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Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
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\title{Supercompactness and Measurable Limits of Strong Cardinals}
\author{Arthur W.~Apter
\thanks{The author wishes to thank James Cummings and Joel Hamkins for
helpful discussions on the subject matter of this paper.
The author also wishes to thank the referee for his comments.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\}
\date{June 5, 1998\\
(revised April 27, 1999)}
\begin{document}
\maketitle
\begin{abstract}
In this paper, two theorems concerning measurable limits
of strong cardinals and supercompactness are proven.
This generalizes earlier work, both individual and
joint with Shelah.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
A well-known result of Menas \cite{Men}
says that if $\ga < \gk$ and $\gk$ is the
$\ga^{{\hbox{\rm th}}}$
measurable limit of strongly compact cardinals,
then $\gk$ is strongly compact but isn't
$2^\gk$ supercompact. That this result is
best possible was demonstrated by Shelah and the
author in \cite{AS2}, in which the consistency,
relative to a supercompact limit of
supercompact cardinals,
of the least measurable limit $\gk$ of either
strongly compact or supercompact cardinals
satisfying $2^\gk > \gk^+$ and $\gk$ is
$< 2^\gk$ supercompact was shown.
Note now that Menas' aforementioned result
easily generalizes to the following.
\begin{lemma} \label{l1}
Suppose $\varphi(x)$ is a formula in the language
of set theory with one free variable, and suppose
$V \models ``$ZFC + $\gk$ is the
$\ga^{{\hbox{\rm th}}}$
measurable limit of cardinals $\gd$ satisfying
property $\varphi(\gd)$ where $\ga < \gk$''.
Then there is no elementary embedding
$j : V \to M$ witnessing the measurability of $\gk$
so that
$M \models ``\gk$ is measurable''.
\end{lemma}
\begin{proof}
If not, then let $j : V \to M$ be such an embedding.
By elementarity,
$M \models ``j(\gk) > \gk$ is the
$j(\ga^{{\hbox{\rm th}}}) = \ga^{{\hbox{\rm th}}}$
measurable limit of cardinals $\gd$ so that $\varphi(\gd)$''.
On the other hand, if $\gb < \ga < \gk$ and $\gg < \gk$ are so that
$V \models ``\gg$ is the $\gb^{{\hbox{\rm th}}}$
measurable limit of cardinals so that $\varphi(\gd)$'',
then again by elementarity and the fact $\gk$
is the critical point of $j$,
$M \models ``j(\gg) = \gg$ is the
$j(\gb^{{\hbox{\rm th}}}) = \gb^{{\hbox{\rm th}}}$
measurable limit of cardinals $\gd$ so that
$\varphi(\gd)$''. Thus, since
$M \models ``\gk$ is measurable'',
$M \models ``\gk$ is the
$\ga^{{\hbox{\rm th}}}$
measurable limit of cardinals $\gd$ so that
$\varphi(\gd)$'', a contradiction.
\end{proof}
A straightforward consequence of Lemma \ref{l1}
is that if $\gk$ and $\varphi$ are as above, then not only is
$\gk$ not $2^\gk$ supercompact, but $\gk$
isn't $\gk + 2$ strong. Thus, unlike
Menas' aforementioned result, it is
impossible for an arbitrary measurable limit
of strong cardinals to be strong. However,
one can reasonably inquire as to the
possibility of an arbitrary measurable limit
of strong cardinals being somewhat supercompact.
The purpose of this paper is to show that
if $\varphi(\gd)$ expresses the property
``$\gd$ is a strong cardinal'', then it
is possible to prove results analogous
to the ones of \cite{AS2} and \cite{A2}.
Specifically, we prove the following two theorems.
\begin{theorem} \label{t1}
Let $V \models ``$ZFC + GCH + $\gk$ is the least
limit of strong cardinals so that $\gk$ is $< \gl$
supercompact''. Let $h : \gk + 1 \to \gl + 1$
be a function satisfying the following three conditions.
\begin{enumerate}
\item $h(\gd) = 0$ if $\gd$ isn't an inaccessible cardinal.
\item For $\gd$ an inaccessible cardinal, $h(\gd) = \gl_\gd$
has the properties
that $\gl_\gd > \gd^+$, $\gl_\gd$ is either inaccessible
or is the successor of a cardinal of cofinality $> \gd$,
and $\gl_\gd$ is below the least strong cardinal above $\gd$.
\item There is $j : V \to M$ witnessing the $< \gl$
supercompactness of $\gk$ with
$j(h)(\gk) = h(\gk) = \gl = \gl_\gk$.
\end{enumerate}
There is then a partial ordering $\FP \in V$ with
$V^\FP \models ``$ZFC + $\gk$ is the least measurable
limit of strong cardinals + For every cardinal
$\gd \le \gk$ which is an inaccessible limit of
strong cardinals and every cardinal
$\gg \in [\gd, \gl_\gd)$, $2^\gg = \gl_\gd$ +
$\gk$ is $< \gl$ supercompact''.
\end{theorem}
\begin{theorem} \label{t2}
Let $V \models ``$ZFC + GCH + $\gk$ is the least
supercompact cardinal + No cardinal $\gl > \gk$
is measurable''. Let $h : \gk + 1 \to {\hbox{\rm Ord}}$
satisfy the following four conditions.
\begin{enumerate}
\item $h(\gd) = 0$ if $\gd$ isn't an inaccessible cardinal.
\item For $\gd$ an inaccessible cardinal, $h(\gd) = \gl_\gd$
has the properties that
$\gl_\gd > \gd^+$, $\gl_\gd$ is either inaccessible or is the
successor of a cardinal of cofinality $> \gd$, and $\gl_\gd$
is below the least strong cardinal above $\gd$.
