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\title{On Tall Cardinals and Some Related Generalizations
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal, hypermeasurable cardinal,
tall cardinal, indestructibility,
Magidor iteration of Prikry forcing, Easton support iteration
of Prikry type forcings, core model,
Rudin-Keisler ordering of ultrafilters.}}
%Gitik iteration}}
\author{Arthur W.~Apter\thanks{The
first 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
\\
\\
Moti Gitik\thanks{The second author's research
was partially supported by ISF Grant 234/08.}\\
Department of Mathematics\\
Tel Aviv University\\
69978 Tel Aviv, Israel\\
http://www.math.tau.ac.il/$\sim$gitik\\
gitik@post.tau.ac.il}
\date{December 30, 2012\\
(revised July 3, 2013)}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
We continue the study
of tall cardinals and related notions
begun by Hamkins in \cite{H09}
and answer three of his questions
posed in that paper.
%We answer three questions of Hamkins on tall cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and preliminaries}\label{s1}
We begin with the following definitions.
\begin{definition}\label{d1}
{\bf (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 $\gl$.
%for every elementary
%embedding $j^* : V \to M^*$ with critical point $\gk$
%such that $j^*(\gk) > \gl$, $(M^*)^{\gd^+} \not\subseteq M^*$.
$\gk$ is {\em strongly tall} if for every ordinal $\gl \ge \gk$,
there is an elementary embedding witnessing that
$\gk$ is $\gl$ tall which is generated by
a $\gk$-complete measure on some set.
\end{definition}
\noindent The first part of the next definition
(i.e., {\em tall with bounded closure}) is
due to Hamkins and is found in \cite[Section 5]{H09}.
\begin{definition}\label{d2}
$\gk$ is {\em tall with bounded closure} if
$\gk$ is not a tall cardinal, yet there is
a cardinal $\gd$ with $\go \le \gd < \gk$ such
that for all ordinals $\gl \ge \gk$,
there is an elementary embedding $j : V \to M$
with critical point $\gk$ such that $j(\gk) > \gl$
and $M^\gd \subseteq M$.
Let $\go \le \gd < \gk$ be a fixed cardinal.
$\gk$ is {\em tall with bounded closure $\gd$} if
for every ordinal $\gl \ge \gk$,
there is an elementary embedding $j : V \to M$
with critical point $\gk$ such that $j(\gk) > \gl$,
$M^\gd \subseteq M$, and $M^{\gd^+} \not\subseteq M$.
\end{definition}
In \cite{H09}, Hamkins made a systematic study of
tall cardinals and some related notions
and posed the following questions.
\begin{enumerate}
\item\label{q1}(implicit to \cite[Section 4]{H09}) Is it possible to
construct a model containing
infinitely many tall cardinals in which the measurable and
tall cardinals coincide precisely?
\item\label{q2}(\cite[Question 5.5]{H09}) Is it possible
to construct a model containing a tall cardinal with bounded closure?
%with a cardinal $\gk$ exhibiting
%tallness with closure bounded below $\gk$?
%Is it possible to construct a model containing
%a tall cardinal having closure $\go$ but not closure $\go_1$?
\item\label{q3}(\cite[Question 2.12]{H09}) Are strong tallness
and strong compactness equivalent? Are they equiconsistent?
\end{enumerate}
The purpose of this paper is to answer Questions \ref{q1} and
\ref{q2} affirmatively and Question \ref{q3} negatively.
Specifically, we prove the following eight theorems,
along with a corollary to one of them.
Theorem \ref{t1} addresses Question \ref{q1}.
Theorems \ref{t2} -- \ref{t2d} address Question \ref{q2}.
Theorem \ref{t4} addresses Question \ref{q3}.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact + No cardinal
$\gl > \gk$ is measurable''. There is then a partial ordering
$\FP \subseteq V$ such that $V^\FP \models ``$ZFC + $\gk$ is
supercompact + No cardinal $\gl > \gk$ is measurable + For
every $\gd < \gk$, $\gd$ is measurable iff $\gd$ is tall''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $\gk$ is a strong cardinal.
Then $\gk$ is a tall cardinal
having bounded closure $\go$.
%Then for every $\gl > \gk$, there is
%an elementary embedding $j' : V \to M'$ with
%critical point $\gk$ such that
%$j'(\gk) > \gl$, $(M')^\go \subseteq M'$, and
%$(M')^{\go_1} \not\subseteq M'$.
%witnessing the $\gl$ tallness of $\gk$ such that
%$M$ is closed under $\go$ sequences yet $M$ is
%not closed under $\go_1$ sequences.
%Con(ZFC + There is a strong cardinal) $\implies$
%Con(ZFC + There is a tall cardinal having closure $\go$
%but not closure $\go_1$).
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V = {\cal K}$ and
$V \models ``$ZFC + $\gk$ is strong +
$\eta > \gk$ is such that $o(\eta) = \go_1$ +
No cardinal $\gd > \eta$ is measurable''.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``\gk$ is a tall cardinal having
bounded closure $\go$ + There are neither any
tall cardinals nor any tall cardinals having
bounded closure $\gd$ for $\go_1 \le \gd < \gk$''.
%$V^\FP \models ``\gk$ is a tall cardinal having
%closure $\go$ but not closure $\go_1$ + There
%are no tall cardinals having closure $\go_1$''.
\end{theorem}
\begin{theorem}\label{t2a}
The following conditions are equivalent:
\begin{enumerate}
\item\label{i1} There is an elementary embedding
$j : V \to M$ with critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet
$(j(\gk))^{\go_1} \not\subseteq M$.
\item\label{i2} There exists a Rudin-Keisler increasing
sequence of ultrafilters over $\gk$ having length $\go_1$.
\item\label{i3} There exists an elementary embedding
$j : V \to M$ such that $M^\go \subseteq M$ and a sequence
$\la \eta_\ga \mid \ga < \go_1 \ra$ of ordinals below
$j(\gk)$ such that for every $\ga < \go_1$ and for every
$f : [\gk]^\ga \to \gk$, it is the case that
$\eta_\ga \neq j(f)(\la \eta_\gb \mid \gb < \ga \ra)$.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{t2b}
Suppose that there is no sharp for a strong cardinal
(i.e., that $o$ pistol does not exist).
If there is an elementary embedding
$j : V \to M$ with critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$, then either $o(\gk) \ge \go_1$ in
${\cal K}$, or $\gk$ is measurable in ${\cal K}$ and
$\{\nu < \gk \mid o^{\cal K}(\nu) \ge \go_1\}$ is
unbounded in $\gk$.
\end{theorem}
\begin{theorem}\label{t2c}
Suppose that $o(\gk) \ge \go_1$ in ${\cal K}$.
Then there is a generic extension $V$ of ${\cal K}$
with an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
%$(j(\gk))^\go \subseteq M$
$M^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$.
\end{theorem}
\begin{theorem}\label{t2d}
Suppose that $\gk$ is a measurable cardinal
in ${\cal K}$ and $\{\gn < \gk \mid o^{{\cal K}}(\gn) \ge \go_1\}$
is unbounded in $\gk$. Then there is a generic extension
$V$ of ${\cal K}$ and
an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
\end{theorem}
%\begin{theorem}\label{t2d}
%
%Suppose that $\gk$ is a measurable cardinal
%in $V^*$ and $\{\gn < \gk \mid o(\gn) \ge \go_1\}$
%is unbounded in $\gk$ in $V^*$.
%Then there is a generic extension $V$ of $V^*$ and
%an elementary embedding $j : V \to M$ having
%critical point $\gk$ such that
%$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
%$(j(\gk))^{\go_1} \not\subseteq M$.
%
%\end{theorem}
\begin{theorem}\label{t4}
The following theories are equiconsistent:
a) ZFC + There is a strong cardinal and a proper class
of measurable cardinals.
b) ZFC + There is a strongly tall cardinal.
%Con(ZFC + There is a strong cardinal and a proper class
%of measurable cardinals) $\implies$
%Con(ZFC + There is a strongly tall cardinal).
\end{theorem}
We take this opportunity to make a few remarks
concerning Theorems \ref{t1} -- \ref{t4}.
%Theorem \ref{t1} is due jointly to both authors.
%The remaining theorems and corollary are due to the second author.
Theorem \ref{t1} provides a positive answer to Question
\ref{q1}, since in $(V_\gk)^{V^\FP}$, there is a proper
class of tall cardinals, and the tall
and measurable cardinals precisely coincide.
As we will show, however, the use of a supercompact cardinal is
unnecessary in order to construct a model witnessing a
positive answer to Question \ref{q1}.
We prove Theorem \ref{t1} in this form, though, because
we feel it is of independent interest to show that
the tall and measurable cardinals can coincide precisely
below a supercompact cardinal.
In addition, Question \ref{q1} is a direct
analogue of a famous question concerning
strongly compact and measurable cardinals,
which we will discuss at greater length
in Section \ref{s2} after the proof of Theorem \ref{t1}.
Theorem \ref{t2} shows that any strong cardinal
is in fact tall with bounded closure $\go$.
%always has $\gl$ tallness embeddings for any ordinal $\gl$
%witnessing a bounded degree of closure.
Theorem \ref{t3}
provides a positive answer to Question \ref{q2}, and
Theorems \ref{t3} -- \ref{t2d} address the consistency
strength of the existence of a tall cardinal $\gk$
exhibiting tallness with closure bounded below $\gk$.
Theorem \ref{t4} exactly pins down the consistency
strength of the existence of a strongly tall cardinal
and shows that it is much weaker than the consistency
strength of a strongly compact cardinal.
Before beginning the proofs of our theorems, we briefly mention
some preliminary information and terminology.
%Before beginning the proofs of
%Theorems \ref{t1} and
%\ref{p1}, we briefly
%mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in the usual interval notation.
If $\gk \ge \go$ is a regular cardinal and $\gl$
is an arbitrary ordinal, then
$\add(\gk, \gl)$ is the standard partial ordering
for adding $\gl$ Cohen subsets of $\gk$.
When forcing, $q \ge p$ will mean that
{\it $q$ is stronger than $p$}.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
The abuse of notation mentioned above will be compounded
by writing $x \in V^\FP$ instead of $\dot x \in V^\FP$.
Any term for trivial forcing will
always be taken as a term for the
partial ordering $\{\emptyset\}$.
If $\varphi$ is a formula in the forcing language
with respect to $\FP$ and $p \in \FP$, then
$p \decides \varphi$ means that
{\it $p$ decides $\varphi$}.
From time to time within the course of our
discussion, we will refer to
partial orderings $\FP$ as being
%{\it Gitik iterations}.
{\it Easton support iterations of Prikry type forcings}.
By this we will mean an Easton support iteration
as first given by the second author in \cite{G86},
to which we refer readers for a discussion
of the basic properties of
and terminology associated with such an iteration.
As in \cite{GS}, we will say that
the partial ordering $\FP$
is {\em $\gk^+$-weakly closed
and satisfies the Prikry property} 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, i.e.,
there is an upper bound for every
increasing chain of conditions having length $\gk$.
\end{enumerate}
Key to the proof of Theorem \ref{t1} is
the following result due to the second
author and Shelah. It is a corollary of
the work of \cite[Section 2]{GS}.
\begin{theorem}\label{t5}
Suppose $V \models ``$ZFC + GCH + $\gd < \gk$ are such that
$\gd$ is a regular cardinal and $\gk$ is a strong cardinal''.
There is then a $\gd^+$-weakly closed partial ordering $\FI(\gd, \gk)$
satisfying the Prikry property having cardinality $\gk$
such that $V^{\FI(\gd, \gk)} \models ``\gk$ is a strong cardinal whose
strongness is indestructible under $\gk^+$-weakly
closed partial orderings satisfying the Prikry property''.
\end{theorem}
We mention that we are assuming some
familiarity with the large cardinal
notions of measurability, measurable cardinals of
high Mitchell order, tallness,
hypermeasurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult \cite{J},
\cite{Mi79}, \cite{Mi74},
or \cite{SRK}. %for further details.
In addition, we are assuming some familiarity with
basic inner model and core model theory,
as presented in \cite{Z} and \cite{Mi10}.
In particular, ${\cal K}$ will always denote the core model.
Finally, we are assuming some familiarity with
the Rudin-Keisler ordering on ultrafilters, for which
we refer readers to \cite{G88}.
\section{Models where the measurable and tall
cardinals coincide precisely}\label{s2}
%We turn to the proofs of our theorems, beginning
We begin with the proof of Theorem \ref{t1}, which we
restate for the convenience of readers.
\setcounter{theorem}{+0}
\begin{theorem}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact + No cardinal
$\gl > \gk$ is measurable''. There is then a partial ordering
$\FP \subseteq V$ such that $V^\FP \models ``$ZFC + $\gk$ is
supercompact + No cardinal $\gl > \gk$ is measurable + For
every $\gd < \gk$, $\gd$ is measurable iff $\gd$ is tall''.
\end{theorem}
\begin{proof}
We start with the key fact that a Magidor iteration \cite{Ma} of
Prikry forcing preserves the tallness of a strong cardinal.
\begin{lemma}\label{l1}
Suppose $\gd < \gk$ and
$V \models ``\gk$ is a strong cardinal''.
%Let $A \subseteq \gk$ be an unbounded set of
%measurable cardinals.
Let ${\FP(\gd, \gk)}$ be the Magidor iteration of Prikry
forcing which adds a Prikry sequence to every
%$\gd \in A$. Then
measurable cardinal in the open interval $(\gd, \gk)$. Then
$V^{\FP(\gd, \gk)} \models ``\gk$ is a tall cardinal''.
\end{lemma}
\begin{proof}
Let $\gl > \gk$ be an arbitrary strong limit cardinal
of cofinality at least $\gk$.
By the proof of \cite[Theorem 4.1]{H09}, we may
take $j : V \to M$ to be an elementary embedding
witnessing the $\gl$ tallness of $\gk$
generated by the $(\gk, \gl)$-extender
${\cal E} = \la E_a \mid a \in [\gl]^{< \go} \ra$
such that $M \models ``\gk$ is not measurable''.
Since $M^\gk \subseteq M$, it is the case that $\cal E$
is {\em $\gk^+$-directed}, i.e., for each $\gk$ sequence
$\la E_i \mid i < \gk \ra$ of measures from $\cal E$,
there is some $E \in {\cal E}$ such that for each $i < \gk$,
$E_i <_{\rm RK} E$ (so $E$ projects onto $E_i$
as in the Rudin-Keisler ordering). To see this, let
$j_i : {\rm Ult}(V, E_i) \to M$ be the canonical elementary
embedding of ${\rm Ult}(V, E_i)$ into $M$, and let
$\gt_i = j_i([{\rm id}]_{E_i})$. Because $M^\gk \subseteq M$,
$\gt =_{\rm df} \la \gt_i \mid i < \gk \ra \in M$. Consequently, there
must be some $E \in {\cal E}$ such that for some
$\gs \in {\rm Ult}(V, E)$ and $j_E : {\rm Ult}(V, E) \to M$
the canonical elementary embedding, $\gt = j_E(\gs)$.
However, this just means that for every $i < \gk$,
$E_i <_{\rm RK} E$.
For each $E \in {\cal E}$, $E = E_b$ for $b \in [\gl]^{< \go}$,
let $k_E : V \to {\rm Ult}(V, E) = M_E$
be the canonical elementary embedding.
Note that since the canonical elementary embedding
$\ell_E : M_E \to M$ is such that
${\rm cp}(\ell_E) > \gk$ and $M \models ``\gk$ is not measurable'',
$M_E \models ``\gk$ is not measurable'' as well. Therefore, if
we consider now $E^*$ defined in $V^{\FP(\gd, \gk)}$ by
$p \forces ``\dot x \in \dot E^*$'' iff there is $q \in k_E({\FP(\gd, \gk)})$,
$q \ge k_E(p)$ such that $|k_E(p) - q| = 0$ (where
$|\ \ \ |$ is the distance function from \cite{Ma}),
$k_E(p) \rest \gk = q \rest \gk = p$, and
$q \forces ``b \in k_E(\dot x)$'', then because
$M_E \models ``\gk$ is not measurable'', the arguments of
\cite[Theorem 2.5]{Ma} show that $E^*$ is well-defined and is a
$\gk$-additive ultrafilter extending $E$.
It is routine (although tedious) to verify that
${\cal E}^* = \la E^*_a \mid a \in [\gl]^{< \go} \ra \in V^{\FP(\gd, \gk)}$
is hence a $(\gk, \gl)$-extender extending $\cal E$
which is $\gk^+$-directed.
(Note that $\gk^+$-directedness follows because by its definition,
projection maps in the sense of the Rudin-Keisler ordering
between members of ${\cal E}^*$ remain projection maps
in the same sense.)
To show that in fact ${\cal E}^*$ witnesses that
$V^{\FP(\gd, \gk)} \models ``\gk$ is $\gl$ tall'', it suffices
to show that for
$M_* = {\rm Ult}(V^{\FP(\gd, \gk)}, {\cal E}^*)$,
$M^\gk_* \subseteq M_*$.
%$M_{{\cal E}^*} = {\rm Ult}(V, {\cal E}^*)$,
%$M^\gk_{{\cal E}^*} \subseteq M_{{\cal E}^*}$.
To see this, suppose
$\la a_i \mid i < \gk \ra \in V^{\FP(\gd, \gk)}$ is a
$\gk$ sequence of members of $M_*$. There must be
$E^*_i \in {\cal E}^*$ and $a'_i \in M_i =
{\rm Ult}(V^{\FP(\gd, \gk)}, E^*_i)$ such that for
$\ell^*_i : M_i \to M_*$ the canonical elementary embedding,
$\ell^*_i(a'_i) = a_i$. Since ${\cal E}^*$ is $\gk^+$-directed,
let $E^* \in {\cal E}^*$ be such that for each $i < \gk$,
$E^*_i <_{\rm RK} E^*$. Consider $M_{E^*} =
{\rm Ult}(V^{\FP(\gd, \gk)}, E^*)$, with $j_{E^*} : M_{E^*} \to M_*$
the canonical elementary embedding. Note that there is
a canonical elementary embedding $j^*_i : M_i \to M_{E^*}$
generated by the projection of $E^*$ to $E^*_i$ and that
$j_{E^*} \circ j^*_i : M_i \to M_*$ is an elementary embedding.
Also, since $M_{E^*}$ is the ultrapower via a measure,
it is $\gk$-closed with respect to $V^{\FP(\gd, \gk)}$. Therefore,
$a'' = \la a''_i \mid i < \gk \ra \in M_{E^*}$, where
for $i < \gk$, $a''_i = j^*_i(a'_i)$. However,
$j_{E^*}(a'') = \la a_i \mid i < \gk \ra \in M_*$, so
$M^\gk_* \subseteq M_*$.
Thus, $V^{\FP(\gd, \gk)} \models ``\gk$ is $\gl$ tall''.
Since $\gl$ was arbitrary, this completes the
proof of Lemma \ref{l1}.
