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\title{Some Remarks on Tall Cardinals and Failures of GCH
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal, hypermeasurable cardinal,
tall cardinal.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
the referee for helpful comments and
suggestions 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/aapter\\
awapter@alum.mit.edu}
\date{February 13, 2013\\
(revised October 26, 2013)}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
We investigate two global GCH patterns which
are consistent with the existence of a tall cardinal
and also present some related open questions.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with the following definition due to
Hamkins \cite{H09}.
Suppose $\gk$ is a cardinal and $\gl \ge \gk$ is
an arbitrary ordinal.
$\gk$ is {\em $\gl$ tall} if
there is an elementary embedding $j : V \to M$
with critical point $\gk$ such that $j(\gk) > \gl$
and $M^\gk \subseteq M$. $\gk$ is {\em tall} if $\gk$
is $\gl$ tall for every ordinal $\gl$.
In \cite{H09}, Hamkins made a systematic study of
tall cardinals and established many of their basic properties.
He also made the interesting observation
\cite[page 18]{H09} that ``strongness is to tallness as
supercompactness is to strong compactness'' and established
in \cite{H09} many results that either support this
thesis directly or are analogues of conjectures
believed true about strongly compact and supercompact cardinals.
In particular, \cite[Corollary 3.2]{H09} shows the
consistency relative to a strong cardinal of a tall
cardinal $\gk$ with GCH holding at and below $\gk$ yet
failing above $\gk$. This provides a negative solution %answer
to an analogue of a
question about strongly compact cardinals attributed
to Woodin \cite[Question 22.22, page 310]{K},
which asks if $\gk$ is strongly compact and GCH
holds everywhere below $\gk$, then does GCH hold everywhere?
Note that the answer %to Woodin's question
remains unknown
in the context of ZFC (although as shown in \cite{A00},
a negative solution may be obtained when the Axiom of
Choice is false).
In addition, it is possible to invert Woodin's question and ask
if $\gk$ is strongly compact and GCH fails
everywhere below $\gk$, then must GCH fail somewhere
at or above $\gk$ (or is this even consistent)?
%at every regular cardinal $\gd < \gk$, then must GCH fail at some
%regular cardinal $\gd \ge \gk$?
Once again, an answer
%to this question
remains unknown in the context of ZFC
(although as shown in \cite{A12b}, a negative solution
to a weaker version of this question may
be obtained when the Axiom of Choice is false).
Of course, if $\gk$ is either supercompact or strong,
then an easy reflection argument shows
that the answer to the appropriate analogue of the
first of the above questions must be yes.
If $\gk$ is strong, then once again,
an easy reflection argument shows that the
answer to the appropriate analogue of the
second of the above questions
must also be yes (and the fact that it is
relatively consistent for $\gk$ to be strong and for
GCH to fail everywhere below $\gk$ will be addressed
in the proof of Theorem \ref{t1}).
If $\gk$ is supercompact, then $\gk$ is also
strongly compact, so by Solovay's theorem \cite{S},
GCH must hold at any singular strong limit cardinal above $\gk$.
Another easy reflection argument then shows that there
must be unboundedly many in $\gk$ singular strong
limit cardinals at which GCH holds.
This establishes that the theory
``ZFC + $\gk$ is supercompact + GCH fails everywhere
below $\gk$'' is inconsistent.
The purpose of this paper is to show that as with Woodin's original
question, it is possible to obtain negative answers to versions
of this second question for tall cardinals. In particular, we have
the following theorem.
\begin{theorem}\label{t1}
Con(ZFC + There is a supercompact cardinal with infinitely
many inaccessible cardinals above it) $\implies$
Con(ZFC + There is a tall cardinal $\gd$ such that GCH
fails everywhere below $\gd$ yet holds for every cardinal
$\gg \ge \gd$).
\end{theorem}
\noindent If we weaken our requirements to
%the existence of a model containing
a tall cardinal $\gk$ in which GCH
fails only at every regular cardinal below $\gk$ yet
holds for every cardinal $\gd \ge \gk$, then it is possible
to obtain this cardinal pattern
%the existence of such a model
from only a strong cardinal.
Specifically, we have the following theorem.
\begin{theorem}\label{t2}
Con(ZFC + There is a strong cardinal) $\implies$
Con(ZFC + There is a tall cardinal $\gk$ such that GCH
fails at every regular cardinal
below $\gk$ yet holds for every cardinal $\gd \ge \gk$).
\end{theorem}
As corollaries to the proofs of
Theorems \ref{t1} and \ref{t2}, we will be able to force
%over the models witnessing the conclusions of these theorems
and obtain analogous cardinal patterns
in which our tall cardinal
$\gk$ is also the least measurable cardinal.
