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%\title{Identity Crises and Strong Compactness IV: Tall versus Strong Cardinals
\title{Tall, Strong, and Strongly Compact Cardinals
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal, tall cardinal, non-reflecting stationary set of ordinals, indestructibility.
%indestructibility, Magidor iteration of Prikry forcing, Gitik iteration of forcings
%satisfying the Prikry property.
%Easton support iteration of Prikry type forcings.
}}
%Gitik iteration}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank Stamatis Dimopoulos for helpful
email correspondence on the subject matter of this paper.}\\
%\thanks{The
% author's research was partially
% supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{\today}
%\date{July 23, 2017}\date{August 2, 2017}\date{August 5, 2017}
\date{August 12, 2017}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
We construct three models in which there %is an identity crisis
are different relationships
among the classes of strongly compact, strong, and non-strong tall cardinals.
In the first two of these models, the strongly compact and strong cardinals
coincide precisely, and every strongly compact/strong cardinal is a limit of
non-strong tall cardinals.
In the remaining model, the strongly compact cardinals are
precisely characterized as the measurable limits of strong cardinals,
and every strongly compact cardinal is a limit of non-strong tall cardinals.
These results extend and generalize those of %the ones
of \cite{AC2} and \cite{A03}.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with some definitions. %and terminology.
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 $\gl$.
Hamkins made a systematic study of tall cardinals in \cite{H09}.
In particular, among many other results,
he showed that every cardinal which is either strong or strongly compact
is in addition tall,
%gave a proof of a theorem he said
%was essentially due to Gitik that the theories
%``ZFC + There is a tall cardinal'' and ``ZFC + There is a strong cardinal''
%are equiconsistent,
and also produced models of ZFC %containing
with many different varieties of
non-strong tall cardinals.
Turning now to the main narrative,
in \cite{AC2} and \cite{A03}, the following theorems were proven.
\begin{theorem}\label{t1}{\bf (\cite[Theorem 1]{AC2})}
Con(ZFC + There is a proper class of supercompact cardinals) $\implies$
Con(ZFC + There is a proper class of strongly compact cardinals +
No strongly compact cardinal $\gk$ is $2^\gk = \gk^+$ supercompact +
$\forall \gk[\gk$ is strongly compact iff $\gk$ is a strong cardinal]).
\end{theorem}
\begin{theorem}\label{t2}{\bf (\cite[Theorem 1]{A03})}
Suppose $V \models ``$ZFC + %$\K \neq \emptyset$
$\K$ is the proper class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC +
$\gk$ is
strongly compact iff $\gk$ is a
measurable limit of strong cardinals +
The strongly
compact cardinals are the elements of
$\K$ together with their measurable
limit points''.
Further, in $V^\FP$,
any $\gk \in \K$ which was a supercompact
limit of supercompact cardinals in $V$
remains supercompact.
\end{theorem}
Since Theorems \ref{t1} and \ref{t2} were proven prior to
Hamkins' research leading to his paper \cite{H09}, the issue
of tall cardinals was not considered in either \cite{AC2} or \cite{A03}.
In particular, these theorems do not address
the question of whether it is possible to construct models of ZFC witnessing
the same conclusions in which
each strongly compact cardinal is also a limit of non-strong tall cardinals.
%there are in addition non-strong tall cardinals.
%\noindent Left unaddressed by these theorems, however, is
%the question of whether it is possible to construct models of ZFC witnessing
%similar conclusions in which there are in addition non-strong tall cardinals.
The purpose of this paper is to produce such universes.
Specifically, we will prove the following three theorems.
\begin{theorem}\label{t3}
Con(ZFC + There is a proper class of supercompact cardinals) $\implies$
Con(ZFC + There is a proper class of strongly compact cardinals +
No strongly compact cardinal $\gk$ is $2^\gk = \gk^+$ supercompact +
$\forall \gk[\gk$ is strongly compact iff $\gk$ is a strong cardinal] + Every
strongly compact cardinal is a limit of (non-strong) tall cardinals).
\end{theorem}
\begin{theorem}\label{t3a}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gl > \gk$ is measurable''. Then there is a
partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + %GCH +
$\gk$ is both the
only strong and only strongly compact cardinal + $\gk$ is not
$2^\gk = \gk^+$ supercompact + Every measurable
cardinal is tall + No cardinal $\gl > \gk$ is measurable''.
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + %$\K \neq \emptyset$
$\K$ is the proper class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC +
$\gk$ is
strongly compact iff $\gk$ is a
measurable limit of strong cardinals +
%iff $\gk$ is a measurable limit of non-strong tall cardinals +
%Every strongly compact cardinal is a limit of tall cardinals +
The strongly
compact cardinals are the elements of
$\K$ together with their measurable
limit points''.
Further, in $V^\FP$,
every strongly compact cardinal is a limit of non-strong tall cardinals.
Finally, in $V^\FP$,
any $\gk \in \K$ which was a supercompact
limit of supercompact cardinals in $V$
remains supercompact.
\end{theorem}
%\begin{theorem}\label{t4a}
%Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact +
%No cardinal $\gl > \gk$ is measurable''. Then there is a
%partial ordering $\FP \in V$ such that
%$V^\FP \models ``$ZFC + GCH + $\gk$ is both the only
%strongly compact cardinal and only measurable limit of strong cardinals + Every measurable
%cardinal is tall + No cardinal $\gl > \gk$ is measurable''.
%\end{theorem}
Thus, the models witnessing the conclusions of Theorems \ref{t3} and \ref{t4}
have the same characterizations of the strongly compact cardinals as do
the models witnessing the conclusions of Theorems \ref{t1} and \ref{t2}, except
that each strongly compact cardinal is in addition a limit of non-strong tall cardinals.
The model witnessing the conclusions of Theorem \ref{t3a}
is an analogue of the model witnessing the conclusions of Theorem \ref{t1},
except in a universe with a restricted number of large cardinals.
However, it has the additional feature that each measurable
cardinal is also tall.
Further, as in \cite{AC2} and \cite{A03}, we will concentrate
on the proper class versions of Theorems \ref{t3} and \ref{t4},
and not discuss the (easier) analogues of these theorems
when the class of supercompact cardinals is actually a set.
%The models witnessing the conclusions of Theorems \ref{t3a} and \ref{t4a}
%are analogues of the models witnessing the conclusions of Theorems \ref{t1}
%and \ref{t2}, except in universes with a restricted number of large cardinals.
%However, they have the additional feature that each measurable
%cardinal is also tall.
The structure of this paper is as follows.
Section \ref{s1} contains our introductory comments and
preliminary information concerning notation and terminology.
Section \ref{s2} contains the proofs of Theorems \ref{t3} -- \ref{t4}.
Section \ref{s3} contains our concluding remarks.
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 $\varphi$}.
%When forcing, $q \ge p$ means that
%{\em $q$ is stronger than $p$}.
%and $p \decides \varphi$ means that
%{\em $p$ decides $\varphi$}.
%For $\gk$ a regular cardinal and $\gl$ an ordinal,
%$\add(\gk, \gl)$ is the standard partial ordering for adding
%$\gl$ many Cohen subsets of $\gk$.
%For $\ga < \gb$ ordinals,
%$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
%$(\a, \b)$ are as in standard interval notation.
%$[\ga, \gb]$ and $(\ga, \gb]$ are as in standard interval notation.
%If $G$ is $V$-generic over $\FP$,
%we will abuse notation slightly
%and use both $V[G]$ and $V^\FP$
%to indicate the universe obtained by forcing with $\FP$.
%If $\FP$ is a reverse Easton iteration
%such that at stage $\ga$, a nontrivial
%forcing is done adding a subset
%of $\gd$, then we will say that
%$\gd$ is in the field of $\FP$.
%We will, from time to time, confuse terms with the sets
%they denote and write $x$ when we actually mean $\dot x$ or $\check x$.
The partial ordering
$\FP$ is {\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even
%(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\em ${\prec} \gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga\mid \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even
%(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\em $(\gk, \infty)$-distributive} if
given a sequence $\la D_\ga \mid \ga < \gk \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$ is dense open as well.
Note that if $\FP$ is $\gk$-strategically closed,
then $\FP$ is $(\gk, \infty)$-distributive.
Further, if $\FP$ is $(\gk, \infty)$-distributive and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$ \FP$ is ${<}\gk$-strategically closed
%if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
Suppose that $\gk < \gl$ are regular cardinals.
A partial ordering $\FP(\gk, \gl)$ that will be used
throughout the course of this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP(\gk, \gl)$ is
defined as
$\{ p \mid \ $For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, such that $\gb \in S_p$ implies
$\gb > \gk$ and cof$(\gb) = \gk \}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP(\gk, \gl)$ (see
\cite{Bu}), in $V[G]$,
if we assume $\gl$ is inaccessible in $V,$ %GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p \mid p\in G \} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced, the
bounded subsets of $\gl$ are the same as those in $V$,
%and cardinals, cofinalities, and GCH
and cofinalities have been preserved.
It is also virtually immediate that $\FP(\gk, \gl)$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu}) that
$\FP(\gk, \gl)$ is ${\prec} \gl$-strategically closed.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of our theorems.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\it admits a gap at $\gd$}.