\end{proof}
Since the proof of Theorem \ref{t1}
requires that we force over a ground model
$V$ satisfying certain indestructibility properties
for strongness, we next show that this is possible
in the following lemma.
\begin{lemma}\label{l2}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact + No cardinal
$\eta > \gk$ is measurable''. Let ${\cal C} = \{\gd < \gk \mid
\gd$ is a strong cardinal which is not a limit of
strong cardinals$\}$''. There is then a partial ordering
$\FI \in V$ such that $V^\FI \models ``$ZFC + $\gk$ is
supercompact + No cardinal $\eta > \gk$ is measurable +
For every $\gd \in {\cal C}$, $\gd$ is a strong cardinal whose
strongness is indestructible under
$\gd^+$-weakly closed partial orderings satisfying the Prikry property''.
\end{lemma}
\begin{proof}
Let $\la \gd_\ga \mid \ga < \gk \ra$ enumerate in increasing order the
members of ${\cal C}$. %together with their limit points.
For every $\ga < \gk$, let $\gg_\ga = (\sup_{\gb < \ga} \gd_\gb)^+$,
where $\gg_0 = \go$.
%where $\gg_\ga = \go$ if $\ga = 0$.
%Let $\la \gg_\ga \mid \ga < \gk \ra$ enumerate in
%increasing order $\{\go\}$ together with the successors of
%the limit points of members of ${\cal C}$.
$\FI = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
is defined as the
%Gitik style Easton support iteration of
Easton support iteration of Prikry type forcings of
length $\gk$ such that $\FP_0 = \{\emptyset\}$.
For every $\ga < \gk$,
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
%which, at any stage $\ga < \gk$, forces
$\FI(\gg_\ga, \gd_\ga)$ of Theorem \ref{t5} as defined in
$V^{\FP_\ga}$. Note that this definition makes sense,
since inductively, it is the case that
$\card{\FP_\ga} < \gd_\ga$.
By the Hamkins-Woodin results \cite{HW},
$V^{\FP_\ga} \models ``\gd_{\ga}$ is a strong cardinal'',
meaning that $\FP_{\ga + 1}$ may be correctly defined.
Note also that the only strong cardinals on which
$\FI$ acts nontrivially are those strong cardinals
which are not limits of strong cardinals in $V$.
In other words, if $V \models ``\gd$ is a strong
cardinal which is a limit of strong cardinals'', then
$\FI$ acts trivially on $\gd$.
By its definition,
$V^{\FP_{\ga + 1}} \models ``\gd_{\ga}$ is a strong
cardinal whose strongness is indestructible under
$\gd^+_\ga$-weakly closed partial orderings satisfying
the Prikry property''. Factor $\FI$ as
$\FI = \FP_{\ga + 1} \ast \dot \FP^{\ga + 1}$.
Since also by its definition,
$\forces_{\FP_{\ga + 1}} ``\dot \FP^{\ga + 1}$ is
$\gd^+_\ga$-weakly closed and satisfies
the Prikry property'',
$V^{\FP_{\ga + 1} \ast \dot \FP^{\ga + 1}} = V^\FI \models
``\gd_{\ga}$ is a strong
cardinal whose strongness is indestructible under
$\gd^+_\ga$-weakly closed partial orderings satisfying
the Prikry property''. Consequently, since $\ga$ was arbitrary,
$V^\FI \models
``$For every $\gd \in {\cal C}$, $\gd$ is a strong cardinal whose
strongness is indestructible under
$\gd^+$-weakly closed partial orderings satisfying the Prikry property''.
To show that $V^\FI \models ``\gk$ is supercompact'',
we follow the proof of \cite[Lemma 2.1]{A06a}.
Let $\gl \ge \gk^+ = 2^\gk$
be any regular cardinal.
Take $j : V \to M$ as an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk$.
By \cite[Lemma 2.1]{AC2},
in $M$, $\gk$ is a limit of
strong cardinals. In addition, since
$V \models ``$No cardinal $\eta > \gk$ is measurable'',
$M \models ``$No cardinal $\eta \in (\gk, \gl]$ is measurable''.
Hence, in $M$,
$j({\FI})$ is forcing equivalent to ${\FI} \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gl$.
We may now apply the argument of \cite[Lemma 1.5]{G86}.
Specifically, let $G$ be $V$-generic over ${\FI}$.
Since GCH in $V$ implies that
$V \models ``2^{\gl} = \gl^+$'', we may let
$\la \dot x_\ga \mid \ga < \gl^+ \ra$ be an
enumeration in $V$ of all of the
canonical ${\FI}$-names of subsets of
$P_\gk(\gl)$.
Because $\FI$ is $\gk$-c.c$.$ and $M^\gl \subseteq M$,
$M[G]^\gl \subseteq M[G]$.
By \cite[Lemmas 1.4 and 1.2]{G86}, we may therefore
define an increasing sequence
$\la p_\ga \mid \ga < \gl^+ \ra$
of elements of $j({\FI})/G$
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
reverse Easton iteration, except that
at coordinates at which, e.g., %something like
Prikry forcing or some variant or
generalization thereof occurs in $p_\ga$,
measure 1 sets are shrunk and stems are not
extended. For a more precise definition,
readers are urged to consult \cite{G86}.}
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]{G86} 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
${\FI}$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j({\FI})/G}
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x)$''.
(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, $\forces_{{\FI}} ``\gk$ is
$\gl$ supercompact''.
Since $\gl$ was arbitrary,
$V^\FI \models ``\gk$ is supercompact''.
Finally, since $\FI$ may be defined so that
$\card{\FI} = \gk$, $V^\FI \models ``$No cardinal
$\eta > \gk$ is measurable''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We assume now that our ground model, which
with an abuse of notation we relabel as $V$,
has the properties of the model $V^\FI$
constructed in Lemma \ref{l2}. Given this,
and adopting the notation of Lemma \ref{l1},
let $\FP(\gg_\ga, \gd_\ga)$ for every $\ga < \gk$
be the Magidor iteration of Prikry forcing from \cite{Ma}
%as defined in $V$
which adds a Prikry
sequence to every measurable cardinal in
the open interval $(\gg_\ga, \gd_\ga)$.
The partial ordering
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
with which we force
is defined as the
%Gitik style Easton support iteration of
Easton support iteration of Prikry type forcings of
length $\gk$ such that $\FP_0 = \{\emptyset\}$.
For every $\ga < \gk$,
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
%which, at any stage $\ga < \gk$, forces
$\FP(\gg_\ga, \gd_\ga)$ of Lemma \ref{l1} as defined in
$V$ (and {\em not as defined in $V^{\FP_\ga}$}).
To see that this makes sense, i.e., that
$V^{\FP_\ga} \models ``\FP(\gg_\ga, \gd_\ga)$ as defined in
$V$ is $\gg^+_\ga$-weakly closed and satisfies the Prikry property'',
we note that by their definitions, the cardinality of $\FP_\ga$
is less than the least measurable cardinal in the open interval
$(\gg_\ga, \gd_\ga)$. Consequently, by the results of \cite{LS},
for any $\gd \in (\gg_\ga, \gd_\ga)$, $\gd$ is measurable in $V$ iff
$\gd$ is measurable in $V^{\FP_\ga}$, and
every normal measure $\mu^*$ over $\gd$ in $V^{\FP_\ga}$
has the form $\{x \subseteq \gd \mid \exists y \in \mu[y \subseteq x]\}$,
where $\mu \in V$ is some normal measure over $\gd$.
This means that informally, every normal measure $\mu^*$
used in the Magidor iteration of Prikry forcing
$\FP^*(\gg_\ga, \gd_\ga)$ as defined in
$V^{\FP_\ga}$ which adds a Prikry sequence to every measurable
cardinal in the open interval $(\gg_\ga, \gd_\ga)$
may be replaced by its ground model counterpart $\mu$.
More formally, let
$p = \la \la s_\gb, A_\gb \ra \mid \gb < \gd_\ga \ra \in
\FP^*(\gg_\ga, \gd_\ga)$,
where $s_\gb$ is a finite sequence of ordinals
and $\la A_\gb \mid \gb < \gd_\ga \ra$ is a sequence of terms
for measure 1 sets which are forced to be members of
the appropriate normal measure. %$\mu^*_\gb$.
We proceed inductively.
Let $\FR_\gb = \FP^*(\gg_\ga, \gd_\ga) \rest \gb$, i.e.,
$\FR_\gb$ is the Magidor iteration defined up to stage $\gb$
in $V^{\FP_\ga}$.
By the work of \cite{Ma}, for some normal measure
$\mu^*_\gb \in V^{\FP_\ga \ast \dot \FR_\gb}$,
$\forces_{\FP_\ga \ast \dot \FR_\gb} ``A_\gb \in \mu^*_\gb$''.
By the results of \cite{LS}, there must exist
some normal measure $\mu_\gb \in V^{\FR_\gb}$
and some term $B_\gb$ such that
$\forces_{\FP_\ga} ``B_\gb \in \mu_\gb$ and $B_\gb \subseteq A_\gb$''.
By replacing each $A_\gb$ with $B_\gb$, we inductively define a condition
$q = \la \la s_\gb, B_\gb \ra \mid \gb < \gd_\ga \ra \in
\FP(\gg_\ga, \gd_\ga) \subseteq \FP^*(\gg_\ga, \gd_\ga)$ such that
$q \ge_{\FP^*(\gg_\ga, \gd_\ga)} p$.
Thus, $\FP(\gg_\ga, \gd_\ga)$ is dense in
$\FP^*(\gg_\ga, \gd_\ga)$, a partial ordering which is
$\gg^+_\ga$-weakly closed and satisfies the Prikry property
in $V^{\FP_\ga}$. It therefore immediately follows that
$V^{\FP_\ga} \models ``\FP(\gg_\ga, \gd_\ga)$ as defined in
$V$ is $\gg^+_\ga$-weakly closed and satisfies the Prikry property''.
%, taking as our inductive hypothesis that
%every term $A_\gb$ may be replaced by a term $B_\gb$ such that
%Essentially the same argument from \cite{LS} which shows that
\begin{lemma}\label{l3}
$V^\FP \models ``$Any $\gd \in {\cal C}$ is a tall cardinal''.
\end{lemma}
\begin{proof}
Suppose $\gd \in {\cal C}$. It is then the case that for some $\ga < \gk$,
$\gd = \gd_\ga$.
%Consequently, by the definition of
%$\FP$, since each of its components is an element of $V$,
Because each component of $\FP$ is an element of $V$,
it is possible to write
$\FP = (\prod_{\gb < \ga} \FP(\gg_\gb, \gd_\gb)) \times
\FP(\gg_\ga, \gd_\ga) \times
(\prod_{\gb > \ga} \FP(\gg_\gb, \gd_\gb)) = \FP^0 \times
\FP^1 \times \FP^2$, where
$\FP^0, \FP^1, \FP^2 \in V$ and the ordering on $\FP^0$ and $\FP^2$
is the one used in an
Easton support iteration of Prikry type forcings.
%Gitik style Easton support iteration of
%weakly closed forcings satisfying the Prikry property.
Since by our observations above, $\FP^2$ is in fact $\gd^+$-weakly
closed and satisfies the Prikry property,
by the indestructibility properties of $V$,
$V^{\FP^2} \models ``\gd$ is a strong cardinal''.
Further, by the fact $\FP^2$ is $\gd^+$-weakly closed,
the definition of $\FP^1 = \FP(\gg_\ga, \gd_\ga)$
as the Magidor iteration of Prikry forcing adding
a Prikry sequence to each measurable cardinal
in the open interval $(\gg_\ga, \gd_\ga)$ is the
same in both $V$ and $V^{\FP^2}$.
By Lemma \ref{l1}, this means that
$V^{\FP^2 \times \FP^1} \models ``\gd$ is a tall
cardinal''. Finally, because $\card{\FP^0} < \gd$, by
\cite[Theorem 2.13]{H09},
$V^{\FP^2 \times \FP^1 \times \FP^0} = V^\FP \models
``\gd$ is a tall cardinal''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$Any measurable cardinal is either
a member of ${\cal C}$ or a limit of members of ${\cal C}$''.
\end{lemma}
\begin{proof}
Since $\card{\FP} = \gk$ and $V \models ``$No cardinal
$\eta > \gk$ is measurable'', by the results of \cite{LS},
$V^\FP \models ``$No cardinal $\eta > \gk$ is measurable'' as well.
We may thus assume that $\gd < \gk$ and
$V^\FP \models ``\gd$ is measurable'', since
$V^\FP \models ``\gk$ is supercompact and a limit of
members of ${\cal C}$''. If in addition
$V^\FP \models ``\gd$ is neither a member of ${\cal C}$
nor a limit of members of ${\cal C}$'', then let $\ga < \gk$ be such that
$\ga$ is least with $\gd_\ga > \gd$. Because in both $V$ and
$V^\FP$, $\gd$ is not a limit of members of ${\cal C}$, it must be
the case that $\gd \in (\gg_\ga, \gd_\ga)$.
As in the proof of Lemma \ref{l3}, write
$\FP = (\prod_{\gb < \ga} \FP(\gg_\gb, \gd_\gb)) \times
\FP(\gg_\ga, \gd_\ga) \times
(\prod_{\gb > \ga} \FP(\gg_\gb, \gd_\gb)) = \FP^0 \times
\FP^1 \times \FP^2$.
%Because $\FP^1 = \FP(\gg_\ga, \gd_\ga)$ and
The work of \cite{Ma} shows that
$V^{\FP^1} \models ``$There are no measurable
cardinals in the open interval $(\gg_\ga, \gd_\ga)$'',
i.e., that
$V^{\FP(\gg_\ga, \gd_\ga)} \models ``$There are no measurable
cardinals in the open interval $(\gg_\ga, \gd_\ga)$''.
Since as we observed in the proof of Lemma \ref{l3},
$\FP^1$ retains its properties in $V^{\FP^2}$,
it is also the case that
$V^{\FP^2 \times \FP^1} \models
``$There are no measurable
cardinals in the open interval $(\gg_\ga, \gd_\ga)$''.
Because $\card{\FP^0} < \gd$, the results of \cite{LS} again imply that
$V^{\FP^2 \times \FP^1 \times \FP^0} = V^\FP \models
``$There are no measurable
cardinals in the open interval $(\gg_\ga, \gd_\ga)$''.
This contradiction completes the proof of Lemma \ref{l4}.
\end{proof}
By \cite[Corollary 2.7]{H09}, any measurable
limit of tall cardinals is also a tall cardinal.
In addition, the same argument as found in the
proof of Lemma \ref{l2} shows that
$V^\FP \models ``\gk$ is supercompact''.
These facts, together with Lemmas \ref{l1} -- \ref{l4},
complete the proof of Theorem \ref{t1}.
\end{proof}
As we mentioned when making our
introductory comments in Section \ref{s1},
it is completely unnecessary to use a
supercompact cardinal in order to construct
a model in which the tall and measurable
cardinals precisely coincide. An inaccessible
limit of strong cardinals is more than enough for this purpose.
To see this, suppose $\gk$ is an inaccessible
limit of strong cardinals instead of a supercompact cardinal.
Suppose further that the partial orderings $\FI$ and $\FP$
of Theorem \ref{t1} are both defined as they were in
our original proof, i.e., as Easton support iterations
of length $\gk$.
%The partial orderings $\FI$ and $\FP$ are both
%Easton support iterations of length $\gk$, so
By \cite[Lemma 0.6]{A00}, in $V^{\FI \ast \dot \FP}$,
$\gk$ remains inaccessible. Thus, the proofs we gave
above show that in $(V_\gk)^{V^{\FI \ast \dot \FP}}$,
there is a proper class of tall cardinals, and
the tall and measurable cardinals precisely coincide.
As we also mentioned in our introductory comments in
Section \ref{s1}, a famous question (essentially
due to Magidor) asks whether it is possible
to construct a model of ZFC containing infinitely
many strongly compact cardinals in which the
measurable and strongly compact cardinals
precisely coincide. To date, this question
remains open, and has defied every effort
to obtain a positive answer.
We were able to prove Theorem \ref{t1}
because the work of \cite{GS} shows
that it is possible to do Prikry
forcing above a strong cardinal while
preserving strongness. However, as is
fairly well known (see, e.g., \cite[Section 4]{Ma} and
\cite[Lemma 3.1]{A06b}), adding
a Prikry sequence above a strongly compact
cardinal destroys strong compactness.
Thus, the methods of this paper cannot be used
to provide a positive answer to Magidor's question.
\section{Tall cardinals with bounded degrees of closure}\label{s3}
Having completed the proof of Theorem \ref{t1},
we turn now to the proofs of Theorems \ref{t2} -- \ref{t2d}.
We begin with the proof of Theorem \ref{t2}, which
we again restate.
\setcounter{theorem}{+1}
\begin{theorem}\label{t2}
Suppose $\gk$ is a strong cardinal.
Then $\gk$ is a tall cardinal
having bounded closure $\go$.
%Then for every $\gl > \gk$, there is
%an elementary embedding $j' : V \to M'$ with
%critical point $\gk$ such that
%$j'(\gk) > \gl$, $(M')^\go \subseteq M'$, and
%$(M')^{\go_1} \not\subseteq M'$.
%witnessing the $\gl$ tallness of $\gk$ such that
%$M$ is closed under $\go$ sequences yet $M$ is
%not closed under $\go_1$ sequences.
%Con(ZFC + There is a strong cardinal) $\implies$
%Con(ZFC + There is a tall cardinal having closure $\go$
%but not closure $\go_1$).
\end{theorem}
\begin{proof}
Suppose $V \models ``$ZFC + $\gk$ is a strong cardinal''.
%Let $\gd > \gl > \gk$ be arbitrary strong limit cardinals, and
%such that ${\rm cof}(\gl) \ge \go_1$,
%and let $\gd > \gl$
%be an arbitrary strong limit cardinal such that
%${\rm cof}(\gd) \ge \gk$.
Let $\gl > \gk$ be a strong limit cardinal.
Let $j : V \to M$ be an elementary embedding
%witnessing the $\gd$ strongness of $\gk$
such that $M \supseteq H(\gl^{+ \go_1})$
which is generated by a $(\gk, \gd)$-extender ${\cal E}$
for the appropriate strong limit cardinal $\gd > \gl$.
%We may assume that $M^\gk \subseteq M$.
Consider ${\cal E}' = {\cal E} \rest \gl^{+ \go_1}$,
with $j' : V \to M'$ the elementary embedding generated
by ${\cal E}'$ and $k : M' \to M$ the canonical elementary embedding.
Since $k \circ j' = j$, $j(\gk) > \gl$, and
${\rm cp}(k) > \gl$, $j'(\gk) > \gl$.
Because ${\cal E}' \not\in M'$,
$M'$ is not $\go_1$ closed.
To see this, let $\ga < \go_1$, and define
${\cal E}_\ga = {\cal E} \rest \gl^{+ \ga}$.
Let $\gb < \go_1$ be such that
${\cal E}_\ga \in H(\gl^{+ \gb})$. Note that
${\cal E}_\ga \in M'$, since $H(\gl^{+ \gb}) \subseteq M'$ and
${\cal E}_\ga \in H(\gl^{+ \gb})$.
Hence, if $(M')^{\go_1} \subseteq M'$, then
$\la {\cal E}_\gg \mid \gg < \go_1 \ra \in M'$,
thereby allowing us to recover ${\cal E}'$ within $M'$.