It will also be possible to show that relative
to the appropriate assumptions, it is the case
that our witnessing models contain a proper
class of strong cardinals.
We very briefly mention that we are assuming a basic knowledge
of large cardinals and forcing, for which we
refer readers to \cite{J, K}.
A basic knowledge of Hamkins' paper \cite{H09}
is also helpful. In particular,
by \cite[Theorem 2.10]{H09}, any strong cardinal
is also a tall cardinal.
For any regular cardinal $\gd$
and any ordinal $\ga$,
$\add(\gd, \ga)$ is the usual partial ordering for adding
$\ga$ Cohen subsets of $\gd$.
The partial ordering $\FP$ is {\em $\gd$-directed closed}
if for any directed $D \subseteq \FP$ such that
$\card{D} < \gd$, there is some $p \in \FP$ extending
each member of $D$.
$\FP$ is {\em $(\gd, \infty)$-distributive}
if for any $\gd$ sequence $\la D_\ga \mid \ga < \gd \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gd} D_\ga$ is dense open as well.
Any partial ordering $\FP$ which is $\gd^+$-directed
closed is automatically $(\gd, \infty)$-distributive.
For $\gl > \gd$, {\em $\gd$ is strong up to $\gl$}
if $\gd$ is $\ga$ strong for every $\ga < \gl$.
If $G \subseteq \FP$ is $V$-generic, we will abuse
notation somewhat by using both $V^\FP$ and $V[G]$
interchangeably.
The following fact is basic and will be used in several
of our proofs.
\begin{fact}\label{f1}
For every cardinal $\gd$, there is a (possibly proper class)
$\gd^+$-directed closed reverse Easton iteration $\FP(\gd)$ such that
after forcing with $\FP(\gd)$, GCH holds for all cardinals
at and above $\gd$.
\end{fact}
\begin{sketch}
Define the (possibly proper class) reverse Easton iteration
$\FP(\gd) = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga \in {\rm Ord} \ra$, where $\FP_0 = \add(\gd^+, 1)$.
For each ordinal $\ga$, if
$\forces_{\FP_\ga} ``$There is a cardinal greater than $\gd$
violating GCH'', then
$\forces_{\FP_\ga} ``\dot \FQ_\ga = \dot \add(\gg^+, 1)$ where
$\gg$ is the least cardinal greater than $\gd$ violating GCH''.
If this is not the case, i.e., if there is some
$p \in \FP_\ga$ such that
$p \forces_{\FP_\ga} ``$All cardinals greater than $\gd$
satisfy GCH'', then we stop our construction and
define $\FP(\gd) = \FP_\ga/p$.
Since by its definition, for any cardinal $\gg$,
$\add(\gg^+, 1)$ is $\gg^+$-directed closed, %by its definition,
$\FP(\gd)$ is $\gd^+$-directed closed.
The arguments found in the proof of
\cite[Theorem 2.1]{A12a} then show that
$\FP(\gd)$ is as desired.
In particular, after forcing with $\add(\gg^+, 1)$
for some $\gg$, all cardinals less than or equal to
$\gg^+$ are preserved, $2^\gg = \gg^+$, $2^\gg$ of
the ground model is collapsed to $\gg^+$, and all
cardinals greater than or equal to $(2^\gg)^+$ of
the ground model are preserved.
\end{sketch}
\section{The Proofs of Theorems \ref{t1} and \ref{t2}
and Related Results}\label{s2}
We turn now to the proofs of our theorems, beginning
with the proof of Theorem \ref{t1}.
\begin{proof}
Let $\ov V \models ``$ZFC + $\gk$ is supercompact +
There are infinitely many inaccessible cardinals
greater than $\gk$''. Without loss of generality,
we assume that $\ov V \models {\rm GCH}$ as well.
By work of Foreman and Woodin \cite{FW},
for any fixed integer $n \ge 1$, we may
assume that $\ov V$ has been generically extended
to a model $V'$ of ZFC in which the following hold:
\begin{enumerate}
\item\label{i1} $\gk$ is $\beth_n(\gk)$ supercompact.
\item\label{i2} GCH fails everywhere below $\gk$.
\item\label{i3} $2^\gk = \gl$ where $\gl$
is weakly inaccessible.
%For every $\gd \in [\gk, \gl)$,
%$2^\gd = \gl$, where $\gl$ is the least weakly
%inaccessible cardinal greater than $\gk$.
