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\title{Indestructible Strong Compactness
and Level by Level Equivalence with No
Large Cardinal Restrictions
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
indestructibility, Gitik iteration, %Prikry forcing,
Magidor iteration of Prikry forcing,
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.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{February 20, 2015}
\date{August 10, 2015}
\begin{document}
\maketitle
\begin{abstract}
We construct a model for the level by level equivalence
between strong compactness and supercompactness with an
arbitrary large cardinal structure in which the least
supercompact cardinal $\gk$ has its strong compactness
indestructible under $\gk$-directed closed forcing.
%We show that indestructibility for strong
%compactness of the least supercompact cardinal $\gk$
%under $\gk$-directed closed forcing
%is compatible with level by level equivalence
%between strong compactness and supercompactness
%in a universe of unrestricted size.
This is in analogy to and generalizes \cite[Theorem 1]{A07}, but
without the restriction that no cardinal is supercompact
up to an inaccessible cardinal.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{A07}, the following theorem was proven.
\begin{theorem}\label{t1}
Suppose $V \models
``$ZFC + There
is a supercompact cardinal''.
There is then a model $\ov V \models
``$ZFC +
There is a supercompact cardinal $\gk$ +
Level by level equivalence between strong
compactness and supercompactness holds''
in which the strong compactness of $\gk$
is indestructible
under $\gk$-directed closed forcing.
\end{theorem}
In $\ov V$, it is the case that
no cardinal is supercompact up to an
inaccessible cardinal. Consequently,
$\gk$ of
necessity must be the only supercompact
cardinal in $\ov V$,
and $\ov V$ does not contain a measurable
cardinal above $\gk$.
Thus, $\ov V$ has a rather restricted
large cardinal structure. This raises
the following
\bigskip
\noindent Question: Is it possible to prove an
analogue to Theorem \ref{t1}, but in a universe with
no restrictions on its large cardinal structure?
\bigskip
The purpose of this paper is to answer the above Question
in the affirmative. Specifically, we will prove the following theorem,
which also extends and generalizes \cite[Theorem 1]{A09} as well as
Theorem \ref{t1} mentioned above.
\begin{theorem}\label{t2}
Suppose $V \models
``$ZFC + $\K \neq \emptyset$ is the (possibly proper)
class of supercompact cardinals + $\gk$ is the least
supercompact cardinal''.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models
``$ZFC + $\K$ is the
class of supercompact cardinals''.
In $V^\FP$,
level by level equivalence between strong
compactness and supercompactness holds, and
the strong compactness of $\gk$
is indestructible
under $\gk$-directed closed forcing.
\end{theorem}
\noindent We observe that since $V \models ``\gk$ is the
least supercompact cardinal + $\K$ is the class of
supercompact cardinals'' and
$V^\FP \models ``\K$ is the class of supercompact cardinals'',
it automatically follows in Theorem \ref{t2} that
$V^\FP \models ``\gk$ is the least supercompact cardinal''.
Note that \cite[Theorem 5]{AH4} shows 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.
Thus, in any universe in which
level by level equivalence between
strong compactness and supercompactness
holds and there is an indestructibly
supercompact cardinal, there must of
necessity be a restricted number of
large cardinals.
This is in sharp contrast to our Theorem \ref{t2},
where we have level by level equivalence between
strong compactness and supercompactness holding in a universe with
an arbitrary large cardinal structure, together with the least
supercompact cardinal $\gk$ having its {\em strong compactness}
indestructible under any $\gk$-directed closed forcing notion.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{A07}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
For $\ga \le \gb$ ordinals,
$[\ga, \gb]$ and $(\ga, \gb]$
are as in standard interval notation.
%is the usual closed interval of ordinals between $\ga$ and $\gb$.
For $\gk$ a cardinal, the
partial ordering $\FP$ is
{\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
If $\gk$ is a regular cardinal, $\add(\gk, 1)$
is the standard partial ordering for adding a
single Cohen subset of $\gk$.
