$V^\FP \models ``$No cardinal $\gl > \gk$ is
$2^\gl$ supercompact + Level by level
equivalence between strong compactness
and supercompactness holds above $\gk$''.
As $V^\FP \models ``\gk$ is a
strong cardinal'',
any failure of level by level
equivalence between strong
compactness and supercompactness at $\gk$
would have to be reflected below $\gk$,
contradicting Lemma \ref{l2}.
These remarks, together with Lemmas
\ref{l1} - \ref{l3}, complete the proof of
Theorem \ref{t1}.
\end{proof}
We observe that by the Gap Forcing
Theorem of \cite{H2} and \cite{H3},
since $\gk$ is the least strong
cardinal in $V$, $\gk$ is the
least strong cardinal in
$V^\FP$ as well.
\section{Concluding Remarks}\label{s3}
We conclude this paper by
noting that by assuming our initial
model $V_0$ contains a supercompact cardinal,
we are able to ensure that the
large cardinal structure in
terms of supercompactness below
our indestructible strong
cardinal is non-trivial.
Indeed, if the supercompact cardinal
$\gk_0$ has large cardinals such as
inaccessibles, Mahlos, weakly compacts,
Ramseys, etc$.$ above it,
then our methods of proof show that in
$V_0$,
$V$, and consequently in
$V^\FP$,
for any $\gg < \gk$, there are cardinals
$\gd < \gl$, $\gd, \gl \in (\gg, \gk)$
such that $\gd$ is $\gl$ supercompact
and $\gl$ is inaccessible, Mahlo,
weakly compact, Ramsey, etc.
It is possible, however, to force over
a canonical inner model for a strong
cardinal $\gk$ with, e.g., the
partial ordering of Theorem 4.10 of
\cite{H4} to obtain a model in which
$\gk$ is an indestructible strong cardinal.
In this model, no cardinal $\gd$
is $\gd^+$ strongly compact, and
level by level equivalence between
strong compactness and supercompactness
holds trivially.
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\end{thebibliography}
\end{document}