\end{enumerate}
\noindent Then, by forcing over $V'$
with $\FP(\gl) = \FP(2^\gk)$,
we may further assume that $V'$ has been generically
extended to a model $V$ in which $\gk$ is $2^\gk$
supercompact, properties (\ref{i2}) and (\ref{i3})
of $V'$ remain true, and GCH holds
for all cardinals greater than or equal to
$\gl$.\footnote{Strictly
speaking, it is not necessary to force over
$V'$ to obtain $V$ where GCH holds at and above
$\gl = 2^\gk$ in order to prove
Theorem \ref{t1} as stated. This is only done to show that
in the model $V^*$ witnessing the conclusions
of Theorem \ref{t1}, there is a proper class
of strong cardinals. This issue will be discussed
further in the paragraph immediately following
the proof of Proposition \ref{p1}.}
This follows by Fact \ref{f1}
(and uses in particular that
$V' \models ``\FP(\gl)$ is $\gl^+$-directed closed'').
We henceforth work over $V$.
By the proof of \cite[Lemma 2.1]{AC2} (see also
the proof of \cite[Proposition 26.11]{K}), since
%$\gk$ is $\beth_4(\gk)$ supercompact and hence
$\gk$ is $2^\gk$ supercompact,
$\{\gd < \gk \mid \gd$ is strong up to $\gk\}$ is
unbounded in $\gk$. Thus, $V_\gk \models ``$There
is a proper class of strong cardinals''. Consequently,
we may let $\gd < \gk$ be such that
$V_\gk \models ``\gd$ is a strong cardinal''.
Consider $(\FP(\gd))^{V_\gk}$, which
we henceforth write as $\FP(\gd)$.
%$\FP(\gd) \subseteq V_\gk$.
%Let $\FP(\gd) \subseteq V_\gk$ be as in Fact \ref{f1}.
%be the partial ordering defined in Fact \ref{f1}.
Since $V_\gk \models ``\FP(\gd)$ is
$\gd^+$-directed closed and
$\gd$ is a tall cardinal'', by \cite[Theorem 3.1]{H09}
(which says that any tall cardinal $\gd$ automatically
has its tallness indestructible under
$(\gd, \infty)$-distributive forcing),
$(V_\gk)^{\FP(\gd)} = V^* \models ``\gd$
is a tall cardinal''. By
Fact \ref{f1} and the fact that
$V \models ``$GCH fails everywhere below $\gk$'',
$V^* \models ``$ZFC + GCH fails everywhere
below $\gd$ yet holds for every cardinal $\gg \ge \gd$''.
This completes the proof of Theorem \ref{t1}.
\end{proof}
\begin{pf}
Theorem \ref{t2} is proven similarly.
Suppose $\ov V \models ``$ZFC + $\gk$ is a strong cardinal''.
By passing to the appropriate inner model (see, e.g., \cite{Z}),
we may assume that $\ov V \models {\rm GCH}$ as well.
Consequently, by work of Friedman and Honzik \cite[Theorem 3.17]{FH},
we may assume that $\ov V$ has been generically extended
to a model $V$ of ZFC such that $V \models ``\gk$ is a
strong cardinal + For
every regular cardinal $\gd$, $2^\gd = \gd^{++}$''.
Consider once again $(\FP(\gk))^V$
(or as above, just $\FP(\gk)$).
%$\FP(\gk) \subseteq V$.
Then as in the proof of Theorem \ref{t1},
$V^{\FP(\gk)} \models ``\gk$ is a tall cardinal
such that GCH fails at every regular cardinal
below $\gk$ yet holds for every cardinal $\gd \ge \gk$''.
This completes the proof of Theorem \ref{t2}.
\end{pf}
In Theorems \ref{t1} and \ref{t2}, our tall
cardinals $\gd$ and $\gk$ are strong in their respective
universes over which we force with
$\FP(\gd)$ and $\FP(\gk)$. Therefore,
in the models witnessing the conclusions
of Theorems \ref{t1} and \ref{t2}, $\gd$ and $\gk$
are both quite large in size (e.g., each is a measurable
limit of measurable cardinals).
Consider what happens if
we first force with the Magidor iteration of Prikry forcing \cite{Ma}
which destroys every measurable cardinal below
either $\gd$ or $\gk$. The work of \cite{Ma}
shows that this partial ordering has size $2^\gd$ or $2^\gk$.
A theorem of Gitik \cite[Lemma 2.1]{AG} shows that
since $\gd$ and $\gk$ are initially strong cardinals,
forcing with this partial ordering preserves the
tallness of either $\gd$ or $\gk$.
