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%
% ------------------------------------------------------------------------------
%
\title{Singular Failures of GCH and Level by Level Equivalence
% and the Level by Level
% Equivalence between Strong Compactness
% and Supercompactness
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
Gitik iteration, $\gk$-weakly closed
partial ordering satisfying the Prikry condition,
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{January 12, 2014}
\begin{document}
\maketitle
\begin{abstract}
We construct a model
for the level by
level equivalence
between strong compactness
and supercompactness in
which below the least
supercompact cardinal $\gk$,
there is an unbounded set
of singular cardinals which witness
the only failures of GCH in the universe.
In this model, the structure
of the class of supercompact
cardinals can be arbitrary.
%This result is best possible.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{A06}, the following theorem was proven.
\begin{theorem}\label{t0}
%Suppose we start with a model
Assume the existence of a model
for ZFC + GCH containing at least
one supercompact cardinal in which level
by level equivalence between
strong compactness and supercompactness
holds.
There is then a forcing extension,
containing exactly the same
supercompact cardinals and preserving
level by level equivalence between
strong compactness and supercompactness,
in which GCH fails on a stationary
subset of the least supercompact cardinal
composed of singular strong limit cardinals
of cofinality $\go$.
%In particular, this forcing extension
%has a stationary set of violations of SCH
%below the least supercompact cardinal.
\end{theorem}
In any model $V^*$ witnessing the conclusions of
Theorem \ref{t0} constructed in \cite{A06},
there are many regular cardinals at which GCH fails (and
in particular, there are many inaccessible cardinals
at which GCH fails).
%In particular, $V^* \models ``2^\gk = \gk^{++}$ where
%$\gk$ is the least supercompact cardinal''.
This is since $V^*$ is built by forcing
over either a model witnessing the conclusions of
\cite[Theorem 1]{A03} or a modification
of this model, both of which contain many inaccessible
cardinals at which GCH fails.
This raises the following
\bigskip\noindent Question:
Is it possible to
construct a model for the level by level equivalence
between strong compactness and supercompactness
%containing at least one supercompact cardinal
in which GCH fails precisely on a stationary
subset of the least supercompact cardinal composed
entirely of singular cardinals?
More weakly, is it possible to
construct a model for the level by level equivalence
between strong compactness and supercompactness
%containing at least one supercompact cardinal
in which GCH fails precisely on an unbounded
subset of the least supercompact cardinal composed
entirely of singular cardinals?
\bigskip The purpose of this paper is to answer
the weaker version of the
above Question in the affirmative. More specifically,
we prove the following theorem.
\begin{theorem}\label{t1}
%Suppose we start with a model
Assume the existence of a model
for ZFC + GCH containing at least
one supercompact cardinal in which level
by level equivalence between
strong compactness and supercompactness
holds.
There is then a forcing extension,
containing exactly the same
supercompact cardinals and preserving
level by level equivalence between
strong compactness and supercompactness,
in which GCH fails precisely on an unbounded
subset $A$ of the least supercompact cardinal
composed of singular cardinals
of cofinality $\go$.
%In particular, this forcing extension
%has a stationary set of violations of SCH
%below the least supercompact cardinal.
\end{theorem}
In the model witnessing the conclusions
of Theorem \ref{t1}, it is of course the case
that every limit cardinal is automatically
a strong limit cardinal. Therefore, $A$ is
composed entirely of strong limit cardinals
and consequently also witnesses failures of SCH.
We note that by Solovay's theorem
of \cite{S}, GCH must hold at any
singular strong limit cardinal above
a strongly compact cardinal.
Thus, as in \cite{A06}, any failures of GCH that occur
on singular strong limit cardinals
must of necessity take place below
the least strongly compact cardinal.
Further, by Silver's theorem \cite{Si},
if GCH fails at a singular strong limit
cardinal $\gd$ of uncountable cofinality,
then it fails at many
singular strong limit cardinals below $\gd$.
In addition, any set having measure
one with respect to a normal measure
over a measurable cardinal must of
course concentrate on regular cardinals.
Therefore, one cannot improve
Theorem \ref{t1} by having
violations of
GCH above the least supercompact cardinal,
or by having $A$ be composed entirely of singular
cardinals of uncountable cofinality, or by
changing ``unbounded'' to normal measure one.
%It is for these two reasons that
%Theorem \ref{t1} gives a result
%that is best possible in terms of
%possible failures of SCH that are
%consistent with the level by level
%equivalence between strong compactness
%and supercompactness.
We take this opportunity to point out that
although the proofs of our new Theorem \ref{t1} and
\cite[Theorem 1]{A06} (Theorem \ref{t0} of this paper) are quite similar,
there are stark differences both in the theorems proven
and the design of the forcing conditions used in each case.
In \cite[Theorem 1]{A06}, the goal is to create a
stationary set of singular failures of GCH below the
least supercompact cardinal $\gk$ in a model satisfying
level by level equivalence between strong compactness
and supercompactness, with no thought to
regulating precisely the GCH pattern below $\gk$.