To prove that $M'$ is $\go$ closed, it suffices to show that
every countable set of generators of ${\cal E}'$ is
a member of $M'$. To see this, let
$\la x_n \mid n < \go \ra$ be a countable
sequence of elements of $M'$.
Then there are functions $\la f_n \mid n < \go \ra$ and a sequence
$\la a_n \mid n < \go \ra$ of generators of ${\cal E}'$
such that $x_n = j(f_n)(a_n)$. Let $a$ be a generator of ${\cal E}'$
coding $\la a_n \mid n < \go \ra$, with $\pi_n(a) = a_n$.
By hypothesis, $a \in M'$. Since
$\la j(f_n) \mid n < \go \ra \in M'$ and $\la \pi_n \mid n < \go \ra$
is definable in $M'$, $\la x_n \mid n < \go \ra \in M'$ as well.
The proof of Theorem \ref{t2} will thus
be complete once we have established that
every countable set of generators of ${\cal E}'$ is
a member of $M'$.
Consequently, let $a$ be such a set,
and let $\gt < \go_1$ be such that $a \in H(\gl^{+ \gt})$.
For ${\cal E}'' = {\cal E}' \rest \gl^{+ \gt + 1}$ and
$M'' = {\rm Ult}(V, {\cal E}'')$, it is then the case that
$H(\gl^{+ \gt}) \subseteq M''$, from which it immediately
follows that $a \in M''$. Since for $i : M'' \to M'$
the canonical elementary embedding, $i(a) = a$, we have
that $i(a) = a \in M'$.
This completes the proof of Theorem \ref{t2}.
\end{proof}
For the convenience of readers, we
also restate Theorem \ref{t3} before giving its proof.
\setcounter{theorem}{+2}
\begin{theorem}\label{t3}
Suppose $V = {\cal K}$ and
$V \models ``$ZFC + $\gk$ is strong +
$\eta > \gk$ is such that $o(\eta) = \go_1$ +
No cardinal $\gd > \eta$ is measurable''.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``\gk$ is a tall cardinal having
bounded closure $\go$ + There are neither any
tall cardinals nor any tall cardinals having
bounded closure $\gd$ for $\go_1 \le \gd < \gk$''.
%$V^\FP \models ``\gk$ is a tall cardinal having
%closure $\go$ but not closure $\go_1$ + There
%are no tall cardinals having closure $\go_1$''.
\end{theorem}
\begin{proof}
%Turning now to the proof of Theorem \ref{t3}, suppose
Suppose $V = {\cal K}$ and
$V \models ``$ZFC + $\gk$ is strong + $\eta > \gk$
is such that $o(\eta) = \go_1$ + No cardinal
$\gd > \eta$ is measurable''.
%Without loss of generality, we assume that $\eta$
%is the least such cardinal.
%We assume in
%addition that $V = {\cal K}$ and that there are no
%measurable cardinals in ${\cal K}$ above $\eta$.
%Since standard techniques (see \cite{Z}) show that if
%$V' \models ``\gl$ is tall'', then
%$({\cal K})^{V'} \models ``\gl$ is strong'',
%we know that $V \models ``$No cardinal in
%the half-open interval $(\gk, \eta]$ is tall''.
%By truncating the universe at the appropriate
%ordinal if necessary, we may also assume that
%$V \models ``$No cardinal $\gl > \eta$ is tall''.
We define an Easton support iteration of Prikry type forcings
%a Gitik iteration
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga \le \eta \ra$
of length $\eta$ as follows.
For $\ga < \eta$, $\dot \FQ_\ga$ is a term for
trivial forcing, unless $\ga$ is a measurable cardinal in $V$
such that $o(\ga) < \go_1$. If this is the case, then
$\dot \FQ_\ga$ is a term for the forcing of \cite{G86}
which adds a Magidor sequence
(see \cite{Ma78}) of order type $\go^{o(\ga)}$ to $\ga$.
For $\ga = \eta$, $\dot \FQ_\ga$ is a term for the
Prikry type forcing from \cite{G86} which changes
the cofinality of $\eta$ to $\go_1$ without adding
over $V^{\FP_\eta}$ any new bounded subsets of $\eta$.
This partial ordering can be defined in $V^{\FP_\eta}$
so as to be $\eta^+$-c.c$.$ and have cardinality $\eta^+$.
Consequently, forcing with $\dot \FQ_\eta$ over
$V^{\FP_\eta}$ adds no new countable sets of ordinals.
This is since if $A \in V^{\FP_\eta \ast \dot \FQ_\eta} = V^\FP$ were
a new countable set of ordinals, then because
$V^{\FP_\eta} \models ``\FQ_\eta$ is $\eta^+$-c.c.'',
there is a set $B \in V$ such that $B \supseteq A$ and
$\card{B} = \eta$. However, since no new countable subsets
are added to $B$, this is impossible.
Let $G$ be $V$-generic over $\FP$, with
$G \rest \eta = G_\eta$.
We claim that $V' = V[G]$ is our desired model in which
$\gk$ is a tall cardinal having closure $\go$ but not
closure $\go_1$ and in which there are no tall cardinals
having closure $\go_1$.
We begin by showing that
$V' \models ``\gk$ is not a tall cardinal having closure $\go_1$''.
If this is not true, then choose some $\gl > \eta$, and let
$j' : V' \to M'$ be such that
${\rm cp}(j') = \gk$, $j'(\gk) > \gl$, and
$(M')^{\go_1} \subseteq M'$.
Consider $j = j' \rest {\mathcal K}$.
%By work of Zeman and Schindler \cite{Z},
Note that
$j$ is given as an iterated ultrapower of ${\mathcal K}$
using extenders at and above $\gk$
(see \cite{Sch} and \cite{Z}).
Then $\eta$ is regular in $({\cal K})^{M'}$,
since it is regular in ${\cal K}$ and
$({\cal K})^{M'}$ is an iterated ultrapower of
${\cal K}$ by its extenders.
By elementarity, $M'$ is a generic extension of
$j({\cal K}) = ({\cal K})^{M'}$ by $j(\FP$). In addition,
by its definition, forcing with $\FP$ does not change
the cofinality of any cardinal below $\gk$ to $\go_1$.
Hence, by elementarity, forcing with $j(\FP)$ does not
change the cofinality of any cardinal below $j(\gk)$ to $\go_1$ in $M'$.
However, since $V' \models ``{\rm cof}(\eta) = \go_1$'' and
$(M')^{\go_1} \subseteq M'$,
$M' \models ``{\rm cof}(\eta) = \go_1$'' as well.
This immediately contradicts that $j(\gk) > \gl > \eta$.
To show that $V' \models ``\gk$ is a tall cardinal having closure $\go$'',
let $\gl > \eta$ be a regular cardinal. Let ${\cal E}$ be a
$(\gk, \gl)$-extender, with
$k : V \to M$ the corresponding elementary embedding.
By the arguments of \cite{GS},
in $V[G_\eta]$, ${\cal E}$ extends to a $(\gk, \gl)$-extender
${\cal E}^*$. Let $k^* : V[G_\eta] \to M^*$ be the
corresponding elementary embedding. Since no new $\go$ sequences
and no new subsets of $\gk$ are added to $V[G_\eta]$
after forcing with $\FQ_\eta$,
${\cal E}^*$ remains an extender in $V'$
with well-founded ultrapower. Let $k' : V' \to M'$ be the
corresponding elementary embedding. Then
$k' \rest V[G_\eta] = k^*$, since no new subsets of
$\gk$ were added to $V[G_\eta]$ after forcing with $\FQ_\eta$.
Hence, $k'(\gk) > \gl$. Also, since we
may assume that $M^\gk \subseteq M$, this is preserved to
$V[G_\eta]$, i.e.,
$(M^*)^\gk \subseteq M^*$. As $V[G_\eta]$ and $V'$
have the same countable sets of ordinals, in $V'$,
$(M')^\go \subseteq M'$. Consequently, because
$\gl$ was an arbitrary regular cardinal,
$V' \models ``\gk$ is a tall cardinal having closure $\go$''.
It remains to show that
$V' \models ``$There are no tall cardinals having closure $\go_1$''.
Because $\FP$ may be defined so that $\card{\FP} = \eta^+$,
by the results of \cite{LS} and the fact that
$V = {\cal K}$ and ${\cal K}$ contains no measurable
cardinals above $\eta$,
$V' \models ``$There are no measurable cardinals greater than $\eta$''.
It thus suffices to show that
$V' \models ``$No $\gd \in (\gk, \eta)$ is a tall cardinal
having closure $\go_1$''. To see this, suppose to the contrary that
$V' \models ``\gd \in (\gk, \eta)$ is a tall cardinal having
closure $\go_1$''.
Take $i : V' \to N$ with $i(\gd) > \eta$ and $N^{\go_1} \subseteq N$.
As above (see \cite{Sch} and \cite{Z}), $i^* = i \rest {\cal K}$ is
given as an iterated ultrapower of ${\cal K}$ using
extenders at and above $\gd$.
In addition, because
${\cal K} \models ``\eta$ is regular'' and
$i^*({\cal K})$ is an inner model of ${\cal K}$,
$i^*({\cal K}) \models ``\eta$ is regular'' as well.
Since $V' \models ``{\rm cof}(\eta) = \go_1$'',
$N$ must be a generic extension of $i^*({\cal K})$, and
$N^{\go_1} \subseteq N$ in $V'$,
this means that
$N \models ``{\rm cof}(\eta) = \go_1$''.
However, as $\eta < i(\gd)$, by reflection, it follows that
in $V'$, unboundedly many ${\cal K}$-regular cardinals below $\gd$
have their cofinalities changed to $\go_1$.
By the definition of $\FP$, this is impossible.
This contradiction completes the proof of Theorem \ref{t3}.
\end{proof}
%\noindent
Theorems \ref{t2} and \ref{t3} raise the question of
classifying the consistency strength of the existence
of embeddings witnessing a bounded degree
of closure, which we address now.
We deal here with $\go_1$, but the same arguments
actually apply to any regular $\gd \le \gk$.
We begin with Theorem \ref{t2a},
%the following theorem giving the
which gives the equivalence of three conditions
for the existence of such elementary embeddings.
\begin{theorem}
%\label{t2a}
The following conditions are equivalent:
\begin{enumerate}
\item\label{i1} There is an elementary embedding
$j : V \to M$ with critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$.
\item\label{i2} There exists a Rudin-Keisler increasing
sequence of ultrafilters over $\gk$ having length $\go_1$.
\item\label{i3} There exists an elementary embedding
$j : V \to M$ such that $M^\go \subseteq M$ and a sequence
$\la \eta_\ga \mid \ga < \go_1 \ra$ of ordinals below
$j(\gk)$ such that for every $\ga < \go_1$ and for every
$f : [\gk]^\ga \to \gk$, it is the case that
$\eta_\ga \neq j(f)(\la \eta_\gb \mid \gb < \ga \ra)$.
\end{enumerate}
\end{theorem}
\begin{proof}
To show that $(\ref{i1}) \implies (\ref{i2})$, let
$a \subseteq j(\gk)$, $\card{a} = \ha_1$ be such that
$a \not\in M$. Let $\la \eta_\ga \mid \ga < \go_1 \ra$
be an increasing enumeration of $a$. Define a
$\gk$-complete ultrafilter $U_\ga$ over $V_\gk$ by
$$x \in U_\ga \hbox{ iff } \la \eta_\gb \mid \gb < \ga \ra
\in j(x).$$
Clearly, if $\gg < \ga$, then $U_\gg \le_{\rm RK} U_\ga$.
We claim that for every $\gg < \go_1$, there is some $\ga$,
$\gg < \ga < \go_1$ such that $U_\gg <_{\rm RK} U_\ga$ (i.e.,
the inequality is strict). To see this, suppose otherwise.
Then there is $\gg < \go_1$ such that for every $\ga$ with
$\gg < \ga < \go_1$, we have that $U_\gg =_{\rm RK} U_\ga$.
For every $\ga$ with $\gg < \ga < \go_1$, we fix a function
$f_\ga : [\gk]^\gg \to [\gk]^\ga$ witnessing
$U_\gg =_{\rm RK} U_\ga$. We have
$j(f_\ga)(\la \eta_\gb \mid \gb < \gg \ra) =
\la \eta_\gb \mid \gb < \ga \ra$. But
$j(\la f_\ga \mid \gg < \ga < \go_1 \ra) =
\la j(f_\ga) \mid \gg < \ga < \go_1 \ra$, so
$\la j(f_\ga) \mid \gg < \ga < \go_1 \ra \in M$.
Since $\la \eta_\gb \mid \gb < \gg \ra$ is a countable
sequence of ordinals below $j(\gk)$,
$\la \eta_\gb \mid \gb < \gg \ra \in M$.
Consequently, $\la j(f_\ga)(\la \eta_\gb \mid \gb < \gg \ra)
\mid \gg < \ga < \go_1 \ra \in M$, from which it follows that
%$\la \eta_\ga \mid \gg < \ga < \go_1 \ra \in M$.
%This immediately implies that
$\la \eta_\ga \mid \ga < \go_1 \ra \in M$.
This contradicts that $a \not\in M$, thereby proving
$(\ref{i1}) \implies (\ref{i2})$.
To show that $(\ref{i2}) \implies (\ref{i1})$, let
$\la U_\ga \mid \ga < \go_1 \ra$ be a Rudin-Keisler
increasing sequence of ultrafilters over $\gk$.
For $\ga < \go_1$, denote by
$j_\ga : V \to M_\ga$ the ultrapower embedding
generated by $U_\ga$, and for $\ga < \gb < \go_1$,
denote by $j_{\ga, \gb} : M_\ga \to M_\gb$ the
elementary embedding generated by a projection of
$U_\gb$ to $U_\ga$. Let
$\la \la M, i_\ga \ra \mid \ga < \go_1 \ra$ be the direct
limit of the system
$\la \la M_\ga, j_{\ga, \gb} \ra \mid \ga < \gb < \go_1 \ra$, where
$i_\ga : M_\ga \to M$. It is then the case that $M$ and the
limit embedding $j : V \to M$ are as desired.
To see this, we first
note that $M^\go \subseteq M$. This follows since if
$x \subseteq M$ is countable, then for some $\ga < \go_1$,
$x$ has a preimage in $M_\ga$. However, $(M_\ga)^\gk \subseteq
M_\ga$, since $M_\ga$ is the ultrapower by an ultrafilter over $\gk$.
We now define $a \subseteq j(\gk)$
with $\card{a} = \ha_1$ such that $a \not\in M$
by $a = \{i_\ga([{\rm id}]_{U_\ga}) \mid \ga < \go_1\}$.
To see that $a$ is as desired, %$a \not\in M$,
assume to the contrary that
$a \in M$. It must then be true that for some $\gb < \go_1$
and some $b \in M_\gb$, $i_\gb(b) = a$. But then for every
$\ga \ge \gb$, $U_\ga$ must be Rudin-Keisler equivalent to
$U_\gb$, which is impossible.
Since each $i_\ga([{\rm id}]_{U_\ga}) < j(\gk)$,
this completes the proof of $(\ref{i2}) \implies (\ref{i1})$.
To show that $(\ref{i2}) \implies (\ref{i3})$,
we use the previous construction.
The set $a$ just defined is as desired, since %the sequence
$\la U_\ga \mid \ga < \go_1 \ra$ is a strictly increasing
Rudin-Keisler sequence of ultrafilters.
Finally, to show that $(\ref{i3}) \implies (\ref{i2})$,
we use $\la \eta_\ga \mid \ga < \go_1 \ra$ to define the
$U_\ga$ s as in $(\ref{i1}) \implies (\ref{i2})$ and argue as in
$(\ref{i1}) \implies (\ref{i2})$.
Even if $a \in M$, the argument remains valid.
This completes the proof of Theorem \ref{t2a}.
\end{proof}
We remark that in general,
$(j(\gk))^\go \subseteq M$ does not imply that
$M^\go \subseteq M$. To see this,
suppose $\gk < \gl$ are both measurable cardinals.
We construct $j : V \to M$ by first taking an
ultrapower via a measure over $\gk$, and then taking
an iterated ultrapower $\go$ many times by a measure
over $\gl$. It will then be the case that
$(j(\gk))^\go \subseteq M$ but $(j(\gl))^\go \not\subseteq M$.
In addition, an argument using the work of \cite{GS}
shows that it is impossible to
replace the condition of Theorem \ref{t2a}(\ref{i3}) with
$\eta_\ga \neq j(f)(\xi_1, \ldots, \xi_n)$ whenever
$n < \go$, $\xi_1, \ldots, \xi_n < \eta_\ga$, and
$f : [\gk]^n \to \gk$.
\begin{corollary}\label{c1a}
Suppose that $\gk$ is a ${\cal P}^\gl(\gk)$ hypermeasurable
cardinal for $\gl \ge 2$ of cofinality different
from $\go$. Then there is an elementary embedding
$j : V \to M$ having critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$. Moreover, the embedding may be
constructed so that ${\cal P}^\gl(\gk) \subseteq M$.
\end{corollary}
\begin{proof}
Let $i : V \to N$ be an elementary embedding witnessing the
${\cal P}^\gl(\gk)$ hypermeasurability of $\gk$.
Since $\gl \ge 2$, $i$ clearly witnesses the
${\cal P}^2(\gk)$ hypermeasurability of $\gk$ as well.
Hence, ${\cal P}^2(\gk) \subseteq N$,
from which it follows that $i(\gk) > (2^\gk)^+$.
We define by induction an increasing sequence
$\la \eta_\ga \mid \ga < \go_1 \ra$ of ordinals below
$(2^\gk)^+$ satisfying the assumptions of
Theorem \ref{t2a}(\ref{i3}) as follows.
Begin by setting $\eta_0 = \gk$.
Assume now that $\la \eta_\gb \mid \gb < \ga \ra$
has been defined. To define $\eta_\ga$, we first note that
$\la \eta_\gb \mid \gb < \ga \ra \in N$, since
${\cal P}^2(\gk) \subseteq N$. Let
$$U_\ga = \{x \subseteq [\gk]^\ga \mid \la \eta_\gb \mid
\gb < \ga \ra \in i(x)\},$$
with $i_\ga : V \to N_\ga$ the corresponding ultrapower embedding.
Note that $2^\gk < i_\ga(\gk) < (2^\gk)^+$.
Consider $k_\ga : N_\ga \to N$ defined by setting
$k_\ga([f]_{U_\ga}) = i(f)(\la \eta_\gb \mid \gb < \ga \ra)$.
Let $\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra \in N_\ga$
list all subsets of $\gk$. Then since $k_\ga \rest \gk + 1$
is the identity, we will have $k_\ga(A_\xi) = A_\xi$. But then
$k_\ga(\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra) =
\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra$, since
${\cal P}(\gk) \subseteq N_\ga$. In particular,
$(2^\gk)^{N_\ga} \ge 2^\gk$ and
$k_\ga \rest 2^\gk + 1$ is the identity. Thus,
${\rm cp}(k_\ga) = ((2^\gk)^+)^{N_\ga}$.
We now set $\eta_\ga = ((2^\gk)^+)^{N_\ga}$,
thereby completeing our construction.