If we then force with either $\FP(\gd)$ or $\FP(\gk)$,
since the Magidor iteration of Prikry forcing
preserves both cardinals and the sizes of power sets
(see \cite{A12c} for a discussion of these facts),
we have the following two corollaries to Theorems \ref{t1} and \ref{t2}.
\begin{corollary}\label{c1}
Con(ZFC + There is a supercompact cardinal with infinitely
many inaccessible cardinals above it) $\implies$
Con(ZFC + There is a tall cardinal $\gd$ such that GCH
fails everywhere below $\gd$ yet holds for every cardinal
$\gg \ge \gd$ + $\gd$ is the least measurable cardinal).
\end{corollary}
\begin{corollary}\label{c2}
Con(ZFC + There is a strong cardinal) $\implies$
Con(ZFC + There is a tall cardinal $\gk$ such that GCH
fails at every regular cardinal
below $\gk$ yet holds for every cardinal $\gd \ge \gk$ +
$\gk$ is the least measurable cardinal).
\end{corollary}
It is in fact the case that the model $V^*$
witnessing the conclusions of Theorem \ref{t1}
(and hence, by the L\'evy-Solovay results
\cite{LS}, the model witnessing the conclusions
of Corollary \ref{c1} as well) contains a
proper class of strong cardinals.
In particular, we have the following proposition.
\begin{proposition}\label{p1}
$V^* \models ``$There is a proper class of strong cardinals''.
\end{proposition}
\begin{proof}
With a slight abuse of notation, write
$\FP(\gd)$ for $(\FP(\gd))^{V_\gk}$.
%$\FP(\gd)$ as defined over $V_\gk$.
It is then the case that $\FP(\gd) \in V$.
We will show that $V^{\FP(\gd) \ast \dot \add(\gk^+, 1)} \models
``\gk$ is $2^\gk = \gk^+$ supercompact''.
This suffices, since as in the
proof of Theorem \ref{t1}, it is then true that
$\{\gd < \gk \mid \gd$ is strong up to $\gk\}$ is unbounded in
$\gk$ in $V^{\FP(\gd) \ast \dot \add(\gk^+, 1)}$. Because
$\forces_{\FP(\gd)} ``\dot \add(\gk^+, 1)$ is $\gk^+$-directed closed'',
%for any $\gd < \gk$ such that
%$V^{\FP(\gd) \ast \dot \add(\gk^+, 1)} \models ``\gd$ is $\gk$ strong'',
%the $(\gd, \gk)$-extender ${\cal E} = \la E_a \mid a \in [\gk]^{< \go} \ra
%\in V^{\FP(\gd) \ast \dot \add(\gk^+, 1)}$ witnessing this fact
%is also a member of $V^{\FP(\gd)}$. Thus,
$\{\gd < \gk \mid \gd$ is strong up to $\gk\}$ is unbounded in
$\gk$ in $V^{\FP(\gd)}$ as well. From this, it immediately follows that
in $V^* = (V_\gk)^{\FP(\gd)}$, there is a proper class of
strong cardinals.
To see this, we use an argument found in the proof of
\cite[Theorem 2.1]{A12a}, quoting verbatim when appropriate.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$ in $V$
generated by a supercompact ultrafilter over
$P_\gk(\gl)$. In particular, $M^{\gl} \subseteq M$.
We use a standard lifting argument
to show that $j$ lifts in $V^{\FP(\gd) \ast \dot \add(\gk^+, 1)}$ to
$j : V^{\FP(\gd) \ast \dot \add(\gk^+, 1)}
\to M^{j(\FP(\gd) \ast \dot \add(\gk^+, 1))}$. Specifically, let
$G_0$ be $V$-generic over $\FP(\gd)$, and let
$G_1$ be $V[G_0]$-generic over
$\add(\gk^+, 1)$.
Observe that $j(\FP(\gd) \ast \dot \add(\gk^+, 1)) =
\FP(\gd) \ast \dot \add(\gk^+, 1) \ast
\dot \FQ \ast \dot \add(j(\gk^+), 1)$.
Working in $V[G_0][G_1]$,
we first note that since $\FP(\gd) \ast \dot \add(\gk^+, 1)$
is $(2^\gk)^+ = \gl^+$-c.c.,
%and $\add(\gk^+, 1)$ is $\gk^+$-directed closed in $V[G_0]$,
$M[G_0][G_1]$ remains $\gl$ closed with respect to $V[G_0][G_1]$.
This means that $\FQ$ is $\gl^+$-directed closed
in both $M[G_0][G_1]$ and $V[G_0][G_1]$.
Since $M[G_0][G_1] \models ``\card{\FQ} = j(\gk)$'',
the number of dense open subsets of $\FQ$ present
in $M[G_0][G_1]$ is $(2^{j(\gk)})^M$.