\item If $\gd$ is $< \gl_\gd$ supercompact, there is
$j_\gd : V \to M$ witnessing the $< \gl_\gd$ supercompactness
of $\gd$ with
$j_\gd(h)(\gd) = h(\gd) = \gl_\gd$.
\item For all cardinals $\gg \ge h(\gk)$, there is
$j_{\gk, \gg} : V \to M$ witnessing the $\gg$
supercompactness of $\gk$ with
$j_{\gk, \gg}(h)(\gk) = h(\gk) = \gl_\gk$.
\end{enumerate}
There is then a partial ordering $\FP \in V$ with
$V^\FP \models ``$ZFC + $\gk$ is the least
supercompact cardinal + For every cardinal
$\gd \le \gk$ which is an inaccessible limit of
strong cardinals and every cardinal
$\gg \in [\gd, \gl_\gd)$, $2^\gg = \gl_\gd$ +
If $\gd$ is a measurable limit of strong
cardinals, then $\gd$ is $< \gl_\gd$ supercompact''.
\end{theorem}
Theorems \ref{t1} and \ref{t2}
should be compared with Theorem 3
of \cite{AS2} and Theorem 1 of \cite{A2}.
We will see, however, in Lemma \ref{l2}
that the least measurable limit of strong cardinals
is well below the least measurable limit of
supercompact cardinals, and in fact every
supercompact cardinal has a normal measure
concentrating on limits of strong cardinals
which are somewhat supercompact.
Thus, the results of this paper
are quite different from the corresponding
results of \cite{AS2} and \cite{A2}, and
Theorem \ref{t2} is non-trivial.
The structure of this paper is as follows.
Section \ref{s1} contains our introductory
comments and preliminary remarks concerning
notation, terminology, etc.
Section \ref{s2} contains a discussion of
forcing notions found in
\cite{AS1}, \cite{AS2}, and
\cite{GS}, all of which will be used in the
construction of the partial orderings used in
the proofs of Theorems \ref{t1} and \ref{t2}.
Section \ref{s3} contains a discussion of the proof of Theorem \ref{t1}.
Section \ref{s4} contains a discussion of the proof of Theorem \ref{t2}.
Section \ref{s5} contains our concluding remarks.
Before giving the proofs of our theorems, we very briefly
mention some preliminary information.
We will use in this paper, with
one exception, the notation, terminology, and
conventions from \cite{AS2}.
The one exception is that here,
if $\gk < \gl$ are regular cardinals, then
${{\hbox{\rm Add}}}(\gk, \gl)$ is the standard
partial ordering for adding $\gl$ Cohen
subsets to $\gk$.
We mention that we are assuming familiarity with the
notions of
measurability, strongness, superstrongness, Woodinness,
strong compactness, and supercompactness.
Interested readers may consult \cite{Ka} and
\cite {AS2} for
further details.
However, unlike in \cite{Ka}, we will say the cardinal
$\gk$ is $\gl$ strong for $\gl > \gk$ if there is
$j : V \to M$ having critical point $\gk$ so that
$j(\gk) > \gl$ and
$V_\gl \subseteq M$.
Finally, we mention that since ideas and notions from
\cite{AS1} and \cite{AS2} are used throughout the
course of this paper, it would be most helpful to
readers if copies of these papers were kept close at hand.
\section{Forcing Notions from \cite{AS1}, \cite{AS2},
and \cite{GS}}\label{sec}\label{s2}
In order to define in a meaningful way the iterations
to be used in the proofs of Theorems \ref{t1} and \ref{t2},
we first recall the definitions and properties of the
fundamental building blocks of these iterations.
Fix $\gg < \gd < \gl$, $\gl > \gd^+$ regular cardinals in our
ground model $V$, with $\gd$ inaccessible and $\gl$ either
inaccessible or the successor of a cardinal of cofinality
$> \gd$. We recall now the partial orderings
$\FP^0_{\gd, \gl}$,
$\FP^1_{\gd, \gl}[S]$,
and
$\FP^2_{\gd, \gl}[S]$
of Section 4 of \cite{AS2}.
We assume GCH holds for all
cardinals $\gk \ge \gd$.
As in Section 4 of \cite{AS2},
the first notion of forcing $\FP^0_{\gd, \gl}$ is just
the standard notion of forcing
for adding a non-reflecting stationary set of ordinals of cofinality
$\gg $ to $\gl$.
Specifically, $\FP^0_{\gd,\gl} = \{ p$ : 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, so that $\gb \in S_p$ implies
$\gb > \gd$ and cof$(\gb) = \gg \}$,
ordered by $q \ge p$
($q$ extends $p$)
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 $\FP^0_{\gd, \gl}$ (see
\cite{Bu} or \cite{KiM}), in $V[G]$,
since GCH holds in $V$ for all cardinals
$\gk \ge \gd$, a non-reflecting stationary
set $S=S[G]=\cup\{S_p:p\in G \} \subseteq \gl$
of ordinals of cofinality $\gg $ has been introduced, the
bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH at cardinals
$\gk \ge \gd$ have been preserved.
It is also virtually immediate that $\FP^0_{\gd, \gl}$
is $\gg$-directed closed, and it can be shown
(see \cite{Bu} or \cite{KiM}) that
$\FP^0_{\gd, \gl}$
is $\prec \gl$-strategically closed.
Work now in $V_1 = V^{\FP^0_{\gd, \gl}}$, letting $\dot S$
be a term always forced to denote the above set $S$.
$\FP^2_{\gd, \gl}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ non-stationary).
Specifically, $\FP^2_{\gd, \gl} [S] = \{ p$ : For
some successor ordinal $\ga < \gl$,
$p : \ga \to \{0,1\}$ is a characteristic function of
$C_p$, a club subset of $\ga$, so that
$C_p \cap S = \emptyset \}$,
ordered by $ q \ge p $ iff $C_q$ is an end extension of $C_p$.
