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%
% ------------------------------------------------------------------------------
%
\title{Indestructibility under Adding Cohen
Subsets and Level by Level Equivalence
% An Indestructibility Theorem for
% Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal, indestructibility,
Gitik iteration of Prikry-like forcings,
level by level equivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.
In addition, the author wishes
to thank the two referees for many
helpful comments, suggestions,
and corrections which have been
incorporated into the current
version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{January 7, 2008\\
(revised May 30, 2008 and July 20, 2008)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model for the
level by level equivalence between
strong compactness and supercompactness
in which the least supercompact
cardinal $\gk$ has its strong compactness
indestructible under
%both trivial forcing and
adding arbitrarily many Cohen subsets.
There are no restrictions on the large
cardinal structure of our model.
%In this model, the large cardinal structure is arbitrary.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Since the notion of level by level
equivalence between strong
compactness and supercompactness is
central to this paper, we
begin with its definition.
Suppose $V$ is a model of ZFC
in which for all regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
Such a universe will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
%Any model witnessing level by
%level equivalence between strong
%compactness and supercompactness
%also witnesses the Kimchi-Magidor
%property \cite{KM} that the classes
%of strongly compact and supercompact
%cardinals coincide precisely,
%except at measurable limit points.
The exception
is provided by a theorem of Menas \cite{Me},
who showed that if $\gl \ge \gk$
and $\gk$ is a measurable limit
of cardinals $\gd$ each of which is $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact,
but need not be $\gl$ supercompact.
We will therefore say that {\em $\gk$
is a witness to a failure of level
by level equivalence between strong compactness
and supercompactness} iff there is a regular
$\gl > \gk$ such that
$\gk$ is $\gl$ strongly compact but is not
a measurable limit of cardinals $\gd$
which are $\gl$ supercompact, and
$\gk$ is not $\gl$ supercompact.
%We will also say that $\gk$ is a witness to level by
%level equivalence between strong compactness and supercompactness
%iff for every regular cardinal $\gl > \gk$,
%$\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$ supercompact.
Note that inductively, $\gk$
witnesses a failure of level by level
equivalence between strong compactness
and supercompactness iff
$\gl > \gk$ is regular,
$\gk$ is $\gl$ strongly compact but is not
a measurable limit of cardinals $\gd$
which are $\gl$ strongly compact, and
$\gk$ is not $\gl$ supercompact.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
In \cite{A06}, the following question was posed:
Is it possible to construct a model for the
level by level equivalence between strong
compactness and supercompactness in which
the least supercompact cardinal $\gk$ has
its supercompactness indestructible
%under trivial forcing\footnote{As the
%referee has pointed out, any strongly
%compact or supercompact cardinal is
%automatically indestructible under
%trivial forcing.} and
under partial orderings
of the form $\add(\gk, \gl)$, where
$\gl$ is some
ordinal?\footnote{For $\gd$ a regular cardinal and $\gg$
an ordinal,
$\add(\gd, \gg)$ is the
standard partial ordering for adding
$\gg$ many Cohen subsets of $\gd$.
To avoid trivialities, we always assume
that $\gg \ge 1$.}
Note that earlier indestructibility theorems
for the least supercompact cardinal
in conjunction with level by level
equivalence between strong compactness and
supercompactness may be found
in the papers \cite{AH02}, \cite{A03},
\cite{A06}, and \cite{A07}.
The purpose of this paper is to
establish a result which
provides a partial answer to
the aforementioned
question. Specifically, we prove the
following theorem.
\begin{theorem} \label{t1}
Suppose
$V \models ``$ZFC + $\gk$ is the least
supercompact cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk$ is the
least supercompact cardinal +
Level by level equivalence
between strong compactness and
supercompactness holds''.
In $V^\FP$, $\gk$'s
strong compactness is indestructible
%under both trivial forcing and
under partial orderings
of the form $\add(\gk, \gl)$, where
$\gl$ is an arbitrary ordinal.
\end{theorem}
We take this opportunity to
note that \cite[Theorem 1]{A07}
gives the existence of a model
containing a supercompact cardinal
$\gk$ in which both level by level
equivalence between strong compactness
and supercompactness holds and
$\gk$'s strong compactness is
indestructible under {\em arbitrary}
$\gk$-directed closed forcing.
However, in this model, the large
cardinal structure is severely restricted,
as no cardinal is supercompact up to an
inaccessible cardinal. In particular,
$\gk$ is the only supercompact cardinal,
and there are no inaccessible cardinals above $\gk$.
(As mentioned in \cite{A07},
a modification of the forcing presented
allows for a witnessing model which can contain
inaccessibles above $\gk$, but no measurables.
Whether a further modification is possible
allowing measurables above $\gk$ is unknown.)