In $V$,
since $M$
is given via an ultrapower by a supercompact
ultrafilter over $P_\gk(2^\gk)$,
this is calculated as
$\card{\{f \mid f : [2^\gk]^{< \gk} \to 2^\gk\}} =
\card{\{f \mid f : 2^\gk \to 2^\gk\}} = 2^{2^\gk} = 2^\gl$.
Since $V \models ``2^\gl = \gl^+$''
%``2^{2^\gk} = (2^\gk)^+ = \gl^+$''
and $\gl^+$ is preserved from $V$ to
$V[G_0][G_1]$, we may let
$\la D_\ga \mid \ga < \gl^+ \ra \in V[G_0][G_1]$
enumerate the dense open subsets of $\FQ$ present in $M[G_0][G_1]$.
We may now use the fact that $\FQ$ is
$\gl^+$-directed closed in $V[G_0][G_1]$ to meet each $D_\ga$
and thereby construct in $V[G_0][G_1]$
an $M[G_0][G_1]$-generic
object $H_0$ over $\FQ$. Our construction guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast H_0$,
so $j$ lifts in $V[G_0][G_1]$ to
$j : V[G_0] \to M[G_0][G_1][H_0]$.
It remains to lift $j$ in $V[G_0][G_1]$ through
$\add(\gk^+, 1)$.
Because $V[G_0] \models ``\card{\add(\gk^+, 1)}
= 2^\gk = (2^\gk)^{V} = \gl$'',
$M[G_0][G_1][H_0] \models ``\card{\add(j(\gk^+), 1)}
= 2^{j(\gk)} = (2^{j(\gk)})^M$''.
Therefore, since
$M[G_0][G_1][H_0]$ remains $\gl$ closed
with respect to $V[G_0][G_1]$,
$M[G_0][G_1][H_0] \models ``\add(j(\gk^+), 1)$ is
$j(\gk^+)$-directed closed'', and
$j(\gk^+) > j(\gk) > \gl$, there is a master condition
$q \in \add(j(\gk^+), 1)$ for $j '' \{p \mid p \in G_1\}$.
Further, the number of dense open subsets of
$\add(j(\gk^+), 1)$ present in $M[G_0][G_1][H_0]$ is
$(2^{2^{j(\gk)}})^M$.
This is calculated in $V$ as
$\card{\{f \mid f : [2^\gk]^{< \gk} \to 2^{2^\gk}\}} =
\card{\{f \mid f : 2^\gk \to (2^\gk)^+\}} =
\card{\{f \mid f : \gl \to \gl^+\}} = 2^\gl =
(2^\gk)^+ = \gl^+$.
Working in $V[G_0][G_1]$,
since $\add(j(\gk^+), 1)$ is $\gl^+$-directed
closed in both $M[G_0][G_1][H_0]$ and $V[G_0][G_1]$,
we may consequently use
the arguments of the preceding paragraph to construct
an $M[G_0][G_1][H_0]$-generic object $H_1$
over $\add(j(\gk^+), 1)$ containing $q$.
Since by the definition of $H_1$,
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast H_0 \ast H_1$,
$j$ lifts in $V[G_0][G_1]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][H_0][H_1]$.
As $V[G_0][G_1] \models ``\card{\gl} = \gk^+$'',
this means that
$V^{\FP(\gd) \ast \dot \add(\gk^+, 1)} \models
``\gk$ is $2^\gk = \gk^+$ supercompact''.
This completes the proof of Proposition \ref{p1}.
\end{proof}
We take this opportunity to observe that if we did not
wish to show that $V^* \models ``$There is a proper
class of strong cardinals'', it would be unnecessary to
force over $V'$ with $\FP(\gl)$.
The proof of Proposition \ref{p1} requires
a sufficient amount of GCH above $\gl$, which
is why we needed to generically extend $V'$ to $V$.
%to some model witnessing enough GCH above $\gl$. %to $V$.
(We could have, of course, only forced exactly the amount
of GCH required to allow the arguments of Proposition \ref{p1}
to go through, as opposed to forcing GCH to hold for
all cardinals greater than or equal to $\gl$.)
We turn our attention now to proving a version of
Theorem \ref{t2} in which our witnessing model
contains a proper class of strong cardinals.
One might expect to proceed by starting with
a model containing a proper class of strong cardinals
in which GCH holds, then use \cite[Theorem 3.17]{FH} to
obtain a model containing a proper class of
strong cardinals in which $2^\gd = \gd^{++}$
for every regular cardinal $\gd$, and then force GCH
to hold on a proper class of cardinals above a fixed
strong cardinal $\gk$. The problem is that the
usual argument for the preservation of a strong cardinal
$\gl$ after a reverse Easton iteration (as found, e.g., in
\cite[Theorem 4.10]{H4}) requires that $2^\gl = \gl^+$
in the model over which the forcing has been done.