By $(\ref{i3}) \implies (\ref{i2})$ of Theorem \ref{t2a},
we can let $\la U^*_\ga \mid \ga < \go_1 \ra$ be a strictly
increasing Rudin-Keisler sequence of ultrafilters over $\gk$.
Denote by $\gr_{\gb, \ga}$ the projection from
$U^*_\gb$ onto $U^*_\ga$, whenever $\ga \le \gb < \go_1$.
Then, for every countable sequence $a$ of elements of
${\cal P}^\gl(\gk)$, define
$$E_a = \{x \subseteq V_\gk \mid a \in i(x)\}.$$
This definition makes sense, since by our assumption that
${\rm cof}(\gl) \neq \go$, we may also assume that
$({\cal P}^\gl(\gk))^\go \subseteq {\cal P}^\gl(\gk)$.
If $a$ is a subsequence of $b$,
denote by $\pi_{\b, a}$ the projection of $E_b$
onto $E_a$. Let ${\cal E} = \la \la E_a, \pi_{b, a} \mid
a, b \in [{\cal P}^\gl(\gk)]^{\ha_0}, a$ is a subsequence of $b \ra$
be the corresponding extender, with $i_{\cal E} : V \to N_{\cal E}$ the
associated embedding. Then because $i$ witnesses the
${\cal P}^\gl(\gk)$ hypermeasurability of $\gk$,
$i_{\cal E}$ does as well. In particular,
$N_{\cal E} \supseteq {\cal P}^\gl(\gk)$ and
$(N_{\cal E})^\go \subseteq N_{\cal E}$.
%Let $E$ be an extender witnessing the
%${\cal P}^\gl(\gk)$ hypermeasurability of $\gk$, and
Consider now $\la E_a \times U^*_\ga \mid
a \in [{\cal P}^\gl(\gk)]^{\ha_0}$, $\ga < \go_1 \ra$
with projections $\la \la \pi_{b, a}, \gr_{b, a} \ra \mid a$
is a subsequence of $b$, $\ga \le \gb < \go_1 \ra$.
It is a directed system whose limit model $M$
will be as desired. To see this, use
%let $j_E : V \to M_E$ be the extender embedding generated by $E$. Then use
$i_{\cal E}(\la U^*_\ga \mid \ga < \go_1 \ra)$ over
$N_{\cal E}$ as in $(\ref{i2}) \implies (\ref{i1})$ of Theorem \ref{t2a} to
obtain $M$. Because
%${\cal E}$ witnesses the ${\cal P}^\gl(\gk)$ hypermeasurability of $\gk$,
${\cal P}^\gl(\gk) \subseteq N_{\cal E}$, %In particular,
$i_{\cal E}(\gk) > \card{{\cal P}^\gl(\gk)}$.
Since the embedding generated by
$\la U^*_\ga \mid \ga < \go_1 \ra$ over $V$ has critical point $\gk$,
the embedding generated
by $i_{\cal E}(\la U^*_\ga \mid \ga < \go_1 \ra)$ from $N_{\cal E}$ to $M$ has
critical point $i_{\cal E}(\gk)$. Consequently,
${\cal P}^\gl(\gk) \subseteq M$.
This completes the proof of Corollary \ref{c1a}.
\end{proof}
We turn our attention now to addressing the strength of
the existence of an elementary embedding
$j : V \to M$ with critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$.
We will prove three theorems in this regard,
beginning with the following.
%In particular, we have the following theorem.
\begin{theorem}%\label{t2b}
Suppose that there is no sharp for a strong cardinal
(i.e., that $o$ pistol does not exist).
If there is an elementary embedding
$j : V \to M$ with critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$, then either $o(\gk) \ge \go_1$ in
${\cal K}$, or $\gk$ is measurable in ${\cal K}$ and
$\{\nu < \gk \mid o^{\cal K}(\nu) \ge \go_1\}$ is
unbounded in $\gk$.
\end{theorem}
\begin{proof}
Suppose otherwise, i.e., that $\gk$ is a measurable
cardinal in ${\cal K}$, $o(\gk) < \go_1$ in ${\cal K}$, and
$\{\nu < \gk \mid o^{\cal K}(\nu) \ge \go_1\}$ is
bounded in $\gk$.
Consider $j^* = j \rest {\cal K}$. Then
$j^* : {\cal K} \to ({\cal K})^M$ is an iterated ultrapower
of ${\cal K}$. Hence, each $x \in ({\cal K})^M$ is of the form
$j^*(f)(\gk_{\ga_1}, \ldots, \gk_{\ga_n})$ for some
$f : [\gk]^n \to {\cal K}$ in ${\cal K}$ and some
critical points $\gk_{\ga_1}, \ldots, \gk_{\ga_n}$ of the iteration.
Note that by our assumptions, only measures are involved in
the iteration up to $j^*(\gk)$. We ignore the iteration
above $j^*(\gk)$, if any, since it is irrelevant for the
arguments below. Then it is possible to find an increasing
sequence of critical points (generators of $j^*$) not in $M$
which has all initial segments in $M$. Consider the
least possible $\gd$ which is the limit of such a sequence.
Let $\gth$ be a generator of $j^*$. Then at some stage
during the iteration, $\gth$ was a measurable cardinal, and
a measure over $\gth$ was applied. We may assume that
always the smaller measures in the Mitchell order are
applied before the larger ones. Denote by
${\rm meas}(\gth)$ the final image of $\gth$
during the continuation of the iteration to the final
model $({\cal K})^M$. Then ${\rm meas}(\gth)$ is a
measurable cardinal in $({\cal K})^M$, and
${\rm meas}(\gth) \le j^*(\gk)$. There are
$f_\gth : [\gk]^{n_\gth} \to \gk + 1$ in ${\cal K}$
and a sequence of generators
$\gr_{1, \gth} < \cdots < \gr_{n_\gth, \gth} < \gth$ such that
$j^*(f)(\gr_{1, \gth}, \ldots, \gr_{n_\gth, \gth}) =
{\rm meas}(\gth)$.
\begin{lemma}\label{l2b1}
Let $\la \gr_\gb \mid \gb < \ga \ra$ be a sequence
of generators corresponding to the same measurable
cardinal $\gl$, i.e., for every $\gb, \gg < \ga$,
${\rm meas}(\gr_\gb) = {\rm meas}(\gr_\gg)$.
Then $\ga < \go_1$.
\end{lemma}
\begin{proof}
By our assumptions, the measurable cardinals of
order at least $\go_1$ are bounded below $\gk$.
Hence, there are only countably many normal measures over $\gk$.
Consequently, if $\ga \ge \go_1$, the same measure
was used during the iteration uncountably many times.
However, since $M^\go \subseteq M$, it is impossible
to use the same measure even $\go + 1$ many times.
This completes the proof of Lemma \ref{l2b1}.
\end{proof}
\begin{lemma}\label{l2b2}
$\card{j^*(\gk)}^{\cal K} = (\gk^+)^{\cal K} = \gk^+$.
\end{lemma}
\begin{proof}
Suppose that $\card{j^*(\gk)}^{\cal K} \ge (\gk^{++})^{\cal K}$.
Since by our assumptions, there are only countably many
normal measures over $\gk$ in ${\cal K}$,
some of these measures must be used more than $\go$ many times
in the iteration. This is impossible, however, since
$(j(\gk))^\go \subseteq M$.
This completes the proof of Lemma \ref{l2b2}.
\end{proof}
It now immediately follows that $\gd$ must be singular
in ${\cal K}$ with cofinality less than $\gk$.
This is since $\gk \le \gd \le j^*(\gk)$, by Lemma \ref{l2b2},
$\card{j^*(\gk)}^{\cal K} = (\gk^+)^{\cal K} = \gk^+$,
and both $\gk$ and $\gk^+$ remain regular in $V$.
Suppose that $\gd$ is a regular cardinal in $({\cal K})^M$.
\begin{lemma}\label{l2b3}
There is a continuous, increasing sequence
$\la \gk_\ga \mid \ga < \go_1 \ra$ of critical
points of the iteration such that:
\begin{enumerate}
\item\label{i1a} $\gd = \bigcup_{\ga < \go_1} \gk_\ga$.
\item\label{i2a} For every $f : \gk \to \gk$ such that $f \in {\cal K}$,
and for every $\ga < \go_1$, it is the case that
$(j^*(f) '' \gk_\ga) \cap [\gk_\ga, \gd) = \emptyset$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose otherwise. Let $\la \gt_\ga \mid \ga < \go_1 \ra$
be a continuous, increasing sequence
of critical points of the iteration
having limit $\gd$.
Consider the set
$$S = \{\ga < \go_1 \mid \exists g: \gk \to \gk,
g \in {\cal K} \hbox{ {\rm such that }}
(j^*(g) '' \gt_\ga) \cap [\gt_\ga, \gd) \neq \emptyset\}.$$
Then $S$ is stationary, for if not,
pick a club $C \subseteq \go_1$ such that
$C \cap S = \emptyset$. Consider
$\la \gt_\ga \mid \ga \in C \ra$. This sequence
satisfies clause (\ref{i2a}) above, so there will be
a club $C' \subseteq C$ where $\la \gt_\ga \mid \ga \in C' \ra$
satisfies clause (\ref{i1a}) as well, contrary to our assumptions.
Therefore, by applying Fodor's Theorem to the function
$f(\ga) =$ The least $\gb < \ga$ with
$(j^*(g) '' \gt_\gb) \cap [\gt_\ga, \gd) = \emptyset\}$
for some $g : \gk \to \gk$, $g \in {\cal K}$,
there are $\ga^* < \go_1$ and a stationary
$S^* \subseteq S$ such that for every $\ga \in S^*$,
there exists $g_\ga : \gk \to \gk$, $g_\ga \in {\cal K}$
such that
$(j^*(g_\ga) '' \gt_{\ga^*}) \cap [\gt_\ga, \gd) \neq
\emptyset$.
We now argue that there is a set $E \in {\cal K}$ consisting
of functions from $\gk$ to $\gk$ such that
$\card{E}^{\cal K} < \gk$ and
$E \supseteq \{g_\ga \mid \ga < \go_1\}$.
This follows from the following
\begin{claim}\label{cl1}
Let $A \subseteq \gk^+$, $\card{A} < \gk$. Then there is
$B \in {\cal K}$, $\card{B}^{\cal K} < \gk$ such that
$B \supseteq A$.
\end{claim}
\begin{proof}
There is $\eta < \gk^+$, $\eta \supseteq A$.
Let $t_\eta \in {\cal K}$ be a bijection between
$\gk$ and $\eta$. Consider $x = {t^{-1}_\eta} '' A$.
Then $x \subseteq \gk$ and $\card{x} < \gk$.
Hence, there is $\xi < \gk$ such that $x \subseteq \xi$.
The set $B = t_\eta '' \xi$ is as desired.
%This proves Claim \ref{cl1}.
\end{proof}
Consider next $j^*(E) = {j^*} '' E$. This set is in $({\cal K})^M$.
But now the set
$$x = \{\sup((j^*(g) '' \gt_{\ga^*}) \cap \gd) \mid g \in j^*(E)\}$$
is in $({\cal K})^M$. In addition, $x$ is unbounded in $\gd$
and has cardinality less than $\gk$ in $({\cal K})^M$.
But this means that $\gd$ is singular in $({\cal K})^M$,
contrary to our assumptions.
This completes the proof of Lemma \ref{l2b3}.
\end{proof}
It is now possible to infer that there must be some
%$\gk_\gg < \gd$ such that the measures originating
%from $\gk_\gg$ are used unboundedly often below $\gd$.
$\ga_0 < \go_1$ such that the measures originating
from $\gk_{\ga_0}$ are used unboundedly often below $\gd$.
This is since by Lemma \ref{l2b3}, $\ga \le \gb < \go_1$ implies that
${\rm meas}(\gk_\ga) \ge {\rm meas}(\gk_\gb) \ge \gd$.
This follows because the fact that $\gk_\ga$ is
a critical point of the iteration $j^*$ implies that
${\rm meas}(\gk_\ga) = \min{\{j^*(f) '' \gk_\ga \mid
f: \gk \to \gk, f \in {\cal K}\}}$.
For any $\ga < \go_1$, $\gk_\ga < \gd$, and by
clause (\ref{i2a}) of Lemma \ref{l2b3},
$\gd \le {\rm meas}(\gk_\ga)$.
Therefore, the sequence
$\la {\rm meas}(\gk_\ga) \mid \ga < \go_1 \ra$ is non-increasing.
This means that ${\rm meas}(\gk_\ga)$ should stabilize, i.e.,
there are $\ga_0 < \go_1$ and $\mu \ge \gd$ such that
for every $\ga$ with $\ga_0 \le \ga < \go_1$,
$\mu = {\rm meas}(\gk_\ga)$.
However, by our assumptions, there are only countably
many normal measures over $\gk_{\ga_0}$, which means that
one of them should be used $\ha_1$ many times in the
iteration. By Lemma \ref{l2b1}, this is impossible.
Thus, $\gd$ cannot be regular in $({\cal K})^M$.
Suppose now that $\gd$ is singular in $({\cal K})^M$.
%Then for the same reasons that ${\rm cof}(\gd) < \gk$ in ${\cal K}$,
Then ${\rm cof}(\gd) < \gk$ in $({\cal K})^M$.\footnote{Note that
$({\rm cof}(\gd))^{({\cal K})^M} = ({\rm cof}(\gd))^{\cal K}
\ge {\rm cof}(\gd) = ({\rm cof}(\gd))^{M} = \go_1$,
since otherwise, say if ${\rm cof}(\gd) < \gep =
({\rm cof}(\gd))^M$, then necessarily,
$\gep$ must have cofinality ${\rm cof}(\gd) = \go_1$.
However, $M \models ``\gep$ is regular'' and
$\gep < \gk$. This means that $j(\gep) = \gep$,
a contradiction.}
Hence, clearly,
$\gd$ is not a generator and $\gd < j^*(\gk)$. This means we can
pick a function $g_\gd : [\gk]^{n_\gd} \to \gk$ in ${\cal K}$
and a sequence of generators $\xi_{1, \gd} < \cdots
< \xi_{n_\gd, \gd} < \gd$ such that
$j^*(g_\gd)(\xi_{1, \gd}, \ldots, \xi_{n_\gd, \gd}) = \gd$.
Let $({\rm cof}(\gd))^{({\cal K})^M} = \gep$.
Fix a closed, cofinal sequence
$\la \gs_i \mid i < \gep \ra \in ({\cal K})^M$ for $\gd$.
Let $\la \eta_\gt \mid \gt < \go_1 \ra$ be an increasing
sequence of generators unbounded in $\gd$ such that
$\la \eta_\gt \mid \gt < \go_1 \ra \not\in M$ and
$\eta_0 > \xi_{n_\gd, \gd}$.
We would like to use Mitchell's Covering Lemma
\cite[Theorem 4.19, page 1566]{Mi10} to cover
inside $M$ either the sequence $\la \eta_\gt \mid \gt < \go_1 \ra$
or a final segment
$s' = \la \eta_\gt \mid \gg < \gt < \go_1 \ra$
of this sequence by a set $z$ of size below $\gk$.
Then ${\cal P}(z) = {\cal P}(z)^M$, so $s' \in M$.
However, because $\eta_\gt < j(\gk)$ for $\gt < \go_1$ and
$(j(\gk))^\go \subseteq M$,
$s = \la \eta_\gt \mid \gt \le \gg \ra \in M$.
Thus, $s^\frown s' = \la \eta_\gt \mid \gt < \go_1 \ra \in M$,
a contradiction.
If the sequence $\la {\rm meas}(\eta_\gt) \mid \gt < \go_1 \ra \in M$
and moreover, there is a set $A \in ({\cal K})^M$ of size
$\gep' < \gk$ in $({\cal K})^M$ which covers
$\la \eta_\gt \mid \gt < \go_1 \ra$, then working inside $M$,
for sufficiently large $\gth$, we pick an elementary submodel
$N \prec H_\gth$ such that $N^\go \subseteq N$,
$\card{N} = \gep' + 2^{\ha_0}$, and
$\gd, \la \gs_i \mid i < \gep \ra,
\la {\rm meas}(\eta_\gt) \mid \gt < \go_1 \ra,
A \in N$. Then by Mitchell's Covering Lemma,
%\cite[Theorem 4.19, page 1566]{Mi10},
there are
$\gz < \gd$, $h \in ({\cal K})^M$, and a system of
indiscernibles $C$ such that $N \cap \gd \subseteq h[\gz;C]$.
Moreover, for every limit $i < \gep$ with $\gz < \gs_i$, we
have that all but boundedly many indiscernibles for measures
in $\gs_i \cap A$ are in $C$. Now, using a regressive function,
we will obtain that all but boundedly many indiscernibles
for measures in $A$ are in $C$. In particular, a final segment
of $\la \eta_\gt \mid \gt < \go_1 \ra \in C$, and we are done.
In general, we need not have
$\la {\rm meas}(\eta_\gt) \mid \gt < \go_1 \ra \in M$.
We can compensate for this by working a little harder.
Specifically, we define a tree $T$ and begin
by putting $\la \eta_\gt \mid \gt < \go_1 \ra$
at the first level of $T$. Set
$${\rm Succ}_T(\la \eta_\gt \ra) = \{\gr_{\eta_\gt, k} \mid
k < n_{\eta_\gt}\}$$ for every $\eta_\gt \neq \gk$.
If $\eta_\gt = \gk$, then set ${\rm Succ}_T(\la \eta_\gt \ra) =
\emptyset$. The next level (and all further levels as well) are
defined similarly. Thus, if $\gr_{\eta_\gt, k} = \gk$, then set
${\rm Succ}(\la \eta_\gt, \gr_{\eta_\gt, k} \ra) = \emptyset$.
Otherwise set
$${\rm Succ}_T(\la \eta_\gt, \gr_{\eta_\gt, k} \ra) =
\{\gr_{\gr_{\eta_\gt, k_m}} \mid m < n_{\eta_{\gr_\gt, k}}\}.$$
The tree $T$ will be well founded, since ordinals
along its branches are decreasing.
Consider the set of nodes
$S = \{\gr \mid \exists t \in T[t^\frown\gr \in T$ and
all immediate successors (and consequently all successors)
correspond to $j^*(\gk)\}$. Note that $\card{S} \le
\card{T} = \go_1$. In addition, %note that
by the definition of $T$, generators
corresponding to $j^*(\gk)$ (which are just measures
that started originally from $\gk$) appear only at terminal
nodes $(\gk)$, or possibly at nodes one step before
terminal ones. Further, $x = \{{\rm meas}(\gr) \mid \gr \in S\}
\in M$. To see this, observe that
each ${\rm meas}(\gr)$ is of the form
$j^*(f_\gr)(\xi_{1, \gr}, \ldots, \xi_{n_\gr, \gr})$, with
$\xi_{1, \gr}, \ldots, \xi_{n_\gr, \gr}$ being generators
for $j^*(\gk)$.
We have already shown that the number of generators
for $j^*(\gk)$ is at most countable.
Note that the total number of functions $f_\gr$
which are used in $T$ has size at most $\go_1$
(and the total number of functions relevant for $x$
is at most countable).