It is again well-known (see \cite{MekS}) that for $H$
$V_1$-generic over $\FP^2_{\gd, \gl}[S]$, a club set
$C = C[H] = \cup \{C_p : p \in H \}
\subseteq \gl$ which is disjoint to $S$ has been introduced,
the bounded subsets of $\gl$
are the same as those in $V_1$,
and cardinals, cofinalities, and GCH
for cardinals $\gk \ge \gd$ have been preserved.
The following lemma is proven in both \cite{AS1} and \cite{AS2}.
\begin{lemma}\label{lem} (Lemma 1 of \cite{AS1} and \cite{AS2}.)
$\forces_{\FP^0_{\gd, \gl}} ``
\clubsuit({\dot S})$'', i.e., $V_1 \models ``$There is a
sequence $\langle x_\alpha : \alpha \in S \rangle$ so that
for each $\alpha \in S$, $ x_\alpha \subseteq \alpha$ is
cofinal in
$\alpha$, and for any $A \in
{[\gl]}^{\gl}$, $\{\alpha \in S : x_\alpha
\subseteq A \}$ is stationary''.
\end{lemma}
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\ga : \ga \in S \ra$.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gd, \gl}
[S] $.
First, since each element of
$S$ has cofinality $\gg$, the proof of Lemma 1 of \cite{AS1} and \cite{AS2}
shows each $x \in X$ can be assumed to be
so that order type$(x) = \gg$. Then,
$\FP^1_{\gd, \gl}[ S]$ is defined as the set of all
4-tuples $\la w, \ga, \bar r, Z \ra$ satisfying the
following properties.
\begin{enumerate}
\item $w \in {[\gl]}^{< \gd}$.
\item $\ga < \gd$.
\item $ \bar r = \la r_i : i \in w \ra$ is a
sequence of functions from $\ga$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\ga$.
\item $Z \subseteq \{x_\gb : \gb \in S\}$
is a set so that if $z \in Z$, then for some
$y \in {[w]}^\gg$, $y \subseteq z$ and $z - y$
is bounded in the $\gb$ so that $z = x_\gb$.
\end{enumerate}
\noindent As in \cite{AS1}, the definition of $Z$ implies
$|Z| < \gd$.
The ordering on $\FP^1_{\gd, \gl}[S]$ is given by
$\la w^1, \ga^1, \bar r^1, Z^1 \ra \le
\la w^2, \ga^2, \bar r^2, Z^2 \ra$ iff the following hold.
\begin{enumerate}
\item $w^1 \subseteq w^2$.
\item $\ga^1 \le \ga^2$.
\item If $i \in w^1$, then $r^1_i
\subseteq r^2_i$.
\item $Z^1 \subseteq Z^2$.
\item If $z \in Z^1 \cap {[w^1]}^\gg$ and
$\ga^1 \le \ga < \ga^2$, then $|\{i \in z :
r^2_i(\ga) = 0\}| = |\{i \in z : r^2_i(\ga) = 1\}| = \gg$.
\end{enumerate}
The intuition behind the definition of $\FP^1_{\gd, \gl}[S]$
just given is essentially the same as in \cite{AS1}
and Section 4 of \cite{AS2}. Specifically,
we wish to be able simultaneously to make $2^\gd = \gl$,
destroy the measurability of $\gd$, and be able to
resurrect the $< \gl$ supercompactness of $\gd$ if
necessary. $\FP^1_{\gd, \gl}[S]$ has been designed so as to
allow us to do all of these things.
The proof that $V^{\FP^1_{\gd, \gl}[S]}_1 \models
``\gd$ is non-measurable'' is as in Lemma 3 of \cite{AS1}.
In particular, the argument of Lemma 3 of \cite{AS1} will show
that $\gd$ can't carry a $\gg$-additive uniform ultrafilter.
We can then carry through the proof of Lemma 4 of \cite{AS1} to show
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$ is equivalent to
${\hbox{\rm Add}}(\gl, 1) \ast \dot {\hbox{\rm Add}}(\gd, \gl)$.
The proofs of Lemma 5 of \cite{AS1}
and Lemma 6 of \cite{AS2} will then show
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ preserves cardinals
and cofinalities, is $\gl^+$-c.c.,
is $< \gd$-strategically closed, and is so that
$V^{\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]} \models
``2^\gk = \gl$ for every cardinal $\gk \in [\gd, \gl)$''.
Although the above definition of
$\FP^1_{\gd, \gl}[S]$
(occasionally to be referred to as
the ``simpler version'')
is perfectly adequate for our purposes, as mentioned at
the end of \cite{AS2}, it will not suffice to prove Theorem
3 of \cite{AS2}. In order to do so, a more complicated form of
$\FP^1_{\gd, \gl}[S]$
is required.
The more complicated version of
$\FP^1_{\gd, \gl}[S]$
can then be used in the proofs of
Theorems \ref{t1} and \ref{t2}
of this paper.
Interested readers may consult Section 1 of
\cite{AS2} for further details on the
definition of this partial ordering,
but we will not give its definition
here.
We will, however, structure our proofs so that
they are equally applicable if either the
simpler or more complicated version of
$\FP^1_{\gd, \gl}[S]$ is used.
As in \cite{AS2}, in order to define the iteration
used in the proof of Theorem \ref{t1}, we need to
use in addition to the above partial orderings
an indestructibility partial ordering $\FP^*$.
Unlike \cite{AS2}, however, instead of using
Laver's partial ordering of \cite{L} for making
the supercompactness of $\gd$ indestructible
under $\gd$-directed closed forcing, we use
a version of the Gitik-Shelah partial ordering
$\FP^*$ of \cite{GS} for making the strongness of
$\gd$ indestructible under $\gd$-weakly closed
partial orderings satisfying the Prikry property.