This will, of course, not be the case in the
approach just suggested, and is the reason
Proposition \ref{p1} is used to show that
$V^* \models ``$There is a proper class of strong
cardinals''. It is, however, possible
to proceed in a different fashion, by using
stronger assumptions. Specifically, we have the
following result.
\begin{theorem}\label{t3}
Con(ZFC + There is a cardinal $\gl$ such that $\gl$ is $2^\gl$
supercompact and $2^\gd = \gd^{++}$ for every regular
cardinal $\gd \le \gl$) $\implies$ Con(ZFC + There is a tall cardinal
$\gk$ such that GCH fails at every regular cardinal below
$\gk$ yet holds for every cardinal $\gd \ge \gk$ +
There is a proper class of strong cardinals).
\end{theorem}
\noindent Note that a model witnessing the hypotheses
of Theorem \ref{t3} may be obtained starting with
a model for ``ZFC + There exists a supercompact cardinal''
(or even weaker assumptions --- for the
optimal hypotheses, see \cite{CoMa}) by using Menas' techniques
from \cite[Theorem 18]{Me76}.
\begin{proof}
Let $\ov V \models
``$ZFC + There is a cardinal $\gl$ such that $\gl$ is $2^\gl$
supercompact and $2^\gd = \gd^{++}$ for every regular
cardinal $\gd \le \gl$''.
As in the proof of Theorem \ref{t1},
we may begin by forcing with $\FP(\gl^{++}) = \FP((2^\gl))$
to generically extend $\ov V$ to a model $V$ such that
$V \models ``$ZFC + $\gl$ is $2^\gl = \gl^{++}$ supercompact +
GCH fails for every regular cardinal $\gd \le \gl$ +
GCH holds for every regular cardinal $\gd \ge \gl^{++}$''.
Let $\gk < \gl$ be such that
$V \models ``\gk$ is strong up to $\gl$''.
As in the proof of Proposition \ref{p1}, we slightly abuse
notation and write $\FP(\gk)$ for $(\FP(\gk))^{V_\gl}$.
%$\FP(\gk)$ as defined over $V_\gl$.
If we now force over $V$ with
$\FP(\gk) \ast \dot \add(\gl^+, 1)$, the arguments used
in the proofs of Theorems \ref{t1} and \ref{t2} and
Proposition \ref{p1} show that
$V^* = (V_\gl)^{\FP(\gk)} \models ``$ZFC + $\gk$ is a
tall cardinal + GCH fails at every regular cardinal
below $\gk$ yet holds for every cardinal $\gd \ge \gk$ +
There is a proper class of strong cardinals''.
This completes the proof of Theorem \ref{t3}.
\end{proof}
In analogy to Corollary \ref{c2},
we have the following corollary to Theorem \ref{t3}.
\begin{corollary}\label{c3}
Con(ZFC + There is a cardinal $\gl$ such that $\gl$ is $2^\gl$
supercompact and $2^\gd = \gd^{++}$ for every regular
cardinal $\gd \le \gl$)
$\implies$ Con(ZFC + There is a tall cardinal
$\gk$ such that GCH fails at every regular cardinal below
$\gk$ yet holds for every cardinal $\gd \ge \gk$ + $\gk$ is
the least measurable cardinal +
There is a proper class of strong cardinals).
\end{corollary}
\section{Concluding Remarks}\label{s3}
We conclude with some open questions and related
remarks raised by the results and proofs of
this paper. In particular:
\begin{enumerate}
\item\label{q1} Are the theories ``ZFC + $\gk$ is strongly compact +
GCH holds everywhere below $\gk$ yet fails for some regular
cardinal $\gd > \gk$'', ``ZFC + $\gk$ is strongly compact +
GCH fails everywhere below $\gk$ yet holds for all regular cardinals
$\gd \ge \gk$'', and
``ZFC + $\gk$ is strongly compact +
GCH fails for all regular cardinals
below $\gk$ yet holds for all regular cardinals
$\gd \ge \gk$'' consistent?
As we have already noted, by Solovay's theorem \cite{S},
if $\gk$ is strongly compact, then
GCH must hold at any singular strong limit
cardinal above $\gk$. Consequently, if GCH holds at every regular cardinal
above $\gk$, then GCH must hold at {\em every} cardinal above $\gk$
(since all singular cardinals above $\gk$ are then
strong limit cardinals as well).
\item\label{q2} Is the theory ``ZFC + $\gk$ is strongly compact + GCH
fails everywhere below $\gk$'' consistent?
Note that in this question, we are not imposing any constraints
on the size of $2^\gd$ for cardinals $\gd \ge \gk$.