If we let $\la t_i \mid i < \go_1 \ra$ be an enumeration
of all of these functions in $V$, then
$j(\la t_i \mid i < \go_1 \ra) = \la j^*(t_i) \mid i < \go_1 \ra \in M$.
Hence, by using the generators for $x$ and the sequence
$\la j^*(t_i) \mid i < \go_1 \ra$, we may now infer that $x \in M$.
Let us now cover $\la j^*(t_i) \mid i < \go_1 \ra$
by a set of size less than $\gk$ in $({\cal K})^M$.
We will do this by covering $\la t_i \mid i < \go_1 \ra$
by a set in ${\cal K}$ of cardinality less than $\gk$
and then applying $j^*$ to our covering set.
In particular, we argue in $V$ as follows:
$2^\gk = \gk^+$ in ${\cal K}$ (and in $V$ as
well since $o(\gk) < \go_1$). This means that
we can code in ${\cal K}$ functions by ordinals
less than $\gk^+$. Thus, there is $\gg < \gk^*$ such that
the codes for $\la t_i \mid i < \go_1 \ra$ are all
below $\gg$. Pick a bijection $h : \gg \to \gk$ in
${\cal K}$. There is $\gg^* < \gk$ such that the
images of the codes of the $t_i$ s are below $\gg^*$.
Then $h '' \gg^*$ will be the desired covering of
the set of the codes of the $t_i$ s.
Now, back in $M$, let $B = j^*(h '' \gg^*)$.
For sufficiently large $\gth$, we pick an elementary submodel
$N \prec H_\gth$ such that $N^\go \subseteq N$,
$\card{N} = \gg^* + 2^{\ha_0}$, and
$\gd, \gk, j^*(\gk), B \in N$.
Then as before, by Mitchell's Covering Lemma,
%\cite[Theorem 4.19, page 1566]{Mi10},
there are
$\gz < \gd$, $h \in ({\cal K})^M$, and a system of
indiscernibles $C$ such that $N \cap \gd \subseteq h[\gz;C]$.
Moreover, for every limit $i < \gep$ with $\gz < \gs_i$, we
have that all but boundedly many indiscernibles for measures
in $\gs_i \cap N$ are in $C$. Now, using a regressive function,
we will obtain that all but boundedly many indiscernibles
for measures in $N$ are in $C$. In particular,
again as before, a final segment
of $\la \eta_\gt \mid \gt < \go_1 \ra \in C$, and we are done.
This completes the proof of Theorem \ref{t2b}.
\end{proof}
\begin{theorem}%\label{t2c}
Suppose that $o(\gk) \ge \go_1$ in ${\cal K}$.
Then there is a generic extension $V$ of ${\cal K}$
with an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
%$(j(\gk))^\go \subseteq M$
$M^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$.
\end{theorem}
\begin{proof}
By the Theorem of \cite{G88}, assuming that
$o(\gk) \ge \go_1$ in ${\cal K}$, it is possible
to force over ${\cal K}$ to obtain a strictly
increasing Rudin-Keisler sequence of ultrafilters
over $\gk$ having length $\go_1$.
Theorem \ref{t2c} then follows by the proof of
Theorem \ref{t2a}, $(\ref{i2}) \implies (\ref{i1})$.
\end{proof}
\begin{theorem}%\label{t2d}
Suppose that $\gk$ is a measurable cardinal
in ${\cal K}$ and $\{\gn < \gk \mid o^{{\cal K}}(\gn) \ge \go_1\}$
is unbounded in $\gk$. Then there is a generic extension
$V$ of ${\cal K}$ and
an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
\end{theorem}
\begin{proof}
Fix a normal measure $U$ over $\gk$.
For each $\nu < \gk$, let $\nu^*$ be the least cardinal
above $\nu$ with $o(\nu) = \go_1$. Let
${\vec W}(\nu^*) = \la W(\nu^*, \xi) \mid \xi < \go_1 \ra$
witness that $o(\nu^*) = \go_1$, i.e.,
${\vec W}(\nu^*)$ is an increasing sequence in the
Mitchell ordering $\triangleleft$ \cite{Mi74} %$\lhd$
of normal measures over $\gk$.
We now turn ${\vec W}(\nu^*)$ into a Rudin-Keisler
increasing sequence of ultrafilters. Let
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
be an Easton support iteration of Prikry type forcings
of length $\gk$,
%where $\dot \FQ_\ga$ is a term for trivial forcing unless $\ga \in A$.
%Under these circumstances,
where for every $\ga < \gk$,
$\dot \FQ_\ga$ is a
term for the forcing of \cite[Section 2]{G88}
(see also \cite{G10})
which adds either a Prikry or Magidor sequence to every
measurable cardinal $\gg \in (\ga, \ga^*)$.
Note that for all such $\gg$, $o(\gg) < \go_1$ by
the definition of $\ga^*$. This extends
${\vec W}(\ga^*) = \la W(\ga^*, \xi) \mid \xi < \go_1 \ra$
into a Rudin-Keisler increasing commutative sequence
${\vec W'}(\ga^*) = \la W'(\ga^*, \xi) \mid \xi < \go_1 \ra$
of $\ga^*$ complete ultrafilters over $\ga^*$.
Let $G$ be ${\cal K}$-generic over $\FP = \FP_\gk$.
We claim that in $V = {\cal K}[G]$, %$\gk$ is $\gl$ tall with
there is an elementary embedding $j : V \to M$
having critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
%closure $\go$ but not closure $\go_1$ as witnessed by some
%$j' : V' \to M'$.
To see this, fix some $\xi < \go_1$.
Define an ultrafilter $U_{\xi}$ over $\gk^2$ in ${\cal K}$ by
$$x \in U_{\xi} \hbox{ {\rm iff} }
\{\nu < \gk \mid \{\gz < \nu^* \mid
(\nu, \gz) \in x\} \in W(\nu^*, \xi)\} \in
U.$$
The ultrapower by $U_{\xi}$ is the ultrapower by
$U$ followed by the ultrapower by
$(j_{U}(\vec W))(\gk^*, \xi)$.
Let $j_{\xi} : {\cal K} \to M_{\xi}$ be the corresponding
elementary embedding.
%Then the diagram $$j_{\gd, \xi} : V \to^{j_\gd} M_\gd \to$$
Then we can write
$j_{\xi} = j_{(j_{U}(\vec W))(\gk^*, \xi)} \circ j_{U}$
and obtain a commutative system of embeddings.
%from ${\cal K}$ into $M$ and $M$ into $M_{\xi}$.
Consider now what happens in $V$.
By using the argument found in the proof of Lemma \ref{l2}
for the construction of the supercompact ultrafilter ${\cal U}$,
we may extend the ultrafilter $U_{\xi}$ of ${\cal K}$ to an
ultrafilter $U_{\xi}'$ of $V$ by constructing an
increasing sequence of conditions successively deciding the statements
``$(\gk, \xi) \in j_{\xi}(\dot x)$'' for all suitable
canonical names $\dot x$.
Because $o(\gz) = \xi$, by the definition of $\FP$,
for a typical $(\nu, \gz)$, a Magidor sequence of
order type $\go^\xi$ was added to $\gz$.
Also, by elementarity, in the ultrapower by $U'_\xi$,
the same thing is true. Thus, let
$j_{U'_\xi} : {\cal K} \to M'_\xi \simeq V^{\gk^2}/U'_\xi$.
Let $[{\rm id}]_{U'_\xi} = \la \gk, \tilde{\xi} \ra$.
Then $M'_\xi$ has a Magidor sequence of order type
$\go^\xi$ for $\tilde{\xi}$ over its ground model
${\cal K}^{M'_\xi}$.
Let $\gr < \xi$. Set $\gs_{\xi, \gr}(\nu, \gz) =
(\nu, \gz_\gr)$, where $\gz_\gr$ is the $\gr^{\rm th}$
member of the Magidor sequence added to $\gz$.
Note that by their definitions,
$\gs_{\xi, \gr}$ will project the extension
$U_{\xi}'$ of $U_{\xi}$ to the extension
$U_{\gr}'$ of $U_{\gr}$.
Consequently,
$$\la \la U_{\xi}' \mid \xi < \go_1 \ra,
\la \gs_{\xi, \gr} \mid \gr \le \xi < \go_1 \ra \ra$$
forms a Rudin-Keisler commutative sequence.
We check that it is strictly increasing.
By Theorem \ref{t2a}, $(\ref{i2}) \implies (\ref{i1})$,
this will suffice to prove Theorem \ref{t2d}.
To do this, we suppose otherwise. Then there are
$\gr < \xi < \go_1$ such that $U'_\gr =_{\rm RK} U'_\xi$.
Let $f : \gk^2 \to \gk^2$ be a witnessing isomorphism.
Then in the ultrapower by $U'_\xi$ we will have
$j_{U'_\xi}(f)(\gk, \tilde{\xi_\gr}) = (\gk, \tilde{\xi})$ since
$U'_\gr = \{x \subseteq \gk^2 \mid \la \gk, \tilde{\xi_\gr} \ra \in
j_{U'_\xi}(x)\}$ because of the projection map $\gs_{\xi, \gr}$.
%We have now the following
By the next claim (Claim \ref{cl2}),
we will be able to assume that $f$ is the
identity in the first coordinate and is strictly
increasing in the second coordinate once the first
one has been fixed, i.e., if $\gt < \gt' < \nu^*$ and
$f(\nu, \gt) = (\ga, \gb)$, $f(\nu, \gt') = (\ga', \gb')$, then
$\nu = \ga = \ga'$ and $\gt < \gb < \gb'$.
\begin{claim}\label{cl2}
There is $f' : \gk^2 \to \gk^2$ such that
\begin{enumerate}
\item $[f']_{U_\xi'} = [f]_{U_\xi'}$.
\item For every inaccessible $\nu < \gk$ and
$\gt < \gt' < \nu^*$, if $f'(\nu, \gt) = (\ga, \gb)$ and
$f'(\nu, \gt') = (\ga', \gb')$, then $\nu = \ga = \ga'$ and
$\gt < \gb < \gb'$.
\end{enumerate}
\end{claim}
\begin{proof}
Without loss of generality, we assume that for
every inaccessible $\nu$ and every $\gt < \nu^*$,
it is the case that $f(\nu, \gt) < \nu^*$.
Therefore, for any inaccessible cardinal $\nu < \gk$,
we may define in $V$ the set
$C_\nu = \{\gt < \gn^* \mid$ For all $\gs < \gt$, the
second coordinate of $f(\nu, \gs)$ is less than
$\gt\}$, which is a club subset of $\nu^*$.
Note that the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c. Hence, there is a club $E_\nu \in {\cal K}$,
$E_\nu \subseteq C_\nu$. Consequently, by normality,
we have that $E_\nu \in W(\nu^*, \gth)$ for every
$\gth < \go_1$. It then follows that $E_\nu$ and
$C_\nu$ will each be in $W'(\nu^*, \gth)$. Thus,
for every $\gth < \go_1$, the set
$x = \{(\nu, \gt) \in \gk^2 \mid \gt \in E_\nu\} \in U_\gth$.
However, $U_\gth'$ extends $U_\gth$, so in particular,
$x \in U_\xi'$. This means that $f \rest x$ is as desired.
This completes the proof of Claim \ref{cl2}.
\end{proof}
For every inaccessible cardinal $\nu < \gk$ and every
$\gt \in [\nu, \nu^*)$, set $f_\nu(\gt) =$ The second coordinate of
$f(\nu, \gt)$. Then $j_{U_\xi'}(f)_\gk(\tilde{\xi_\gr}) =
\tilde{\xi}$.
Pick a set $A_\nu \in W(\nu^*, \gr) - W(\nu^*, \xi)$.
Let $g_\nu = f_\nu \rest A_\nu$. Note that each $g_\nu$
is strictly increasing. Also, in the ultrapower by
$U_\xi'$, $g_\gk(\tilde{\xi_\gr}) = \tilde{\xi}$.
For $\nu < \gk$ an inaccessible cardinal, define
$h_\nu \in {\cal K}$ by
$h_\nu(\gt) = \{\mu \mid \exists p \in \FP_{\nu^*}
[p \forces ``g_\nu(\gt) = \mu$''$]\}$.
By its definition, %not only do we have that
$h_\nu : A_\nu \to {\cal P}(\nu^*)$, and for every
$\gt \in A_\nu$, $g_\nu(\gt) \in h_\nu(\gt)$ and
$\min(h_\nu(\gt)) > \gt$.
\begin{claim}\label{cl3}
There is $B_\nu \in W(\nu^*, \xi)$ such that
$B_\nu \cap \bigcup \rge(h_\nu) = \emptyset$.
\end{claim}
\begin{proof}
If not, then $\bigcup \rge(h_\nu) \in W(\nu^*, \xi)$.
Consequently,
$\nu^* \in j_{W(\nu^*, \xi)}(\bigcup \rge(h_\nu))$.
So there is $\gt \in j_{W(\nu^*, \xi)}(A_\nu)$ such that
$\nu^* \in (j_{W(\nu^*, \xi)}(h_\nu))(\gt)$. But
$\min((j_{W(\nu^*, \xi)}(h_\nu))(\gt)) > \gt$, so
$\nu^* > \gt$. Then $(j_{W(\nu^*, \xi)}(h_\nu))(\gt) = h_\nu(\gt)$.
This is since $\nu^*$ is the critical point of the embedding
$j_{W(\nu^*, \xi)}$ and $\card{h_\nu(\gt)} < \nu^*$.
(This last fact follows because
as we have already observed, the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c.) But $h_\nu(\gt) \subseteq \nu^*$, so
$\nu^* \not\in h_\nu(\gt)$.
This contradiction completes the proof of Claim \ref{cl3}.
\end{proof}
We now look at what happens at $\gk$ in the ultrapower
by $U_\xi'$. It is the case that
$\tilde{\xi} \in B_\gk$. To see this, let
$z = \{(\nu, \gz) \mid \gz \in B_\nu\}$.
We have that $z \in U_\xi \subseteq U_\xi'$.
Hence $(\gk, \tilde{\xi}) \in j_{U_\xi'}(z)$,
so $\tilde{\xi} \in B_\gk$. Then $\tilde{\xi} =
g_\gk(\tilde{\xi_\gr}) \in h_\gk(\tilde{\xi_\gr})$ and
$B_\gk \cap \bigcup(\rge(h_\gk)) = \emptyset$.
This is impossible. This completes the proof of Theorem \ref{t2d}.
%This impossibility thereby completes the proof of Theorem \ref{t2d}.
\end{proof}
We conclude Section \ref{s3} by noting that
it is possible to prove Theorem \ref{t2c} by
forcing over an arbitrary model $V^*$ of ZFC
in which $\gk$ has a coherent sequence of measures
of length at least $\go_1$.
In addition, it is possible to prove Theorem \ref{t2d}
by forcing over an arbitrary model $V^*$ of ZFC in which
$\{\nu < \gk \mid$ There is a coherent sequence of measures
over $\nu$ of length at least $\go_1\}$ is unbounded in $\gk$.
In order to minimize the technical
details involved, however, we force over ${\cal K}$ instead.
\section{The consistency strength of
strongly tall cardinals}\label{s4}
Recall that
$\gk$ is {\em strongly tall} if for every ordinal $\gl \ge \gk$,
there is an elementary embedding witnessing that
$\gk$ is $\gl$ tall which is generated by
a $\gk$-complete measure on some set.
We address the consistency strength of
strongly tall cardinals with the following theorem.
\begin{theorem}
The following theories are equiconsistent:
a) ZFC + There is a strong cardinal and a proper class
of measurable cardinals.
b) ZFC + There is a strongly tall cardinal.
%Con(ZFC + There is a strong cardinal and a proper class
%of measurable cardinals) $\implies$
%Con(ZFC + There is a strongly tall cardinal).
\end{theorem}
\begin{proof}
%Turning now to the proof of Theorem \ref{t4},
We begin with the proof of Theorem \ref{t4}(a).
Suppose $V \models ``$ZFC + $\gk$ is strong +
There is a proper class of measurable cardinals''.
Assume without loss of generality that
$V \models {\rm GCH}$ as well. Fix a proper class
$\la \gl_\ga \mid \ga \in {\rm Ord} \ra$
satisfying the following properties.
\begin{enumerate}
\item $\gl_0 > \gk$.
\item $\ga < \gb$ implies that $\gl_\ga < \gl_\gb$.
\item If $\ga$ is a limit ordinal, then $\gl_\ga =
\bigcup_{\gb < \ga} \gl_\gb$.
\item For every $\ga$, $\gl_{\ga + 1}$ is a measurable cardinal.
\end{enumerate}
We now define the partial ordering $\FP$ used in
the proof of Theorem \ref{t4}(a).
Let $\ga$ be an ordinal. Let $\FQ_\ga$ be the
reverse Easton iteration of length $\gl_{\ga + 1}$
which does trivial forcing except at regular
cardinals in the half-open interval $(\gl_\ga, \gl_{\ga + 1}]$.
At such a stage $\gd$, the forcing used is $\add(\gd, \gl_\ga)$.
$\FP$ is taken as the Easton support product
$\prod_{\ga \in {\rm Ord}} \FQ_\ga$.
Let $G$ be $V$-generic over $\FP$.
Standard arguments show that $V[G] \models {\rm ZFC}$.
The proof of Theorem \ref{t4}(a) will be complete
once we have shown that
$V[G] \models ``\gk$ is strongly tall''.
Towards this end, let $\gl \ge \gk$ be a regular cardinal.
Choose $\ga$ such that $\gl_\ga > \gl$ and
${\rm cof}(\ga) >> \gl$.
Work for the time being in $V$. Fix a
$(\gk, \gl)$-extender ${\cal E}$, with $j : V \to M$
the corresponding ultrapower embedding. It is then the case that
${\rm cp}(j) = \gk$, $\gl_\ga > j(\gk) > \gl$, and
$M^\gk \subseteq M$. For every $a \in [\gl]^{\le \gk}$, set
$U_a = \{x \subseteq V_\gk \mid a \in j(x)\}$. Let
$j_a : V \to M_a$ be the corresponding ultrapower embedding.
If $a$ is a subsequence of $b$, then we denote by
$\pi_{b,a}$ the obviously defined projection of
$U_b$ onto $U_a$ and let $k_{a,b} : M_a \to M_b$ be the
corresponding elementary embedding between the ultrapowers. Then
$$\la \la M_a \mid a \in [\gl]^{\le \gk} \ra,
\la k_{a,b} \mid a, b \in [\gl]^{\le \gk},
a\ {\hbox{\rm is a subsequence of }} b \ra \ra$$
is a $\gk^+$-directed system having limit
$\la M, \la j_a \mid a \in [\gl]^{\le \gk} \ra \ra$.
Let $W \in V$ be a normal ultrafilter over $\gl_{\ga + 1}$, with
$i : V \to N$ the corresponding ultrapower embedding.
For every $a \in [\gl]^{\le \gk}$, let $W_a = U_a \times W$,
with $i_a : V \to N_a$ the corresponding ultrapower embedding.
Note that $i_a$ may be obtained either by first applying
$U_a$ and then $j_a(W)$ (which is actually $W$) or by first
applying $W$ and then $U_a$.