%In Hamkins' terminology of \cite{H2, H3}, $\FP$ is {\it mild
%with respect to a cardinal $\gk$} iff every set of ordinals $x$ in
%$V^\FP$ of size less than $\gk$ has a ``nice'' name $\tau$
%in $V$ of size less than $\gk$, i.e., there is a set $y$ in $V$,
%$|y| <\gk$, such that any ordinal forced by a condition in $\FP$
%to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
%Finally, if $\FP$ is mild with respect to $\gk$ and $\gk$ is
%$\gl$ strongly compact in $V[G]$ for any $\gl \ge \gk$, then
%$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
%\noindent
An immediate corollary of Theorem \ref{tgf} is that forcing with
a partial ordering $\FP$ admitting a gap at some regular cardinal $\gd$ creates
no new measurable, tall, strong, or supercompact cardinals above $\gd$.
In particular, if $\gd = \go$, then forcing with $\FP$ creates no new
measurable, tall, strong, or supercompact cardinals.
In addition, by \cite[Corollary 13]{H2}, if $\gk$ is $\gl$ strong
in $V^\FP$ via $j^*$ where
$\FP$ admits a gap at some regular cardinal $\gd < \gk$
and $\gl$ is either a successor ordinal or has cofinality greater than $\gd$,
then $\gk$ was $\gl$ strong in the ground model as witnessed by $j^* \rest V$.
%In addition, by \cite[Corollary 13]{H2}, if $\gk$ is $\gl$ strong after forcing with
%$\FP$ admitting a gap at some regular cardinal $\gd < \gk$
%and $\gl$ is either a successor ordinal or has cofinality greater than $\gd$,
%then $\gk$ was $\gl$ strong in the ground model.
We mention that we are assuming some
familiarity with the large cardinal
notions of measurability,
%measurable cardinals of high Mitchell order, hypermeasurability,
tallness, strongness, strong compactness, and supercompactness.
Interested readers may consult \cite{J}
or \cite{H09}. %\cite{SRK}. %for further details.
%We note only that we will say {\em $\gk$ is supercompact
%(strongly compact) up to the cardinal $\gl$} if $\gk$ is
%$\gd$ supercompact (strongly compact) for every $\gd < \gk$.
\section{The Proofs of Theorems \ref{t3} -- \ref{t4}}\label{s2}
We turn now to the proofs of Theorem \ref{t3} -- \ref{t4},
starting with the proof of Theorem \ref{t3}.
\begin{proof}
%\begin{proof}
In analogy to \cite{AC2}, we first prove Theorem \ref{t3} for one cardinal.
In particular, starting from a model for ``ZFC + $\gk$ is supercompact'',
we will force and construct a model where $\gk$ is both the least strong and
least strongly compact cardinal in which $\gk$ is also a limit of (non-strong) tall cardinals.
In this model, it will in addition
be the case that $\gk$ is not $2^\gk = \gk^+$ supercompact.
Before beginning the proof, however, we give some intuition and motivation for
the definition of our forcing conditions. In \cite{AC2}, in order to construct
the requisite models, it was only necessary %to force
to add non-reflecting stationary sets of ordinals of the appropriate cofinality
to rid ourselves of %get rid of %destroy
each strong cardinal $\gd < \gk$.
This is not sufficient in the current situation, since the forcing just described
will not ensure that there are non-strong tall cardinals below $\gk$.
We will therefore %force to
destroy all ground model strong
cardinals which are themselves limits of ground model strong cardinals,
after first adding a Cohen subset of $\go$ %in order
to create a
gap at $\ha_1$. %so that Theorem \ref{tgf} can be applied.
This will guarantee by Theorem \ref{tgf} that all strong cardinals below $\gk$ have been
eliminated, and that $\gk$ has become a limit of non-strong tall cardinals.
Getting specific,
suppose $V \models ``$ZFC + $\gk$ is supercompact''.
Without loss of generality, by first doing a preliminary forcing if necessary,
we assume in addition that $V \models {\rm GCH}$.
By \cite[Lemma 2.1]{AC2} and the succeeding remarks, the $V$-strong
cardinals below $\gk$ which are limits of $V$-strong cardinals are
unbounded in $\gk$.
%we may let
%$\la \gd_\ga \mid \ga < \gk \ra$ be an enumeration of the $V$-strong cardinals
%which are themselves limits of strong cardinals.
We may therefore let $A = \la \gd_\ga \mid \ga < \gk \ra$ be
an enumeration of this set.
%the $V$-strong cardinals below $\gk$ which are themselves limits of $V$-strong cardinals.
The partial ordering $\FP^\gk$ we use in the
proof of Theorem \ref{t1} for one cardinal
is defined analogously as in \cite{AC2}.
It is the Easton support iteration
$\la \la \FP^\gk_\ga, \dot \FQ^\gk_\ga \ra \mid \ga < \gk \ra$,
where $\FP_0^\gk = %is the partial ordering
\add(\omega, 1)$ and
$\forces_{\FP^\gk_\ga} ``\dot \FQ^\gk_\ga = \dot \FP(\go, \gd_\ga)$''.
%adds a non-reflecting stationary set of ordinals of
%cofinality $\omega$ to $\gd_\ga$''.
Since by its definition, $\card{\FP^\gk} = \gk$,
$V^{\FP^\gk} \models ``%2^\gk = \gk^+$ (and in fact,
2^\gd = \gd^+$ for every cardinal $\gd \ge \gk$''.
\begin{lemma}\label{l1}
$V^{\FP^\gk} \models ``$No cardinal $\gd < \gk$ is a strong cardinal''.
\end{lemma}
\begin{proof}
The proof is quite different from and subtler than \cite[Lemma 2.2]{AC2},
its analogue in \cite{AC2}.
It is motivated by ideas due to Hamkins found in \cite{H98} and \cite{HS}.
By its definition, we may write $\FP^\gk = \add(\go, 1) \ast \dot \FQ$, where
$\forces_{\add(\go, 1)} ``\dot \FQ$ is $\ha_1$-strategically closed''.
By our remarks immediately following Theorem \ref{tgf},
we may consequently infer that if
$V^{\FP^\gk} \models ``\gd$ is a strong cardinal'', then
$V \models ``\gd$ is a strong cardinal'' as well.
Therefore, to prove Lemma \ref{l1}, it suffices to show that if
$V \models ``\gd < \gk$ is strong'', then
$V^{\FP^\gk} \models ``\gd$ is not a strong cardinal''.
This is clearly true if $V \models ``\gd$ is a strong cardinal which is a
limit of strong cardinals''. This is
since under these circumstances, by the definition of $\FP^\gk$,
$V^{\FP^\gk} \models ``$There is
$S \subseteq \gd$ which is a non-reflecting stationary set of ordinals of
cofinality $\go$ and thus $\gd$ is not weakly compact''.
Hence, to complete the proof of Lemma \ref{l1}, we must show that if
$V \models ``\gd$ is a strong cardinal which is not a limit of strong cardinals'', then
$V^{\FP^\gk} \models ``\gd$ is not a strong cardinal''.
To do this, suppose to the contrary that
$V^{\FP^\gk} \models ``\gd$ is a strong cardinal''. Because
$V \models ``\gd$ is not a limit of strong cardinals'', we may write
%$\FP^\gk = \FP^\gk_\gd \ast \dot \FQ_\gd \ast \dot \FR$, where $\card{\FP_\gd} < \gd$,
$\FP^\gk = \FR^* \ast \dot \FR^{**} \ast \dot \FR$, where $\card{\FR^*} < \gd$,
$\FR^*$ adds a Cohen subset of
$\go$ and also adds non-reflecting stationary sets of ordinals of cofinality $\go$ to
each cardinal below $\gd$ which is a $V$-strong limit of $V$-strong cardinals,
$\dot \FR^{**}$ is a term for the partial ordering adding a non-reflecting stationary
set of ordinals of cofinality $\go$ to the least $V$-strong cardinal $\gd' > \gd$ which is a
limit of $V$-strong cardinals, and $\dot \FR$ is a term for the
rest of $\FP^\gk$. Since
$\forces_{\FR^* \ast \dot \FR^{**}} ``\dot \FR$ is $\gs$-strategically closed for
$\gs$ the least inaccessible cardinal above $\gd'$'', it is the case that
$V^{\FP^\gk} = V^{\FR^* \ast \dot \FR^{**} \ast \dot \FR} \models ``\gd$ is $\gd' + 2$ strong'' iff
$V^{\FR^* \ast \dot \FR^{**}} \models ``\gd$ is $\gd' + 2$ strong''.
The proof of Lemma \ref{l1} will therefore be complete if we can show that
$V^{\FR^* \ast \dot \FR^{**}} \models ``\gd$ is not $\gd' + 2$ strong''.
Towards this end, let $G^*$ be $V$-generic over $\FR^*$ and $G^{**}$ be
$V[G^*]$-generic over $\FR^{**}$.
Since $V[G^*][G^{**}] \models ``\gd$ is $\gd' + 2$ strong'',
we may let $j^* : V[G^*][G^{**}] \to M[j^*(G^*)][j^*(G^{**})]$ be an elementary embedding having
%we may let $j^* : V[G^*][G^{**}] \to M^*$ be an elementary embedding having
critical point $\gd$ which witnesses the $\gd' + 2$ strongness of $\gd$ such that
$M[j^*(G^*)][j^*(G^{**}] \subseteq V[G^*][G^{**}]$,
$M[j^*(G^*)][j^*(G^{**})]^\gd \subseteq M[j^*(G^*)][j^*(G^{**})]$ in
$V[G^*][G^{**}]$, and
$(V_{\gd' + 2})^{V[G^*][G^{**}]} \in M[j^*(G^*)][j^*(G^{**}]$.