%$\FP$ is $\gk$-strategically closed
%if in the two person game in which
%the players construct an increasing sequence
%$\la p_\ga \mid \ga \le \gk \ra$, where
%player I plays odd stages and player II
%plays even and limit stages (always
%choosing the trivial condition at stage 0),
%player II has a strategy which ensures
%the game can always be continued.
%$\FP$ is ${<}\gk$-strategically closed
%if $\FP$ is $\gd$-strategically closed
%for every cardinal $\gd < \gk$.
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 $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
For $p \in \FP$ and $\varphi$
a formula in the forcing language
with respect to $\FP$,
$p \decides \varphi$ means that
{\em $p$ decides $\varphi$}.
We recall for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
Suppose $V$ is a model of ZFC
%containing supercompact cardinals
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 model will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
The exception is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact, the $\gk$ is $\gl$
strongly compact but need not be $\gl$ supercompact.
%We will also say that {\em $\gk$
%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.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
The partial ordering $\FP$ which
will be used in the proof of
Theorem \ref{t2} is a {\em Gitik iteration}.
By this we will mean an Easton support iteration
as first given by Gitik in \cite{G},
to which we refer readers for a discussion
of the basic properties of
and terminology associated with such an iteration.
For the purposes of this paper,
%at any stage $\gd$ at which
%a nontrivial forcing is done in a Gitik iteration,
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
$\gd$-directed closed partial ordering and
$\dot \FQ^1_\gd$ is a term for either
trivial forcing or a
Magidor iteration \cite{Ma} of Prikry forcing
(although other types of partial orderings
may be used in the general case --- see
\cite{G} for additional details).
We assume familiarity with the
large cardinal notions of
measurability, strongness, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} %and \cite{SRK}
for further details.
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.
Also, we say that {\em $\gk$ is supercompact (strong)
up to an inaccessible cardinal $\gl$} if
$\gk$ is $\gd$ supercompact ($\gd$ strong)
for every $\gd < \gl$.
%(although $\gk$ need not be $\gl$ supercompact).
For any cardinal $\gd$, we adopt as our notation that
$\gd'$ is the least strong cardinal greater than $\gd$
in our ground model $V$.
A measurable cardinal
$\gk$ is said to have {\em trivial Mitchell rank} if
there is no normal measure ${\cal U}$ over $\gk$ with associated
elementary embedding $j : V \to M$ such that $M \models ``\gk$
is measurable''.
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose $V \models ``$ZFC + $\K \neq \emptyset$ is the
(possibly proper) class of supercompact cardinals +
$\gk$ is the least supercompact cardinal''.
Without loss of generality, by first forcing GCH and
then doing the forcing of \cite{AS97a}, we assume in addition that
$V \models ``$GCH + Level by level equivalence between strong
compactness and supercompactness holds''.
The partial ordering $\FP$ used in the proof of Theorem \ref{t2} is
defined as $\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd < \gk \ra$,
the Gitik iteration of length $\gk$
which possibly does nontrivial forcing only at
those stages $\gd < \gk$ which are in $V$ measurable limits of strong
cardinals. At such a $\gd$, we let $\dot \FQ_\gd = \dot \FQ^0_\gd \ast
\dot \FQ^1_\gd$, where $\dot \FQ^0_\gd$ is a term for
the lottery sum of all partial orderings in $V^{\FP_\gd}$ which are
$\gd$-directed closed and have rank below $\gd'$.
%the least $V$-strong cardinal $\gd'$ above $\gd$.
If trivial forcing is selected in
the stage $\gd$ lottery, we do nothing, i.e.,
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``$If
$V^{\FP_\gd \ast \dot \FQ^0_\gd} = V^{\FP_\gd}$, then
$\dot \FQ^1_\gd$ is trivial forcing''.
%the forcing at stage $\gd$ is trivial.
If this is not the case, then let $\FR_\gd$ be the partial ordering
selected by the stage $\gd$ lottery.