In the current situation, we are both seeking
and obtaining just such an exact control.
This requires that much greater care be taken in the
construction of the partial orderings employed both in
the proof of Theorem \ref{t1} and its generalization (Theorem \ref{t2})
%the proof sketch of Theorem \ref{t2}
given at the end of the paper.
Specifically, whereas the proof of \cite[Theorem 1]{A06}
only requires an iteration of Cohen forcing followed
by an iteration of Prikry forcing, the forcing conditions
used in this paper are much more intricate.
In particular, for both Theorems \ref{t1} and \ref{t2},
we must iterate very complicated partial orderings
originally due to Gitik (see both
\cite{G02} and \cite[Section 2]{G10}) and make
sure that the relevant definitions can in fact be presented correctly.
This is especially true in the proof of Theorem \ref{t2}, where
each of the two cases found in the definition of the forcing conditions
must be handled quite carefully.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{A06}.
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,
%$\add(\gk, \gk^{++})$ is the
%standard partial ordering for adding
%$\gk^{++}$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
$(\ga, \gb]$ and $[\ga, \gb]$
are as in standard interval notation.
%is the usual half-open
%interval of ordinals which doesn't
%include $\ga$ but includes $\gb$.
If $A$ is any set of ordinals, then
$A'$ is {\em the set of limit points of $A$}.
%$\gk$-directed closed if every directed set of conditions
%of size less than $\gk$ has an upper bound.
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 $\gk$ a cardinal, the
partial ordering $\FP$ is
{\em $\gk$-closed} if for any $\gd < \gk$,
any increasing chain of conditions of length
$\gd$ has an upper bound.
As in \cite{GS}, we will say that
the partial ordering $\FP$
is {\em $\gk$-weakly closed
and satisfies the Prikry condition} if
it meets the following criteria.
\begin{enumerate}
\item $\FP$ has two partial
orderings $\le$ and $\le^*$ with
$\le^* \ \subseteq \ \le$.
\item For every $p \in \FP$
and every statement $\varphi$
in the forcing language
with respect to $\FP$, there
is some $q \in \FP$ such that
$p \le^* q$ and $q \decides \varphi$.
%($q$ decides $\varphi$).
\item The partial ordering
$\le^*$ is $\gk$-closed.
\end{enumerate}
For more details on these definitions,
readers are urged to consult
\cite{GS} or \cite{G10}.
Throughout the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\em Gitik iterations}.
By this we will mean an Easton support iteration
as first given by Gitik in \cite{G}
(and elaborated upon further in
\cite{GS} and \cite{G10}),
where at any stage $\gd$ at which
a nontrivial forcing is done,
we assume the partial ordering
$\FQ_\gd$ with which we force is
$\eta$-weakly closed for some $\eta < \gd$
and satisfies the Prikry condition.
%This is an
%Easton support iteration of
%length $\gk$ with the usual ordering,
%except, roughly speaking, the stems of
%Prikry conditions are extended
%nontrivially only finitely often ---
For additional details and explanations,
%a more precise definition
%and an explanation of associated
%terminology, including what it
%means for one condition to be
%an Easton extension of another,
see \cite{G} or \cite{G10}.
By \cite[Lemmas 1.4 and 1.2]{G},
if the first stage in the definition of
$\FP$ at which a nontrivial forcing is done
is $\eta_0$-weakly closed,
%if $\gd_0$ is the first stage in the
%definition of $\FP$ at which a nontrivial forcing is done,
%and $\forces_{\FP_{\gd_0}} ``\dot \FQ_{\gd_0}$ is $\eta_0$-weakly closed'',
then forcing with
$\FP$ adds no bounded subsets to $\eta_0$.
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}.
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.
Note that 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 $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
When this situation occurs, the
terminology we will henceforth
use is that {\em $\gk$ is a witness
to the Menas exception at $\gl$}.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
We assume familiarity with the
large cardinal notions of
measurability, strongness, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} and \cite{K} for further details.
We just mention that a cardinal $\gk$
will be said to be {\em supercompact
up to a strong cardinal $\gl$} if
$\gk$ is $\gd$ supercompact
for every $\gd < \gl$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal +
Level by level equivalence between
strong compactness and supercompactness holds''.
Let $A_0 = \{ \gd < \gk \mid \gd$ is
the limit of an $\go$ sequence of strong cardinals$\}$.
%and let $A_1 = A_0'$.
%\{\gd \in A_0 \mid \gd$ is a limit point of $A_0\}$.
$A = A_0 - A_0'$
will be the unbounded subset of $\gk$ on
which we will force failures of GCH.
Our partial ordering $\FP$ may therefore be
informally described as
the Gitik iteration of length $\gk$ which, for
$\gd \in A$, does Gitik's forcing of
\cite{G02} (see also \cite[Section 2]{G10})
for forcing $2^\gd = \gd^{++}$
while preserving GCH elsewhere
without either collapsing cardinals or
adding bounded subsets of $\gd$
by using either long or short extenders.