Observe that it is also possible to write
$\FR^* \ast \dot \FR^{**} = \add(\go, 1) \ast \dot \FS$, where
$\forces_{\add(\go, 1)} ``\dot \FS$ is $\ha_1$-strategically closed''.
Thus, by Theorem \ref{tgf} and the succeeding remarks, %immediately following,
$j^*$ must lift some elementary embedding
$j : V \to M$ witnessing the $\gd' + 2$ strongness of $\gd$ in $V,$ where
$M \subseteq V$, $V_{\gd' + 2} \in M$, and $j(\gd) > \gd' + 2$.
Further, as $\card{\FR^*} < \gd$ and
$\gd$ is the critical point of both $j$ and
$j^*$, $j(\FR^*) = \FR^*$ and $j^*(G^*) = G^*$, i.e.,
$j^* : V[G^*][G^{**}] \to M[G^*][j^*(G^{**})]$.
%Putting the previous two sentences together,
%Because
%$(V_{\gd' + 2})^{V[G^*][G^{**}]} \in M[G^*][[j^*(G^{**})]$ %$V_{\gd' + 2} \subseteq M$,
%and $\FR^* \in (V_{\gd' + 1})^{V[G^*][G^{**}]}$,
%we may consequently infer that
Because $V_{\gd' + 2} \subseteq M$,
$(V_{\gd' + 1})^{V[G^*]} = (V_{\gd' + 1})^{M[G^*]}$.
Thus, as $\FR^{**} \in (V_{\gd' + 1})^{V[G^*]}$,
$\FR^{**} \in (V_{\gd' + 1})^{M[G^*]}$.
Therefore, since $M[G^*] \subseteq V[G^*]$,
$G^{**}$ is also %both $V[G^*]$- and
$M[G^*]$-generic over $\FR^{**}$, so that in particular,
$G^{**}$ is not a member of either $V[G^*]$ or $M[G^*]$.
However, because
%$G^{**} \in (V_{\gd' + 2})^{V[G^*][G^{**}]}$ and
%$j^*$ witnesses the
%$\gd' + 2$ strongness of $\gd$ in $V[G^*][G^{**}]$,
$(V_{\gd' + 2})^{V[G^*][G^{**}]} \in M[G^*][[j^*(G^{**})]$
and $G^{**} \in (V_{\gd' + 2})^{V[G^*][G^{**}]}$,
$G^{**} \in M[G^*][[j^*(G^{**})]$.
Note that by elementarity, as
$\forces_{\FR^*} ``\dot \FR^{**}$ adds a non-reflecting stationary
set of ordinals of cofinality $\go$ to a measurable cardinal $\gd' > \gd$'',
in $M,$
$\forces_{\FR^*} ``j(\dot \FR^{**})$ adds a non-reflecting stationary
set of ordinals of cofinality $\go$ to a measurable cardinal
$j(\gd') > j(\gd) > \gd' + 2 > \gd'$''.
Hence, in $M,$
$\forces_{\FR^*} ``j(\dot \FR^{**})$ is $\gs$-strategically closed for
$\gs$ the least inaccessible cardinal above $\gd'$''.
Therefore, because $j^*(G^{**})$ is
$M[G^*]$-generic over $j(\FR^{**})$,
$G^{**} \in M[G^*]$.
This contradiction completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^{\FP^\gk} \models ``\gk$ is a limit of non-strong tall cardinals''.
\end{lemma}
\begin{proof}
Since the set $A$ defined above is unbounded in $\gk$, the set
$B = \{\gd < \gk \mid \gd$ is a $V$-strong cardinal which is not a limit of
$V$-strong cardinals$\}$ is unbounded in $\gk$ as well. We show that
$V^{\FP^\gk} \models ``$Every $\gd \in B$ is a tall cardinal''.
This will suffice, since by Lemma \ref{l1},
$V^{\FP^\gk} \models ``$No $\gd \in B$ is a strong cardinal''.
Towards this end, fix $\gd \in B$. With the same meaning as in the proof of Lemma \ref{l1}, write
$\FP^\gk = \FR^* \ast \dot \FR^{**} \ast \dot \FR$. Since
$\card{\FR^*} < \gd$, by the Hamkins-Woodin results \cite{HW}, %L\'evy-Solovay results \cite{LS},
$V^{\FR^*} \models ``\gd$ is a strong cardinal''. As we have already noted,
it then immediately follows that $V^{\FR^*} \models ``\gd$ is a tall cardinal''.
By \cite[Theorem 3.1]{H09}, $\gd$'s tallness is indestructible under
$(\gd, \infty)$-distributive forcing. Because by its definition,
$\forces_{\FR^*} ``\dot \FR^{**} \ast \dot \FR$ is $(\gd, \infty)$-distributive'',
$V^{\FR^* \ast \dot \FR^{**} \ast \dot \FR} = V^{\FP^\gk} \models
``\gd$ is a tall cardinal''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^{\FP^\gk} \models ``$No cardinal $\gd < \gk$ is strongly compact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3} is essentially the same as the proof of \cite[Lemma 2.3]{AC2}.
Since it is relatively brief, we include it for completeness.
Specifically, by the definition of $\FP^\gk$
and the fact $A$ is unbounded in $\gk$,
$V^{\FP^\gk} \models ``$There are unboundedly in $\gk$ many cardinals $\gd < \gk$
containing a non-reflecting stationary set of ordinals of cofinality $\go$''.
Hence, by \cite[Theorem 4.8]{SRK} and the succeeding remarks,
$V^{\FP^\gk} \models ``$No cardinal $\gd < \gk$ is strongly compact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^{\FP^\gk} \models ``\gk$ is strongly compact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l4} is essentially the same as the proof of \cite[Lemma 2.4]{AC2}.
The argument is originally due to Magidor, but was unpublished by him.
For completeness and ease of presentation, we provide a sketch,
and refer readers to \cite{AC2} for any missing details.
Specifically,
let $\gl > \gk$ be an arbitrary regular cardinal, with $j : V \to M$ an
elementary embedding witnessing the $\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter over $P_\gk(\gl)$ such that
$V \models ``\gk$ is not $\gl$ supercompact''. Since $\gl \ge \gk^+ = 2^\gk$,
we know that $M \models ``\gk$ is measurable''.
We may therefore let
$k : M \to N$ be an elementary embedding
generated by a normal measure ${\cal U} \in M$ over $\gk$
such that $N \models ``\gk$ is not measurable''.
The elementary
embedding $i = k \circ j$ witnesses the
$\gl$ strong compactness of $\gk$ in $V$.
It will follow that %We show now
$i$ lifts in $V^{\FP^\gk}$
to an elementary embedding
$i : V^{\FP^\gk} \to N^{i(\FP^\gk)}$
witnessing the $\gl$ strong compactness of $\gk$.
To see this, we begin with a few observations.
First, note that by \cite[Lemma 2.1]{AC2} and the succeeding remarks,
in both $V$ and $M$, $\gk$ is a strong cardinal which is a limit of strong cardinals.
%Further, as was observed in the proof of \cite[Lemma 2.4]{AC2}, since
Further, as was observed in \cite[proof of Lemma 2.4, page 31, fourth paragraph]{AC2}, since
$M \models ``\gk$ is not $\gl$ supercompact'',
$M \models ``$No cardinal in the half-open interval $(\gk, \gl]$ is
a strong cardinal''.
The previous two sentences consequently imply
that by the definition of $\FP$,
$j(\FP^\gk) = \FP^\gk \ast \dot \FP(\go, \gk) \ast \dot \FR$,
%where $\dot \FP(\go, \gk)$ is a term for the partial ordering
%adding a non-reflecting stationary set of ordinals of
%cofinality $\go$ to $\gk$, and
where the first ordinal at which
$\dot \FR$ is forced to do nontrivial forcing is %well
above $\gl$.
This means we may write $i(\FP^\gk) =
\FP^\gk \ast \dot \FQ^1 \ast \dot \FQ^2$,
where $\dot \FQ^1$ is forced to act nontrivially on
ordinals in the interval $(\gk, k(\gk)]$, and $\dot \FQ^2$
is forced to act nontrivially on ordinals in the interval
$(k(\gk), k(j(\gk))) = (k(\gk), i(\gk))$.
%$j(\FP^\gk) = \FP^\gk \ast \dot \FP(\go, \gk) \ast \dot \FR$,
Now, take $G_0$ to be
$V$-generic over $\FP^\gk$, and build in
$V[G_0]$ generic objects $G_1$ and $G_2$
for $\FQ^1$ and $\FQ^2$ respectively.
The construction of $G_1$ uses that
by GCH and
the fact that $k$ is given by an
ultrapower embedding, we may let
$\la D_\ga \mid \ga < \gk^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of $\FQ^1$ present
in $N[G_0]$.
Since $N \models ``\gk$ is not measurable'',
the first nontrivial stage of forcing in $\FQ^1$ occurs %well
above $\gk$. This implies that
$N[G_0] \models ``\FQ^1$ is ${\prec} \gk^+$-strategically closed''.
Because $N[G_0]$ remains
$\gk$-closed with respect to $V[G_0]$,
by the ${\prec} \gk^+$-strategic
closure of $\FQ^1$ in both $N[G_0]$ and $V[G_0]$,
we may work in $V[G_0]$ and
meet each $D_\ga$ in order to construct $G_1$.
%Since $j '' G_0 \subseteq G_0 \ast G_1$,
%we may lift $j$ in $V[G_0]$ to
%$j : V[G_0] \to M[G_0][G_1]$.