Define a $\gk^+$-directed system
$$\la \la N_a \mid a \in [\gl]^{\le \gk} \ra,
\la \ell_{a,b} \mid a, b \in [\gl]^{\le \gk},
a\ {\hbox{\rm is a subsequence of }} b \ra \ra$$
in the obvious manner. Let
$\la N_{\cal E}, \la i_a \mid a \in [\gl]^{\le \gk} \ra \ra$
be its limit, with $i_{\cal E} : V \to N_{\cal E}$ the
corresponding embedding. Note that $N_{\cal E}$ may be viewed
as the ultrapower by ${\cal E} \times W$ or as the iterated
ultrapower by ${\cal E}$ and then by $j(W) = W$ or as the
ultrapower by first applying $W$ and then ${\cal E}$.
Consider now $V[G]$. Let $W'$ be an extension of $W$ in
$V[G]$ and $i' : V[G] \to N[G']$, $i' \supseteq i$ the
corresponding ultrapower embedding.
Write $\FP = \FP_{{<} \ga} \times \FP_\ga \times \FP_{{>} \ga}$,
with $\FP_{< \ga} = \prod_{\gb < \ga} \FQ_\gb$,
$\FP_\ga = \FQ_\ga$, and $\FP_{> \ga} = \prod_{\gb > \ga} \FQ_\gb$.
Since $\FP$ is defined as a product forcing, the order of the
products just given can be changed. In addition,
$i(\FP_{{<} \ga}) = \FP_{{<} \ga}$,
and $W$ is not affected by $\FP_{{>} \ga}$ because of
its closure. Let $G_{{<} \ga} = G \rest \FP_{< \ga}$,
$G_\ga = G \rest \FP_\ga$, and $G_{{>} \ga} = G \rest \FP_{> \ga}$.
Then $G' \rest \FP_{< \ga} = G_{{<} \ga}$,
$G' \rest \FP_\ga = G_\ga$, and
$G' \rest \FP_{{>} \ga}$ is generated by $i '' G_{{>} \ga}$. Denote by
$G^{{>} \ga}$ the part of $G'$ in the interval
$(\gl_{\ga + 1}, i(\gl_{\ga + 1})]$. Let
$\la f_\xi \mid \xi < \gl_\ga \ra$ be the Cohen functions from
$\gl_{\ga + 1}$ to $\gl_{\ga + 1}$ added by $G_\ga$ over
$\gl_{\ga + 1}$, with $\la f'_\xi \mid \xi < i(\gl_\ga) \ra$ the
corresponding functions from $i(\gl_{\ga + 1})$ to $i(\gl_{\ga + 1})$
added by $G'$.
Consider now $i_{\cal E} : V \to N_{\cal E}$. We extend it to
$i'_{\cal E} : V[G] \to N_{\cal E}[G_{\cal E}]$ as follows. We first
generate $G^{{<} \ga}_{\cal E}$, the part of $G_{\cal E}$ below
$\gl_\ga$, by $j '' G_{{<} \ga}$. We then use
$G^{{>} \ga}$ to generate the part of $G_{\cal E}$ in the interval
$(\gl_{\ga + 1}, i(\gl_{\ga + 1})]$. Finally, the part of
$G_{\cal E}$ above $i(\gl_{\ga + 1})$ is generated by
$i_{\cal E} '' G_{{>} \ga}$.
For every $a \in [\gl]^{\le \gk}$, let
$W_a' = \{x \subseteq V_\gk \times \gl_{\ga + 1} \mid
(a, \gl_{\ga + 1}) \in i'_{\cal E}(x)\}$
be the extension of $W_a$ in $V[G]$ and
$i_a' : V[G] \to N_a[G_a']$, $i_a' \supseteq i_a$ the
corresponding ultrapower embedding.
Then the $\gk^+$-directed system
$$\la \la N_a \mid a \in [\gl]^{\le \gk} \ra,
\la \ell_{a,b} \mid a, b \in [\gl]^{\le \gk},
a\ {\hbox{\rm is a subsequence of }} b \ra \ra$$
extends in the obvious fashion
to the $\gk^+$-directed system
$$\la \la N_a[G_a'] \mid a \in [\gl]^{\le \gk} \ra,
\la \ell_{a,b}' \mid a, b \in [\gl]^{\le \gk},
a\ {\hbox{\rm is a subsequence of }} b \ra \ra$$ with limit
$\la N_{{\cal E}}[G_{{\cal E}}], \la i_a' \mid
a \in [\gl]^{\le \gk} \ra \ra$ and
$i_{{\cal E}}' : V[G] \to N_{{\cal E}}[G_{{\cal E}}]$ the
corresponding embedding.
Fix $a \in [\gl]^{\le \gk}$. Let the Cohen functions
added by $G_a'$ over $i_a(\gl_{\ga + 1})$ corresponding to
$\la f_\xi \mid \xi < \gl_\ga \ra$ be denoted by
$\la f_{a, \xi}' \mid \xi < i_a(\gl_\ga) \ra$.
We come now to the crucial point of the construction.
For every $\gz \in a$, let $\gz_a$ be the ordinal
represented in $M_a$ by the coordinate $\gz$, i.e.,
$j_a(\gz_a) = \gz$. We change the value of
$f'_{a, i_a(\gz)}(\gl_{\ga + 1})$ to $\gz_a$ and let
$f''_{a, i_a(\gz)}$ be the resulting function.
Since we have changed only one value, $f''_{a, i_a(\gz)}$
remains Cohen generic. Note that the number of changes
made is at most $\gk$, which is small relative to
$\gl_{\ga + 1}$. Consequently, after all of the
changes have been made to $G_{{\cal E}}$, the resulting set
$G_{{\cal E}}''$ remains
$N_a[G_{{\cal E}} \rest i_a(\gl_{\ga + 1})]$-generic. Let
$i_a'' : V[G] \to N_a[G_a'']$ be the corresponding embedding.
Note that since the generic set has been changed,
$i_a'' \neq i_a'$. Regardless, we have a $\gk^+$-directed system
$$\la \la N_a[G_a''] \mid a \in [\gl]^{\le \gk} \ra,
\la \ell_{a,b}'' \mid a, b \in [\gl]^{\le \gk},
a\ {\hbox{\rm is a subsequence of }} b \ra \ra$$ with limit
$\la N_{{\cal E}}[G_{{\cal E}}''], \la i_a'' \mid
a \in [\gl]^{\le \gk} \ra \ra$ and
$i_{{\cal E}}'' : V[G] \to N_{{\cal E}}[G_{{\cal E}}'']$ the
corresponding embedding.
Since $G_{{\cal E}}'' \in N_{{\cal E}}[G_{{\cal E}}]$,
$N_{{\cal E}}[G_{{\cal E}}''] = N_{{\cal E}}[G_{{\cal E}}]$.
Using $\gk^+$-directedness, it follows that
$(N_{{\cal E}}[G_{{\cal E}}])^\gk \subseteq N_{{\cal E}}[G_{{\cal E}}]$.
By using an appropriate coding of $[\gl]^{\le \gk}$ in $V$,
any ultrafilter of the form $U_a$ for
$a \in [\gl]^{\le \gk}$ may be replaced by an ultrafilter
of the form $U_{\{\gz\}}$ for some $\gz < \gl$.
Consequently, any system defined using
$\la U_a \mid a \in [\gl]^{\le \gk} \ra$ may be
replaced by a system defined from this coding using only
$\la U_{\{\gz\}} \mid \gz < \gl \ra$, i.e.,
the two systems will have the same direct limit.
Let $W^* = \{z \subseteq \gl_{\ga + 1} \mid
\gl_{\ga + 1} \in i_{{\cal E}}''(z)\}$. By its definition,
and using the fact mentioned in the preceding paragraph,
$W^*$ extends $W$ and projects onto $W_a$ for every
$a \in [\gl]^{\le \gk}$.
Let $i^* : V[G] \to N^*$ be the corresponding
elementary embedding and
$k^* : N^* \to N_{{\cal E}}[G_{{\cal E}}'']$ be
the standard embedding which forms a commutative diagram, i.e.,
$k^*([g]_{W^*}) = i_{{\cal E}}''(g)(\gl_{\ga + 1})$.
Because $i_{\cal E}''$ is definable in $V[G]$,
$i^*$ is definable in $V[G]$ as well.
The next claim is used to
finish the proof of Theorem \ref{t4}(a).
\begin{claim}\label{c1}
$k^*$ is a map onto $N_{{\cal E}}[G_{{\cal E}}'']$
and hence $N^* = N_{{\cal E}}[G_{{\cal E}}'']$.
\end{claim}
\begin{proof}
Note that every element of $N_{{\cal E}}[G_{{\cal E}}'']$
is of the form $i_{{\cal E}}''(h)(\gz, \gl_{\ga + 1})$ for
some $h : \gk \times \gl_{\ga + 1} \to V[G]$ and $\gz < \gl$.
%Therefore, if we
Fix $\gz < \gl$ and consider $f_\gz$. We have
$$k^*([f_\gz]_{W^*}) = (i_{{\cal E}}''(f_\gz))(\gl_{\ga + 1}) = \gz$$
because of the change we made to the value of the Cohen function. Then
for any $h : \gk \times \gl_{\ga + 1} \to V[G]$,
$$i_{{\cal E}}''(h)(\gz, \gl_{\ga + 1}) =
i_{{\cal E}}''(h)((i_{{\cal E}}''(f_\gz))(\gl_{\ga + 1}), \gl_{\ga + 1}) =
i_{{\cal E}}''(t)(\gl_{\ga + 1}) = k^*([t]_{W^*}),$$
where $t(\gr) = h(f_\gz(\gr), \gr)$ for every $\gr < \gl_{\ga + 1}$.
\end{proof}
Because $W^* \supseteq W$ and $W^*$ projects onto
$U_a$ for every $a \in [\gl]^{\le \gk}$,
$i^*(\gk) > \gl_{\ga + 1}$. Since
$N^* = N_{{\cal E}}[G_{{\cal E}}'']$,
$N_{{\cal E}}[G_{{\cal E}}''] = N_{{\cal E}}[G_{{\cal E}}]$,
and
$(N_{{\cal E}}[G_{{\cal E}}])^\gk \subseteq N_{{\cal E}}[G_{{\cal E}}]$,
$i^*$ maps $V[G]$ into a $\gk$-closed inner model.
Consequently, because for any ordinal $\ga$,
$\gl_{\ga + 1}$ is a measurable cardinal in $V$,
$V[G] \models ``\gk$ is strongly tall''.
This completes the proof of Theorem \ref{t4}(a).
\end{proof}
\begin{pf}
Having completed the proof of Theorem \ref{t4}(a),
we turn now to the proof of Theorem \ref{t4}(b).
Suppose $\gk$ is a strongly tall cardinal and that
there is no inner model with two strong cardinals.
We show this implies that there are arbitrarily large
measurable cardinals in ${\cal K}$ (which of course
can be assumed to contain one strong cardinal).
Suppose $\gth > \gk$. Let $\gl >> \gth$ be a
strong limit cardinal. Let $U$ be a $\gk$-complete
uniform ultrafilter on some cardinal $\gd$ with
corresponding elementary embedding $j : V \to M$
such that $j(\gk) > \gl$. Then $\gd \ge \gth$, and
uniformity and $\gk$-completeness together imply that
${\rm cof}(\gd) \ge \gk$.
Let $\FP_U$ be Prikry tree forcing defined
with respect to $U$ (see \cite{G10} for the exact definition).
Force with $\FP_U$ over $V$. Then as with ordinary Prikry forcing,
$V$ and $V^{\FP_U}$ have the same bounded subsets of $\gk$, and
$\gd$ has cofinality $\go$ in $V^{\FP_U}$.
Therefore, if $\gd$ was regular in $V$,
work of Schindler \cite{Sch} shows that
$\gd$ is measurable in ${\cal K}$.
Suppose now that $V \models ``{\rm cof}(\gd) = \eta < \gd$''.
Let $\la \gd_n \mid n < \go \ra$ be the cofinal $\go$
sequence added by $\FP_U$.
Observe that there is no set $x \in V$ of cardinality
less than $\gd$ in $V$ covering $\{\gd_n \mid n < \go\}$.
To see this, suppose otherwise. Without loss of generality,
we can assume that $x \subseteq \gd$. By the uniformity of
$U$, it is the case that $x \not\in U$. This, however, implies that a
final segment of the $\gd_n$ s will be in the compliment of $x$, an
immediate contradiction. Hence, by applying covering arguments
to Schindler's core model \cite{Sch},
for every $\gt < \gd$, there is
a measurable cardinal in ${\cal K}$ above $\gt$.
In particular, there is a measurable cardinal in ${\cal K}$
above $\gth$.
This completes the proofs of both Theorem \ref{t4}(b) and
Theorem \ref{t4}.
\end{pf}
\section{Concluding remarks}\label{s5}
We conclude by posing some questions and making some related comments.
These are as follows:
\begin{enumerate}
\item\label{q1a} Is it possible to obtain a model of ZFC in
which the first $\go$ strongly compact and measurable
cardinals precisely coincide? More generally, is it possible to
obtain a model of ZFC in which there are infinitely
many (including possibly even proper class many) strongly
compact cardinals, and the measurable and strongly
compact cardinals precisely coincide?
\item\label{q2a} Is $V = {\cal K}$ really needed in
the hypotheses of Theorem \ref{t3}, or is it
possible to construct a model
with a tall cardinal having bounded closure $\go$
in which there are neither any
tall cardinals nor any tall cardinals having
bounded closure $\gd$ for $\go_1 \le \gd < \gk$
%witnessing the conclusions of Theorem \ref{t3}
by forcing
over an arbitrary model $V$ of ZFC satisfying the
current assumptions?
\item\label{q3a} Is the existence of $\eta > \gk$ with
$o(\eta) = \go_1$ really needed in order to construct
a model
with a tall cardinal having bounded closure $\go$
in which there are neither any
tall cardinals nor any tall cardinals having
bounded closure $\gd$ for $\go_1 \le \gd < \gk$?
%witnessing the conclusions of Theorem \ref{t3}?
In particular, are the theories
``ZFC + There is a tall cardinal with bounded closure $\go$''
and
``ZFC + There is a tall cardinal with bounded closure''
equiconsistent?
Are the theories
``ZFC + There is a strong cardinal''
and
``ZFC + There is a tall cardinal with bounded closure''
equiconsistent?
\item\label{q4a} Suppose that there is no $\eta > \gk$ with
$o(\eta) = \go_1$ in ${\cal K}$. Is it possible to have an
elementary embedding $j : V \to M$ such that
${\rm cp}(j) = \gk$, $j(\gk) \ge \gk^{++}$, and
$M^\go \subseteq M$, yet for no elementary embedding
$j' : V \to M'$ with ${\rm cp}(j') = \gk$ and $j'(\gk) \ge \gk^{++}$
is it the case that $(M')^{\go_1} \subseteq M'$?
\item\label{q5a} The same question as Question \ref{q4a},
except that we require in addition that
$\gk^{++} = (\gk^{++})^M$. %= (\gk^{++})^{M'}$?
\end{enumerate}
Question \ref{q1a} and its generalized
version are variants of Magidor's question posed
at the end of Section \ref{s2}. Also, in Theorem \ref{t3},
it is possible to eliminate the assumption of
no measurable cardinals above $\eta$.
%by truncating ${\cal K}$ appropriately.
Note that assuming the existence in ${\cal K}$ of $\eta > \gk$ with
$o(\eta) = \go_1$, it is possible first to use the
construction given in the proof of Theorem \ref{t3} and then force with
$\add(\go, 1) \ast \dot {\rm Coll}(\gk^+, {<} \gl)$.
Here, $\gl \ge \eta^+$ is a fixed regular cardinal, and
${\rm Coll}(\gk^+, {<} \gl)$ is the standard L\'evy
collapse which makes $\gl = \gk^{++}$.
By the results of \cite{LS}, the model obtained after
forcing with $\add(\go, 1)$ also witnesses the
conclusions of Theorem \ref{t3}.
Since forcing with ${\rm Coll}(\gk^+, {<} \gl)$
will add no new subsets of $\gk$, the relevant
extender is not affected, and in our final model $V$,
there is an elementary embedding $j : V \to M$ such that
${\rm cp}(j) = \gk$, $j(\gk) \ge \gk^{++}$,
$M^\go \subseteq M$, and $M^{\go_1} \not\subseteq M$.
By Hamkins' results of \cite{H03}, because
$\add(\go, 1) \ast \dot {\rm Coll}(\gk^+, {<} \gl)$
``admits a closure point at $\go$'' (see \cite{H03}
for a definition of this terminology), there is no
elementary embedding $j' : V \to M'$ such that
${\rm cp}(j') = \gk$, $j'(\gk) \ge \gk^{++}$, and
$(M')^{\go_1} \subseteq M'$.
With a little more work (i.e., by using a preparatory
forcing similar to the one given in \cite{G89}),
it is possible to ensure also that
$\gk^{++} = (\gk^{++})^M$.
It is unclear at all, however, whether an assumption beyond
$o(\gk) = \gk^{++}$ is really needed. In fact,
this prompts us to ask the related question
\begin{enumerate}
\setcounter{enumi}{+5}
\item\label{q6a} Suppose $o(\gk) = \gk^{++}$.
Is it possible to force in
$V^\FP$ an $\go$-directed but not
$\go_1$-directed sequence
$\la \la U_\ga, \pi_{\ga, \gb} \ra \mid \ga \le \gb < \gk^{++} \ra$
of ultrafilters over $\gk$ such that there is no
$\gk$-directed sequence of ultrafilters of length $\gk^{++}$
in $V^\FP$?
\end{enumerate}
We end by conjecturing that it is possible and that
methods from \cite{G89} may be relevant.
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\begin{lemma}\label{mps}
$V^\FP \models ``\gk$ is supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A06a}.
Let $\gl \ge \gk^+ = 2^\gk$
be any regular cardinal.
Take $j : V \to M$ as an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk$
such that $M \models ``\gk$
is not $\gl$ supercompact''.
By \cite[Lemma 2.1]{AC2},
in $M$, $\gk$ is a
Mahlo limit of
strong cardinals. This means
by the definition of $\FP$ that
it is possible to opt for trivial forcing in
the stage $\gk$ lottery
held in $M$ in the
definition of $j(\FP)$.
Further, $M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is
strong''. This is since
otherwise, in $M$,
$\gk$ is supercompact
up to a strong cardinal,
so by the proof of \cite[Lemma 2.4]{AC2},
$\gk$ is supercompact in $M$.
Consequently, in $M$, above the appropriate condition,
$j({\FP})$ is forcing equivalent to ${\FP} \ast \dot \FQ$, where
the first nontrivial stage in $\dot \FQ$
takes place well after $\gl$.
We may now apply the argument of \cite[Lemma 1.5]{G86}.
Specifically, let $G$ be $V$-generic over ${\FP}$.
%By the definition of ${\FP}$,
%$j '' G = G$.
Since GCH in $V$ implies that
$V \models ``2^{\gl} = \gl^+$'', 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
$P_\gk(\gl)$.