%By its definition, this partial ordering is
%$\gd$-weakly closed and satisfies the Prikry condition.
%with respect to the appropriately chosen sequence
%$\la \gd_i \mid i < \go \ra$ of $V$-strong cardinals which
%is cofinal in $\gd$.
%At all other stages,
The iteration acts trivially otherwise, i.e.,
whenever $\gd \not\in A$.
It is necessary to define $\FP$ more formally, which we do as follows:
%To do this, let $\la \gd_\ga \mid \ga < \gk \ra$
%enumerate $A$ in increasing order.
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
is the Gitik iteration of length $\gk$ such that
$\FP_0 = \{\emptyset\}$.
$\dot \FQ_\gd$ is a term for trivial forcing unless $\gd \in A$.
In order to define $\dot \FQ_\gd$
for $\gd \in A$, let $\la \gk_{n, \gd} \mid n < \go \ra$
be an increasing sequence of $V$-strong cardinals such that
$\sup(\la \gk_{n, \gd} \mid n < \go \ra) = \gd$.
Since $\gd \not\in A_0'$, we may assume without
loss of generality that
$\sup(\{\gk_{n, \gg} \mid \gg \in A$, $n < \go$, and $\gg < \gd\})
< \gk_{0, \gd}$.
%$\sup(\{\gk_{0, \gg} \mid \gg \in A$ and $\gg < \gd\})
%< \gk_{0, \gd}$.
%$\sup(\{\gg \in A \mid \gg < \gk_{0, \gd}\})
%< \gk_{0, \gd}$.
%and that
%$\gk_{0, \gd}$ is the least strong cardinal
%greater than $\sup(\{\gd \in A \mid \gd < \gk_{0, \gd}\})$.
It will then inductively follow that
$\card{\FP_\gd} < \gk_{0, \gd}$, which means that
$\forces_{\FP_\gd} ``$GCH holds for a final segment
of cardinals which starts below $\gk_{0, \gd}$''
(and in fact,
$\forces_{\FP_\gd} ``$GCH holds for all cardinals
greater than or equal to $\card{\FP_\gd}^+$'').
%$\forces_{\FP_\gd} ``$GCH holds for all cardinals
%greater than or equal to $\card{\FP_\gd}^+$''.
This means that there is (more than) enough GCH to allow
the coding and $\Delta$-system arguments of \cite{G02}
or \cite[Section 2]{G10}
to be used so that $\dot \FQ_\gd$ may be taken as a term for
Gitik's forcing of \cite{G02} or \cite[Section 2]{G10}
for forcing $2^\gd = \gd^{++}$
while preserving GCH elsewhere
without either collapsing cardinals or
adding bounded subsets of $\gd$
by using either long or short extenders.
By the arguments of either \cite{G02} or \cite[Section 2]{G10},
for any $\gd \in A$, $\dot \FQ_\gd$ is a term for a
$\gk_{0, \gd}$-weakly closed partial ordering satisfying
the Prikry condition. Consequently, by \cite[Lemmas 1.4 and 1.2]{G}
and the fact $\FP$ is a Gitik iteration,
$V^\FP \models ``$Every $\gd \in A$
is a singular strong limit cardinal
of cofinality $\go$ such that $2^\gd = \gd^{++}$''.
%, i.e., $V^\FP \models ``$Every $\gd \in A$
%is a singular strong limit cardinal
%of cofinality $\go$ violating SCH''.
Because $\FP$ may be defined so that
$\card{\FP} = \gk$, by the results of
\cite{LS}, $V^\FP \models ``\K - \{\gk\}$
is the class of supercompact cardinals
above $\gk$''.
In addition, the usual Easton arguments in tandem with the arguments
of either \cite{G02} or \cite[Section 2]{G10} show that
$V^\FP \models ``$GCH fails precisely on the members of $A$''.
Thus, the proof of
Theorem \ref{t1} is completed
by the following two lemmas.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
\end{lemma}
\begin{proof}
We combine the proofs of
\cite[Lemma 2.3]{A06} and \cite[Lemma 2.1]{A06}.
Since $V^\FP \models ``$GCH fails on
an unbounded set of singular strong
limit cardinals below $\gk$'', by
Solovay's theorem of \cite{S},
$V^\FP \models ``$There are no
strongly compact cardinals below $\gk$''.
Thus, the proof of Lemma \ref{l1}
will be complete once we have shown that
$V^\FP \models ``\gk$ is supercompact''.
To do this,
%note that by the definition of $\FP$, $V \models ``$GCH holds for all
%cardinals greater than or equal to $\gk$''. Consequently,
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.
This means that
$j(\FP) = \FP \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well above $\gl$.
%The remainder of the proof of Lemma \ref{l1} is now
%as in the proof of Lemma \ref{l1}.