In Theorem \ref{t1} of this paper, although
we have far less indestructibility
(and therefore prima facie a seemingly
weaker result), there are no restrictions
placed on the large cardinal structure of
our witnessing model, which can contain
supercompact limits of supercompact cardinals, etc.
Thus, because of our methods of proof,
in the presence of level by level
equivalence between strong compactness
and supercompactness,
we have a tradeoff.
%there is a tradeoff.
On the one hand, it is possible to have
full indestructibility
for strong compactness for the least supercompact cardinal
and a rather impoverished large cardinal structure.
On the other hand, it is possible
to have limited indestructibility
for strong compactness for the least supercompact cardinal
and a rich large cardinal structure.
%emphasize that there is a real difference between Theorem \ref{t1} and
%\cite[Theorem 1]{A07}.
We now very briefly give some
preliminary information
concerning notation and terminology.
%For anything left unexplained,
%readers are urged to consult \cite{A03},
%\cite{A01a},
%\cite{AS97a}, or \cite{AS97b}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$},
and $p \decides \varphi$ means that
{\em $p$ decides $\varphi$}.
%For $\gk$ a regular cardinal and $\gl$ an ordinal,
%$\add(\gk, \gl)$ is the standard partial ordering for adding
%$\gl$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
%$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
%$(\a, \b)$ are as in standard interval notation.
$[\ga, \gb]$ and
$(\ga, \gb]$ are as in standard interval notation.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
%If $\FP$ is a reverse Easton iteration
%such that at stage $\ga$, a nontrivial
%forcing is done adding a subset
%of $\gd$, then we will say that
%$\gd$ is in the field of $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
The partial ordering
$\FP$ is {\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is {\em $\gk$-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
%(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
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 $\delta < \gk$.
As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra : \FP \in {\cal A}$
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
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. 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 expressions ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
The partial ordering $\FP$ which
will be used in the proof of
Theorem \ref{t1} is a {\em Gitik
iteration of Prikry-like forcings}.
As such, we take this opportunity
to mention some of the basic
properties of this sort of iteration.
We begin our discussion
with a preliminary definition found in \cite{GS}.
Suppose $\gd$ is inaccessible.
A partial ordering $\FQ$ is
%$\la \FQ, \le, \le^* \ra$ is
{\em $\gd$-weakly closed and satisfies the Prikry
property} if it meets the following criteria.
%$\le^* \ \subseteq \ \le$ satisfies the following properties.
\begin{enumerate}
\item $\FQ$ has two partial orderings
$\le$ and $\le^*$, with
$\le^* \ \subseteq \ \le$.
\item For any condition $p \in \FQ$ and any formula
$\varphi$ in the forcing language with respect to
$\FQ$, there is some
$q \ge^* p$ such that $q \decides \varphi$.
\item For each $\gg < \gd$,
any $\le^*$ increasing chain of
elements of $\FQ$ of length
$\gg$ has an upper bound.
%For any increasing chain
%$p_0 \le^* p_1 \le^* \cdots \le^* p_\ga \le^* \cdots
%(\ga < \gg < \gd)$
%with respect to $\le^*$ of elements of $\FQ$,
%there is an upper bound $q$ for the whole chain.
\end{enumerate}
Given the above definition,
a {\em Gitik iteration $\FP$
of Prikry-like forcings
having length $\gk$} is an
Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra
: \ga < \gk \ra$ in which at each
nontrivial stage $\gd$
(which for our purposes will always
be inaccessible), the forcing used is
$\gd$-weakly closed and satisfies
the Prikry property. In our situation,
each component $\dot \FQ_\gd$ of the
iteration used at a nontrivial
stage $\gd$ has the form
$\dot \FQ^0_\gd \ast \dot \FQ^1_\gd \ast \dot \FQ^2_\gd$,
where $\dot \FQ^0_\gd$ is a term for a
certain $\gd$-directed closed partial ordering,
$\dot \FQ^1_\gd$ is a term for either
trivial forcing or Prikry forcing, and
$\dot \FQ^2_\gd$ is a term for a certain
$\gd$-strategically closed partial ordering.
(That a $\gd$-strategically closed
partial ordering may be used as
a component in a Gitik iteration
of Prikry-like forcings was shown in \cite{A01}.)
The ordering $\le$ on $\FP$ is the standard
%a modification of the usual
ordering used when an Easton support iteration is defined
except,
roughly speaking, the stems of
Prikry conditions are extended
nontrivially only finitely often.
Its definition is given in
%\cite{G, GS, AG},
\cite{G}, \cite{GS}, and \cite{AG},
but for concreteness, we repeat it here.
Specifically, let $p, q \in \FP$, $p = \la \dot p_\ga : \ga < \gk \ra$,
$q = \la \dot q_\ga : \ga < \gk \ra$.