By \cite[Lemmas 1.4 and 1.2]{G86} and the
fact that $M^{\gl} \subseteq M$, we may
define 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
reverse Easton iteration, except that
at coordinates at which Prikry forcing occurs in $p_\ga$,
measure 1 sets are shrunk and stems are not
extended. For a more precise definition,
readers are urged to consult \cite{G86}.}
every initial segment of
the sequence is in $M$, 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]{G86} 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,
%$V^\FP \models ``\gk$ is supercompact''.
this completes the proof of Lemma \ref{mps}.
\end{proof}
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 {\it 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 $\gk$ is a measurable cardinal, $\gl > \gk$
is arbitrary, and ${\cal E} = \la E_a \mid a \in [\gl]^{< \go} \ra$
is a $(\gk, \gl)$-extender.
Say that $\cal E$
is {\em $\gk^+$-directed} if for each $\gk$ sequence
$\la E_i \mid i < \gk \ra$ of measures from $\cal E$,
there is some $E \in {\cal E}$ such that for each $i < \gk$,
$E_i <_{\rm RK} E$ (so $E$ projects onto $E_i$
as in the Rudin-Keisler ordering).
We have the following
general fact.
\begin{lemma}\label{mps}
Suppose
\end{lemma}
If $\FP$ is
an arbitrary partial ordering
and $\gk$ is a regular cardinal,
%$\FP$ is {\it $\gk$-distributive}
%if for every sequence
%$\la D_\ga : \ga < \gk \ra$
%of dense open subsets of $\FP$,
%$\bigcap_{\ga < \gk} D_\ga$
%is dense open.
%$\FP$ is {\it $\gk$-closed}
%if every increasing chain of elements
%$\la p_\ga : \ga < \gk \ra$ of $\FP$
%has an upper bound $p \in \FP$.
$\FP$ is {\it $\gk$-directed closed}
if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga \mid \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha \mid \alpha < \delta \rangle$
is {\it directed} if
every two elements
$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$.
%If in addition, any directed subset of
%$\FP$ of size $\gk$ has an upper bound,
%then $\FP$ is said to be
%{\it ${\le} \gk$-directed closed}.
$\FP$ is {\it $\gk$-strategically closed} if in the
two person game
of length $\gk + 1$
in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga \le \gk\rangle$, where player I plays odd
stages and player
II plays even stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\it ${<}\gk$-strategically closed} if
$\FP$ is $\gd$-strategically closed for every
$\gd < \gk$.
Note that if $\FP$ is
$\gk$-directed closed, then $\FP$ is
${<}\gk$-strategically closed
(so since $\add(\gk, 1)$ is $\gk$-directed
closed, $\add(\gk, 1)$ is ${<}\gk$-strategically
closed as well).
We adopt Hamkins' terminology of \cite{H3, H2, H03}
and say that {\it $x \subseteq \gk$
is a fresh subset of $\gk$
with respect to $\FP$} if
$\FP$ is nontrivial forcing,
$x \in V^\FP$, $x \not\in V$, yet
$x \cap \ga \in V$ for every $\ga < \gk$.
%Note that
%if $\FP$ is $\gk$-strategically closed,
%then $\FP$ is $\gk$-distributive.
%Also, if $\FP$ is
%$\gk$-strategically closed and
%$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$\FP$ is {\it ${\prec}\gk$-strategically closed} if in the
%two person game in which the players construct an increasing
%sequence
%$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
%stages and player
%II plays even and limit stages (again choosing the
%trivial condition at stage 0),
%then player II has a strategy which
%ensures the game can always be continued.
%For the purposes of this paper,
%at any stage $\gd$ at which
%a nontrivial forcing is done in a Gitik iteration,
%we assume the partial ordering
%$\FQ_\gd$ with which we force is
%either $\gd$-directed closed or is
%Prikry forcing defined with respect to
%a normal measure over $\gd$
%(although other types of partial orderings
%may be used in the general case --- see
%\cite{G86} for additional details).
%By Lemmas 1.2 and 1.4 of \cite{G86},
%if $\gd_0$ is the first stage in the
%definition of $\FP$ at which a nontrivial
%forcing is done, then forcing with
%$\FP$ adds no bounded subsets to $\gd_0$.
This is since if it were, then for $\ga < \go_1$ and
${\cal E}_\ga = {\cal E} \rest \gl^{+ \ga}$,
${\cal E}_\ga \in M'$, since $H(\gl^{+ \ga + 1}) \subseteq M'$ and
${\cal E}_\ga \in H(\gl^{+ \ga + 1})$.
Hence, $\la {\cal E}_\ga \mid \ga < \go_1 \ra \in M'$,
thereby allowing us to recover ${\cal E}'$ within $M'$.
We note that in the proof of Theorem \ref{t2} just given,
in order to obtain the desired elementary
embedding $j : V \to M$
witnessing the $\gl$ tallness of $\gk$
such that $M$ is closed under $\go$ sequences
yet $M$ is not closed under $\go_1$ sequences,
we need to restrict an elementary embedding
witnessing more than $\gl$ strongness.
%We remark that the proof just given
%required restricting an extender witnessing greater strength
%in order to create the appropriate embedding into an
%uses an elementary embedding of the same degree of
%strongness to create in a generic extension an
%elementary embedding into an
%inner model which is $\go$ closed yet which isn't
%$\go_1$ closed.
In general, an elementary embedding witnessing
only measurability cannot be used for this purpose,
i.e., cannot be used to create %in a generic extension
an elementary embedding into an inner model which
is $\go$ closed yet which isn't $\go_1$ closed.
To see this, assume that $\gk$ is a measurable
cardinal and $V = L[U]$. Assume further that
$V'$ is a set forcing extension of $V$ and that there is
an elementary embedding $j : V' \to M$ such that
$M$ is $\go$ closed yet $M$
witnesses no additional degree of closure. %is not $\gk$ closed.
Because $V = L[U]$,
standard arguments (see, e.g., \cite{Z}) show that
$j \rest V$ must be an iterated ultrapower.
However, this iteration cannot be infinite, for if it were, then
by the $\omega$ closure of $M$, the sequence
$\la \gk_n \mid n \leq \go \ra \in M$.
Because $\gk_\go < j(\gk)$, $\gk_\go$ must be measurable in $M$.
This is impossible, since
$\la \gk_n \mid n < \go \ra \in M$ must be a Prikry sequence
through $\gk$. Thus, the iteration must be finite, meaning that
$M$ must be $\gk$ closed with respect to $V'$.
%Even though the above paragraph shows that
%in general, it is not possible to
%take an arbitrary elementary embedding and use
%it to construct an $\go$ closed but not
%$\go_1$ closed ultrapower,
Somewhat surprisingly, given the
results of the above paragraph, it is possible
for certain values of $\gl$ to start with
an elementary embedding witnessing the $\gl$
strongness of $\gk$ and force to construct an
elementary embedding having critical point $\gk$
into an inner model
which is $\go$ closed yet which isn't
$\go_1$ closed. Specifically, we have the following theorem.
Theorems \ref{t2} and \ref{t3} raise the following
\bigskip\noindent
\noindent Question: Is the consistency strength of a tall
cardinal having closure exactly $\go$ with no tall cardinals
having closure greater than or equal to $\go_1$ above
the consistency strength of a strong cardinal?
\bigskip
We now define %$a \subseteq j(\gk)$
$a$ with $\card{a} = \ha_1$ such that $a \not\in M$
by $a = \{i_\ga([{\rm id}]_{U_\ga}) \mid \ga < \go_1\}$.
To see that $a$ is as desired, %$a \not\in M$,
assume to the contrary that
$a \in M$. It must then be true that for some $\gb < \go_1$
and some $b \in M_\gb$, $i_\gb(b) = a$. But then for every
$\ga \ge \gb$, $U_\ga$ must be Rudin-Keisler equivalent to
$U_\gb$, which is impossible.
Next, let
$b = \{\gt_\ga \mid \ga < \go_1\}$, where for some fixed
$\gb < \go_1$,
$i_\gb(\gt_\ga) = i_\ga([{\rm id}]_{U_\ga})$. Let
$f_\ga : \gk \to \gk$ be such that $[f_\ga]_{U_\gb} = \gt_\ga$.
Since $f_\ga$ projects $U_\gb$ onto $U_\ga$,
$b$ is as desired.
\begin{corollary}\label{c1a}
Suppose that $\gk$ is a ${\cal P}^\g2(\gk)$ hypermeasurable
cardinal. Then there is an elementary embedding
$j : V \to M$ having critical point $\gk$ such that
$(j(\gk))^\go \subseteq M$ yet $(j(\gk))^{\go_1}
\not\subseteq M$.
\end{corollary}
\begin{proof}
Let $i : V \to N$ be an elementary embedding witnessing the
${\cal P}^2(\gk)$ hypermeasurability of $\gk$.
Because ${\cal P}^2(\gk) \subseteq N$, $i(\gk) > (2^\gk)^+$.
We define by induction an increasing sequence
$\la \eta_\ga \mid \ga < \go_1 \ra$ of ordinals below
$(2^\gk)^+$ satisfying the assumptions of
Theorem \ref{t2a}(\ref{i3}) as follows.
Begin by setting $\eta_0 = \gk$.
Assume now that $\la \eta_\gb \mid \gb < \ga \ra$
has been defined. To define $\eta_\ga$, we first note that
$\la \eta_\gb \mid \gb < \ga \ra \in N$, since
${\cal P}^2(\gk) \subseteq N$. Let
$$U_\ga = \{x \subseteq [\gk]^\ga \mid \la \eta_\gb \mid
\gb < \ga \ra \in i(x)\},$$
with $i_\ga : V_\ga \to N_\ga$ the corresponding ultrapower embedding.
Note that $2^\gk < i_\ga(\gk) < (2^\gk)^+$.
Consider $k_\ga : N_\ga \to N$ defined by setting
$k_\ga([f]_{U_\ga}) = i(f)(\la \eta_\gb \mid \gb < \ga \ra)$.
Let $\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra \in N_\ga$
list all subsets of $\gk$. Then since $k_\ga \rest \gk + 1$
is the identity, we will have $k_\ga(A_\xi) = A_\xi$. But then
$k_\ga(\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra) =
\la A_\xi \mid \xi < (2^\gk)^{N_\ga} \ra$, since
${\cal P}(\gk) \subseteq N_\ga$. In particular,
$(2^\gk)^{N_\ga} \ge 2^\gk$ and
$k_\ga \rest 2^\gk + 1$ is the identity. Thus,
${\rm cp}(k_\ga) = ((2^\gk)^+)^{N_\ga}$.
We now set $\eta_\ga = ((2^\gk)^+)^{N_\ga}$,
thereby completeing our construction.
By $(\ref{i3}) \implies (\ref{i2})$ of Theorem \ref{t2a},
we can let $\la U_\ga \mid \ga < \go_1 \ra$ be a strictly
increasing Rudin-Keisler sequence of ultrafilters over $\gk$.
Let $E$ be an extender witnessing the
${\cal P}^2(\gk)$ hypermeasurability of $\gk$, and consider
$\la E \times U_\ga \mid \ga < \go_1 \ra$.
It is a directed system whose limit model $M$
will be as desired. To see this, let
$j_E : V \to M_E$ be the extender embedding
generated by $E$. Then use
$j_E(\la U_\ga \mid \ga < \go_1 \ra)$ over
$M_E$ as in $(\ref{i2}) \implies (\ref{i1})$ of Theorem \ref{t2a} to
obtain $M$. Because $E$ witnesses the
${\cal P}^2(\gk)$ hypermeasurability of $\gk$,
${\cal P}^2(\gk) \subseteq M_E$.
In particular, $j_E(\gk) > 2^{2^\gk}$.
Since the embedding generated by
$\la U_\ga \mid \ga < \go_1 \ra$ over $V$ has critical point $\gk$,
the embedding generated
by $j_E(\la U_\ga \mid \ga < \go_1 \ra)$ from $M_E$ to $M$ has
critical point $j_E(\gk)$. Consequently,
${\cal P}^2(\gk) \subseteq M$.
This completes the proof of Corollary \ref{c1a}.
\end{proof}
Further, $\card{x}^V \le \go_1$. Let
$\card{x}^M = \gep$. Then $\gep < \gk$, since otherwise,
${\rm cof}^{{\cal K}}(\gep)$ will be either $\gk$ or
$\gk^+$. This means that in $V$, either $\gk$ or
$\gk^+$ has cofinality $\go_1$, which is impossible.
Work now in $M$. By the Mitchell Covering Lemma
\cite[Theorem 4.51]{Mi10}, there are $\gs < \gk$,
$h \in ({\cal K})^M$, and a maximal set of
indiscernibles $C \in M$ for $x$ such that $x \subseteq h[\gs ; C]$.
Maximality implies that a final segment
$s' = \la \eta_\gt \mid \gg < \gt < \go_1 \ra$ of the sequence
$\la \eta_\gt \mid \gt < \go_1 \ra$ is contained in $C$.
But then $s' \in M$, since $\card{C} \le \gep < \gk$ in $M$ and
${\cal P}(\gep) \subseteq M$.
However, because $\eta_\gt < j(\gk)$ for $\gt < \go_1$ and
$(j(\gk))^\go \subseteq M$,
$s = \la \eta_\gt \mid \gt \le \gg \ra \in M$.
Thus, $s^\frown s' = \la \eta_\gt \mid \gt < \go_1 \ra \in M$,
a contradiction.
so since $(j(\gk))^\go \subseteq M$,
all relevant generators are in $M$. It is the case that
$\la f
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 $\gk$ an inaccessible
cardinal, we will say that
the partial ordering $\FP$ is
%will be said to be
{\it ${\prec}\gk$-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 $\gd < \gk$.
For more details on these definitions,
readers are urged to consult
\cite{GS}.\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.}
Throughout the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik style iterations of
Prikry-like forcings.}
By this we will mean an iteration
as first given by Gitik in \cite{G}
(and elaborated upon further in
\cite{GS}),
where at any stage $\gd$ at which
a nontrivial forcing is done,
we assume the partial ordering
$\FQ_\gd$ with which we force is
${\prec}\gd$-weakly closed
and satisfies the Prikry condition.
By Lemmas 1.2 and 1.4 of \cite{G},
if $\gd_0$ is the first stage in the
definition of $\FP$ at which a nontrivial
forcing is done, then forcing with
$\FP$ adds no bounded subsets to $\gd_0$.
\begin{theorem}%\label{t2d}
Suppose that $\gk$ is a measurable cardinal
in ${\cal K}$ and $\{\gn < \gk \mid o^{{\cal K}}(\gn) \ge \go_1\}$
is unbounded in $\gk$. Then there is a generic extension
$V$ of ${\cal K}$ and
an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
\end{theorem}
\begin{proof}
Fix a normal measure $U$ over $\gk$.
For each $\nu < \gk$, let $\nu^*$ be the least cardinal
above $\nu$ with $o(\nu) = \go_1$. Let
${\vec W}(\nu^*) = \la W(\nu^*, \xi) \mid \xi < \go_1 \ra$
witness that $o(\nu^*) = \go_1$, i.e.,
${\vec W}(\nu^*)$ is an increasing sequence in the
Mitchell ordering $\triangleleft$ \cite{Mi74} %$\lhd$
of normal measures over $\gk$.
We now turn ${\vec W}(\nu^*)$ into a Rudin-Keisler
increasing sequence of ultrafilters. Let
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
be an Easton support iteration of Prikry type forcings
of length $\gk$,
%where $\dot \FQ_\ga$ is a term for trivial forcing unless $\ga \in A$.
%Under these circumstances,
where for every $\ga < \gk$,
$\dot \FQ_\ga$ is a
term for the forcing of \cite[Section 2]{G88}
(see also \cite{G10})
which adds either a Prikry or Magidor sequence to every
measurable cardinal $\gg \in (\ga, \ga^*)$.
Note that for all such $\gg$, $o(\gg) < \go_1$ by
the definition of $\ga^*$. This extends
${\vec W}(\ga^*) = \la W(\ga^*, \xi) \mid \xi < \go_1 \ra$
into a Rudin-Keisler increasing commutative sequence
${\vec W'}(\ga^*) = \la W'(\ga^*, \xi) \mid \xi < \go_1 \ra$
of $\ga^*$ complete ultrafilters over $\ga^*$.
Let $G$ be ${\cal K}$-generic over $\FP = \FP_\gk$.
We claim that in $V = {\cal K}[G]$, %$\gk$ is $\gl$ tall with
there is an elementary embedding $j : V \to M$
having critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
%closure $\go$ but not closure $\go_1$ as witnessed by some
%$j' : V' \to M'$.
To see this, fix some $\xi < \go_1$.
Define an ultrafilter $U_{\xi}$ over $\gk^2$ in ${\cal K}$ by
$$x \in U_{\xi} \hbox{ {\rm iff} }
\{\nu < \gk \mid \{\gz < \nu^* \mid
(\nu, \gz) \in x\} \in W(\nu^*, \xi)\} \in
U.$$
The ultrapower by $U_{\xi}$ is the ultrapower by
$U$ followed by the ultrapower by
$(j_{U}(\vec W))(\gk^*, \xi)$.
Let $j_{\xi} : {\cal K} \to M_{\xi}$ be the corresponding
elementary embedding.
%Then the diagram $$j_{\gd, \xi} : V \to^{j_\gd} M_\gd \to$$
Then we can write
$j_{\xi} = j_{(j_{U}(\vec W))(\gk^*, \xi)} \circ j_{U}$
and obtain a commutative system of embeddings.
%from ${\cal K}$ into $M$ and $M$ into $M_{\xi}$.
Consider now what happens in $V$.
By using the argument found in the proof of Lemma \ref{l2}
for the construction of the supercompact ultrafilter ${\cal U}$,
we may extend the ultrafilter $U_{\xi}$ of ${\cal K}$ to an
ultrafilter $U_{\xi}'$ of $V$ by constructing an
increasing sequence of conditions successively deciding the statements
``$(\gk, \xi) \in j_{\xi}(\dot x)$'' for all suitable
canonical names $\dot x$.
Because $o(\gz) = \xi$, by the definition of $\FP$,
for a typical $(\nu, \gz)$, a Magidor sequence of
order type $\go^\xi$ was added to $\gz$.
Also, by elementarity, in the ultrapower by $U'_\xi$,
the same thing is true. Thus, let
$j_{U'_\xi} : {\cal K} \to M'_\xi \simeq V^{\gk^2}/U'_\xi$.
Let $[{\rm id}]_{U'_\xi} = \la \gk, \tilde{\xi} \ra$.
Then $M'_\xi$ has a Magidor sequence of order type
$\go^\xi$ for $\tilde{\xi}$ over its ground model
${\cal K}^{M'_\xi}$.
Let $\gr < \xi$. Set $\gs_{\xi, \gr}(\nu, \gz) =
(\nu, \gz_\gr)$, where $\gz_\gr$ is the $\gr^{\rm th}$
member of the Magidor sequence added to $\gz$.
Note that by their definitions,
$\gs_{\xi, \gr}$ will project the extension
$U_{\xi}'$ of $U_{\xi}$ to the extension
$U_{\gr}'$ of $U_{\gr}$.