We may now show that
$V^\FP \models ``\gk$ is $\gl$ supercompact''
as in the proof of \cite[Lemma 2.1]{A06}.
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 $p_\ga$ are extended. For a more precise
definition, readers are urged to consult either
\cite{G} or \cite{G10}.}
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, $\forces_{\FP} ``\gk$ is
$\gl$ supercompact''.
Since $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds''.
\end{lemma}
\begin{proof}
We modify the proof of \cite[Lemma 2.4]{A06},
quoting verbatim when appropriate.
Since $\FP$ may be defined so that
$\card{\FP} = \gk$, and since
$V \models ``$Level by level equivalence
between strong compactness and
supercompactness holds'', by the results
of \cite{LS},
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds above $\gk$''.
By Lemma \ref{l1},
$V^\FP \models ``$Level by level equivalence
between strong compactness and supercompactness
holds at $\gk$''.
Thus, the proof of Lemma \ref{l2}
will be complete once we have shown 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 such that
$V^\FP \models ``\gd$ is $\gl$ strongly
compact and $\gl$ is regular''. Let
$\gg = \sup(\{\ga < \gd \mid \ga$ is
a nontrivial stage of forcing$\})$, and write
%be the supremum of the nontrivial
%stages of forcing below $\gd$, and write
$\FP = \FP_\gg \ast \dot \FP^\gg$.
%By the definition of $\FP$, let
%$\eta > \gd$, $\eta < \gk$ be the
%least cardinal
%at which $\dot \FP^\gg$ is forced to
%do a nontrivial forcing.
By \cite[Lemmas 1.4 and 1.2]{G} and the definition of $\FP$,
$\forces_{\FP_\gg} ``$Forcing with $\dot \FP^\gg$ adds
no bounded subsets to $\gg^*$, the least $V$-strong cardinal above $\gg$''.
We assume for the time being that $\gl < \gg^*$.
Therefore, we may infer that
$\forces_{\FP_\gg} ``\gd$ is $\gl$
strongly compact'' iff
$\forces_{\FP} ``\gd$ is $\gl$
strongly compact'', i.e.,
$V^{\FP_\gg} \models ``\gd$ is
$\gl$ strongly compact''.
We consider now two cases. \bigskip
\noindent Case 1: $\gg < \gd$.
In this situation, by the
definition of $\FP$,
$\card{\FP_\gg} < \gd$.
Thus, by the results of \cite{LS},
$V^{\FP_\gg} \models ``\gd$ is $\gl$ strongly compact''
iff
$V \models ``\gd$ is $\gl$ strongly compact''.
Since $V \models ``$Level by level equivalence
between strong compactness and supercompactness holds'',
either $V \models ``\gd$ is $\gl$ supercompact'', or
$V \models ``\gd$ is a witness to the Menas
exception at $\gl$''. Again by the results of
\cite{LS},
either $V^{\FP_\gg} \models ``\gd$ is $\gl$ supercompact'', or
$V^{\FP_\gg} \models ``\gd$ is a witness to the Menas
exception at $\gl$''. Regardless of which
of these occurs, $\gd$ does not witness a
failure of level by level equivalence
between strong compactness and supercompactness. \bigskip
\noindent Case 2: $\gg = \gd$.
If this occurs,
then by the definition of
$\FP$, it must be the case that
$\card{\FP_\gd} = \gd$.
Note that since $\gd$
is measurable in $V^{\FP_\gd}$,
$\gd$ must be Mahlo in $V^{\FP_\gd}$
and thus also Mahlo in $V$. Consequently,
$\FP_\gd$ is the direct limit of
$\la \FP_\ga \mid \ga < \gd \ra$, and
$\FP_\gd$ satisfies $\gd$-c.c$.$ in $V$.
This means that
since $\FP_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FP_\gd}$ as well
(this follows because $\gd$ is 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 $V^{\FP_\gd}$),
(the proof of)
\cite[Lemma 8]{A97} (see in particular
the argument found starting in
\cite[third paragraph of page 111]{A97}) or (the proof of)
\cite[Lemma 3]{AC1} tells us that every $\gd$-additive
uniform ultrafilter over a cardinal
$\gb \ge \gd$ present in
$V^{\FP_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gb$ 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
$\gb \in [\gd, \gl]$ 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
$\gb \in [\gd, \gl]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gl$ strongly
compact in $V$.
Observe now that $\gd$ cannot
witness in $V$ the Menas exception
at $\gl$. The reason is that
if this were the case, then
$\gd$ would have to be a limit of
cardinals which are $\gl$ supercompact in $V$.
However, by the definition of
$\FP$, any such cardinal $\gb$
would have to be in $V$ supercompact up
to a strong cardinal, which as we have
already observed,
%in the proof of Lemma \ref{l1},
implies that
$\gb$ is supercompact in $V$.