We recall now the definition of this
property of partial orderings.
\begin{definition}\label{d1}
A partial ordering $\la \FQ, \le, \le^* \ra$ is
$\gd$-weakly closed and satisfies the Prikry
property if
$\le^* \subseteq \le$ is so that the following hold.
\begin{enumerate}
\item For any condition $p \in \FQ$ and any formula
$\varphi$ in the forcing language with respect to
$\FQ$, there is $q \ge^* p$ so that $q \decides \varphi$.
\item For any increasing chain
$p_0 \le^* p_1 \le^* \cdots \le^* p_\ga \le^* \cdots (\ga < \gd)$
with respect to $\le^*$ of elements of $\FQ$,
there is an upper bound $q$ for the whole chain.
\end{enumerate}
\end{definition}
As noted in \cite{GS}, $\gd$-closed partial orderings,
Prikry forcing defined on a measurable cardinal $\gg > \gd$,
and appropriate versions of Magidor's forcing of
\cite{Ma} for changing the cofinality of a large enough
measurable cardinal $\gg > \gd$ to some uncountable cardinal
or Radin forcing \cite{Ra} based on a large enough measurable
cardinal $\gg > \gd$, are all $\gd$-weakly closed and satisfy the
Prikry property. We observe, however, that any
$\gd$-strategically closed partial ordering
$\la \FQ, \le \ra$ also meets these criteria.
To see this, let $\cal S$ be a strategy for player II,
and define $p \le^*q$ iff $p \le q$ and there is a
play of the game of length $\ga < \gd$ so that if
player II starts with $p$, playing according to
$\cal S$ results in player II playing $q$ $\ga$ steps later.
If $p \le^* q$ and
there is a play of the game taking player II $\ga$
steps to go from $p$ to $q$ via $\cal S$
and $q \le^* r$ and
there is a play of the game taking player II $\gb$
steps to go from $q$ to $r$ via $\cal S$, then
$p \le^* r$ since
there is a play of the game taking player II
$\ga + \gb$ steps to go from $p$ to $r$ via ${\cal S}$.
Thus, $\le^*$ is well-defined.
By the definition of strategic closure,
any increasing chain
$p_0 \le^* p_1 \le^* \cdots \le^* p_\ga \le^* \cdots (\ga < \gd)$
has an upper bound. Since for $\varphi$ a formula
in the forcing language with respect to
$\FQ$ and $p \in \FQ$, there is $q \ge^* p$ so that
$q \decides \varphi$ (player II plays $p$, player I
plays $q' \ge p$ so that $q' \decides \varphi$,
and player II plays $q \ge q'$ given according to
$\cal S$), any $\gd$-strategically closed partial
ordering is also $\gd$-weakly closed and satisfies the
Prikry property. We therefore define our indestructibility
forcing $\FP^*$ for a strong cardinal $\gd$ as the
partial ordering of \cite{GS} so that when a
non-trivial forcing is done at a stage
$\gg < \gd$, the forcing is $\gg$-strategically closed.
This means $\FP^*$ will make the strongness of $\gd$
indestructible under forcing with $\gd$-strategically
closed partial orderings, but the strongness of
$\gd$ won't necessarily be indestructible under
forcing with partial orderings such as
Prikry, Magidor, or Radin forcing.
(We could if we wished not place such restrictions
on the non-trivial components of $\FP^*$ at a
stage $\gg < \gd$, but this would make the
definition of the forcing conditions $\FP$
used in the proof of Theorem \ref{t1} somewhat
more complicated.)
We conclude this section by noting that when the
proofs of Theorems \ref{t1} and \ref{t2} are given,
it will be irrelevant as to whether the
simpler or more complicated version of
$\FP^1_{\gd, \gl}[S]$ is used.
When the simpler version is used,
we will assume without loss of
generality that each element of $S$ has cofinality
$\omega$. Thus, we will write without fear of
ambiguity
$\FP^0_{\gd, \gl}$,
$\FP^1_{\gd, \gl}[S]$, and
$\FP^2_{\gd, \gl}[S]$.
\section{The Proofs of Theorem \ref{t1} and a Preliminary Lemma}\label{s3}
Before beginning the proof of Theorem \ref{t1}, we
prove the following result of \cite{AC},
which shows that any supercompact cardinal must have
a normal measure concentrating on strong cardinals.
\begin{lemma}\label{l2}
Let $\gk$ be $2^\gk$ supercompact and strong. Assume
$j : V \to M$ is an elementary embedding
witnessing the $2^\gk$ supercompactness
of $\gk$, and let $\mu$ be the normal measure over
$\gk$ associated with $j$. Then
$\{ \gd < \gk : \gd$ is a strong cardinal$\} \in \mu$.
\end{lemma}
\begin{proof}
We first show, for $j$ and $\mu$ as in the statement of
Lemma \ref{l2}, that
$\{\gd < \gk : \gd$ is $\gk$ strong$\} \in \mu$.
(See also the proof of Proposition 26.11 of \cite{Ka}.)
To see this, note that since $M^{2^\gk} \subseteq M$,
$j \rest V_{\gk + 1} \in M$. Thus, as in
\cite{A3}, page 203, or \cite{C}, Section 2.1,
Facts 1 - 3, there is
${\cal E} \in M$ a $(\gk, j(\gk))$ extender and
$k : M \to {{\hbox{\rm Ult}}}(M, {\cal E})$
so that $\gk$ is the critical point of $k$ and
$M$ and ${{\hbox{\rm Ult}}}(M, {\cal E})$ agree
through rank $j(\gk)$. This means
$M \models ``\gk$ is superstrong with target $j(\gk)$'', so
$M \models ``\gk$ is $j(\gk)$ strong''. Hence, by reflection,
$\{\gd < \gk : \gd$ is $\gk$ strong$\} \in \mu$.