Then $q \ge p$ iff
$q$ extends $p$ with respect to the
usual Easton support iteration
ordering, but in addition, for some finite
$A \subseteq {\hbox{\rm support}}(p)$ and all
$\gb \in {\hbox{\rm support}}(p) - A$,
for $\dot q_\gb = \la \dot r_\gb', \dot s_\gb', \dot t_\gb' \ra$,
$\dot p_\gb = \la \dot r_\gb, \dot s_\gb, \dot t_\gb \ra$,
$q \rest \gb \ast \dot r_\gb' \in \FP_\gb \ast \dot \FQ^0_\gb$,
$p \rest \gb \ast \dot r_\gb \in \FP_\gb \ast \dot \FQ^0_\gb$,
$q \rest \gb \ast \dot r_\gb' \forces_{\FP_\gb \ast \dot \FQ^0_\gb}
``$If $\dot s_\gb'$ and $\dot s_\gb$ are conditions
with respect to Prikry forcing,
then $\dot s_\gb'$ and $\dot s_\gb$
have the same stems''.
%then $| \dot s_\gb' - \dot s_\gb| = 0$,
%where $|\ \ \ |$ is the distance function of \cite{Ma}''.
(Intuitively, this means that
$s_\gb'$ is obtained from $s_\gb$
``by shrinking measure 1 sets''.)
%(Intuitively, $s_\gb' \ge s_\gb$ and $|s_\gb' - s_\gb| = 0$
%means that $s_\gb'$ is obtained from $s_\gb$
%``by shrinking measure 1 sets''.)
Further, if $A = \emptyset$ in the above definition,
then $q$ is called an {\em Easton extension of $p$}.
By \cite[Lemma 1.4]{G},
for any $p \in \FP$ and any formula
$\varphi$ in the language of
forcing with respect to $\FP$,
there is some Easton extension $q$ of $p$
such that $q \decides \varphi$.
By \cite[Lemma 1.2]{G}, if $\gd$ is
the least cardinal %in the iteration
acted upon nontrivially and
for some $\gg < \gd$,
$\la p_\ga : \ga < \gg \ra$ is a
sequence of elements of $\FP$ such that
$\ga < \gb$ implies $p_\gb$ is
an Easton extension of $p_\ga$,
%$\le^*$ increasing chain of elements of $\FP$,
then there is some condition
$p \in \FP$ which is an Easton extension
of each $p_\ga$.
Therefore, in analogy to Prikry forcing,
forcing with $\FP$ adds no new bounded
subsets of $\gd$. Thus,
\cite[Lemmas 1.2 and 1.4]{G} show that $\FP$ is
$\gd$-weakly closed and satisfies the
Prikry property, with $\le^*$ defined as
Easton extension.
We assume familiarity with the
large cardinal notions of
measurability, strongness, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} for further details.
%We do mention, however, that
%the cardinal $\gk$ is ${<} \gl$
%strongly compact or ${<} \gl$ supercompact if
%$\gk$ is $\gg$ strongly compact or $\gg$
%supercompact for every $\gg < \gl$.
We do wish to point out explicitly,
however, that an {\em indestructibly
supercompact cardinal $\gk$} is
one as in \cite{L}, i.e.,
a supercompact cardinal which
remains supercompact after
$\gk$-directed closed forcing.
%We do note, however, that we will say
We say that {\em $\gk$ is supercompact (or strongly compact)
up to the cardinal $\gl$} if
$\gk$ is $\gg$ supercompact
(or $\gg$ strongly compact) for every
$\gg < \gl$.
Finally, for any ordinal $\gd$, $\gd'$
will denote the least strong cardinal
in the ground model $V$ greater than $\gd$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V \models ``$ZFC + $\gk$ is the least
supercompact cardinal''.
By first forcing GCH and then
forcing as in \cite{AS97a},
we also assume in addition that
$V \models ``$GCH + Level by level
equivalence between strong compactness
and supercompactness holds''.
The goal now is to preserve level by
level equivalence between strong compactness
and supercompactness and the fact $\gk$
is the least supercompact cardinal, while
forcing indestructibility of the strong
compactness of $\gk$ under partial
orderings of the form $\add(\gk, \gl)$,
where $\gl$ is an arbitrary ordinal.
The partial ordering $\FP$ used in the
proof of Theorem \ref{t1} is a length $\gk$
Gitik iteration of Prikry-like forcings
$\la \la \FP_\ga, \dot \FQ_\ga \ra
: \ga < \gk \ra$.
Specifically, $\FP_0$ is
trivial forcing $\{\emptyset\}$.
%the partial ordering for adding a Cohen subset of $\go$.