This is a contradiction, since
$\gb < \gk$, and $\gk$ is the
least supercompact cardinal in $V$.
Thus, by the level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gd$ is $\gl$ supercompact''.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$ such that
$M \models ``\gd$ is not $\gl$ supercompact''.
Write
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$.
As in Lemma \ref{l1}, $M \models ``$No cardinal
$\gb \in (\gd, \gl]$ is strong''.
We may consequently infer that the first
nontrivial stage in $\dot \FQ$
%ordinal in the domain of $\dot \FQ$
is well above $\gl$.
Hence, since in analogy to the
proof of Lemma \ref{l1}, $2^\gl = \gl^+$
in both $V$ and $V^{\FP_\gd}$,
%once again using the fact that
%$V \models ``2^\gl = \gl^+$'',
we may apply the same argument as
given in the proof of Lemma \ref{l1}
to infer that
$V^{\FP_\gd} \models ``\gd$ is
$\gl$ supercompact''.
We have now shown that Lemma \ref{l2} is
true if $\gl < \gg^*$. We consequently assume
that $\gl \ge \gg^*$. In this situation, it
is then the case that regardless of whether
we are in Case 1 or Case 2, some cardinal
below $\gk$ is supercompact in $V$ up to
a strong cardinal and hence is fully supercompact
in $V$. This contradicts that $\gk$ is the least
$V$-supercompact cardinal and therefore
completes the proof of Lemma \ref{l2}.
\end{proof}
Lemmas \ref{l1} -- \ref{l2} complete the proof of Theorem \ref{t1}.
\end{proof}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we make several remarks.
First, we note that
Gitik's forcing of \cite{G02} or \cite[Section 2]{G10}
may be modified to produce
%other values for $2^\gl$ whenever $\gl \in A$.
failures of GCH different from $2^\gl = \gl^{++}$ on
the set $A$ from Theorem \ref{t1}. For details, readers
are referred to either of these papers.
If the forcing is modified so that $2^\gl > \gl^{++}$
for $\gl \in A$, however, GCH will not fail precisely
on the members of $A$, since $2^{\gl^+} > \gl^{++}$
whenever $\gl \in A$. Because of the nature of our
iteration, though, GCH will continue to hold at
all (strongly) inaccessible cardinals,
even with the modification just described.
As our construction shows, $A$ contains none of
its limit points. This raises the question of whether
it is possible to prove a version of Theorem \ref{t1}
where $A$ contains some of its limit points.
A modification of the construction just given shows
that this is indeed the case. Specifically, we have the
following theorem.
\begin{theorem}\label{t2}
%Suppose we start with a model
Assume the existence of a model
for ZFC + GCH containing at least
one supercompact cardinal in which level
by level equivalence between
strong compactness and supercompactness
holds.
There is then a forcing extension,
containing exactly the same
supercompact cardinals and preserving
level by level equivalence between
strong compactness and supercompactness,
in which GCH fails precisely on an unbounded
subset $A$ of the least supercompact cardinal
composed of singular cardinals
of cofinality $\go$.
In addition, the supremum of any $\go$ sequence
of consecutive members of $A$ is a member of $A$ as well.
%which contains an unbounded subset of its limit points.
%In particular, this forcing extension
%has a stationary set of violations of SCH
%below the least supercompact cardinal.
\end{theorem}
\begin{sketch}
%An outline of the argument is as follows:
%Specifically,
As before, suppose
$V \models ``$ZFC + GCH + $\K$ is the
class of supercompact cardinals +
$\gk$ is the least supercompact cardinal +
Level by level equivalence between
strong compactness and supercompactness holds''.
Let $A_0 = \{ \gd < \gk \mid \gd$ is
the limit of an $\go$ sequence of strong cardinals$\}$.
%and let $A_1 = A_0''$.
%\{\gd \in A_0 \mid \gd$ is a limit point of limit points of $A_0\}$.
$A = A_0 - A_0''$
will be the unbounded subset of $\gk$
containing some of its limit points on
which we will force failures of GCH.
Our partial ordering $\FP$ may therefore once again be
informally described as
the Gitik iteration of length $\gk$ which, for
$\gd \in A$, does Gitik's forcing of
\cite{G02} (once again, see also \cite[Section 2]{G10})
for forcing $2^\gd = \gd^{++}$
while preserving GCH elsewhere
without either collapsing cardinals or
adding bounded subsets of $\gd$
by using either long or short extenders as appropriate.
%By its definition, this partial ordering is
%$\gd$-weakly closed and satisfies the Prikry condition.
%with respect to the appropriately chosen sequence
%$\la \gd_i \mid i < \go \ra$ of $V$-strong cardinals which
%is cofinal in $\gd$.
%At all other stages,
The iteration acts trivially otherwise, i.e.,
whenever $\gd \not\in A$.
As we did earlier,
it is necessary to define $\FP$ more formally, which we do as follows:
%To do this, let $\la \gd_\ga \mid \ga < \gk \ra$
%enumerate $A$ in increasing order.