The construction of $G_2$ first requires
building an $M$-generic object $G^{*}_2$
for the term forcing partial ordering
$\FT$ associated with $\dot \FR$ and defined
in $M$ with respect to $\FP^\gk \ast
\dot \FP(\go, \gk)$.
%Unlike the argument given in the
%proof of Lemma \ref{l4} for $G^{**}_2$, however,
$G^{*}_2$ is built using the facts that since
$M^\gl \subseteq M$ and the
first nontrivial stage of forcing in $\FT$
occurs %well
above $\gl$, $\FT$ is ${\prec}\gl^+$-strategically
closed in both $M$ and $V$, which means
that the diagonalization argument employed
in the construction of $G_1$ may be applied
in this situation as well.
$k '' G^{*}_2$ now generates an $N$-generic
object $G^{**}_2$ for $k(\FT)$ and an
$N[G_0][G_1]$-generic object $G_2$ for $\FQ^2$
This tells us that $i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$, i.e.,
$V^{\FP^\gk} \models ``\gk$ is $\gl$ strongly compact''.
Since $\gl$ was arbitrary,
this completes the proof sketch of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^{\FP^\gk} \models ``\gk$ is a strong cardinal''.
\end{lemma}
\begin{proof}
%The proof of Lemma \ref{l5} is essentially the same as the proof of \cite[Lemma 2.5]{AC2}.
%Once again,
%for completeness and ease of presentation, we provide a sketch,
%referring readers to \cite{AC2} for any missing details.
Let $\gl > \gk$ be a singular strong limit cardinal whose cofinality
is at least $\gk$, with
%such that $\gl = \ha_\gl$, with
$j : V \to M$ an elementary embedding witnessing the $\gl$ strongness
of $\gk$ generated by a $(\gk, \gl)$-extender such that
$M \models ``\gk$ is not a strong cardinal''.
%Let $i : V \to N$ be the elementary embedding witnessing the
%measurability of $\gk$ generated by the ultrafilter
%$\U = \{x \subseteq \gk \mid \gk \in j(x)\}$.
Since $M \models ``\gk$ is not a strong cardinal'', it follows that
$\gk$ is a trivial stage of forcing in the definition of $j(\FP^\gk)$ in $M$.
The proof that the embedding $j$ lifts
to an embedding $j : V^{\FP^\gk} \to M^{j(\FP^\gk)}$
witnessing the $\gl$ strongness of $\gk$ is a modification of the
one given in %the proof of
\cite[Theorem 4.10]{H4} and \cite[Lemma 4.2]{A03a}
which takes into account that only trivial forcing
occurs at stage $\gk$ in $M$ in the definition of $j(\FP^\gk)$.\footnote{Note
%both $V$ and $M$ in the definitions of
%$\FP^\gk$ and $j(\FP^\gk)$ respectively.\footnote{Note
that it is also possible to use
the argument found in \cite[Lemma 2.5]{AC2} to prove this lemma.
However, since the proof found here is a bit shorter and more direct,
we give it instead.}
%Theorem 4.10 of \cite{H4}.
%We present the argument in a fairly complete fashion here as well,
%For the benefit of readers, we present the argument here as well,
%taking the liberty to quote freely from it.
We will take the liberty to quote freely from the proof of \cite[Lemma 4.2]{A03a}
as appropriate.
We may assume that
$M = \{j(f)(a) \mid a \in {[\gl]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq V\}$. Since
%Since $M \models ``\gk$ is not a strong cardinal'', it follows that
%$\gk$ is a trivial stage of forcing in the definition of $j(\FP^\gk)$ in $M$.
Further, as in \cite[proof of Lemma 2.5, page 32, second paragraph]{AC2},
%the proof of \cite[Lemma 2.5]{AC2},
$M \models ``$There are no strong cardinals in the half-open interval $(\gk, \gl]$''.
Consequently, $j(\FP^\gk) = \FP^\gk \ast \dot \FR$,
where the first ordinal on which $\dot \FR$ is forced to act
nontrivially is above $\gl$.
Suppose $G_0$ is $V$-generic over $\FP^\gk$.
Since
$\FP^\gk$ is an Easton support iteration having length $\gk$,
$\FP^\gk$ is $\gk$-c.c. Thus,
as we may assume that $M^\gk \subseteq M$,
$M[G_0]^\gk \subseteq M[G_0]$ in $V[G_0]$.
%and $H$ is $V[G]$-generic over $\FQ$,
Therefore, $\FR$ is ${\prec} \gk^+$-strategically closed in both
$V[G_0]$ and $M[G_0]$, and $\FR$ is $\gl$-strategically closed in $M[G_0]$.
%As in \cite{H4} and \cite{A03a}, by using a suitable coding
%which allows us to identify finite
%subsets of $\gl$ with elements of $\gl$,
%the definition of $M$ allows us to find some
%$\ga < \gl$ and function $g$ so that
%$\dot \FQ = j(g)(\ga)$.
Let
%(assuming that $\dot \FQ$ has been chosen reasonably). Let
$N = \{i_{G_0}(\dot z) \mid \dot z =
j(f)(\gk, \gl)$ %j(f)(\gk, \ga, \gl)$
for some function $f \in V\}$.
%It is easy to verify that
As in \cite[Theorem 4.10]{H4} and \cite[Lemma 4.2]{A03a}, one may verify that
$N \prec M[G_0]$, that $N$ is closed under
$\gk$ sequences in $V[G_0]$, and that
%$\gk$, $\ga$, $\gl$, $\FQ$, and $\FR$
$\gk$, $\gl$, and $\FR$
are all elements of $N$.
Further, since
$\FR$ is $j(\gk)$-c.c$.$ in $M[G_0]$ and
there are only $2^\gk = \gk^+$ many functions
$f : {[\gk]}^2 \to V_\gk$ in $V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G_0]$ and $V[G_0]$,
we can build an $N$-generic object $G_1$ over $\FR$ in $V[G_0]$
as in the construction of the generic object $G_1$ found in the
proof sketch of Lemma \ref{l4}.
%an $N$-generic object $H'$ over $\FR$
%as follows. Let $\la D_\gs : \gs < \gk^+ \ra$ enumerate in
%$V[G][H]$ the dense open subsets of $\FR$ present in $N$ so that
%every dense open subset of $\FR$ occurring in $N$ appears at an
%odd stage at least once in the enumeration.
%If $\gs$ is an odd ordinal, $\gs = \tau + 1$ for some $\tau$.
%Player I picks $p_\gs \in D_\gs$ extending $q_\tau$
%$\sup(\la q_\gb : \gb < \gs \ra)$
%(initially, $q_{0}$ is the empty condition), and player II responds by picking $q_\gs \ge p_\gs$
%according to a fixed strategy ${\cal S}$ (so $q_\gs \in D_\gs$). If $\gs$ is a limit ordinal, player II
%uses ${\cal S}$ to pick $q_\gs$ extending each $q \in \la q_\gb : \gb < \gs \ra$.
%By the ${\prec} \gk^+$-strategic closure of $\FR$ in $V[G][H]$,
%player II's strategy can be assumed to be a winning one, so
%$\la q_\gs : \gs < \gk^+ \ra$ can be taken as an increasing sequence of conditions with
%$q_\gs \in D_\gs$ for $\gs < \gk^+$.
%Let $H' = \{p \in \FR : \exists \gs < \gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $G_1$ is actually $M[G_0]$-generic over $\FR$.
If $D$ is a dense open subset of $\FR$ in $M[G_0]$, then
$D = i_{G_0}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G_0}(j(f)(\gk, \ga_1, \ldots, \ga_n))$, where
$\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that $j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of $\FR$ in $M[G_0]$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G_0]$ and therefore $\gl$-distributive in
$M[G_0]$, $\ov D$ is a name for a dense open
subset of $\FR$ in $M[G_0]$ which is definable without
the parameters $\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G_0}(\ov D) \in N$.
Thus, since $G_1$
meets every dense open subset
of $\FR$ present in $N$,
%is $N$-generic over $\FR$,
$G_1 \cap i_{G_0}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$G_1 \cap i_{G_0}(\dot D) \neq \emptyset$.
This means $G_1$ is
$M[G_0]$-generic over $\FR$, so in
$V[G_0]$, $j$ lifts to $j : V[G_0] \to M[G_0][G_1]$.
The lifted version of $j$ is an embedding witnessing the
$\gl$ strongness of $\gk$ in $V[G_0]$. This is
since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{V[G_0]} \subseteq M[G_0] \subseteq M[G_0]G_1]$.
As a consequence,
$V[G_0] \models ``\gk$ is $\gl$ strong''.
Since $\gl$ was arbitrary, this completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^{\FP^\gk} \models ``\gk$ is not $2^\gk = \gk^+$ supercompact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6} is essentially the same as the proof of \cite[Lemma 2.6]{AC2}.
Once again,
since it is relatively brief, we include it for completeness.
By Lemmas \ref{l1} and \ref{l5}, $V^{\FP^\gk} \models ``\gk$ is a strong cardinal
such that no cardinal $\gd < \gk$ is a strong cardinal''.
Thus, by \cite[Lemma 2.1]{AC2} and the succeeding remarks,
$V^{\FP^\gk} \models ``\gk$ is not $2^\gk$ supercompact''.
Since as we have already observed, $V^{\FP^\gk} \models ``2^\gk = \gk^+$'',
this completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l1} -- \ref{l6} complete the proof of Theorem \ref{t3} for one cardinal.