Fix now $\gd < \gk$ so that
$V \models ``\gd$ is $\gk$ strong''. We show that if
$\gl > \gk$ is arbitrary,
$V \models ``\gd$ is $\gl$ strong''.
Let $\gl' > \gl$ be so that any extender ${\cal E}$
witnessing the $\gl$ strongness of $\gd$ is such that
${\cal E} \in V_{\gl'}$. By the strongness of $\gk$, let
$j^* : V \to M^*$ be an embedding having critical point
$\gk$ witnessing that $\gk$ is $\gl'$ strong. Since
$V \models ``\gd$ is $\gk$ strong'',
$M^* \models ``j^*(\gd) = \gd$ is
$j^*(\gk) > \gl' > \gl$ strong''. As
$V_{\gl'} \subseteq M^*$ and
$M^* \models ``\gd$ is $\gl$ strong'',
$V \models ``\gd$ is $\gl$ strong''.
This proves Lemma \ref{l2}.
\end{proof}
We observe that in the above proof, it will actually be the case that
$M \models ``\gk$ is a strong limit of strong cardinals''.
This is since
$M \models ``\gk$ is $j(\gk)$ strong and $j(\gk)$ is strong'',
so by the second paragraph of the above proof,
$M \models ``\gk$ is strong''. Further, if
$\gd < \gk$ is so that
$V \models ``\gd$ is strong'', then
$M \models ``j(\gd) = \gd$ is strong''.
Thus, by reflection, we have the more powerful fact that
$A = \{\gd < \gk : \gd$ is a strong limit of strong
cardinals$\} \in \mu$.
By increasing the degree of supercompactness shown by
$j$ and $M$, we can in essence assume that each
$\gd \in A$ witnesses enough supercompactness to
serve as a $\gk$ as in the hypotheses of Theorem \ref{t1}.
This shows that the hypotheses of Theorem \ref{t1}
are weaker in consistency strength than the assumption of
the existence of a supercompact cardinal.
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + GCH +
$\gk$ is the least limit of strong cardinals so that
$\gk$ is $< \gl$ supercompact'', and take
$h : \gk + 1 \to \gl + 1$ as in the statement
of the hypotheses of Theorem \ref{t1}.
The partial ordering $\FP$ used in the proof of
Theorem \ref{t1} is a certain Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga \le \gk \ra$.
In order to specify the components of $\FP$,
we first let
$\la \gd_\ga : \ga \le \gk \ra$
be the continuous, increasing enumeration of the
set of strong cardinals below $\gk$ together
with its limit points. We take as an inductive
hypothesis that the field of $\FP_\ga$ is
$\{\gd_\gb : \gb < \ga \}$ and that if
$\gd_\ga$ is a strong cardinal which isn't a limit
of strong cardinals, then $|\FP_\ga| < \gd_\ga$.
The definition is then as follows:
\begin{enumerate}
\item $\FP_0$ is the forcing adding a single Cohen real.
\item Assuming $\FP_\ga$ has been defined for $\ga$,
we consider the following four cases.
\begin{enumerate}
\item $\gd_\ga$ is a strong cardinal which isn't a limit of
strong cardinals. By the inductive hypothesis, since
$|\FP_\ga| < \gd_\ga$, the L\'evy-Solovay results \cite{LS}
(see also \cite{GS} or \cite{HW}) show that
$V^{\FP_\ga} \models ``\gd_\ga$ is strong''.
It is thus possible in $V^{\FP_\ga}$ to make $\gd_\ga$
Gitik-Shelah indestructible under $\gd_\ga$-strategically
closed forcings. We therefore let
$\dot \FQ_\ga$ be a term for the partial ordering described in
Section \ref{sec} making $\gd_\ga$ Gitik-Shelah indestructible
under $\gd_\ga$-strategically closed forcings so that
$\forces_{\FP_\ga} ``\dot \FQ_\ga$ is defined using
partial orderings that are at least
${(2^{\gg_\ga})}^+$-strategically closed for
$\gg_\ga = \max(\omega, \sup(\{\gd_\gb : \gb < \ga \},
h(\sup(\{\gd_\gb : \gb < \ga \}))))$'', and we define
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$.
Since $\dot \FQ_\ga$ can be chosen so that
$|\FP_{\ga + 1}| = \gd_\ga < \gd_{\ga + 1}$,
by abusing terminology somewhat and considering
$\dot \FQ_\ga$ as a term for a partial ordering
whose field is $\gd_\ga$, the inductive hypothesis
is easily preserved.
\item $\gd_\ga < \gk$ is a regular limit of
strong cardinals and $\gd_\ga$ isn't
$< h(\gd_\ga)$ supercompact. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
with $\dot \FQ_\ga$ a term for
$\FP^0_{\gd_\ga, h(\gd_\ga)} \ast
\FP^1_{\gd_\ga, h(\gd_\ga)}[\dot S_{h(\gd_\ga)}]$, where
$\dot S_{h(\gd_\ga)}$ is a term for the
non-reflecting stationary subset of $h(\gd_\ga)$ introduced by
$\FP^0_{\gd_\ga, h(\gd_\ga)}$. Since
$\gd_{\ga + 1}$ must be a strong cardinal which isn't a limit of
strong cardinals, by the conditions on $h$,
$|\FP_{\ga + 1}| < \gd_{\ga + 1}$,
so the inductive hypothesis is once again preserved.
\item $\gd_\ga$ is a regular limit of
strong cardinals and $\gd_\ga$ is $< h(\gd_\ga)$ supercompact.
By the leastness of $\gk$, this means $\gd_\ga = \ga = \gk$.
Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, with
$\dot \FQ_\ga$ a term for
$\FP^0_{\gd_\ga, h(\gd_\ga)} \ast
(\FP^1_{\gd_\ga, h(\gd_\ga)}[\dot S_{h(\gd_\ga)}] \times
\FP^2_{\gd_\ga, h(\gd_\ga)}[\dot S_{h(\gd_\ga)}])$, where
$\dot S_{h(\gd_\ga)}$ is as above.
\item $\gd_\ga$ is a singular limit of strong cardinals. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the trivial partial ordering
$\{\emptyset\}$.
\end{enumerate}
\end{enumerate}
Note that if $\ga \ne \gk$ is a limit ordinal, then
$\sup(\{\gd_\gb : \gb < \ga \}) = \gd < \gd'$, where
$\gd'$ is the least strong cardinal $> \gd$.
It is this fact that preserves the inductive hypothesis
at limit ordinals $\ga < \gk$ and at successor stages
$\ga + 1$ when $\gd_\ga$ is a singular limit of
strong cardinals.
The intuitive motivation behind the above definition is
much the same as in \cite{AS2}. Specifically, at any
inaccessible limit $\gd < \gk$ of strong cardinals,
we must force to ensure that $\gd$ becomes non-measurable
and is so that $2^\gd = h(\gd)$. At $\gk$, however,
we must force so as simultaneously to make
$2^\gk = \gl$ while first destroying and then
resurrecting the $< \gl$ supercompactness of $\gk$.
The forcing will preserve the strongness of every
$V$-strong cardinal below $\gk$ which isn't a limit of strong
cardinals and will ensure there are no measurable limits
of strong cardinals below $\gk$.
\begin{lemma}\label{l3}
$V^\FP \models ``$If $\gd < \gk$ is a
$V$-strong cardinal which isn't a limit
of $V$-strong cardinals, then $\gd$ is strong''.
\end{lemma}
\begin{proof}
Let $\gd < \gk$ be a $V$-strong cardinal which
isn't a limit of $V$-strong cardinals. Write
$\FP = \FR_\gd \ast \dot \FR^\gd$, where
$\FR_\gd$ is the portion of $\FP$ whose field is all cardinals
$\le \gd$ and $\dot \FR^\gd$ is a term for the rest of $\FP$.
By case (a) in clause (2) of the inductive definition of $\FP$,
$V_1 = V^\FP \models ``\gd$ is strong and is indestructible
under $\gd$-strategically closed forcings''.
And, since for each appropriate $\gd' > \gd$,
$\FP^0_{\gd', h(\gd')}$ is at least $\gd$-strategically closed,
the definition of the iteration ensures that
$\forces_{\FR_\gd} ``\dot \FR^\gd$ is
$\gd$-strategically closed''. Thus,
$\forces_{\FR_\gd \ast \dot \FR^\gd = \FP}
``\gd$ is strong''. This proves Lemma \ref{l3}
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$The only strong cardinals below
$\gk$ are those which are strong in $V$ and
aren't limits of strong cardinals in $V$,
and every cardinal $\gd < \gk$ which is
an inaccessible limit of strong cardinals is so that
$2^\gg = \gl_\gd$ for all cardinals
$\gg \in [\gd, \gl_\gd)$''.
\end{lemma}
\begin{proof}
If we write
$\FP = \FP_0 \ast \dot \FR$, then by definition,
$|\FP_0| = \omega < \aleph_1$ and
$\forces_{\FP_0} ``\dot \FR$ is $\aleph_1$-strategically closed''.
Therefore, $\FP$ is a ``gap forcing'' in the sense of
Hamkins \cite{H1}, \cite{H2}, and \cite{H3},
so by Hamkins' results of \cite{H2} and \cite{H3},
any cardinal $\gd$ which is strong in
$V^\FP$ must have been strong in $V$.
Thus, the only possibilities for a strong cardinal
$\gd < \gk$ are a $V$-strong cardinal which isn't a limit of
$V$-strong cardinals or a $V$-strong cardinal which is a
limit of $V$-strong cardinals. By Lemma \ref{l3},
the former type of strong cardinal remains strong in $V$.
If $\gd < \gk$ is an inaccessible limit of $V$-strong
cardinals, then as in Lemma \ref{l3}, write
$\FP = \FR_\gd \ast \dot \FR^\gd$.
By the definition of $\FP$,
$V^{\FR_\gd} \models ``\gd$ isn't measurable and
$2^\gg = \gl_\gd$ for all cardinals
$\gg \in [\gd, \gl_\gd)$''.
Further, by the definition of $\FP$,
$\forces_{\FR_\gd} ``\dot \FR^\gd$ is
at least $< \gd'$-strategically closed for $\gd'$
the least inaccessible above $\gd$'', so
$V^{\FR_\gd \ast \dot \FR^\gd} = V^\FP \models
``\gd$ isn't measurable and
$2^\gg = \gl_\gd$ for all cardinals
$\gg \in [\gd, \gl_\gd)$''.
This proves Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is the least measurable limit of
strong cardinals and is so that $\gk$ is $< \gl$
supercompact and $2^\gg = \gl$ for all cardinals
$\gg \in [\gk, \gl)$''.
\end{lemma}
\begin{proof}
By Lemma \ref{l4} and its proof,
$V^\FP \models ``\gk$ is a limit of strong
cardinals and no (inaccessible) limit
of strong cardinals below $\gk$ is
measurable''. Therefore, by the definition of
$\FP$, the proofs of Lemma 9 of
\cite{AS1} and \cite{AS2}, Lemma 4 of \cite{AS2},
the remarks following Lemma 5 of \cite{AS2},
and the remarks in Section 4 of \cite{AS2},
regardless of the choice of the components of
$\FP$,
$V^\FP \models ``\gk$ is the least measurable limit of
strong cardinals, $\gk$ is $< \gl$ supercompact, and
$2^\gg = \gl$ for all cardinals $\gg \in [\gk, \gl)$''.