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
is the Gitik iteration of length $\gk$ such that
$\FP_0 = \{\emptyset\}$.
$\dot \FQ_\gd$ is a term for trivial forcing unless $\gd \in A$.
In order to define $\dot \FQ_\gd$
for $\gd \in A$, we assume first that $\gd$ is not a
limit point of $A$, i.e., that $\gd \not\in A_0'$.
The definition is then as given in the proof of Theorem \ref{t1}. More
specifically, let $\la \gk_{n, \gd} \mid n < \go \ra$
be an increasing sequence of strong cardinals such that
$\sup(\la \gk_{n, \gd} \mid n < \go \ra) = \gd$.
Since $\gd \not\in A_0'$, we may assume without
loss of generality that
$\sup(\{\gk_{n, \gg} \mid \gg \in A$, $n < \go$, and $\gg < \gd\})
< \gk_{0, \gd}$.
%$\sup(\{\gg \in A \mid \gg < \gk_{0, \gd}\})
%< \gk_{0, \gd}$.
%and that
%$\gk_{0, \gd}$ is the least strong cardinal
%greater than $\sup(\{\gd \in A \mid \gd < \gk_{0, \gd}\})$.
It will then as before inductively follow that
$\card{\FP_\gd} < \gk_{0, \gd}$, which means that
$\forces_{\FP_\gd} ``$GCH holds for a final segment
of cardinals which starts below $\gk_{0, \gd}$''
(and in fact,
$\forces_{\FP_\gd} ``$GCH holds for all cardinals
greater than or equal to $\card{\FP_\gd}^+$'').
This means that there is once again (more than) enough GCH to allow
the coding and $\Delta$-system arguments of \cite{G02}
or \cite[Section 2]{G10}
to be used so that $\dot \FQ_\gd$ may be taken as a term for
Gitik's forcing of \cite{G02} or \cite[Section 2]{G10}
for forcing $2^\gd = \gd^{++}$
while preserving GCH elsewhere
without either collapsing cardinals or
adding bounded subsets of $\gd$
by using either long or short extenders.
If $\gd \in A$ is also a limit point of $A$, then since
$\gd \not\in A_0''$,
$\sup(\{\gg \in A \mid \gg < \gd$ is a limit point
of A$\}) = \eta < \gd$. If we let
$\la \gd_n \mid n < \go \ra$ be the first $\go$
members of $A$ greater than $\eta$, by the definition of $A$,
it must now be the case that $\sup(\la \gd_n \mid n < \go \ra) = \gd$.
It must further be the case that $\la \gk_{0, \gd_n} \mid n < \go \ra$
is such that $\sup(\la \gk_{0, \gd_n} \mid n < \go \ra) = \gd$.
It will then follow inductively that for each $n < \go$,
$\card{\FP_{\gd_n}} < \gk_{0, \gd_n}$, which means that
$\forces_{\FP_{\gd_n}} ``$GCH holds for a final
segment of cardinals which starts below $\gk_{0, \gd_n}$''
(and in fact,
$\forces_{\FP_{\gd_n}} ``$GCH holds for all cardinals
greater than or equal to $\card{\FP_{\gd_n}}^+$''). As before,
by the arguments of either \cite{G02} or \cite[Section 2]{G10},
for any $n < \go$, $\dot \FQ_{\gd_n}$ is a term for a
$\gk_{0, \gd_n}$-weakly closed partial ordering satisfying
the Prikry condition. Consequently, by \cite[Lemmas 1.4 and 1.2]{G},
the fact $\FP$ is a Gitik iteration,
the arguments of \cite{LS},
the arguments of either \cite{G02} or \cite[Section 2]{G10},
and the usual Easton arguments,
$V^{\FP_\gd} \models ``$The only cardinals in the open interval
$(\eta, \gd)$ at which GCH fails are the first $\go$ members of $A$
greater than $\eta$ + For each $n < \go$, $o(\gk_{0, \gd_n})$
is (at least) $\gk_{0, \gd_n}^{+ \go}$''.
This means that there is once again (more than) enough GCH to allow
the coding and $\Delta$-system arguments of \cite{G02}
or \cite[Section 2]{G10}
to be used so that $\dot \FQ_\gd$ may be taken as a term for
Gitik's forcing of \cite{G02} or \cite[Section 2]{G10}
for forcing $2^\gd = \gd^{++}$
while preserving GCH elsewhere
without either collapsing cardinals or
adding bounded subsets of $\gd$
by using short extenders.
Because forcing with $\FP_\gd$
will have destroyed the strongness of any $V$-strong
cardinal below $\gd$, it will not be possible as before
to use long extenders in the definition of $\dot \FQ_\gd$.
By its definition, $\forces_{\FP_\gd} ``\dot \FQ_\gd$ is
$\gk_{0, \gd_0}$-weakly closed and satisfies the Prikry condition''.