\end{proof}
\begin{pf}
To prove Theorem \ref{t3} in the general case, i.e., when there is a
proper class of supercompact cardinals, we will modify the
proof given in \cite[Section 3]{AC2}, quoting verbatim where
appropriate. Suppose
$V \models ``$ZFC + There is a proper class of supercompact cardinals''.
Let $\la \gk_\ga \mid \ga \in {\rm Ord} \ra$ enumerate the supercompact
cardinals in increasing order.
%Let $V \models ``$ZFC + $\la \gk_\ga \mid \ga \in {\rm Ord} \ra$ is the proper class of
%supercompact cardinals''.
Without loss of generality, we assume in addition that
$V \models ``$For every ordinal $\ga$,
each $\gk_\ga$ has
its supercompactness Laver indestructible \cite{L} under
$\gk_\ga$-directed closed forcing + For every ordinal $\ga$,
$2^{\gk_\ga} = \gk^+_\ga$'' and that
by ``cutting off'' the universe
if necessary at the least inaccessible
limit of supercompact cardinals, for
$\gg_0 = \omega$ and
$\gg_\ga = \bigcup_{\gb < \ga} \gk_\gb$
for $\ga > 0$,
$\gg_\ga < \gk_\ga$ is singular if
$\ga$ is a limit ordinal.
For each ordinal $\ga$, let
$\la \gd^\ga_\gb \mid \gb < \gk_\ga \ra$ be
an enumeration of the $V$-strong cardinals which are limits
of $V$-strong cardinals in the interval
$(\gg_\ga, \gk_\ga)$, and let
$\FP^{\gk_\ga} = \la \la \FP^{\gk_\ga}_\gb,
\dot \FQ^{\gk_\ga}_\gb \ra \mid \gb < \gk_\ga \ra$
be the Easton support iteration where
$\FP^{\gk_\ga}_0 = \add(\gg^+_\ga, 1)$ and
$\forces_{\FP^{\gk_\ga}_\gb}
``\dot \FQ^{\gk_\ga}_\gb = \dot \FP(\gg^+_\ga, \gd^\ga_\gb)$''.
%adds a non-reflecting stationary set of ordinals of cofinality $\gg^+_\ga$ to $\gd^\ga_\gb$''.
We define $\FP$ as the
Easton support product
$\prod_{\ga \in {\rm Ord}} \FP^{\gk_\ga}$.
%Since every $\FP^{\gk_\ga}$ is $\gg^+_\ga$-directed closed,
The definition of each $\FP^{\gk_\ga}$
together with the
standard Easton arguments show
$V^\FP \models ``$ZFC + For every ordinal $\ga$, $2^{\gk_\ga} = \gk^+_\ga$''.
For each ordinal $\ga$, write
$\FP = \FP_{< \ga} \times
\FP^{\gk_\ga} \times \FP^{> \ga}$, where
$\FP_{< \ga} = \prod_{\gb < \ga} \FP^{\gk_\gb}$ and
$\FP^{> \ga}$ is the remainder of $\FP$.
By the definition of $\FP$ and the fact the
supercompactness of $\gk_\ga$ is indestructible
under set or class forcing, %not destroying GCH,
$V^{\FP^{> \ga}}_1 \models ``$For every ordinal
$\gb$, $2^{\gk_\gb} = \gk^+_\gb$ +
$\gk_\ga$ is supercompact''.
Further, the cardinals in the open interval
$(\gg_\ga, \gk_\ga)$ which are strong in $V^{\FP^{> \ga}}$
are precisely the same as the cardinals in the open interval
$(\gg_\ga, \gk_\ga)$ which are strong in $V$.
To see this, suppose $\gd \in (\gg_\ga, \gk_\ga)$ is such that
$V^{\FP^{> \ga}} \models ``\gd$ is a strong cardinal''. Since
$\FP^{> \ga}$ is $\gk_\ga$-directed closed,
$V \models ``\gd$ is $\gl$ strong for every $\gl < \gk_\ga$''. Because
$V \models ``\gk_\ga$ is supercompact and hence strong'',
it follows from \cite[proof of Lemma 2.5, page 32, second paragraph]{AC2} that
%as we observed in the proof of \cite[Lemma 2.4]{AC2},
$V \models ``\gd$ is a strong cardinal''.
Now, if $\gd \in (\gg_\ga, \gk_\ga)$ is such that
$V \models ``\gd$ is a strong cardinal'', then again because
$\FP^{> \ga}$ is $\gk_\ga$-directed closed,
$V^{\FP^{> \ga}} \models ``\gd$ is $\gl$ strong for every $\gl < \gk_\ga$''. As
$V^{\FP^{> \ga}} \models ``\gk_\ga$ is supercompact and
therefore strong'', as we just noted,
%the proof of \cite[Lemma 2.4]{AC2} tells us that
$V^{\FP^{> \ga}} \models ``\gd$ is a strong cardinal''.
By the facts that $\FP^{> \ga}$ is $\gk_\ga$-directed closed and
the cardinals in the open interval
$(\gg_\ga, \gk_\ga)$ which are strong in $V^{\FP^{> \ga}}$
are precisely the same as the cardinals in the open interval
$(\gg_\ga, \gk_\ga)$ which are strong in $V$, %Thus,
the definition of $\FP^{\gk_\ga}$ is the same in either $V$ or $V^{\FP^{> \ga}}$.
This means that we can apply the results used in the proof of Theorem \ref{t3} for one cardinal
to show that
$V^{\FP^{> \ga} \times \FP^{\gk_\ga}} \models
``\gk_\ga$ is both strongly compact and strong,
there are no strongly compact or strong
cardinals in the interval
$(\gg_\ga, \gk_\ga)$,
$\gk_\ga$ is a limit of non-strong tall cardinals, and
$\gk_\ga$ is not $2^{\gk_\ga} =
\gk^+_\ga$ supercompact''. Since
$V \models ``|\FP_{< \ga}| < 2^{\gg^+_\ga}$'', the
L\'evy-Solovay results \cite{LS} show that
$V^{\FP^{> \ga} \times \FP^{\gk_\ga} \times \FP_{< \ga}}_1 =
V^\FP \models ``\gk_\ga$ is both strongly compact
and strong,
there are no strongly compact or strong
cardinals in the interval
$(\gg_\ga, \gk_\ga)$,
$\gk_\ga$ is a limit of non-strong tall cardinals, and
$\gk_\ga$ is not $2^{\gk_\ga} =
\gk^+_\ga$ supercompact''.
Therefore, since any cardinal $\gd$
which is strongly compact or strong
and is not a $\gk_\ga$ would have to be such that
$\gd \in (\gg_\ga, \gk_\ga)$,
$V^\FP$ is our desired model.
This proves Theorem \ref{t3} for a
proper class of cardinals.
\end{pf}
\begin{pf}
Turning now to the proof of Theorem \ref{t3a},
suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal $\gl > \gk$ is measurable''.
By first forcing with the partial ordering $\FQ$ used in the proof of
\cite[Theorem 1]{AG14}, we obtain a model $V^\FQ$ such that
$V^\FQ \models ``$ZFC + $\gk$ is supercompact + No cardinal $\gl > \gk$ is measurable +
$\gd$ is measurable iff $\gd$ is tall''.
By the definition of $\FQ$, we may assume in addition that
$V^\FQ \models ``2^\gd= \gd^+$ for every $\gd \ge \gk$''.
Let $V_0 = V^\FQ$.
Suppose $\FP^\gk$ is defined in $V_0$ as in the proof of Theorem \ref{t3}.
Since $\card{\FP^\gk} = \gk$, by the results of \cite{LS}, $V_0^{\FP^\gk} \models ``$No
cardinal $\gl > \gk$ is measurable''.
It also follows that
$V_0^{\FP^\gk} \models %``2^\gk = \gk^+$ (and in fact,
``2^\gd = \gd^+$ for every $\gd \ge \gk$''.
%Standard arguments show that by its definition, $V^\FP \models {\rm GCH}$.
In addition, because $V_0 \models ``2^\gk = \gk^+$'',
the same arguments used in the proof of Theorem \ref{t3} tell us that
$V_0^{\FP^\gk} \models ``\gk$ is both the least strong and least
strongly compact cardinal''.
%Since $\card{\FP^\gk} = \gk$,
%The previous two sentences then allow us to infer
We may consequently immediately infer that
$V_0^{\FP^\gk} \models ``\gk$ is both the only strong and only strongly compact cardinal''.
Therefore, again as in the proof of Theorem \ref{t3},
$V_0^{\FP^\gk} \models ``\gk$ is not $2^\gk = \gk^+$ supercompact''.
The proof of Theorem \ref{t3a} is therefore completed by the following lemma.
\begin{lemma}\label{l7}
$V_0^{\FP^\gk} \models ``$Every measurable cardinal is tall''.
\end{lemma}
\begin{proof}
Suppose $V_0^{\FP^\gk} \models ``\gd$ is measurable''. Since
$V_0^{\FP^\gk} \models ``$No cardinal $\gl > \gk$ is measurable and
$\gk$ is a strong cardinal'', we may assume without loss of generality that $\gd < \gk$.
Further, by the factorization of $\FP^\gk$ given in the
first paragraph of the proof of Lemma \ref{l1}
and the remarks immediately following Theorem \ref{tgf},
it follows that in addition, $V_0 = V^\FQ \models ``\gd$ is measurable''.