The only nontrivial stages of forcing
$\gd < \gk$ occur at cardinals which are measurable
limits of strong cardinals in $V$. At such a stage $\gd$,
%$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$,
$\dot \FQ_\gd$ has the form
$\dot \FQ^0_\gd \ast \dot \FQ^1_\gd \ast \dot \FQ^2_\gd$,
%where for $\gd'$ the least strong cardinal in $V$
%(and $V^{\FP_\gd}$ as well) above $\gd$,
where $\dot \FQ^0_\gd$ is a
term for the lottery sum of both trivial
forcing and all partial orderings
of the form $\add(\gd, \gl)$
having rank below $\gd'$.
If $\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``$Level by
level equivalence between strong compactness
and supercompactness fails at $\gd$'', then
$\dot \FQ^1_\gd$ is a term for Prikry forcing over
$\gd$ defined with respect to some normal measure;
otherwise, $\dot \FQ^1_\gd$ is a term for trivial forcing.
No matter what $\dot \FQ^1_\gd$ is,
$\dot \FQ^2_\gd$ is a term for the partial ordering
adding a non-reflecting stationary set of ordinals
of cofinality $\go$ to $\gd'$.
(Note that a precise definition of the
partial ordering for adding a
non-reflecting stationary set of ordinals
of cofinality $\go$ to a limit
cardinal $\gg$
%a partial ordering which is $\gd$-strategically closed,
may be found in, e.g.,
\cite[Section 1, page 1898]{AC1}. A property of
this partial ordering which we use here
is that it is $\gr$-strategically closed
for every $\gr < \gg$.)
\begin{lemma}\label{l1}
Suppose $\gd \le \gk$ is a measurable
limit of strong cardinals in $V$. Then
$\forces_{\FP_\gd} ``$Level by level equivalence
between strong compactness and supercompactness
holds at $\gd$''.
\end{lemma}
\begin{proof}
We use ideas from the proof of \cite[Lemma 2.1]{A07}.
Since $V \models {\rm GCH}$, by
\cite[Lemma 1.5]{G},
$\forces_{\FP_\gd} ``\gd$ is a measurable
cardinal''.
We consequently assume inductively that for
every cardinal $\gg < \gd$ which is
a measurable limit
of strong cardinals in $V$,
$\forces_{\FP_\gg} ``$Level
by level equivalence
between strong compactness and supercompactness
holds at $\gg$''.
Let $\gl > \gd$ be a regular cardinal in
$V^{\FP_\gd}$ such that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ strongly compact''.
Note that because $\gd$ is measurable
and hence inaccessible in $V$,
$\FP_\gd$ is the direct limit of
$\la \FP_\ga : \ga < \gd \ra$.
In addition,
$\FP_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FP_\gd}$,
since $\gd$ is
measurable and hence Mahlo in
$V^{\FP_\gd}$ and $\FP_\gd$
is a subordering of the
direct limit of
$\la \FP_\ga : \ga < \gd \ra$
as calculated in $V^{\FP_\gd}$.
Hence, by \cite[Theorem 2.1.5]{H}
(see also the proofs of
\cite[Lemma 3]{AC1} or \cite[Lemma 8]{A97}),
every $\gd$-additive
uniform ultrafilter over a regular cardinal
$\gg \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gg$ in $V$.
Therefore, since the $\gl$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FP_\gd}$-regular cardinal
$\gg \in [\gd, \gl]$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FP_\gd}$,
and since the fact that $\FP_\gd$
%is the direct limit of $\la \FP_\ga : \ga < \gd \ra$
is $\gd$-c.c$.$
tells us that the regular cardinals
at or above $\gd$ in
$V^{\FP_\gd}$ are the same
as those in $V$,
the preceding sentence implies
that every $V$-regular cardinal
$\gg \in [\gd, \gl]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gl$ strongly
compact in $V$.
By the fact $V \models ``$Level by
level equivalence between strong compactness
and supercompactness holds'',
$V \models ``$Either $\gd$ is $\gl$
supercompact, or $\gd$ is a measurable
limit of cardinals $\gg$ which
are $\gl$ supercompact''.
If $V \models ``\gd$ is a measurable
limit of cardinals $\gg$ which are
$\gl$ supercompact'', then
because $\gd$ is a measurable
limit of strong cardinals in $V$
and $\gl \ge \gd$, for some
$\gg < \gd$ which is $\gl$ supercompact in $V$,
$V \models
``\gg$ is supercompact up to $\gg'$''.
By \cite[Lemma 1.1]{A02},
$V \models ``\gg$ is
supercompact''\footnote{As one of the referees
has pointed out, the fact that
$V \models ``\gg$ is supercompact''
follows as well from the fact that because
$\gg'$ is strong,
$V_{\gg'}$ is $\Sigma_2$ correct.
This use of $\Sigma_2$ correctness
will also be applicable elsewhere in the paper.},
%and hence $\Sigma_2$ reflecting
which contradicts that
$V \models ``\gg < \gk$ and $\gk$
is the least supercompact
cardinal''.