Consequently,
$$\la \la U_{\xi}' \mid \xi < \go_1 \ra,
\la \gs_{\xi, \gr} \mid \gr \le \xi < \go_1 \ra \ra$$
forms a Rudin-Keisler commutative sequence.
We check that it is strictly increasing.
By Theorem \ref{t2a}, $(\ref{i2}) \implies (\ref{i1})$,
this will suffice to prove Theorem \ref{t2d}.
To do this, we suppose otherwise. Then there are
$\gr < \xi < \go_1$ such that $U'_\gr =_{\rm RK} U'_\xi$.
Let $f : \gk^2 \to \gk^2$ be a witnessing isomorphism.
Then in the ultrapower by $U'_\xi$ we will have
$j_{U'_\xi}(f)(\gk, \tilde{\xi_\gr}) = (\gk, \tilde{\xi})$ since
$U'_\gr = \{x \subseteq \gk^2 \mid \la \gk, \tilde{\xi_\gr} \ra \in
j_{U'_\xi}(x)\}$ because of the projection map $\gs_{\xi, \gr}$.
%We have now the following
By the next claim (Claim \ref{cl2}),
we will be able to assume that $f$ is the
identity in the first coordinate and is strictly
increasing in the second coordinate once the first
one has been fixed, i.e., if $\gt < \gt'$ and
$f(\nu, \gt) = (\ga, \gb)$, $f(\nu, \gt') = (\ga', \gb')$, then
$\nu = \ga = \ga'$ and $\gt < \gb < \gb'$.
\begin{claim}\label{cl2}
There is $f' : \gk^2 \to \gk^2$ such that
\begin{enumerate}
\item $[f']_{U_\xi'} = [f]_{U_\xi'}$.
\item For every inaccessible $\nu < \gk$ and
$\gt < \gt' < \nu^*$, if $f'(\nu, \gt) = (\ga, \gb)$ and
$f'(\nu, \gt') = (\ga', \gb')$, then $\nu = \ga = \ga'$ and
$\gt < \gb < \gb'$.
\end{enumerate}
\end{claim}
\begin{proof}
Without loss of generality, we assume that for
every inaccessible $\nu$ and every $\gt < \nu^*$,
it is the case that $f(\nu, \gt) < \nu^*$.
Therefore, for any inaccessible cardinal $\nu < \gk$,
we may define in $V$ the set
$C = \{\gt < \gn^* \mid$ For all $\gs < \gt$, the
second coordinate of $f(\nu, \gs)$ is less than
$\gt\}$, which is a club subset of $\nu^*$.
Note that the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c. Hence, there is a club $E_\nu \in {\cal K}$,
$E_\nu \subseteq C_\nu$. Consequently, by normality,
we have that $E_\nu \in W(\nu^*, \gth)$ for every
$\gth < \go_1$. It then follows that $E_\nu$ and
$C_\nu$ will each be in $W'(\nu^*, \gth)$. Thus,
for every $\gth < \go_1$, the set
$x = \{(\nu, \gt) \in \gk^2 \mid \gt \in E_\nu\} \in U_\gth$.
However, $U_\gth'$ extends $U_\gth$, so in particular,
$x \in U_\xi'$. This means that $f \rest x$ is as desired.
This completes the proof of Claim \ref{cl2}.
\end{proof}
For every inaccessible cardinal $\nu < \gk$ and every
$\gt \in [\nu, \nu^*)$, set $f_\nu(\gt) =$ The second coordinate of
$f(\nu, \gt)$. Then $j_{U_\xi'}(f)_\gk(\tilde{\xi_\gr}) =
\tilde{\xi}$.
Pick a set $A \in W(\nu^*, \gr) - W(\nu^*, \xi)$.
Let $g_\nu = f_\nu \rest A$. Note that each $g_\nu$
is strictly increasing. Also, in the ultrapower by
$U_\xi'$, $g_\gk(\tilde{\xi_\gr}) = \tilde{\xi}$.
For $\nu < \gk$ an inaccessible cardinal, define
$h_\nu \in {\cal K}$ by
$h_\nu(\gt) = \{\mu \mid \exists p \in \FP_{\nu^*}
[p \forces ``g_\nu(\gt) = \mu$''$]\}$.
By its definition, %not only do we have that
$h_\nu : A_\nu \to {\cal P}(\nu^*)$, and for every
$\gt \in A_\nu$, $g_\nu(\gt) \in h_\nu(\gt)$ and
$\min(h_\nu(\gt)) > \gt$.
\begin{claim}\label{cl3}
There is $B_\nu \in W(\nu^*, \xi)$ such that
$B_\nu \cap \bigcup \rge(h_\nu) = \emptyset$.
\end{claim}
\begin{proof}
If not, then $\bigcup \rge(h_\nu) \in W(\nu^*, \xi)$.
Consequently,
$\nu^* \in j_{W(\nu^*, \xi)}(\bigcup \rge(h_\nu))$.
So there is $\gt \in j_{W(\nu^*, \xi)}(A_\nu)$ such that
$\nu^* \in (j_{W(\nu^*, \xi)}(h_\nu))(\gt)$. But
$\min((j_{W(\nu^*, \xi)}(h_\nu))(\gt)) > \gt$, so
$\nu^* > \gt$. Then $(j_{W(\nu^*, \xi)}(h_\nu))(\gt) = h_\nu(\gt)$.
This is since $\nu^*$ is the critical point of the embedding
$j_{W(\nu^*, \xi)}$ and $\card{h_\nu(\gt)} < \nu^*$.
(This last fact follows because
as we have already observed, the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c.) But $h_\nu(\gt) \subseteq \nu^*$, so
$\nu^* \not\in h_\nu(\gt)$.
This contradiction completes the proof of Claim \ref{cl3}.
\end{proof}
We now look at what happens at $\gk$ in the ultrapower
by $U_\xi'$. It is the case that
$\tilde{\xi} \in B_\gk$. To see this, let
$z = \{(\nu, \gz) \mid \gz \in B_\nu\}$.
We have that $z \in U_\xi \subseteq U_\xi'$.
Hence $(\gk, \tilde{\xi}) \in j_{U_\xi'}(z)$,
so $\tilde{\xi} \in B_\gk$. Then $\tilde{\xi} =
g_\gk(\tilde{\xi_\gr}) \in h_\gk(\tilde{\xi_\gr})$ and
$B_\gk \cap \bigcup(\rge(h_\gk)) = \emptyset$.
This is impossible. This completes the proof of Theorem \ref{t2d}.
%This impossibility thereby completes the proof of Theorem \ref{t2d}.
\end{proof}
\begin{theorem}\label{t2d}
Suppose that $\gk$ is a measurable cardinal
in $V^*$ and $\{\gn < \gk \mid o(\gn) \ge \go_1\}$
is unbounded in $\gk$ in $V^*$.
Then there is a generic extension $V$ of $V^*$ and
an elementary embedding $j : V \to M$ having
critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
\end{theorem}
%\begin{theorem}%\label{t2d}
%
%Suppose that $\gk$ is a measurable cardinal
%in ${\cal K}$ and $\{\gn < \gk \mid o^{{\cal K}}(\gn) \ge \go_1\}$
%is unbounded in $\gk$. Then there is a generic extension
%$V$ of ${\cal K}$ and
%an elementary embedding $j : V \to M$ having
%critical point $\gk$ such that
%$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
%$(j(\gk))^{\go_1} \not\subseteq M$.
%
%\end{theorem}
\begin{proof}
Fix a normal measure $U$ over $\gk$ in our ground model $V^*$.
For each $\nu < \gk$, let $\nu^*$ be the least cardinal
above $\nu$ with $o(\nu) = \go_1$. Let
${\vec W}(\nu^*) = \la W(\nu^*, \xi) \mid \xi < \go_1 \ra$
witness that $o(\nu^*) = \go_1$, i.e.,
${\vec W}(\nu^*)$ is an increasing sequence in the
Mitchell ordering $\triangleleft$ \cite{Mi74} %$\lhd$
of normal measures over $\gk$.
We now turn ${\vec W}(\nu^*)$ into a Rudin-Keisler
increasing sequence of ultrafilters. Let
$\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
be an Easton support iteration of Prikry type forcings
of length $\gk$,
%where $\dot \FQ_\ga$ is a term for trivial forcing unless $\ga \in A$.
%Under these circumstances,
where for every $\ga < \gk$,
$\dot \FQ_\ga$ is a
term for the forcing of \cite[Section 2]{G88}
(see also \cite{G10})
which adds either a Prikry or Magidor sequence to every
measurable cardinal $\gg \in (\ga, \ga^*)$.
Note that for all such $\gg$, $o(\gg) < \go_1$ by
the definition of $\ga^*$. This extends
${\vec W}(\ga^*) = \la W(\ga^*, \xi) \mid \xi < \go_1 \ra$
into a Rudin-Keisler increasing commutative sequence
${\vec W'}(\ga^*) = \la W'(\ga^*, \xi) \mid \xi < \go_1 \ra$
of $\ga^*$ complete ultrafilters over $\ga^*$.
Let $G$ be $V^*$-generic over $\FP = \FP_\gk$.
We claim that in $V = V^*[G]$, %$\gk$ is $\gl$ tall with
there is an elementary embedding $j : V \to M$
having critical point $\gk$ such that
$M^\go \subseteq M$ yet %$M^{\go_1} \not\subseteq M$.
$(j(\gk))^{\go_1} \not\subseteq M$.
%closure $\go$ but not closure $\go_1$ as witnessed by some
%$j' : V' \to M'$.
To see this, fix some $\xi < \go_1$.
Define an ultrafilter $U_{\xi}$ over $\gk^2$ in $V^*$ by
$$x \in U_{\xi} \hbox{ {\rm iff} }
\{\nu < \gk \mid \{\gz < \nu^* \mid
(\nu, \gz) \in x\} \in W(\nu^*, \xi)\} \in
U.$$
The ultrapower by $U_{\xi}$ is the ultrapower by
$U$ followed by the ultrapower by
$(j_{U}(\vec W))(\gk^*, \xi)$.
Let $j_{\xi} : V^* \to M_{\xi}$ be the corresponding
elementary embedding.
%Then the diagram $$j_{\gd, \xi} : V \to^{j_\gd} M_\gd \to$$
Then we can write
$j_{\xi} = j_{(j_{U}(\vec W))(\gk^*, \xi)} \circ j_{U}$
and obtain a commutative system of embeddings.
%from $V^*$ into $M$ and $M$ into $M_{\xi}$.
Consider now what happens in $V$.
By using the argument found in the proof of Lemma \ref{l2}
for the construction of the supercompact ultrafilter ${\cal U}$,
we may extend the ultrafilter $U_{\xi}$ of $V^*$ to an
ultrafilter $U_{\xi}'$ of $V$ by constructing an
increasing sequence of conditions successively deciding the statements
``$(\gk, \xi) \in j_{\xi}(\dot x)$'' for all suitable
canonical names $\dot x$.
Because $o(\gz) = \xi$, by the definition of $\FP$,
for a typical $(\nu, \gz)$, a Magidor sequence of
order type $\go^\xi$ was added to $\gz$.
Also, by elementarity, in the ultrapower by $U'_\xi$,
the same thing is true. Thus, let
$j_{U'_\xi} : V^* \to M'_\xi \simeq V^{\gk^2}/U'_\xi$.
Let $[{\rm id}]_{U'_\xi} = \la \gk, \tilde{\xi} \ra$.
Then $M'_\xi$ has a Magidor sequence of order type
$\go^\xi$ for $\tilde{\xi}$ over its ground model.
%$(V^*)^{M'_\xi}$.
Let $\gr < \xi$. Set $\gs_{\xi, \gr}(\nu, \gz) =
(\nu, \gz_\gr)$, where $\gz_\gr$ is the $\gr^{\rm th}$
member of the Magidor sequence added to $\gz$.
Note that by their definitions,
$\gs_{\xi, \gr}$ will project the extension
$U_{\xi}'$ of $U_{\xi}$ to the extension
$U_{\gr}'$ of $U_{\gr}$.
Consequently,
$$\la \la U_{\xi}' \mid \xi < \go_1 \ra,
\la \gs_{\xi, \gr} \mid \gr \le \xi < \go_1 \ra \ra$$
forms a Rudin-Keisler commutative sequence.
We check that it is strictly increasing.
By Theorem \ref{t2a}, $(\ref{i2}) \implies (\ref{i1})$,
this will suffice to prove Theorem \ref{t2d}.
To do this, we suppose otherwise. Then there are
$\gr < \xi < \go_1$ such that $U'_\gr =_{\rm RK} U'_\xi$.
Let $f : \gk^2 \to \gk^2$ be a witnessing isomorphism.
Then in the ultrapower by $U'_\xi$ we will have
$j_{U'_\xi}(f)(\gk, \tilde{\xi_\gr}) = (\gk, \tilde{\xi})$ since
$U'_\gr = \{x \subseteq \gk^2 \mid \la \gk, \tilde{\xi_\gr} \ra \in
j_{U'_\xi}(x)\}$ because of the projection map $\gs_{\xi, \gr}$.
%We have now the following
By the next claim (Claim \ref{cl2}),
we will be able to assume that $f$ is the
identity in the first coordinate and is strictly
increasing in the second coordinate once the first
one has been fixed, i.e., if $\gt < \gt'$ and
$f(\nu, \gt) = (\ga, \gb)$, $f(\nu, \gt') = (\ga', \gb')$, then
$\nu = \ga = \ga'$ and $\gt < \gb < \gb'$.
\begin{claim}\label{cl2}
There is $f' : \gk^2 \to \gk^2$ such that
\begin{enumerate}
\item $[f']_{U_\xi'} = [f]_{U_\xi'}$.
\item For every inaccessible $\nu < \gk$ and
$\gt < \gt' < \nu^*$, if $f'(\nu, \gt) = (\ga, \gb)$ and
$f'(\nu, \gt') = (\ga', \gb')$, then $\nu = \ga = \ga'$ and
$\gt < \gb < \gb'$.
\end{enumerate}
\end{claim}
\begin{proof}
Without loss of generality, we assume that for
every inaccessible $\nu$ and every $\gt < \nu^*$,
it is the case that $f(\nu, \gt) < \nu^*$.
Therefore, for any inaccessible cardinal $\nu < \gk$,
we may define in $V$ the set
$C = \{\gt < \gn^* \mid$ For all $\gs < \gt$, the
second coordinate of $f(\nu, \gs)$ is less than
$\gt\}$, which is a club subset of $\nu^*$.
Note that the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c. Hence, there is a club $E_\nu \in V^*$,
$E_\nu \subseteq C_\nu$. Consequently, by normality,
we have that $E_\nu \in W(\nu^*, \gth)$ for every
$\gth < \go_1$. It then follows that $E_\nu$ and
$C_\nu$ will each be in $W'(\nu^*, \gth)$. Thus,
for every $\gth < \go_1$, the set
$x = \{(\nu, \gt) \in \gk^2 \mid \gt \in E_\nu\} \in U_\gth$.
However, $U_\gth'$ extends $U_\gth$, so in particular,
$x \in U_\xi'$. This means that $f \rest x$ is as desired.
This completes the proof of Claim \ref{cl2}.
\end{proof}
For every inaccessible cardinal $\nu < \gk$ and every
$\gt \in [\nu, \nu^*)$, set $f_\nu(\gt) =$ The second coordinate of
$f(\nu, \gt)$. Then $j_{U_\xi'}(f)_\gk(\tilde{\xi_\gr}) =
\tilde{\xi}$.
Pick a set $A \in W(\nu^*, \gr) - W(\nu^*, \xi)$.
Let $g_\nu = f_\nu \rest A$. Note that each $g_\nu$
is strictly increasing. Also, in the ultrapower by
$U_\xi'$, $g_\gk(\tilde{\xi_\gr}) = \tilde{\xi}$.
For $\nu < \gk$ an inaccessible cardinal, define
$h_\nu \in V^*$ by
$h_\nu(\gt) = \{\mu \mid \exists p \in \FP_{\nu^*}
[p \forces ``g_\nu(\gt) = \mu$''$]\}$.
By its definition, %not only do we have that
$h_\nu : A_\nu \to {\cal P}(\nu^*)$, and for every
$\gt \in A_\nu$, $g_\nu(\gt) \in h_\nu(\gt)$ and
$\min(h_\nu(\gt)) > \gt$.
\begin{claim}\label{cl3}
There is $B_\nu \in W(\nu^*, \xi)$ such that
$B_\nu \cap \bigcup \rge(h_\nu) = \emptyset$.
\end{claim}
\begin{proof}
If not, then $\bigcup \rge(h_\nu) \in W(\nu^*, \xi)$.
Consequently,
$\nu^* \in j_{W(\nu^*, \xi)}(\bigcup \rge(h_\nu))$.
So there is $\gt \in j_{W(\nu^*, \xi)}(A_\nu)$ such that
$\nu^* \in (j_{W(\nu^*, \xi)}(h_\nu))(\gt)$. But
$\min((j_{W(\nu^*, \xi)}(h_\nu))(\gt)) > \gt$, so
$\nu^* > \gt$. Then $(j_{W(\nu^*, \xi)}(h_\nu))(\gt) = h_\nu(\gt)$.
This is since $\nu^*$ is the critical point of the embedding
$j_{W(\nu^*, \xi)}$ and $\card{h_\nu(\gt)} < \nu^*$.
(This last fact follows because
as we have already observed, the forcing above $\nu^*$ does not
add subsets to $\nu^*$, nothing is done over
$\nu^*$ itself, and $\FP_{\nu^*}$ satisfies
$\nu^*$-c.c.) But $h_\nu(\gt) \subseteq \nu^*$, so
$\nu^* \not\in h_\nu(\gt)$.
This contradiction completes the proof of Claim \ref{cl3}.
\end{proof}
We now look at what happens at $\gk$ in the ultrapower
by $U_\xi'$. It is the case that
$\tilde{\xi} \in B_\gk$. To see this, let
$z = \{(\nu, \gz) \mid \gz \in B_\nu\}$.
We have that $z \in U_\xi \subseteq U_\xi'$.
Hence $(\gk, \tilde{\xi}) \in j_{U_\xi'}(z)$,
so $\tilde{\xi} \in B_\gk$. Then $\tilde{\xi} =
g_\gk(\tilde{\xi_\gr}) \in h_\gk(\tilde{\xi_\gr})$ and
$B_\gk \cap \bigcup(\rge(h_\gk)) = \emptyset$.
This is impossible. This completes the proof of Theorem \ref{t2d}.
%This impossibility thereby completes the proof of Theorem \ref{t2d}.
\end{proof}
Theorem \ref{t3} raises two questions, which we
pose now.
\begin{enumerate}
\item\label{q1a} Is $V = {\cal K}$ really needed in
the hypotheses of Theorem \ref{t3}, or is it
possible to construct a model witnessing the
conclusions of Theorem \ref{t3} by forcing
over an arbitrary model $V$ of ZFC satisfying the
current assumptions?
\item\label{q2a} Is the existence of $\eta > \gk$ with
$o(\eta) = \go_1$ really needed in order to construct
a model witnessing the conclusions of Theorem \ref{t3}?
\end{enumerate}