Since $\FQ$ is the partial ordering of \cite[Theorem 1]{AG14}, we know that
$V_0 \models ``\gd$ is tall'' as well.
%We also know, by the definition of $\FP^\gk$, that
%$V_0 \models ``\gd$ is not a strong cardinal which is a limit of strong cardinals''.
We consider now the following two cases.
\bigskip\noindent Case 1: $\gd$ is not a limit of $V_0$-strong cardinals which are limits of
$V_0$-strong cardinals.
We also know, by the definition of $\FP^\gk$, that
$V_0 \models ``\gd$ is not a strong cardinal which is a limit of strong cardinals''.
%In this situation,
We may therefore use the factorization %write
$\FP^\gk = \FR^* \ast \dot \FR^{**} \ast \dot \FR$ %as
given in the proofs of Lemmas
\ref{l1} and \ref{l2} and employ the same argument found in the second
paragraph of the proof of Lemma \ref{l2} (with the slight modification
that by the arguments of \cite{LS}, since $\card{\FR^*} < \gd$,
forcing with $\FR^*$ preserves the fact that $\gd$ is a tall cardinal) to infer that
$V_0^{\FP^\gk} \models ``\gd$ is a tall cardinal''.
\bigskip\noindent Case 2: $\gd$ is a limit of $V_0$-strong cardinals which are limits of
$V_0$-strong cardinals. It then immediately follows that $\gd$ must be a limit of
$V_0$-strong cardinals which are themselves not limits of $V_0$-strong cardinals.
For any such $\gg$,
by Lemma \ref{l2}, $V_0^{\FP^\gk} \models ``\gg$ is a tall cardinal''. Since
$V_0^{\FP^\gk} \models ``\gd$ is a measurable limit of tall cardinals'', by
\cite[Corollary 2.7]{H09}, $V_0^{\FP^\gk} \models ``\gd$ is a tall cardinal''.
\bigskip
Cases 1 and 2 complete the proof of Lemma \ref{l7}.
\end{proof}
Lemma \ref{l7} and setting $\FP = \FQ \ast \dot \FP^\gk$
complete the proof of Theorem \ref{t3a}.
\end{pf}
\begin{pf}
Turning now to the proof of Theorem \ref{t4}, suppose
$V \models ``$ZFC + $\K$ is the proper class of supercompact cardinals''.
As in \cite{A03} and the proof of Theorem \ref{t3} in the general case,
we also assume without loss of generality that
$V \models ``$Every $\gk \in \K$ has its supercompactness
Laver indestructible under
$\gk$-directed closed forcing + $2^\gk = \gk^+$''.
In analogy to the proof of \cite[Theorem]{A03}, we
start by defining the building blocks used in the definition
%begin with a definition of the building blocks used in the definition
of the partial ordering $\FP$ witnessing the conclusions of Theorem \ref{t4}.
%first prove
%Theorem \ref{t4} for one cardinal. In particular, starting from a model for
%``ZFC + $\gk$ is supercompact
%and is not a limit of supercompact cardinals'', we will force and construct a model where
%$\gk$ is both the least strongly compact and least measurable limit of strong cardinals
%in which $\gk$ is also a limit of (non-strong) tall cardinals.
Before beginning the proof, however, as we did in the above
discussion of Theorem \ref{t3}, we first give some intuition and motivation for
the definition of our forcing conditions.
Suppose $\gk \in \K$ is not a limit of supercompact cardinals.
In \cite{A03}, in order to construct
the requisite model, it was only necessary to force to add
non-reflecting stationary sets of ordinals of the appropriate cofinality
to rid ourselves of %get rid of %destroy
each ground measurable limit of strong cardinals $\gd < \gk$,
while also forcing to preserve each ground model strong cardinal $\gd < \gk$.
As before,
this is not sufficient in the current situation, since the partial orderings %forcing
from \cite{A03}
will not ensure that there are non-strong tall cardinals below $\gk$.
We will therefore force to preserve the strongness of some, although not all,
of the ground model
strong cardinals below $\gk$ which are not limits
of ground model strong cardinals, while also adding non-reflecting
stationary sets of ordinals of the appropriate cofinality to
each ground model measurable limit of strong cardinals $\gd < \gk$.
We will leave alone %not destroy
the remaining ground model strong
cardinals $\gd < \gk$
which are not limits of ground model strong cardinals. %, but will leave them alone.
Such $\gd$ will become tall but not strong.
This will guarantee both that all
measurable limits of strong cardinals below $\gk$ have been
eliminated, and that $\gk$ has become a limit of non-strong tall cardinals.
%Getting specific,
%suppose $V \models ``$ZFC + $\K$ is the class of supercompact cardinals''.
%As in \cite{A03}, we also assume without loss of generality that
%$V \models ``$Every $\gk \in \K$ has its supercompactness indestructible under
%$\gk$-directed closed forcing + $2^\gk = \gk^+$''.
%Once again,
%without loss of generality, by first doing a preliminary forcing if necessary,
%we assume in addition that $V \models {\rm GCH}$.
We continue in analogy to \cite{A03}, quoting verbatim when appropriate,
and working under the assumption that all computations and notions
found in this paragraph are given in $V$.
Fix $\gk \in \K$ which %is supercompact but
is not a limit of supercompact cardinals.
Let $\xi$ be either the successor of the supremum of the supercompact
cardinals below $\gk$ or $\go$ if $\gk$ is the least supercompact cardinal,
and let $\eta$ be the least strong cardinal above $\xi$ in $V$.
By \cite[Lemma 2.1]{AC2} and the succeeding remarks, $\eta \in (\xi, \gk)$.
Let $\la \gd_\ga \mid \ga < \gk \ra$ be the
continuous, increasing enumeration of the
%cardinals below $\gk$
cardinals in the interval $(\eta, \gk)$
which are either strong cardinals or measurable limits of strong cardinals.
For $\ga$ an arbitrary ordinal, define
$\ga^-$ as the immediate ordinal
predecessor of $\ga$ if $\ga$ is a
successor ordinal, and $0$ if $\ga$
is either a limit ordinal or $0$.
For each $\ga < \gk$, let
$\gg_\ga = {(\bigcup_{\gb < \ga} \gd_\gb)}^+$,
where if $\ga = 0$, $\gg_\ga =
{(2^{\eta^+})}^+$.
Also, for each $\ga < \gk$, define
$\theta_\ga$ as the least cardinal
such that
$V \models ``\gd_\ga$ is not $\theta_\ga$
supercompact''.
As in \cite{A03},
$\theta_\ga$ is well-defined
for every $\ga < \gk$.
Further, it must be the case that $\theta_\ga < \gd_{\ga + 1}$.
This is since it follows from the argument found in
\cite[proof of Lemma 2.4, page 31, fourth paragraph]{AC2} that
%as we have previously observed,
if $\gd$ is $\gg$
supercompact for every
$\gg < \gd'$ and $\gd'$ is
strong, then $\gd$ is supercompact.
%(see the proof of \cite[Lemma 2.4]{AC2}),
Thus, $\theta_\ga < \gd_{\ga + 1}$ is true because %follows because
$\gd_{\ga + 1}$ is a strong cardinal.
We now define the partial ordering
$\FP_\gk =
\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
as the Easton support
iteration of length $\gk$ satisfying the
following properties:
\begin{enumerate}
\item\label{i1} $\FP_0 = \add(\eta^+, 1)$. %adds a Cohen subset of $\eta^+$.
\item\label{i2}
Suppose $\gd_\ga$ is not a measurable limit
of strong cardinals in $V$ and $\ga$ is either a limit ordinal or
a successor ordinal of the form $\ga' + 2n +1$
for $n \in \go$. Here, $\ga'$ is either a limit ordinal or $0$.
Then $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for a
Gitik-Shelah partial ordering of \cite{GS}
for the cardinal $\gd_\ga$.
We stipulate that this partial ordering be
defined using only components %partial orderings
that are at least $\rho_\ga$-strategically
closed and $\xi$-directed
closed, and that the
%whose
first nontrivial forcing in the definition be %done at a
$\add(\gl_\ga, 1)$ for $\gl_\ga$
the least measurable cardinal above
$\rho_\ga = \max(\theta_{\ga^-},
\gg_\ga, \xi)$.\footnote{It is also possible
to let
$\dot \FQ_\ga$ be a term for
Hamkins' partial ordering of \cite[Theorem 4.10]{H4}
for the cardinal $\gd_\ga$,
assuming the same restrictions on components
%as when using the partial ordering of \cite{GS}
as just mentioned and that the fast function
forcing employed in Hamkins' definition
is $\xi$-directed closed.}
Under these restrictions, the
realization of $\dot \FQ_\ga$
makes the strongness of
$\gd_\ga$ indestructible under forcing with
$\gd_\ga$-strategically closed partial orderings
which are at least $\xi$-directed closed.
\item\label{i3}
Suppose $\gd_\ga$ is not a measurable limit
of strong cardinals in $V$ and $\ga$ is
a successor ordinal of the form $\ga' + 2n +2$ for $n \in \go$.
Here, %$n \in \go$ and
$\ga'$ is either a limit ordinal or $0$.
Then $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for trivial forcing.
\item\label{i4}
Suppose $\gd_\ga$ is a measurable limit of strong cardinals in $V$.
Then $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga = \dot \FP(\xi, \gd_\ga)$.
%is a term for the
%partial ordering adding a non-reflecting stationary
%set of ordinals of cofinality $\xi$ to $\gd_\ga$. %$\FP_{\xi, \gd_\ga}$.