Thus, $V \models ``\gd$ is $\gl$ supercompact''.
The proof of Lemma \ref{l1}
will therefore be complete once we have shown that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
To do this, fix $j : V \to M$ an
elementary embedding witnessing the
$\gl$ supercompactness of $\gd$.
By choosing $j(\gd)$ to be minimal,
we may assume in addition that
$M \models ``\gd$ is not $\gl$ supercompact''.
Because $\gl > \gd$ and
${\rm cp}(j) = \gd$, by GCH in $V$,
$M \models ``\gd$ is a measurable
limit of strong cardinals''.
Therefore, since
$j(\gd) > \gl > \gd$, our inductive
assumptions in $V$ imply that in $M$,
$\forces_{\FP_\gd} ``\gd$ is a measurable cardinal
%which is a limit of $M$-measurable cardinals
and level by level equivalence
between strong compactness and supercompactness
holds at $\gd$''.
Also,
%if we opt for trivial forcing
at stage $\gd$ in $M$ in the definition of
$j(\FP_\gd)$, $\dot \FQ_\gd =
{(\dot \FQ^0_\gd)}^M \ast
{(\dot \FQ^1_\gd)}^M \ast
{(\dot \FQ^2_\gd)}^M$, where
${(\dot \FQ^0_\gd)}^M$ is a term for the
stage $\gd$ lottery sum performed in
the definition of $j(\FP_\gd)$,
${(\dot \FQ^1_\gd)}^M$
is a term for either trivial forcing
or Prikry forcing, and
${(\dot \FQ^2_\gd)}^M$
is a term for a partial
ordering which adds a non-reflecting stationary
set of ordinals to ${(\gd')}^M$.
Thus, if we opt for trivial forcing in the
stage $\gd$ lottery sum done
in $M^{\FP_\gd}$ in the definition of
$j(\FP_\gd)$, our inductive assumptions
allow us to take as ${(\dot \FQ^1_\gd)}^M$
a term for trivial forcing.
Because $M \models ``\gd$
is not $\gl$ supercompact'', again by
\cite[Lemma 1.1]{A02}, $M \models ``$No
cardinal $\gg \in (\gd, \gl]$ is strong''.
Hence, ${(\gd')}^M > \gl$.
Consequently, above the appropriate condition
in $M$, $j(\FP_\gd)$ is forcing equivalent to
$\FP_\gd \ast \dot \FP^{*}$, where $\dot
\FP^{*}$ is a term for a Gitik iteration
of Prikry-like forcings whose first
nontrivial stage takes place at
${(\gd')}^M$, i.e., well beyond $\gl$.
Since $V \models {\rm GCH}$, we may now apply the
argument found in \cite[Lemma 2.1]{A06a}
(see also \cite[Lemma 1.5]{G})
to show that
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $\gd \le \gk$ is in $V$ a measurable limit of
strong cardinals,
$\gl > \gd$ is regular, and
$V \models ``\gd$ is $\gl$
supercompact'', then
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
We use ideas from the proof of \cite[Lemma 2.2]{A07}.
Since $V \models ``\gd$ is
$\gl$ supercompact'', let
$j : V \to M$ be an elementary
embedding witnessing this fact such that
$M \models ``\gd$ is not $\gl$ supercompact''.
As in Lemma \ref{l1},
$M \models ``\gd$ is a measurable
limit of strong cardinals''.
Therefore,
by Lemma \ref{l1} applied in $M$,
$\forces_{\FP_\gd} ``\gd$ is
a measurable cardinal and level by
level equivalence between strong
compactness and supercompactness
holds at $\gd$''. Consequently,
the argument given in the last paragraph
of the proof of Lemma \ref{l1} now applies to
show that in $V$,
$\forces_{\FP_\gd} ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
Since in Lemma \ref{l2},
$\gl$ can be arbitrary, and since by
\cite[Lemma 2.1]{AC2} and the succeeding remarks,
any supercompact cardinal is a limit
of strong cardinals, it immediately
follows that in $V^\FP$, $\gk$ is supercompact.
%Thus, $\gk$'s strong compactness is
%automatically indestructible under trivial forcing.
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds below $\gk$''.
\end{lemma}
\begin{proof}
The first part of the proof of Lemma \ref{l3}
uses ideas from the proof of \cite[Lemma 2.4]{A07}.
Suppose $\gd < \gk$ is in $V$ a measurable
limit of strong cardinals and
$V^\FP \models ``\gl > \gd$ is
a regular cardinal and
$\gd$ is $\gl$ strongly compact''.
Since $\gd$ is a stage at
which a nontrivial forcing
takes place, we may write
$\FP = (\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd) \ast
(\dot \FQ^2_\gd \ast \dot \FS') =
\FR \ast \dot \FS$.