We may view the iteration $\FP$ as being defined on
consecutive ``blocks''of cardinals
$B_\gg = \la \gg_i \mid i \le \go \ra$
of length $\go + 1$, where each $\gg_i \in A$,
$\gg_\go = \sup_{i < \go} \gg_i$, and
for $i < \go$, $\gg_i \not\in A_0'$.
For each $i < \go$, it is the case that
$\forces_{\FP_{\gg_i}} ``\dot \FQ_{\gg_i}$ is
$\gk_{0, \gg_i}$-weakly closed and satisfies the Prikry condition'', i.e.,
$\forces_{\FP_{\gg_i}} ``\dot \FQ_{\gg_i}$ is
$\gk_{0, \gg_0}$-weakly closed and satisfies the Prikry condition''.
In addition, it is also true that
$\forces_{\FP_{\gg_\go}} ``\dot \FQ_{\gg_\go}$ is
$\gk_{0, \gg_0}$-weakly closed and satisfies the Prikry condition''.
If we now let $\dot \FQ$ be a term for the portion
of the iteration which acts on the members of $B_\gg$,
it is then the case that
$\forces_{\FP_{\gg_0}} ``\dot \FQ$ is $\gk_{0, \gg_0}$-weakly
closed and satisfies the Prikry condition''.
This means that slight modifications of the arguments
found in the paragraph immediately preceding the proof
of Lemma \ref{l1} and in the proofs of Lemmas \ref{l1} and \ref{l2}
remain valid and show that $V^\FP$ is a model of ZFC
containing exactly the same supercompact cardinals as $V$ does
in which GCH fails precisely on the members of $A$ and in which
level by level equivalence between strong compactness and
supercompactness holds.
This completes our sketch of the proof of Theorem \ref{t2}.
\end{sketch}
The construction just given may be modified further, so
that $A$ contains, e.g., limit points which are limits
of limit points, limit points which are
limits of limits of limit points, etc. However, our methods do not
allow for $A$ to be stationary, since we always seem to need to
omit from $A$ certain limit points of high order.
%high Cantor-Bendixson rank.
We thus conclude by asking if it is possible for $A$ to
be stationary in Theorem \ref{t1}, in analogy to \cite[Theorem 1]{A06}.
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If $V \models {\rm GCH}$, then by
Lemma 1.5 of \cite{G}, the definition of
$\FP$ just given is valid.
The situation here, however, is far
less clear. This is since without GCH,
we don't know if it is always possible,
for $\gd \in A$, to define
$\FP_{\gd + 1}$ from $\FP_\gd$.
In other words, we don't know
if it is always possible,
for $\gd \in A$, to show that
$\forces_{\FP_\gd} ``\gd$ is measurable''.
We consequently begin with the following lemma.
\begin{lemma}\label{l1}
$\FP$ is well-defined.
In particular, for every
$\gd \in A$, $\forces_{\FP_\gd} ``\gd$
is $\gd^+$ supercompact
(and hence is measurable)''.
\end{lemma}
\begin{proof}
Assume that for $\gd \in A$,
$\FP_\gd$ has been defined.
We show that it is possible
to define $\FP_{\gd + 1}$.
By the L\'evy-Solovay results \cite{LS},
if $\card{\FP_\gd} < \gd$,
$\forces_{\FP_\gd} ``\gd$ is
$\gd^+$ supercompact (and hence
is measurable)''.
Hence, it is possible to do Prikry
forcing over $\gd$ in $V^{\FP_\gd}$
with respect to
any appropriately chosen normal measure,
which means that
$\FP_{\gd + 1}$ can be defined.
We therefore assume that
$\card{\FP_\gd} = \gd$.
If this is the case, then in
spite of the fact
$V \models ``2^\gd = \gd^{++}$'',
The argument of Lemma 1.5 of
\cite{G} is still applicable. Specifically,
let $G$ be $V$-generic over $\FP_\gd$.
Since the choice of $V$ tells us that
$V \models ``2^{\gd^+} = \gd^{++}$'', we may let
$\la \dot x_\ga \mid \ga < \gd^{++} \ra$ be an
enumeration in $V$ of all of the
canonical $\FP_\gd$-names of subsets of
$P_\gd(\gd^+)$. Then,
let $j : V \to M$ be an elementary
embedding witnessing the $\gd^+$
supercompactness of $\gd$
generated by a supercompact
ultrafilter over $P_\gd(\gd^+)$
such that
$M \models ``\gd$ is not $\gd^+$ supercompact''.
This means that
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gd^+$. Consequently,
by Lemmas 1.4 and 1.2 of \cite{G} and the
fact $M^{\gd^+} \subseteq M$, we may
define an increasing sequence
$\la p_\ga \mid \ga < \gd^{++} \ra$
of elements of $j(\FP_\gd)/G$
such that if $\ga < \gb < \gd^{++}$,
$p_\gb$ is an
Easton extension of $p_\ga$,
every initial segment of
the sequence is in $M$, and for every
$\ga < \gd^{++}$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gd^+ \ra \in j(\dot x_\ga)$''.