Let $\gg = \max(\gd, \card{\FR_\gd})$. $\dot \FQ^1_\gd$ is now a
term for the Magidor iteration %\cite{Ma}
of Prikry forcing
defined in $V^{\FP_\gd \ast \dot \FQ^0_\gd} =
V^{\FP_\gd \ast \dot \FR_\gd}$ which adds a
Prikry sequence to each measurable cardinal in the closed interval
$[\gd, \gg]$, i.e.,
$\forces_{\FP_\gd \ast \dot \FQ^0_\gd} ``$If
$V^{\FP_\gd \ast \dot \FQ^0_\gd} \neq V^{\FP_\gd}$, then
$\dot \FQ^1_\gd$ is the
Magidor iteration of Prikry forcing adding a Prikry sequence to
each measurable cardinal in the closed interval
$[\check \gd, \dot \gg]$, where
$\dot \FR_\gd$ is the partial ordering selected in the stage $\gd$
lottery and
$\dot \gg = \max(\check \gd, \card{\dot \FR_\gd})$''.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A14} and
\cite[Lemma 2.1]{AGS},
quoting verbatim when appropriate.
Let $\gl \ge \gk^+$ be an arbitrary
regular cardinal, and let
$j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of
$\gk$ generated by a supercompact ultrafilter over
$P_\gk(\gl)$ such that
$M \models ``\gk$ is not $\gl$ supercompact''.
It is the case that
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong''.
This is since otherwise, $\gk$
is supercompact up to a strong cardinal in $M$,
and thus, by the proof of \cite[Lemma 2.4]{AC2},
$M \models ``\gk$ is supercompact'', a contradiction.
Further, because $\gl \ge \gk^+ = 2^\gk$, by \cite[Lemma 2.1]{AC2},
$M \models ``\gk$ is a measurable limit of strong cardinals''.
Thus, $\gk$ is a stage in $M$ at which either trivial or
nontrivial forcing might possibly occur.
This means that by forcing above a condition opting
for trivial forcing in the stage $\gk$ lottery held in
$M$ in the definition of $j(\FP)$, we may assume that
$j(\FP)$ is forcing equivalent to $\FP \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well above $\gl$.
We now show that
$V^\FP \models ``\gk$ is $\gl$ supercompact''
as in the proof of \cite[Lemma 2.1]{A14}.
Specifically, we apply the argument
of \cite[Lemma 1.5]{G}. In particular,
let $G$ be $V$-generic over $\FP$.
Since $2^\gl = \gl^+$ in both $V$ and $V[G]$, we may let
%$V \models ``2^{\gl} = \gl^+$'', we may let
$\la \dot x_\ga \mid \ga < \gl^+ \ra$ be an
enumeration in $V$ of all of the
canonical $\FP$-names of subsets of
$P_\gk(\gl)$.
Because $\FP$ is a Gitik iteration of length $\gk$,
$\FP$ is $\gk$-c.c. Consequently, $M[G]$ remains
$\gl$ closed with respect to $V[G]$.
Therefore, by \cite[Lemmas 1.4 and 1.2]{G} and the
fact $M[G]^{\gl} \subseteq M[G]$, we may
define in $V[G]$ an increasing sequence
$\la p_\ga \mid \ga < \gl^+ \ra$
of elements of $j(\FP)/G$
such that if $\ga < \gb < \gl^+$,
$p_\gb$ is an
Easton extension of $p_\ga$,\footnote{Roughly speaking,
this means that $p_\gb$ extends $p_\ga$ as in a usual
Easton support iteration, except that no stems
of any components of $p_\ga$
which are conditions in %either Prikry forcing or a
a Magidor iteration of Prikry forcing
are extended. For a more precise
definition, readers are urged to consult either
\cite{G} or \cite{AG}.}
every initial segment of
the sequence is in $M[G]$, and for every
$\ga < \gl^+$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x_\ga)$''.
The remainder of the argument of
\cite[Lemma 1.5]{G} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_\gk(\gl))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_\gk(\gl))}^{V[G]}$ and
for some $\ga < \gl^+$ and some
$\FP$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j(\FP)/G}
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gk(\gl)$.