\end{enumerate}
%\noindent
Let $A = \{\gk \in \K \mid \gk$ is not a limit of supercompact cardinals$\}$.
We then define the partial ordering $\FP$ used in the proof of Theorem \ref{t4}
as the Easton support product $\prod_{\gk \in A} \FP_\gk$.
%$\prod_{\{\gk \in \K \mid \gk \hbox{\rm \ is not a limit of
%supercompact cardinals}\}} \FP_\gk$.
We note that the definition of $\FP$ is almost the same as the one
found in \cite{A03}, with the only key difference that the strongness of
$\gd_\ga$ for $\gd_\ga$ as in (\ref{i3}) above is not preserved by the iteration $\FP_\gk$.
%Therefore, since
Further, for any $\gk \in A$, there are unboundedly in $\gk$ many $\gd_\ga$
as in (\ref{i2}) above, and %since
as in \cite{A03}, the strongness of such $\gd_\ga$ is
preserved by both $\FP_\gk$ and $\FP$.
Therefore, the exact same arguments as in \cite{A03} virtually unchanged show that
$V^\FP \models ``$ZFC + $\gk$ is strongly compact iff $\gk$ is a measurable
limit of strong cardinals + The strongly compact cardinals are the elements of
$\K$ together with their measurable limit points'' and that in $V^\FP$,
any $\gk \in \K$ which was a supercompact limit of supercompact cardinals in
$V^\FP$ remains supercompact.
This means that the proof of Theorem \ref{t4} is completed by the following lemma.
\begin{lemma}\label{l8}
$V^\FP \models ``$If $\gk \in \K$, then $\gk$ is a limit of
non-strong tall cardinals''.
\end{lemma}
\begin{proof}
It suffices to prove Lemma \ref{l8} for $\gk \in A$.
%which is not a limit of supercompact cardinals.
%To do this, in analogy to the proof of Theorem \ref{t3} and
%We begin by showing that
%$V^{\FP_\gk} \models ``\gk$ is a limit of non-strong tall cardinals''.
This is since if $\gl \in \K \setminus A$, %is a limit of supercompact cardinals,
then $\gl$ is a limit of members of $A$
(which are $V$-supercompact cardinals each of which is
not a limit of $V$-supercompact cardinals), and so $\gl$ is a limit of
non-strong tall cardinals.
To do this, in analogy to the proof of Theorem \ref{t3}
in the general case and in analogy to the proof of \cite[Theorem]{A03}, we can write
$\FP = \FP_{< \gk} \times \FP_\gk \times \FP^{\gk}$, where
$\FP_{< \gk} = \prod_{\gd < \gk, \gd \in A} \FP_\gd$,
$\FP^{\gk} = \prod_{\gd > \gk, \gd \in A} \FP_\gd$, and all products have Easton support.
As in the proof of \cite[Theorem]{A03} and the proof
of Theorem \ref{t3} in the general case, since $\FP^\gk$ is $\gk$-directed closed
and each $\gk \in \K$ has its supercompactness indestructible under
$\gk$-directed closed forcing,
the cardinals less than or equal to $\gk$ in $V^{\FP^\gk}$ which are
supercompact, strong, or measurable limits of strong cardinals are
precisely the same as those in $V$.
In addition, as in the proof of \cite[Theorem]{A03}, the cardinals $\gd < \gk$ in
$V^{\FP^\gk}$ and $V$ which are either strong cardinals or measurable limits
of strong cardinals are precisely the same as those cardinals which are
either $\ga$ strong for every $\ga \in (\gd, \gk)$
(in either $V^{\FP^\gk}$ or $V$)
or are measurable limits of
cardinals $\gg$ which are $\ga$ strong for every $\ga \in (\gg, \gk)$
(in either $V^{\FP^\gk}$ or $V$).
This then allows us to conclude as in the proof of \cite[Theorem]{A03} that
$\FP_\gk$ as defined in $V^{\FP^\gk}$ is the same as $\FP_\gk$
as defined in $V$.
Consequently, we will now show that
$V^{\FP^\gk \times \FP_\gk} \models ``\gk$ is a limit of non-strong
tall cardinals''.
This will be enough to complete the proof of Lemma \ref{l8}.
This is since
$\card{\FP_{< \gk}} < \gk$, so by the results of \cite{LS} and \cite{HW},
if $V^{\FP^\gk \times \FP_\gk} \models ``\gk$ is a limit of non-strong
tall cardinals'', then
$V^{\FP^\gk \times \FP_\gk \times \FP_{< \gk}} = V^\FP \models ``\gk$ is a limit of non-strong
tall cardinals''.
To do this, work in $V^{\FP^\gk} = V_1$.
Let $\gd_\ga$ be as in (\ref{i3}) of the definition of $\FP_\gk$.
In analogy to the proofs of Lemmas \ref{l1} and \ref{l2}, by the
definition of $\FP_\gk$, we may write
$\FP_\gk = \FR^* \ast \dot \FR^{**} \ast \dot \FR$.
Here, $\card{\FR^*} < \gd_\ga$, $\FR^*$ is nontrivial,
$\dot \FR^{**} = \dot \add(\gd', 1)$
%is a term for a partial ordering adding a new subset of
for some measurable cardinal $\gd' > \gd_\ga$, and
$\dot \FR$ is a term for the rest of $\FP_\gk$. Since
$\forces_{\FR^* \ast \dot \FR^{**}} ``\dot \FR$ is $\gs$-strategically closed
for $\gs$ the least inaccessible cardinal above $\gd'$'',
as in the proof of Lemma \ref{l1},
$V^{\FR^* \ast \dot \FR^{**} \ast \dot \FR}_1 = V^{\FP_\gk}_1 \models ``\gd_\ga$ is
$\gd' + 2$ strong'' iff
$V^{\FR^* \ast \dot \FR^{**}}_1 \models ``\gd_\ga$ is $\gd' + 2$ strong''.
The remainder of the argument given in Lemma \ref{l1} then shows that
$V^{\FR^* \ast \dot \FR^{**}}_1 \models ``\gd_\ga$ is not
$\gd' + 2$ strong''.
%\footnote{We use $\gd' + 2$ here, instead of
%$\gd' + 1$ as in the proof of Lemma \ref{l1}, because we need to be
%able to detect the measurability of $\gd'$ in the relevant
%target model. This is something which in general requires an elementary embedding
%witnessing the $\gd' + 2$ strongness of $\gd$.}
The argument given in the second paragraph of the proof of Lemma \ref{l2}
then shows that
$V^{\FR^* \ast \dot \FR^{**} \ast \dot \FR}_1 =
V^{\FP^\gk \times \FP_\gk} \models ``\gd$ is a tall cardinal''.
Putting the previous two sentences together, we now have that for
$\gd_\ga$ as in (\ref{i3}) of the definition of $\FP_\gk$,
$V^{\FP^\gk \times \FP_\gk} \models ``\gd_\ga$ is a non-strong tall cardinal''.
Since there are unboundedly many such $\gd_\ga$ below $\gk$,
this completes the proof of Lemma \ref{l8}.
\end{proof}
Lemma \ref{l8} completes the proof of Theorem \ref{t4}.
\end{pf}
\section{Concluding Remarks}\label{s3}
%In conclusion to this paper, we ask whether it is possible
%to prove versions of Theorems \ref{t3} -- \ref{t4} in which
%there are no non-strong tall cardinals in any of the models %constructed
%witnessing the conclusions of these theorems.
In conclusion to this paper, we ask whether it is possible
to prove versions of Theorems \ref{t3} -- \ref{t4} in which
the only tall cardinals are either strong or strongly compact.
This seems to be an extremely challenging question to answer,
both since every tall cardinal $\gk$ is automatically indestructible under
$(\gk, \infty)$-distributive forcing, and since by \cite[Theorem 1]{AG14},
it is consistent to assume in a model containing a supercompact
cardinal that the statement ``$\gk$ is measurable iff $\gk$ is tall'' is true.
Thus, not only does forcing to add a non-reflecting stationary set of ordinals
of small cofinality %(regardless of cofinality)
above a tall cardinal
%using the standard partial ordering mentioned in Section \ref{s1}
not destroy tallness as it does strong compactness, but the set of tall
cardinals which might need to be eliminated in order to answer this question
could be quite large.
%obtain a model
%with no non-strong tall cardinals could be quite large.
%where the only tall cardinals are either strong or strongly compact could be quite large.
For instance, suppose we wish to force
over a model with a supercompact cardinal $\gl$ and construct a model
%containing no non-strong tall cardinals
in which the least strongly compact cardinal is also the least strong cardinal
and the only tall cardinals are either strong or strongly compact.
If the statement ``$\gk$ is measurable iff $\gk$ is tall'' is true,
then iteratively adding non-reflecting stationary sets of ordinals below
%the supercompact cardinal
$\gl$ to any cardinal
$\gd$ which is either a strong cardinal or a
non-strong tall cardinal will in fact destroy all measurable cardinals
below $\gl$. While this will, under the appropriate circumstances,
preserve the strong compactness of $\gl$, it will not preserve the
strongness of $\gl$, since after the forcing has been done, there
will no longer be any measurable cardinals below $\gl$.
Finding a way around this problem seems to be quite difficult.
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\end{document}
Suppose that in conditions (\ref{i2}) and (\ref{i3}) of
the definition of the partial ordering $\FP_\gk$
used in the proof of Theorem \ref{t4}, $\dot \FQ_\ga$ is a term for trivial forcing.