%In addition, for the remainder of the
%proof of Lemma \ref{l3}, we note
%Note that we are working
We will be working above a condition forcing
the statement ``$\gl > \gd$ is
a regular cardinal and
$\gd$ is $\gl$ strongly compact''.
With a slight abuse of notation,
we assume this condition is the
trivial one.
By \cite[Lemmas 1.4 and 1.2]{G} and the
definition of $\FP$ as a Gitik
iteration of Prikry-like forcings,
$\forces_\FR ``$Forcing with
$\dot \FS$ does not add
bounded subsets of $\gd'$ but
does add a non-reflecting
stationary set of ordinals of
cofinality $\go$ to $\gd'$''.
By
\cite[Theorem 4.8]{SRK} and the succeeding remarks,
if $\gg$ is $\eta$ strongly compact and
$\eta$ is regular, then $\eta$
must reflect stationary sets of ordinals
of cofinality $\go$. Consequently,
$\forces_\FP ``\gd$ is not $\gd'$
strongly compact''. Thus, by
the preceding three sentences,
$\forces_{\FP} ``\gl < \gd'$'', and
$\forces_\FR ``\gd$ is $\gl$
strongly compact''.
Therefore, since $\FR =
\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd$ and
$\dot \FQ^1_\gd$ must be a
term either for trivial
forcing or for Prikry forcing
over $\gd$, $\dot \FQ^1_\gd$
is a term for trivial forcing, and
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd}
``\gd$ is $\gl$ strongly compact''.
Hence, as $\dot \FQ^1_\gd$ is not
a term for Prikry forcing over $\gd$,
by the definition of $\FP$,
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd}
``$Either $\gd$ is $\gl$ supercompact,
or $\gd$ is a measurable limit
of cardinals $\gg$ which are $\gl$ supercompact''.
The preceding then immediately implies that
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd
\ast \dot \FQ^1_\gd \ast \dot \FS}
``$Either $\gd$ is $\gl$ supercompact,
or $\gd$ is a measurable limit
of cardinals $\gg$ which are $\gl$ supercompact'', i.e.,
$V^\FP \models ``\gd$ is not a witness
to a failure of level by level equivalence
between strong compactness and supercompactness''.
Suppose now that $\gd < \gk$ is not in $V$ a measurable
limit of strong cardinals,
%i.e., that $\gd$ is a trivial stage of forcing,
and that in
addition, $V^\FP \models ``\gl > \gd$ is a
regular cardinal and $\gd$ is $\gl$ strongly compact''.
As before, we will be
working above a condition forcing
the statement ``$\gl > \gd$ is
a regular cardinal and
$\gd$ is $\gl$ strongly compact''.
Once again, with a slight abuse of notation,
we assume this condition is the
trivial one.
Let
$A = \{\gg \le \gd : \gg$ is either a
measurable limit of strong cardinals in $V$ or
$\gg = \eta'$ for some $\eta$ which is in $V$ a
measurable limit of strong cardinals$\}$.
Define $\gr$ as the least ordinal
greater than or equal to
the supremum of $A$ such that
for the factorization
$\FP = \FP_\gr \ast \dot \FQ$,
$\forces_{\FP_\gr} ``$For all $\gg \in A$,
forcing with $\dot \FQ$ adds no new subsets of $\gg$''.
First, note that $\card{A} < \gd$. To see this,
assume not, i.e., suppose that $\card{A} = \gd$.
We then have that $\gd \neq \eta'$
for any $\eta < \gd$ which is in $V$ a measurable
limit of strong cardinals.
This is since $\gd = \eta'$ for such an
$\eta$ would immediately imply that
$\gd$ were the largest member of $A$ and $\card{A} < \gd$.
Because $\card{A} = \gd$ and
$\gd$ is not in $V$ a measurable limit of
strong cardinals, $\gd \not\in A$.
From this, it immediately follows
that $\gr = \gd$.
%$\FP_A$ and $\FP_\gd$ have the same meaning.
Since the definition of $\FP$ then
implies that $\forces_{\FP_\gd} ``$Forcing with
$\dot \FQ$ adds no new subsets of $2^\gd$'',
$V^{\FP_\gd} \models ``\gd$ is measurable''.
In addition, as forcing can't
create new Mahlo cardinals and
$V^\FP \models ``\gd$ is measurable and
hence Mahlo'', $V \models ``\gd$ is a
Mahlo limit of nontrivial stages of forcing''.
The argument given in the second paragraph of
the proof of Lemma \ref{l1} therefore applies
and shows that $V \models ``\gd$ is measurable''.
Hence, $\gd$ is in $V$ a measurable limit of
strong cardinals, a contradiction.