Thus, $V^\FP \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk$-directed
closed. Then
$V^{\FP \ast \dot \FQ} \models ``\gk$
is strongly compact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.2]{AGS},
again quoting verbatim when appropriate.
Suppose $\FQ \in V^\FP$
is $\gk$-directed closed.
Let $\gl > \max(\gk, |{\rm TC}(\dot \FQ)|)$ be an
arbitrary regular cardinal large enough so that
$(2^{[\gl]^{< \gk}})^V = \gr =
(2^{[\gl]^{< \gk}})^{V^{\FP \ast \dot \FQ}}$ and
$\gr$ is regular in both $V$ and $V^{\FP \ast \dot \FQ}$,
and let $\gs = \gr^+ = 2^\gr$.
%Let $\gs > \gl > \max(|{\rm TC}(\dot \FQ)|, \gk)$
%be sufficiently large regular cardinals, and let
%Let $\gs > \gl$ be a sufficiently large
%regular cardinal, and take
%be sufficiently large regular cardinals, and let
%be a sufficiently large regular cardinal.
Take $j : V \to M$ as an elementary embedding
witnessing the $\gs$ supercompactness of $\gk$ such that
$M \models ``\gk$ is not $\gs$ supercompact''.
As in Lemma \ref{l1}, by \cite[Lemma 2.1]{AC2}
and the fact $\gs > 2^\gk$,
$\gk$ is a measurable limit of strong cardinals in $M$.
Consequently, by the choice of $\gs$, it is
possible to opt for $\FQ$ in the stage
$\gk$ lottery held in $M$ in the
definition of $j(\FP)$.
Further, as in Lemma \ref{l1}, since
$M \models ``$No cardinal
$\gd \in (\gk, \gs]$ is strong'',
the next nontrivial
forcing in the definition of
$j(\FP)$ takes place well above $\gs$.
Thus, in $M$, above the appropriate condition,
$j(\FP \ast \dot \FQ)$ is forcing equivalent to
$\FP \ast \dot \FQ \ast \dot \FS_\gk
\ast \dot \FR \ast j(\dot \FQ)$, where
$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
is a term for either trivial forcing or a Magidor
iteration of Prikry forcing''.
The remainder of the proof of Lemma \ref{l2}
is as in the proof of \cite[Lemma 2]{AG}.
As in the proof of Lemma \ref{l1},
we outline the argument,
%For concreteness, we provide a sketch
%of the proof,
and refer readers to the
proof of \cite[Lemma 2]{AG} for any
missing details.
%By the last sentence of the
%preceding paragraph, in $M$,
%$j(\FP \ast \dot \FQ)$ is
%forcing equivalent to
%$\FP \ast \dot \FQ \ast \dot \FS_\gk
%\ast \dot \FR \ast j(\dot \FQ)$, where
%$\forces_{\FP \ast \dot \FQ} ``\dot \FS_\gk$
%is a term for either Prikry forcing
%or trivial forcing''. Further, since
%$M \models ``$There are no Mahlo
%cardinals in the interval
%$(\gk, \gs]$'', the next nontrivial
%stage in the definition of
%$j(\FP)$ after $\gk$ takes place
%well above $\gs$. Consequently,
By the last two sentences of the preceding paragraph,
as in \cite[Lemma 2]{AG},
there is a term $\gt \in M$ in the
language of forcing with respect to
$j(\FP)$ such that if
$G \ast H$ is either $V$-generic or
$M$-generic over $\FP \ast \dot \FQ$,
$\forces_{j(\FP)} ``\gt$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''.
In other words, $\gt$ is a term for a
``master condition'' for $\dot \FQ$.