In other words, roughly speaking, $\FP_\gk$ is now redefined as the partial ordering
which adds non-reflecting stationary sets of ordinals of cofinality $\xi$ to
$\gd_\ga$ which are $V$-measurable limits of $V$-strong cardinals and
is trivial everyplace else.
\begin{pf}
To prove Theorem \ref{t3} in the general case, i.e., when there is a
proper class of supercompact cardinals, we will as we did above follow the
proof given in \cite[Section 3]{AC2}, quoting verbatim where
appropriate. Suppose
$V \models ``$ZFC + There is a proper class of supercompact cardinals''.
Let $\la \gk_\ga \mid \ga \in {\rm Ord} \ra$ enumerate the supercompact
cardinals in increasing order.
Let $V \models ``$ZFC + $\la \gk_\ga \mid \ga \in
{\rm Ord} \ra$ is the proper class of
supercompact cardinals''. Without loss of
generality, we assume in addition that
$V \models {\rm GCH}$ and that
by ``cutting off'' the universe
if necessary at the least inaccessible
limit of supercompact cardinals, for
$\gg_0 = \omega$ and
$\gg_\ga = \bigcup_{\gb < \ga} \gk_\gb$
for $\ga > 0$,
$\gg_\ga < \gk_\ga$ is singular if
$\ga$ is a limit ordinal.
Further, as in \cite[Section 3]{AC2},
%by the methods of either \cite{A83} or \cite{A98a}
%(both of which generalize Laver's result of \cite{L}),
we can also assume without loss of
generality that for
$\FR = \add(\omega, 1) \ast
\dot \FR^*$,
$V_1 = V^\FR \models ``$The supercompactness of each
$\gk_\ga$ is indestructible under forcing with $\gk_\ga$-directed
closed set or class partial orderings + $2^{\gk_\ga} = \gk^+_\ga$ +
For unboundedly many regular cardinals $\gd$, $2^\gd = \gd^+$''.
%$V_1 = V^\FR \models ``$GCH +
%The supercompactness of each
%$\gk_\ga$ is indestructible under
%forcing with $\gk_\ga$-directed
%closed set or class partial orderings not destroying GCH''.
Since it will be the case that
$\forces_{\add(\omega, 1)}
``\dot \FR^*$ is $\aleph_1$-strategically
closed'' and
$|\add(\omega, 1)| = \omega$, again by the remarks immediately following Theorem \ref{tgf},
%$\FR$ is a gap forcing
%admitting a very low gap.
%Thus, once again by Hamkins' results of
%\cite{Ha1}, \cite{Ha2}, and \cite{Ha3},
$V_1 \models ``$Any cardinal which is
supercompact or strong must have been
supercompact or strong in $V$''.
Work in $V_1$. For each ordinal $\ga$, let
$\la \gd^\ga_\gb \mid \gb < \gk_\ga \ra$ be
an enumeration of the $V$-strong cardinals which are limits
of $V$-strong cardinals in the interval
$(\gg_\ga, \gk_\ga)$, and let
$\FP^{\gk_\ga} = \la \la \FP^{\gk_\ga}_\gb,
\dot \FQ^{\gk_\ga}_\gb \ra \mid \gb < \gk_\ga \ra$
be the Easton support iteration where
$\FP^{\gk_\ga}_0 = \{\emptyset\}$ and
$\forces_{\FP^{\gk_\ga}_\gb}
``\dot \FQ^{\gk_\ga}_\gb$ adds a
non-reflecting stationary set of ordinals
of cofinality $\gg^+_\ga$ to $\gd^\ga_\gb$''.
We define $\FP$ as the
Easton support product
$\prod_{\ga \in {\rm Ord}} \FP^{\gk_\ga}$.
Since every $\FP^{\gk_\ga}$ is
$\gg^+_\ga$-directed closed,
the definition of each $\FP^{\gk_\ga}$
together with the
standard Easton arguments show
$V^\FP_1 \models ``$ZFC + $2^{\gk_\ga} = \gk^+_\ga$ + For unboundedly many regular
cardinals $\gd$, $2^\gd = \gd^+$''.
%$V^\FP_1 \models {\rm ZFC} + {\rm GCH}$.
For each ordinal $\ga$, write
$\FP = \FP_{< \ga} \times
\FP^{\gk_\ga} \times \FP^{> \ga}$, where
$\FP_{< \ga} = \prod_{\gb < \ga} \FP^{\gk_\gb}$ and
$\FP^{> \ga}$ is the remainder of $\FP$.
By the definition of $\FP$ and the fact the
supercompactness of $\gk_\ga$ is indestructible
under set or class forcing, %not destroying GCH,
$V^{\FP^{> \ga}}_1 \models ``2^{\gk_\ga} = \gk^+_\ga$ +
For unboundedly many regular cardinals
$\gd$, $2^\gd = \gd^+$ +
$\gk_\ga$ is supercompact''. Further, since
$\FR \ast (\dot \FP^{> \ga} \times
\dot \FP^{\gk_\ga}) =
\add(\omega, 1) \ast
(\dot \FR^* \ast
(\dot \FP^{> \ga} \times \dot \FP^{\gk_\ga}))$
is such that
$\forces_{\add(\omega, 1)}
``\dot \FR^* \ast (\dot \FP^{> \ga} \times
\FP^{\gk_\ga})$ is
$\ha_1$-strategically closed'',
the remarks immediately following Theorem \ref{tgf}
%the results of \cite{Ha1}, \cite{Ha2}, and \cite{Ha3}
once more apply to show that any cardinal
which is strong in
$V^{\FP^{> \ga} \times \FP^{\gk_\ga}}_1$
must have been strong in $V$.
Thus, we can apply the results used in the proof of Theorem \ref{t3} for one cardinal
to show that
$V^{\FP^{> \ga} \times \FP^{\gk_\ga}} \models
``\gk_\ga$ is both strongly compact and strong,
there are no strongly compact or strong
cardinals in the interval
$(\gg_\ga, \gk_\ga)$,
$\gk_\ga$ is a limit of non-strong tall cardinals, and
$\gk_\ga$ is not $2^{\gk_\ga} =
\gk^+_\ga$ supercompact''. Since
$V_1 \models ``|\FP_{< \ga}| < 2^{\gg^+_\ga}$'', the
L\'evy-Solovay results \cite{LS} show that
$V^{\FP^{> \ga} \times \FP^{\gk_\ga} \times \FP_{< \ga}}_1 =
V^\FP_1 \models ``\gk_\ga$ is both strongly compact
and strong,
there are no strongly compact or strong
cardinals in the interval
$(\gg_\ga, \gk_\ga)$,
$\gk_\ga$ is a limit of non-strong tall cardinals, and
$\gk_\ga$ is not $2^{\gk_\ga} =
\gk^+_\ga$ supercompact''.
Therefore, since any cardinal $\gd$
which is strongly compact or strong
and is not a $\gk_\ga$ must be such that
$\gd \in (\gg_\ga, \gk_\ga)$,
$V^\FP_1$ is our desired model.
This proves Theorem \ref{t3} for a
proper class of cardinals.
\end{pf}
We now define the partial ordering
$\FP_\gk =
\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
as the Easton support
iteration of length $\gk$ satisfying the
following properties:
\begin{enumerate}
\item\label{i1} $\FP_0$ adds a
Cohen subset of $\eta^+$.
\item\label{i2} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ is not a measurable limit
of strong cardinals and $\ga$ is
a successor ordinal of the form $\ga' + 2n +1$
for $n \in \go$ and $\ga'$ a limit ordinal,
$\dot \FQ_\ga$ is a term for a
Gitik-Shelah partial ordering of \cite{GS}
for the cardinal $\gd_\ga$.
We stipulate that this partial ordering be
defined using only component partial orderings
that are at least $\rho_\ga$-strategically
closed and $\xi$-directed
closed, and that the
%whose
first nontrivial forcing in the definition be %done at a
$\add(\gl_\ga, 1)$ for $\gl_\ga$
the least measurable cardinal above
$\rho_\ga = \max(\theta_{\ga^-},
\gg_\ga, \xi)$.\footnote{It is also possible
to let
$\dot \FQ_\ga$ be a term for
Hamkins' partial ordering of \cite[Theorem 4.10]{H4}
for the cardinal $\gd_\ga$,
assuming the same restrictions on components
%as when using the partial ordering of \cite{GS}
as just mentioned and that the fast function
forcing employed in Hamkins' definition
is $\xi$-directed closed.}
Under these restrictions, the
realization of $\dot \FQ_\ga$
makes the strongness of
$\gd_\ga$ indestructible under forcing with
$\gd_\ga$-strategically closed partial orderings
which are at least $\xi$-directed closed.
\item\label{i3} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ is not a measurable limit
of strong cardinals and $\ga$ is
a successor ordinal of the form $\ga' + 2n +2$
for $n \in \go$ and $\ga'$ a limit ordinal,
$\dot \FQ_\ga$ is a term for trivial forcing.
\item\label{i4} $\FP_{\ga + 1} =
\FP_\ga \ast \dot \FQ_\ga$, where if
$\gd_\ga$ is a measurable limit
of strong cardinals,
$\dot \FQ_\ga = \dot \FP(\xi, \gd_\ga)$.
%is a term for the
%partial ordering adding a non-reflecting stationary
%set of ordinals of cofinality $\xi$ to $\gd_\ga$. %$\FP_{\xi, \gd_\ga}$.
\end{enumerate}