Because $\card{A} < \gd$,
it must therefore be the case that $\FP_\gr$ is
forcing equivalent to a partial ordering
having size less than $\gd$.
This is since otherwise, by the definition of $\FP$,
there must be some $\gg < \gd$
which is in $V$ the largest measurable
limit of strong cardinals below $\gd$.
%the largest nontrivial stage
%of forcing below $\gd$, i.e., $\gg$ is
It is true that $\gd \le \gg'$
(if $\gd > \gg'$, $\card{\FP_\gr} < \gd$).
We cannot then have that $\gd = \gg'$,
since if so,
$V^\FP \models ``\gd$ contains a non-reflecting
stationary set of ordinals of cofinality $\go$
and hence is non-weakly compact''.
As we must thus have that $\gd < \gg'$,
the stage $\gg$ lottery must select a
partial ordering $\FQ^*$ such that
$\card{\FQ^*} \ge \gd$.
It then follows that $2^\gg \ge \gd$ in both
$V^{\FP_\gr}$ and $V^{\FP_\gr \ast \dot \FQ} = V^\FP$,
a contradiction to the fact that
$V^\FP \models ``\gd$ is measurable''.
Now, as $\dot \FQ$ has to be a term for a
Gitik iteration of Prikry-like forcings whose
first nontrivial stage is at
or above $\gd'$, again by
\cite[Lemmas 1.4 and 1.2]{G}, it must be the case that
$\forces_{\FP_\gr} ``$Forcing with $\dot \FQ$ adds
no new bounded subsets of $\gd'$''.
We must therefore have that $\gl < \gd'$, for if not,
then it must be true that in both
$V^{\FP_\gr}$ and $V^{\FP_\gr \ast \dot \FQ} = V^\FP$,
$\gd$ is strongly compact up to $\gd'$.
Since $\FP_\gr$ is forcing equivalent to a partial
ordering having size less than $\gd$, by the
L\'evy-Solovay results \cite{LS},
$V \models ``\gd < \gk$ is strongly compact
up to $\gd'$''. Hence, again by \cite[Lemma 1.1]{A02},
$V \models ``\gd$ is strongly compact'', a contradiction
to the fact that
by level by level equivalence between strong compactness
and supercompactness in $V$,
$V \models ``\gk$ is both the least strongly compact
and least supercompact cardinal''.
Since forcing with $\FQ$ over $V^{\FP_\gr}$
adds no new bounded subsets of $\gd'$,
we may consequently conclude that $\gd$ is $\gl$
strongly compact in both
$V^{\FP_\gr}$ and $V^{\FP_\gr \ast \dot \FQ} = V^\FP$.
Again by the results of \cite{LS},
$V \models ``\gd$ is $\gl$ strongly compact'' as well.
Thus, by level by level equivalence between
strong compactness and supercompactness in $V$,
$V \models ``$Either $\gd$ is $\gl$ supercompact,
or $\gd$ is a measurable limit of cardinals $\gg$
which are $\gl$ supercompact''.
Once more using the results of \cite{LS}
and the fact that
forcing with $\FQ$ over $V^{\FP_\gr}$
adds no new bounded subsets of $\gd'$,
in both
$V^{\FP_\gr}$ and $V^{\FP_\gr \ast \dot \FQ} = V^\FP$,
either $\gd$ is $\gl$ supercompact, or
$\gd$ is a measurable limit of cardinals $\gg$
which are $\gl$ supercompact.
In either case, $\gd$ is not a witness in $V^\FP$
to a failure of level by level equivalence
between strong compactness and supercompactness.
As $\gd$ and $\gl$ were arbitrary,
this completes the proof of Lemma \ref{l3}.
\end{proof}
Because $\FP$ may be defined so that
$\card{\FP} = \gk$, by the results of \cite{LS},
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds above $\gk$''. Since as we have already
observed, $V^\FP \models ``\gk$ is supercompact'',
Lemma \ref{l3} allows us to infer that
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness holds''.
Also, note that it is only in the fifth
paragraph of the proof of Lemma \ref{l3}
that we use that nontrivial components
of our lottery sums are forcings for
adding Cohen subsets, rather than arbitrary
$\gg$-directed closed partial orderings.
\begin{lemma}\label{l4}
$V^\FP \models ``\gk$'s strong compactness
is indestructible under forcing with
partial orderings of the form
$\add(\gk, \eta)$ for $\eta$ an
arbitrary ordinal''.
\end{lemma}
\begin{proof}
We use ideas from the proof of
\cite[Lemma 2.6]{A07}.
Suppose $\FQ \in V^\FP$ is such that
$V^\FP \models ``\FQ = \add(\gk, \eta)$'',
where $\eta$ is an arbitrary ordinal.