Thus, if
$\la \dot A_\ga \mid \ga < \rho < \gs \ra$
enumerates in $V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of
${(P_\gk(\gl))}^{V[G \ast H]}$,
we can define in $M$ a sequence of
$\FP \ast \dot \FQ \ast \dot \FS_\gk$
names of elements of $\dot \FR
\ast j(\dot \FQ)$,
$\la \dot p_\ga \mid \ga \le \rho \ra$,
such that
$\dot p_0$ is a term for
$\la 0, \tau \ra$ (where $0$
represents the trivial condition
with respect to $\FR$),
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_{\ga + 1}$ is a term for an
Easton extension of $\dot p_\ga$ deciding
$`\la j(\gb) \mid \gb < \gl \ra \in
j(\dot A_\ga)$' '', and for
$\eta \le \rho$ a limit ordinal,
$\forces_{\FP \ast \dot \FQ \ast \dot \FS_\gk}
``\dot p_\eta$ is a term for an Easton
extension of each member of the sequence
$\la \dot p_\gb \mid \gb < \eta \ra$''.
%If we then
Define now in $V[G \ast H]$ 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 $q' \in \FS_\gk$
either the trivial condition
(if $\FS_\gk$ is trivial forcing)
%of the form $\la \emptyset, B \ra$
%(if $\FS_\gk$ is Prikry forcing),
or of the form
$\la \la \emptyset, \dot B_\gd \ra \mid \gd \in [\gk, \max(\gk, \card{\FQ})]$
is measurable$\ra$ (if $\FS_\gk$ is a Magidor iteration of
Prikry forcing), in $M$,
$\la r, \dot q, \dot q', \dot p_\rho \ra
\forces ``\la j(\gb) \mid \gb < \gl \ra \in
\dot X$'' for some name $\dot X$ of $X$. %, then
As in \cite[Lemma 2]{AG},
${\cal U}$ is a $\gk$-additive, fine
ultrafilter over
${(P_\gk(\gl))}^{V[G \ast H]}$, i.e.,
$V[G \ast H] \models ``\gk$ is
$\gl$ strongly compact''.
Since $\gl$ was arbitrary,
%and since trivial forcing is
%$\gk$-directed closed,
this completes the proof of Lemma \ref{l2}.
%Since $\gl$ was arbitrary,
%$V^{\FP \ast \dot \FQ} \models ``\gk$ is
%strongly compact''.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level equivalence between strong
compactness and supercompactness holds''.
\end{lemma}
\begin{proof}
Because $V \models ``$Level by level equivalence between
strong compactness and supercompactness holds'' and
$\card{\FP} = \gk$, by the L\'evy-Solovay results \cite{LS},
$V^\FP \models ``$Level by level equivalence between strong
compactness and supercompactness holds above $\gk$''.
By Lemma \ref{l1}, $V^\FP \models ``\gk$ is supercompact'',
which means that $V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness holds
at $\gk$''.
Thus, to complete the proof of Lemma \ref{l3}, it suffices
to show that
$V^\FP \models ``$Level by level equivalence between strong
compactness and supercompactness holds below $\gk$''.
To do this, let $\gd < \gk$ and $\gl > \gd$ be regular such that
$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
Consider now the following two cases.
\bigskip\noindent Case 1: $\gd$ is a stage
in the definition of $\FP$ at which only
trivial forcing can take place, i.e.,
$\gd$ is not in $V$ a measurable limit of
strong cardinals. Let $\gg = \sup(\{\gs < \gd \mid
\gs$ is a stage of forcing in the definition of $\FP$
at which nontrivial forcing might occur (so $\gs$ is in
$V$ a measurable limit of strong cardinals)$\})$.
Write $\FP = \FP_{\gg + 1} \ast \dot \FQ$.
%By the definition of $\FP$ and
By the fact that only trivial forcing
occurs at stage $\gd$ in the definition of $\FP$, $\gg < \gd$.
If $\gg$ is non-measurable, then
$\gg$ must be either singular or inaccessible. Therefore,
by the definition of $\FP$,
$\card{\FP_\gg} < \gd$ and only trivial forcing is possible at stage $\gg$.
This means that $\FP_{\gg + 1}$ is forcing equivalent to $\FP_\gg$ and
$\card{\FP_{\gg + 1}} < \gd$.