The remainder of the argument of
Lemma 1.5 of \cite{G} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_\gd(\gd^+))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_\gd(\gd^+))}^{V[G]}$ and
for some $\ga < \gd^{++}$ and some
$\FP_\gd$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j(\FP_\gd)/G}
``\la j(\gb) \mid
\gb < \gd^+ \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gd(\gd^+)$.
Thus, $\forces_{\FP_\gd} ``\gd$ is
$\gd^+$ supercompact (and hence is measurable)''.
It is therefore possible to do Prikry forcing
over $\gd$ in $V^{\FP_\gd}$ with respect to any
appropriately chosen normal measure,
%over $\gd$ in $V^{\FP_\gd}$,
which means that
$\FP_{\gd + 1}$ can be defined.
This completes the proof of Lemma \ref{l1}.
\end{proof}
We note that the argument given in Lemma \ref{l1}
for the preservation of
(a certain amount of) supercompactness
will be used in the proofs of both
Lemmas \ref{l1} and \ref{l2}.
Also, a cardinal $\gk$
will be said to have trivial Mitchell
rank with respect to $\gk^+$ supercompactness
if there is no embedding $j : V \to M$ witnessing
the $\gk^+$ supercompactness of
$\gk$ for which
$M \models ``\gk$ is $\gk^+$ supercompact''.
An ultrafilter ${\cal U}$ generating
this sort of embedding will be said to
have trivial Mitchell rank with
respect to $\gk^+$ supercompactness as well.
If $\gk$ is $\gk^+$ supercompact,
there will always be a $\gk^+$ supercompact
ultrafilter over $P_\gk(\gk^+)$ having
trivial Mitchell rank with respect to
$\gk^+$ supercompactness.
\begin{lemma}\label{l2}
In both $V$ and $V^\FP$,
$A$ is a stationary subset of $\gk$.
\end{lemma}
\begin{proof}
Since $\gk$ is
supercompact, let
$\U$ be a supercompact
ultrafilter over $P_\gk(\gk^+)$.
Let $B = \{\gd < \gk \mid \gd$ is a strong cardinal$\}$.
Take $j : V \to M$ as the elementary
embedding witnessing the $\gk^+$
supercompactness of $\gk$ generated
by ${\cal U}$.
By \cite[Lemma 2.1]{AC2},
$M \models ``\gk$ is a strong cardinal''.
Thus, $B \in \mu$, where
$\mu$ is the normal measure
over $\gk$ generated by $j$.
For any club $C \subseteq \gk$ and ordinal
$\ga < \gk$, since both $B - \{\ga\},
C - \{\ga\} \in \mu$, $C$ must contain
unboundedly in $\gk$ many strong cardinals.
Since $C$ is club, the sup of any $\omega$ of
these strong cardinals must be a member of
$C$ as well.
This of course immediately
yields that $A$ is in $V$
a stationary subset of $\gk$.
Since $\FP$ satisfies $\gk$-c.c.,
%by Exercise H2, page 247 of
by \cite[Exercise H2, page 247]{Ku}, $A$ remains stationary
in $V^\FP$.
This completes the proof of
Lemma \ref{l2}.
\end{proof}
We may assume without loss of generality that no
$\gk_{n, \ga}$ is a limit of strong cardinals.
This is certainly true for $\gk_{0, \ga}$.
If $n > 0$ is such that $\gk_{n, \ga}$
is a limit of strong cardinals, then
consider the open interval $(\gk_{n - 1, \ga}, \gk_{n, \ga})$, and
replace if necessary $\gk_{n, \ga}$ with the least strong cardinal in
$(\gk_{n - 1, \ga}, \gk_{n, \ga})$. This produces a sequence having the
same supremum as $\la \gk_{n, \ga} \mid n < \go \ra$.
Since for each $n < \go$, we may now infer that
$A \cap \gk_{n, \ga}$ is bounded in $A$, it inductively follows that
Then, 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''.
This means that
$j(\FP) = \FP \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gl$. Consequently,
For $\gk$ an inaccessible
cardinal, we will say that
the partial ordering $\FP$ is
%will be said to be
{\em ${\prec}\gk$-weakly closed and
satisfies the Prikry condition} if
it meets the criteria just given,
except that $\le^*$ is
$\gd$-closed for every $\gd < \gk$.
%it is $\gd$-weakly closed and
%satisfies the Prikry condition
%for every $\gd < \gk$.
\footnote{Readers will
note that Gitik and Shelah use
``$\gk$-closed'' to mean what we
would call
``$\gd$-closed for every $\gd < \gk$,''
which is different from our usage.
Our definition of a partial ordering
being $\gk^+$-weakly closed and
satisfying the Prikry condition,
however, has been presented so
as to coincide with theirs.}