This proves Lemma \ref{l5}.
\end{proof}
Lemmas \ref{l3} - \ref{l5} complete the proof of Theorem \ref{t1}.
\end{proof}
\section{The Proof of Theorem \ref{t2}}\label{s4}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is the least supercompact cardinal
+ No cardinal $\gl > \gk$ is measurable'', and take
$h : \gk + 1 \to {\hbox{\rm Ord}}$ as in the statement of the
hypotheses of Theorem \ref{t2}. As before, $\FP$ is a
certain Easton support iteration
$\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga \le \gk \ra$.
In order to specify the components of $\FP$,
as in the proof of Theorem \ref{t1}, we again let
$\la \gd_\ga : \ga \le \gk \ra$
be the continuous, increasing enumeration of the
set of strong cardinals together with its limit points.
The definition of $\FP$ is then exactly the same as
in the proof of Theorem \ref{t1}, with the exception
that in case (a) of clause (2), $\dot \FQ_\ga$ is
taken as a term for the trivial partial ordering
$\{\emptyset\}$.
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$ is the least supercompact cardinal''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6}
is similar to the last part of
the proof of Lemma 2 of \cite{A2}.
Let $\gg > 2^{h(\gk)}$ be an arbitrary successor cardinal,
and let
$\gl$ be the cardinality of $2^{[\gg]^{< \gk}}$.
Let $j : V \to M$ be an elementary embedding witnessing
the $\gl$ supercompactness of $\gk$ satisfying property (4)
of the hypotheses on $h$.
By the definition of $\FP$ and the properties of $h$,
$j(\FP) = (\FP_\gk \ast
(\dot \FP^0_{\gk, h(\gk)} \ast
(\FP^1_{\gk, h(\gk)}[\dot S_{h(\gk)}]
\times \FP^2_{\gk, h(\gk)}[\dot S_{h(\gk)}]))) \ast \dot \FQ
=
(\FP_\gk \ast \dot \FQ_\gk) \ast \dot \FQ
=
\FP \ast \dot \FQ
=
\FP \ast \dot \FR' \ast \dot \FQ_{j(\gk)}$,
where $\dot \FR'$ is a term for the $M^\FP$ partial ordering
$\FP_{j(\gk)} / \FP$.
Since we have assumed there are no measurable cardinals
above $\gk$ in $V$
and $V \models {\hbox{\rm GCH}}$,
in $M$,
$\forces_{\FP} ``$The field of $\dot \FQ$ is composed of
cardinals $> \gl$''.
Further, by the definition of $\FP$ and the fact
$M^\gl \subseteq M$, it is true that in $V$ and $M$,
$\forces_\FP ``$Both $\dot \FR'$ and $\dot \FR' \ast
\dot \FQ_{j(\gk)}$ are $\gl$-strategically closed and
$\gl$ is the cardinality of $2^{[\gg]^{< \gk}}$''.
And, by Lemma 4 of \cite{AS1} and \cite{AS2} and the succeeding remarks,
$\forces_{\FP_\gk} ``\dot \FQ_\gk$ is forcing equivalent
to a $\gk$-directed closed partial ordering having cardinality
$\le 2^{h(\gk)}$''.
Therefore, the standard arguments
(see, e.g., Lemma 2 of \cite{A1})
in turn show that
$M^{\FP \ast \dot \FR'}$ remains $\gl$ closed with respect to
$V^{\FP \ast \dot \FR'}$ and that if
$G_0$ is $V$-generic over $\FP$ and
$G_1$ is $V[G_0]$-generic over $\FR'$,
in $V[G_0][G_1]$,
we can find a master condition $q$ extending each
$p \in j''G_0$. If
$G_2$ is a $V[G_0][G_1]$-generic object over
$\FQ_{j(\gk)}$ so that $q \in G_2$, in
$V[G_0][G_1][G_2]$,
there is an elementary embedding
$j^* : V[G_0] \to M[G_0][G_1][G_2]$
extending $j$. Since in $V$,
$\forces_\FP ``\dot \FR' \ast \dot \FQ_{j(\gk)}$ is $\gl$-strategically
closed'',
$V[G_0] \models ``\gk$ is $\gg$ supercompact''.
Since once again we can write
$\FP = \FP_0 \ast \dot \FR$, where
$|\FP_0| = \omega < \ha_1$ and
$\forces_{\FP_0} ``\dot \FR$ is $\ha_1$-strategically closed'',
Hamkins' results of \cite{H1}, \cite{H2},
and \cite{H3} show that
any cardinal which is supercompact in $V^\FP$
must have been supercompact in $V$, i.e.,
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
This proves Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``$Every inaccessible limit
$\gd \le \gk$ of strong cardinals is so that
$2^\gg = \gl_\gd$ for all cardinals
$\gg \in [\gd, \gl_\gd)$,
and every measurable limit of strong cardinals is
$< \gl_\gd$ supercompact''.
\end{lemma}
\begin{proof}
First, notice that since Lemma \ref{l6} tells us
$V^\FP \models ``\gk$ is the least supercompact cardinal'',
by Lemma \ref{l2} and the succeeding remarks,
$V^\FP \models ``$There are measurable limits of strong
cardinals below $\gk$''.
Now, let $\gd$ be an inaccessible limit of strong cardinals
in $V^\FP$. Since the factorization of $\FP$ as
$\FP_0 \ast \dot \FR$ given in Lemma \ref{l6}
and the results of \cite{H2} and \cite{H3}
tell us that any strong cardinal in $V^\FP$
must have been strong in $V$, $\gd$ is an
inaccessible limit of strong cardinals in
$V$. This allows us to write $\gd = \gd_\ga$
and consider the following two cases.
\begin{enumerate}
\item $V \models ``\gd$ isn't
$< h(\gd) = \gl_\gd$
supercompact''. In this case, by
case (b) in clause (2) of the inductive
definition of $\FP$ and the remarks in
Section 4 of \cite{AS2}
(when the simpler version is used in
the definition of the components of
$\FP$) or Lemmas 3 and 6 of
\cite{AS2} (when the more complicated
version is used in the definition of
the components of $\FP$),
$V^{\FP_{\ga + 1}} \models ``\gd$ isn't measurable and
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$''.