However, if $\gg$ is measurable, i.e.,
if $\gg$ is a stage at which nontrivial forcing could occur,
then note that inductively, it is the case that
$\card{\FP_\gg} \le \gg$. In particular, $\card{\FP_\gg} < \gd$.
It consequently follows that
$\FP_{\gg + 1}$ is forcing equivalent to a partial ordering
having cardinality less than $\gd$.
%in fact, only trivial forcing actually occurs
%(so $\FP_\gg$ and $\FP_{\gg + 1}$ are
%forcing equivalent both to each other and to a partial ordering
%having cardinality less than $\gd$). %$\card{\FP_{\gg + 1}} < \gd$.
This is since otherwise, nontrivial forcing must be selected in the
stage $\gg$ lottery held in the definition of $\FP$
(because if not,
i.e., if trivial forcing is selected in the stage $\gg$ lottery held
in the definition of $\FP$, then %it must be the case that
$\FP_{\gg + 1}$ is forcing equivalent to $\FP_\gg$, a partial ordering
having cardinality less than $\gd$).
Under these circumstances, we must have that
$\forces_{\FP_\gg} ``\dot \FQ_\gg$ is forcing equivalent to a partial
ordering of the form $\dot \FR_\gg \ast \dot \FQ^1_\gg$ where
$\dot \FR_\gg$ is nontrivial'' and
$\forces_{\FP_\gg \ast \dot \FR_\gg} ``\dot \FQ^1_\gg$ is the Magidor
iteration of Prikry forcing which adds a Prikry sequence to every measurable
cardinal in the closed interval $[\gg, \max(\gg, \dot \FR_\gg)]$''.
Since by hypothesis, %$V^{\FP_\gg} \models
$\forces_{\FP_\gg} ``\card{\dot \FR_\gg \ast \dot \FQ^1_\gg} \ge \gd$'',
we must have that
$\forces_{\FP_\gg} ``\card{\dot \FR_\gg} \ge \gd$'' (because if
$p \forces_{\FP_\gg} ``\card{\dot \FR_\gg} < \gd$'', then by the
definition of the Magidor iteration of Prikry forcing,
$p \forces_{\FP_\gg} ``\card{\dot \FR_\gg \ast \dot \FQ^1_\gg} < \gd$''
as well). It then follows that
$V^{\FP_\gg \ast \dot \FR_\gg \ast \dot \FQ^1_\gg} =
V^{\FP_\gg \ast \dot \FQ_\gg} \models ``\gd$ is non-measurable (because
it either contains a Prikry sequence or is non-measurable in
$V^{\FP_\gg \ast \dot \FR_\gg}$ and hence, by the work of \cite{Ma},
remains non-measurable in $V^{\FP_\gg \ast \dot \FR_\gg
\ast \dot \FQ^1_\gg}$)''. Since
$\forces_{\FP_{\gg + 1}} ``$Forcing with $\dot \FQ$ adds no new subsets of
$2^\gd$'',
$V^{\FP_{\gg + 1} \ast \dot \FQ} = V^\FP \models ``\gd$ is non-measurable'',
a contradiction.
Note that the argument given in the preceding two sentences
actually shows that if nontrivial forcing is
selected at stage $\gg$ in the definition of $\FP$, then
$V^\FP \models ``\gg$ is non-measurable''.
This is since any nontrivial $\gg$-directed closed
forcing selected at stage $\gg$ must
have cardinality at least $\gg$, which has as a consequence that
each occurrence of
$\gd$ can be replaced by an occurrence of $\gg$ to obtain the same
contradiction.
We now know that $\FP_{\gg + 1}$ is forcing equivalent to a
partial ordering having cardinality less than $\gd$. We may also infer that
$\gl < \gd'$. This is since otherwise, if $\gl \ge \gd'$, then
$V^\FP \models ``\gd$ is $\gd'$ strongly compact''. Because
$\forces_{\FP_{\gg + 1}} ``$Forcing with $\dot \FQ$ adds no new subsets of
$2^{[\gd']^{< \gd}}$'', it must be the case that
$\forces_{\FP_{\gg + 1}} ``\gd$ is $\gd'$ strongly compact''. However, by
the results of \cite{LS}, $V \models ``\gd$ is $\gd'$ strongly compact''.