In addition, observe that by \cite[Theorem 18]{Me76}, the theory
``ZFC + $\gk$ is supercompact + GCH fails for every regular
cardinal'' is consistent relative to the theory
``ZFC + $\gk$ is supercompact''.
\item\label{q3} What is the consistency strength of the theories
``ZFC + There is a tall cardinal $\gd$ such that GCH fails
everywhere below $\gd$ yet holds for every cardinal $\gg \ge \gd$'',
``ZFC + GCH fails everywhere + There is a strong cardinal'', and
``ZFC + GCH fails everywhere + There is a proper class
of strong cardinals''?
On \cite[page 35]{FW}, it is stated that
Woodin can obtain a model for the theory
``ZFC + GCH fails everywhere'' (in fact, for the theory
``ZFC + $2^\gd = \gd^{++}$ for every cardinal $\gd$'')
starting from a $\wp^2(\gk)$ hypermeasurable cardinal $\gk$
(also known as a $\gk + 2$ strong cardinal $\gk$).\footnote{Models
of ZFC in which GCH fails everywhere constructed using
strongness hypotheses may also be found in \cite{FG1},
\cite{FG2}, and \cite{Mer}.}
%--- related work along these same lines may in addition be found in
%\cite{FG} and \cite{Mer}).
We conjecture that the consistency of the first
two of the above theories
can be established relative to the existence of a
cardinal $\gl$ which is strong up to a
$\wp^2(\gk)$ hypermeasurable cardinal $\gk$
and the consistency strength of the last theory can be established
relative to a $\wp^2(\gk)$ hypermeasurable cardinal $\gk$ which
is a limit of cardinals $\gl$ which are strong up to $\gk$.
However, it is unclear if these assumptions
will provide equiconsistencies in each case.
\item\label{q4} What is the consistency strength of the theories
``ZFC + There is a tall cardinal $\gd$ such that GCH fails
everywhere below $\gd$ yet holds for every cardinal $\gg \ge \gd$ +
There is a proper class of strong cardinals'' and
``ZFC + There is a tall cardinal $\gk$ such that GCH fails
at every regular cardinal below $\gk$ yet holds for every
cardinal $\gd \ge \gk$ + There is a proper class of strong cardinals''?
We conjecture that these lie somewhere below the consistency
strength of a cardinal $\gl$ which is $2^\gl$ supercompact.
Note that \cite[Corollary 3.14]{H09} (which Hamkins
credits orginally to Gitik) tells us that the theories
``ZFC + There is a strong cardinal'' and
``ZFC + There is a tall cardinal'' are equiconsistent.
This indicates that the hypotheses and conclusion of
Theorem \ref{t2}, namely ``ZFC + There is a strong cardinal''
and ``ZFC + There is a tall cardinal $\gk$ such that
GCH fails at every regular cardinal below $\gk$ yet holds for
every cardinal $\gd \ge \gk$'', are equiconsistent as well.
\item\label{q5} Is the theory ``ZFC + There is a tall
cardinal $\gk$ such that GCH holds everywhere below $\gk$
yet fails at a singular strong limit cardinal above $\gk$''
consistent?
%The methods of \cite{H09} do not seem to be able to answer this question.
Gitik has pointed out \cite{GP} that it is impossible
to do Prikry forcing above a tall cardinal $\gk$ while
preserving $\gk$'s tallness without first doing
some sort of preparation forcing below $\gk$.
Thus, an analogue of \cite[Theorem 3.1]{H09},
which tells us that any tall cardinal $\gd$
is automatically indestructible under
$(\gd, \infty)$-distributive forcing,
does not seem to be valid.
This suggests that obtaining a model for this theory
seems to be quite a difficult task.
\end{enumerate}
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\end{document}
Let $j : V \to M$
be an elementary embedding witnessing the $\beth_4(\gk)$
supercompactness of $\gk$.
To see that $V^{\FP(\gd) \ast \dot \add(\gk^+, 1)} \models
``\gk$ is $2^\gk$ supercompact'', we use a version of
Silver's original argument for the
preservation of large cardinal properties
after forcing with a reverse Easton iteration (as presented, e.g., in
\cite[Theorem 21.4]{J}). Specifically, let $G_0 \ast G_1$
be $V$-generic over $\FP(\gd) \ast \dot \add(\gk^+, 1)$.