Since for the $\dot \FQ$ so that
$\FP_{\ga + 1} \ast \dot \FQ = \FP$,
$\forces_{\FP_{\ga + 1}} ``\dot \FQ$
is $\gz$-strategically closed for
$\gz$ the least inaccessible cardinal
$> \gd = \gd_\ga$'',
$V^{\FP_{\ga + 1} \ast \dot \FQ} = V^\FP \models
``\gd$ isn't measurable and
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$''.
\item $V \models ``\gd$ is
$< h(\gd) = \gl_\gd$
supercompact''. In this case, by reflection,
$V \models ``\gd$ is a limit of inaccessible limits
of strong cardinals''. Thus,
by the nature of the definition of $\FP$,
case (c) in clause (2) of the inductive
definition of $\FP$,
the remarks in Section 4 of \cite{AS2}
(when the simpler version is used in
the definition of $\FP$) or Lemma 4
and the remarks following Lemma 5 of
\cite{AS2} (when the more complicated
version is used in the definition of
$\FP$),
and Lemma 9 of \cite{AS1} and \cite{AS2},
$V^{\FP_{\ga + 1}} \models ``\gd$ is
$< h(\gd) = \gl_\gd$ supercompact and
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$''.
Since once again, for the $\dot \FQ$ so that
$\FP_{\ga + 1} \ast \dot \FQ = \FP$,
$\forces_{\FP_{\ga + 1}} ``\dot \FQ$ is
$\gz$-strategically closed for $\gz$
the least inaccessible $> \gd$'',
$V^{\FP_{\ga + 1} \ast \dot \FQ} = V^\FP \models
``\gd$ is $< h(\gd)$ supercompact and
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$''.
\end{enumerate}
Since by the results of \cite{LS} or
\cite{H1}, \cite{H2}, and \cite{H3},
forcing with
$\FP$ can create no new measurable
cardinals above $\gk$, the preceding two cases
complete the proof of Lemma \ref{l7}.
\end{proof}
Lemmas \ref{l6} and \ref{l7} complete the
proof of Theorem \ref{t2}.
\end{proof}
In conclusion to this section,
we remark that it is Lemma \ref{l2} that allows us to
avoid the use of both the indestructibility forcing
of \cite{GS} and the inductive hypothesis found
in the proof of Theorem \ref{t1}
in the definition of the $\FP$ used in
the proof of Theorem \ref{t2}. It is only
when $\gk$ is changed into a large cardinal
(such as a measurable cardinal which is
$< 2^\gk$ supercompact)
whose properties don't necessarily imply
the existence of the
requisite large cardinals below $\gk$,
as is the case in Theorem 3 of \cite{AS2}
and Theorem \ref{t1} of this paper,
that an indestructibility forcing and associated inductive hypothesis
must be used in the definition of $\FP$
to ensure the generic
extension satisfies the desired conclusions.
\section{Concluding Remarks}\label{s5}
In conclusion to this paper, we note that if
$\varphi(x)$ is as in Lemma \ref{l1} and
$\varphi(\gd)$ expresses a local property such as
``$\gd$ is superstrong'',
``$\gd$ is Woodin'',
``$\gd$ is $\gz$ supercompact for $\gz$ the
least inaccessible $> \gd$'', etc.,
then arguments analogous to those given in the
proof of Theorem \ref{t1} but somewhat simpler
can be used to construct a model from the assumption of
$\gk$ being both $< \gl$ supercompact and the least
measurable limit of cardinals $\gd$ satisfying
$\varphi(\gd)$ and $h : \gk + 1 \to \gl + 1$
satisfying properties analogous to properties (1) - (3) of the
hypotheses of Theorem \ref{t1}.
An outline of the argument is as follows.
The forcing conditions $\FP$ used are defined as in
the proof of Theorem \ref{t1}, except that as
in the proof of Theorem \ref{t2},
in case (a) of clause (2),
$\dot \FQ_\ga$ is a term for the trivial partial ordering
$\{\emptyset\}$.
Let $\gd < \gk$ satisfy $\varphi(\gd)$ in $V$.
For ``reasonable'' $\varphi$, it can be verified that
$\FP$ factors as
$\FQ \ast \dot \FR$, where
$|\FQ| < \gd$ and
$\forces_{\FQ} ``\dot \FR$ is (at least)
$\gz$-strategically closed for $\gz$
the least inaccessible cardinal above $\gd$ so that
$V_\gz \models `\gd$ is a cardinal satisfying $\varphi(\gd)$' iff
$V \models `\gd$ is a cardinal satisfying $\varphi(\gd)$' ''.
By applying first the arguments of \cite{LS} or \cite{HW}
and then the strategic closure properties of $\FR$,
it will then be the case that
$V^\FP \models ``\gd$ is a cardinal satisfying $\varphi(\gd)$''.
Since the results of \cite{H1}, \cite{H2},
and \cite{H3} show that
any cardinal $\gd$ satisfying $\varphi(\gd)$ in $V^\FP$
must have satisfied $\varphi(\gd)$ in $V$,
the definition of $\FP$ ensures there are no
measurable limits of cardinals $\gd$ satisfying
$\varphi(\gd)$ below $\gk$,
and Lemma 9 of \cite{AS1} and \cite{AS2} shows that
$V^\FP \models ``\gk$ is $< \gl$ supercompact'',
$V^\FP$ is as desired.
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\end{document}