As $V \models ``\gd'$ is strong'', again by the proof of
\cite[Lemma 2.4]{AC2}, $V \models ``\gd < \gk$ is strongly compact''.
This contradicts the fact that
$V \models ``\gk$ is the least supercompact cardinal and
level by level equivalence between strong compactness and
supercompactness holds (so in particular, there are no strongly
compact cardinals less than $\gk$)''.
Note that the argument just given showing that
if $V^\FP \models ``\gd$ is $\gd'$ strongly compact'', then
$V \models ``\gd$ is $\gd'$ strongly compact''
remains valid
for any $\gl < \gd'$. Therefore, we may infer that
%still allows us to infer that
$V \models ``\gd$ is $\gl$ strongly compact''.
%, since it is valid for any $\gl \le \gd'$.
Hence, because
level by level equivalence between strong compactness and
supercompactness holds in $V$,
$V \models ``$Either $\gd$ is $\gl$ supercompact, or
$\gd$ is a measurable limit of cardinals which are $\gl$ supercompact''.
The preceding analysis
tells us both that
$\FP_{\gg + 1}$ is forcing equivalent to a
partial ordering having cardinality less than $\gd$ and
$\forces_{\FP_{\gg + 1}} ``$Forcing with $\dot \FQ$ adds no new subsets of
$2^{[\gl]^{< \gd}}$''.
This, together with the results of \cite{LS},
then allow us to infer that in each of
$V^{\FP_{\gg + 1}}$ and $V^{\FP_{\gg + 1} \ast \dot \FQ} = V^\FP$,
either $\gd$ is $\gl$ supercompact, or $\gd$ is a measurable
limit of cardinals which are $\gl$ supercompact.
Thus, in $V^\FP$, $\gd$ cannot witness a failure of level by
level equivalence between strong compactness and supercompactness.
\bigskip\noindent Case 2: $\gd$ is a stage
in the definition of $\FP$ at which nontrivial forcing
can take place, i.e., $\gd$ is in $V$ a measurable
limit of strong cardinals.
As we observed in the proof given in Case 1 above,
if nontrivial forcing is selected at stage $\gd$, then
$V^\FP \models ``\gd$ is non-measurable''.
Since by hypothesis, $V^\FP \models ``\gd$ is
$\gl$ strongly compact'', it must therefore be true that
trivial forcing is selected in the stage $\gd$
lottery held in the definition of $\FP$.
%By the same analysis given in the first paragraph of
%Case 1 when $\gg$ is regular, replacing every occurrence of
%$\gg$ with an occurrence of $\gd$, trivial forcing must be
%selected in the stage $\gd$ lottery held in the definition of $\FP$.
We show that as in Case 1, $\gl < \gd'$. To see this, we note that
if $\gl \ge \gd'$, then the same reasoning as earlier tells us that
$\forces_{\FP_{\gd + 1}} ``\gd$ is $\gd'$ strongly compact''. However,
because trivial forcing is selected at stage $\gd$ in the definition of
$\FP$, meaning that $\FP_{\gd + 1}$ and $\FP_\gd$ are forcing
equivalent, it is actually the case that
$\forces_{\FP_\gd}``\gd$ is $\gd'$ strongly compact''.
Arguing now as in the proof of \cite[Lemma 2.1]{A07}
(and quoting verbatim when appropriate),
note that because $\gd$ is measurable in $V$,
$\FP_\gd$ is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$.
Further, $\FP_\gd$ 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
$\la \FP_\ga \mid \ga < \gd \ra$
as calculated in $V^{\FP_\gd}$.