By the definition of $\FP(\gd) \ast \dot \add(\gk^+, 1)$,
we have that $j(\FP(\gd) \ast \dot \add(\gk^+, 1)) =
\FP(\gd) \ast \dot \add(\gk^+, 1) \ast
\dot \FQ \ast \dot \add(j(\gk^+), 1)$, where
$\forces_{\FP(\gd) \ast \dot \add(\gk^+, 1)}
``\dot \FQ \ast \dot \add(j(\gk^+), 1)$
is $(2^\gk)^+$-directed closed''. If we let
$G_2$ be $V[G_0][G_1]$-generic over $\FQ$,
then standard arguments show that
$M[G_0][G_1][G_2]$ remains $2^\gk$-closed %$(2^\gk)^+$-closed
with respect to $V[G_0][G_1][G_2]$, and
There are many questions raised by the results and
proofs in this paper. We conclude by giving
\begin{theorem}\label{t3}
Con(ZFC + GCH +
There is a cardinal $\gl$ such that $\gl$ is $\gl^{++}$
supercompact) $\implies$ Con(ZFC + There is a tall cardinal
$\gk$ such that GCH fails at every regular cardinal below
$\gk$ yet holds for every cardinal $\gd \ge \gk$ +
There is a proper class of strong cardinals).
\end{theorem}
\begin{proof}
Let $\ov V \models
``$ZFC + GCH + $\gl$ is $\gl^{++}$ supercompact''.
We begin by using ideas found in the proof of
\cite[Lemma 2.1]{A12d} to show that $\ov V$
can be generically extended to a model $V$
such that $V \models
``2^\gd = \gd^{++}$ if $\gd \le \gl$
is a regular cardinal + $2^\gd = \gd^+$ for every cardinal
$\gd \ge \gl^+$ + $\gl$ is $2^\gl = \gl^{++}$ supercompact''.
Specifically, let
$\la \gd_\ga \mid \ga < \gl \ra \in \ov V$
enumerate in increasing order
$\{\gd < \gl \mid \gd$ is either an inaccessible
cardinal or a limit of inaccessible cardinals$\}$.
We define a reverse Easton iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid
\ga \le \gl \ra$ of length $\gl + 1$ which begins
by doing trivial forcing (so $\FP_0 = \{\emptyset\}$).
For $\ga < \gl$, $\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the Easton product
which forces $2^\gd = \gd^{++}$ for every regular
cardinal $\gd \in [\gd_\ga, \gd_{\ga + 1})$.
For $\ga = \gl$, $\dot \FQ_\ga$ is a term for
$\add(\gl, \gl^{++})$. Standard arguments then show that
$V = \ov V^\FP \models ``2^\gd = \gd^{++}$ if $\gd \le \gl$
is a regular cardinal + $2^\gd = \gd^+$ for every cardinal
$\gd \ge \gl^+$''.
To see that $V \models ``\gl$ is $2^\gl = \gl^{++}$ supercompact'',
note that the definition of $\FP$ implies that
$j(\FP) = \FP_\gl \ast ((\dot \add(\gl, \gl^{++}) \times
\dot \FQ^0) \ast \dot \FQ^1) \ast \dot \add(j(\gl), j(\gl^{++})) =
\FP_\gl \ast \dot \FQ \ast \dot \add(j(\gl), j(\gl^{++}))$, where
$\dot \add(\gl, \gl^{++}) \times \dot \FQ^0$ is a term for
the Easton support product $\prod_{\gd \in [\gd_\gl,
\gd_{\gl + 1})} \add(\gd, \gd^{++})$, $\dot \FQ^1$ is a term
for the portion of $j(\FP)$ defined between
$\gd_{\gl + 1}$ and $j(\gl)$, and the first nontrivial component of
$\dot \FQ^0$ is a term for
$(\add(\gl^+, \gl^{+ 3}))^{M[G]}$. Let now
$G$ be $V$-generic over $\FP_\gl$ and $H$ be $V[G]$-generic over
$(\add(\gl, \gl^{++}))^{V[G]}$. Since $\FP_\gl$ is
$\gl$-c.c., standard arguments show that $M[G]$ remains
$\gl^{++}$ closed with respect to $V[G]$ and that
$\FQ^0$ is $\gl^+$-directed closed in both
$M[G]$ and $V[G]$.
If we then let $\gk < \gl$ be the least cardinal such that
$V \models ``\gk$ is $\gl$ strong'' and force with
$\FP(\gk) \ast \dot \add(\gl^+, 1)$, the arguments used
in the proofs of Theorems \ref{t1} and \ref{t2} and
Proposition \ref{p1} show that
$V^* = (V_\gl)^{\FP(\gd)} \models ``$ZFC + $\gk$ is a
tall cardinal + GCH fails at every regular cardinal
below $\gk$ yet holds for every cardinal $\gd \ge \gk$ +
There is a proper class of strong cardinals''.
This completes the proof of Theorem \ref{t3}.
\end{proof}