Using GCH in $V$, let
$\gl > \max(\card{{\rm TC}(\dot \FQ)})$
be an arbitrary regular cardinal large
enough so that
${(2^{[\gl]^{< \gk}})}^V = \rho =
{(2^{[\gl]^{< \gk}})}^{V^{\FP \ast \dot \FQ}}$
and $\rho$ is regular in both
$V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \rho^+ =
2^\rho = 2^{[\rho]^{< \gk}}$. Take as
$j : V \to M$ an elementary embedding
witnessing the $\gs$ supercompactness
of $\gk$
such that $M \models ``\gk$ is not $\gs$ supercompact''.
Since as before,
$M \models ``$No cardinal $\gd \in (\gk, \gs]$
is strong'',
%by the closure properties of $M$,
$M \models ``\card{{\rm TC}(\dot \FQ)}$ is
below the least strong cardinal above $\gk$''.
Thus, because as earlier,
$M \models ``\gk$ is a measurable
limit of strong cardinals'',
$\dot \FQ$ is a term for an allowable choice
in the stage $\gk$ lottery sum performed in
$M$ in the definition of $j(\FP)$.
We hence
%assume without loss of generality
for the rest of this proof force in $M$
above a condition $p^*$ that opts
for $\FQ$ in $M^\FP$, so that above $p^*$,
$j(\FP)$ is forcing equivalent to
$\FP_\gk \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast (\dot \FQ^2_\gk \ast \dot \FR') =
\FP \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast \dot \FR$.
The remainder of the proof of Lemma \ref{l4},
as with the proof of \cite[Lemma 2.6]{A07},
is as in the proof of \cite[Lemma 2]{AG}.
%For concreteness, we provide a sketch of the proof, and
We therefore refer readers to the
proof of \cite[Lemma 2]{AG} for any
missing details.
By the last sentence of the
preceding paragraph, in $M$,
above $p^*$,
$j(\FP \ast \dot \FQ)$ is
forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FQ^1_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FQ^1_\gk$
is a term for either Prikry forcing
or trivial forcing''. Further, since
$M \models ``$No cardinal $\gd \in (\gk, \gs]$
is strong'',
the first nontrivial
stage in the definition of
$\dot \FR$ after $\gk$ takes place
well above $\gs$.
Consequently, as in \cite[Lemma 2]{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is (equivalently) $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga : \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of
${(P_\gk(\gl))}^{V[G][H]}$,
we can define in $M$ a sequence of
$\FP \ast \dot \FQ \ast \dot \FQ^1_\gk$
names of elements of $\FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ^1_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) : \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\gl \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ \ast \dot \FQ^1_\gk}
``\dot p_\gl$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb : \gb < \gl \ra$''.
In $V[G][H]$, we may now define a set
${\cal U} \subseteq 2^{[\gl]^{< \gk}}$ by
$X \in {\cal U}$ iff
$X \subseteq P_\gk(\gl)$ and for some
$\la r, q \ra \in G \ast H$
and some $q' \in \FQ^1_\gk$
either the trivial condition
(if $\FQ^1_\gk$ is trivial forcing)
or of the
form $\la \emptyset, B \ra$
(if $\FQ^1_\gk$ is Prikry forcing), in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) : \gb < \gl \ra \in
j(\dot X)$'' for some name $\dot X$ of $X$.
As in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G][H]}$, i.e.,
$V[G][H] \models ``\gk$ is
$\gl$ strongly compact''.
Since $\gl$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$No cardinal $\gd < \gk$
is strongly compact''.
\end{lemma}
\begin{proof}
By the definition of $\FP$ and the fact
that for any supercompact cardinal $\gk$,
there are unboundedly in $\gk$ many strong
cardinals below $\gk$, $V^\FP \models ``$Unboundedly
in $\gk$ many cardinals $\gd < \gk$ contain
non-reflecting stationary sets of ordinals
of cofinality $\go$''.
By \cite[Theorem 4.8]{SRK} and the
succeeding remarks, $V^\FP \models ``$No
cardinal $\gd < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
%By taking $\ov V = V^\FP$,
Lemmas \ref{l1} - \ref{l5} and the
intervening remarks complete
the proof of Theorem \ref{t1}.
\end{proof}
By \cite[Theorem 5]{AH02}, we know that if
$\gk$ is indestructibly supercompact and
level by level equivalence between strong
compactness and supercompactness holds, then
no cardinal $\gl > \gk$ is $2^\gl$ supercompact.
However, the proof of \cite[Theorem 5]{AH02} does not
rule out a positive answer to the question
posed at the beginning of this paper.
We therefore conclude by reiterating
this question and asking,
more generally, if it is possible to
have a model containing many supercompact
cardinals in which level by level equivalence
between strong compactness and supercompactness
holds and each supercompact cardinal has
its supercompactness indestructible under
the sorts of partial orderings we have just
used.
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\end{document}