Hence, by the proofs of
%Theorem 2.1.5 of \cite{H}
\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 $\gd'$
strong compactness of $\gd$ in
$V^{\FP_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FP_\gd}$-regular cardinal
$\gg \in [\gd, \gd']$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FP_\gd}$,
and since the fact $\FP_\gd$
is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$
tells us 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
$\gg \in [\gd, \gd']$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gd'$ strongly
compact in $V$,
%The same reasoning
%As in the proof of Case 1, we may now infer that
%$V \models ``\gd < \gk$ is strongly compact'',
which gives the same contradiction as in Case 1.
We now know that $\gl < \gd'$.
Further, by the argument just given,
$V \models ``\gd$ is $\gl$ strongly compact''.
Therefore, by the fact that level by level equivalence
between strong compactness and supercompactness holds
in $V$, $V \models ``$Either $\gd$ is $\gl$ supercompact,
or $\gd$ is a measurable limit of cardinals which are
$\gl$ supercompact''.
However, the latter cannot occur, since if it did,
some cardinal $\gg < \gd < \gk$ would have to be $\gg'$ supercompact.
Once more, the proof of \cite[Lemma 2.4]{AC2} yields that
$V \models ``\gg$ is supercompact'', contradicting the fact that
$\gk$ is the least supercompact cardinal in $V$.
This means that $V \models ``\gd$ is $\gl$ supercompact'', so we may
apply the argument found in the proof of Lemma \ref{l1} to show that
in $V^{\FP_\gd} = V^{\FP_{\gd + 1}}$, $\gd$ is $\gl$ supercompact.
Once again, write $\FP = \FP_{\gd + 1} \ast \dot \FQ$.
Because $\forces_{\FP_{\gd + 1}} ``$Forcing with $\dot \FQ$ adds
no new subsets of $2^{[\gl]^{< \gd}}$'',
$V^{\FP_{\gd + 1} \ast \dot \FQ} = V^\FP \models ``\gd$ is $\gl$
supercompact''. Consequently, level by level equivalence between
strong compactness and supercompactness holds at $\gd$ in $V^\FP$.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
By Lemma \ref{l1},
$V^\FP \models ``\gk$ is supercompact''.
If $\gd < \gk$ is such that
$V^\FP \models ``\gd$ is supercompact'', then
in particular, %since $\gd' < \gk$,
$V^\FP \models ``\gd$ is $\gd'$ supercompact''.
Since $V^\FP \models ``\gd$ is $\gd'$ strongly compact''
as well, by the proof of Lemma \ref{l3},
$V \models ``\gd$ is $\gd'$ strongly compact''.
This gives the same contradiction as in Lemma \ref{l3}, so
$V^\FP \models ``\gk$ is the least supercompact cardinal''.
Because by its definition, $\card{\FP} = \gk$, by the
results of \cite{LS},
$V^\FP \models ``\K - \{\gk\}$ is the class of supercompact
cardinals above $\gk$''. Thus, $V^\FP \models ``\K$ is the
class of supercompact cardinals''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} -- \ref{l4} complete the proof of Theorem \ref{t2}.
\end{proof}
As mentioned in Section \ref{s1}, \cite[Theorem 5]{AH4}
tells us that if there are sufficiently large cardinals present in
the universe, then level by level equivalence between strong
compactness and supercompactness is incompatible with an
indestructibly supercompact cardinal. Thus, if the universe $V^\FP$
witnessing the conclusions of Theorem \ref{t2} has a rich enough
large cardinal structure, we automatically know that
$\gk$'s supercompactness is not indestructible by some $\gk$-directed
closed forcing $\FR$. By the proof of
\cite[Theorem 5]{AH4}, $\FR$'s rank is fairly
large. In fact, \cite[Lemma 2.4]{AGS} tells us that no matter the nature
of the large cardinals present, after forcing with $\add(\gk, 1)$,
%(the partial ordering adding a single Cohen subset of $\gk$),
not only is
$\gk$ not supercompact, but it has trivial Mitchell rank.
Thus, we conclude by reiterating a question first asked in \cite{AH4}
and still open, namely whether it is possible to have a universe
where there is an indestructibly supercompact cardinal but
containing relatively few large cardinals
in which level by level equivalence between strong compactness and
supercompactness holds.
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\end{document}