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\title{Strong Compactness, Measurability, and the
Class of Supercompact Cardinals
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly compact
cardinal, non-reflecting stationary set of ordinals}
}
\author{Arthur W.~Apter\thanks{
The author wishes to thank James Cummings for
helpful discussions on the proof of Theorem
\ref{t2}, and the referee for numerous
comments and suggestions which have been
incorporated into this version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{October 14, 1999\\
(revised June 6, 2000, September 10, 2000, and
September 14, 2000)}
\begin{document}
\maketitle
\begin{abstract}
We prove two theorems concerning strong compactness,
measurability, and the class of supercompact cardinals.
We begin by showing, relative to the appropriate hypotheses, that it
is consistent non-trivially for every supercompact cardinal
to be the limit of (non-supercompact) strongly compact
cardinals.
We then show, relative to the existence of a
non-trivial (proper or improper) class of supercompact
cardinals, that it is possible
to have a model with
the same class of supercompact cardinals in which
every measurable cardinal $\gd$ is $2^\gd$ strongly compact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
It is well-known that the
structure of the class of supercompact cardinals
can vary quite a bit.
The work of Magidor \cite{Ma}
shows that it is consistent, relative to a
supercompact cardinal, for the least supercompact
cardinal to be the least strongly compact
cardinal.
The work of Kimchi and Magidor \cite{KM}
shows that it is consistent, relative to a
proper class of supercompact cardinals,
for the classes of supercompact and strongly
compact cardinals to coincide precisely,
except at measurable limit points, where
a result of Menas \cite{Me} shows that this
is impossible.
The work of \cite{A97} shows that, relative to
the existence of a cardinal $\Omega$ which is
an inaccessible limit of
measurable limits of supercompact cardinals,
it is consistent for the classes of supercompact
and non-supercompact strongly compact cardinals
to have, roughly speaking, any conceivable structure
dictated by a ground model function
$f : \Omega \to 2$.
The purpose of this paper is to
explore additional possibilities for
how strong compactness and measurability
can interact with the
class of supercompact cardinals.
Specifically, we prove the following two theorems.
\begin{theorem}\label{t1}
Suppose $V \models {\rm ZFC}$ and
${\mathfrak K} \subseteq V$
is so that
${\mathfrak K} \neq \emptyset$ is the
(possibly proper) class of supercompact
limits of supercompact cardinals.
Assume further that if
${\mathfrak K}$ is a set,
$V \models ``$No cardinal $\gl > \sup({\mathfrak K})$
is supercompact''.
There is then a partial ordering
$\FP \subseteq V$ with $V^\FP \models
``$ZFC +
The only supercompact cardinals are
the elements of ${\mathfrak K}$ +
Every supercompact
cardinal is a limit of (non-supercompact)
strongly compact cardinals''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models {\rm ZFC}$ and
${\mathfrak K} \subseteq V$
is so that
${\mathfrak K} \neq \emptyset$ is the
(possibly proper) class of supercompact
cardinals.
There is then a partial ordering
$\FP \subseteq V$ with $V^\FP \models
``$ZFC +
The only supercompact cardinals are
the elements of ${\mathfrak K}$ +
For every cardinal $\gd$,
$\gd$ is measurable iff $\gd$ is
$2^\gd$ strongly compact''.
\end{theorem}
To a large extent, the proofs of the above two
theorems will rely on Hamkins' work of \cite{H1},
\cite{H2}, and \cite{H3}.
Theorem \ref{t1} extends and generalizes
Corollary 4 of \cite{A97}.
Theorem \ref{t2} provides
instances of models with supercompact
cardinals in which every measurable
cardinal possesses a non-trivial degree of
strong compactness.
Theorem \ref{t2} should be contrasted with
the models constructed in \cite{AS97a}.
In these models, which contain supercompact
cardinals, for regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except possibly
if $\gk$ is a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact.
Note that Theorem 1.1 of \cite{A81}
provides an example of a
universe in which every measurable cardinal
has a non-trivial degree of strong
compactness, but in
which no supercompact cardinals are present.
We take the opportunity now to mention some
preliminary material.
If $\ga < \gb$ are ordinals, then
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
When forcing, $q \ge p$ will mean that
$q$ is stronger than $p$.
If $\FP$ is our partial ordering,
$V^\FP$ and $V[G]$ will be used
interchangeably to denote the generic extension
when forcing with $\FP$.
We may, from time to time, confuse terms with
the sets they denote and write $x$ when we
actually mean $\dot x$,
especially when $x$ is in the ground model $V$.
The partial ordering
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$. $\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed 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 $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
%Note that trivially, if $\FP$ is ${<}\gk$-closed, then $\FP$ is
%${<}\gk$-strategically
%closed and ${\prec}\gk $-strategically closed. The converse of
%both of these facts is false.
Suppose that $\gl$ is a limit ordinal of
uncountable cofinality.
A subset $S \subseteq \gl$ will be called a
{\it non-reflecting stationary set of ordinals}
if $S$ is stationary in $\gl$, yet for no
limit ordinal $\gd < \gl$ of uncountable cofinality
do we have that
$S \cap \gd$ is stationary in $\gd$.
We will use in the proofs of
Theorems \ref{t1} and \ref{t2}
two different versions of a partial
ordering originally due to Jensen
for adding a non-reflecting stationary set
of ordinals to a regular cardinal.
The proofs of the basic properties of
each of these partial orderings can
essentially be found in \cite{Bu},
pages 435--437.
For completeness and comprehensibility, we
will include below in Lemmas \ref{i1} - \ref{i3}
proofs of the basic properties for the partial
ordering used in the proof of Theorem \ref{t2}.
Readers should then be able to transfer these
proofs to the partial ordering used in the
proof of Theorem \ref{t1}.
Suppose now that $\gk < \gl$ are regular cardinals.
The partial ordering
$\FP(\gk, \gl)$ used in the proof
of Theorem \ref{t1} is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically,
$\FP(\gk, \gl) =
\{ p$ : 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
$\gb > \gk$ 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} or \cite{KM}), in $V[G]$,
%if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\cup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced,
and the
bounded subsets of $\gl$ are the same as those in $V$.
%and cardinals, cofinalities, and GCH
%have been preserved.
It is also virtually immediate that $\FP(\gk, \gl)$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu} or \cite{KM}) that
$\FP(\gk, \gl)$
is ${\prec}\gl$-strategically closed.
Suppose now that $\gk$ is a Mahlo cardinal.
The partial ordering $\FP(\gk)$ used in the
proof of Theorem \ref{t2} is the partial
ordering for adding a non-reflecting
stationary set of ordinals of a certain
type to $\gk$.
Specifically, $\FP(\gk) =
\{ p$ : For some
$\ga < \gk$, $p : \ga \to \{0,1\}$ 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 if $\gb < \sup(S_p)$ is inaccessible,
then $S_p - S_p \rest \gb$ is composed of
ordinals of cofinality at least $\gb\}$,
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 $S_p$.
\begin{lemma}\label{i1}
For any cardinal $\gd < \gk$,
$\FP(\gk)$ is $\gd$-strategically closed.
\end{lemma}
\begin{proof}
Let $\gd < \gk$ be given.
Since $\gk$ is a Mahlo cardinal, let
$\gg > \gd$, $\gg < \gk$ be inaccessible.
Consider the two person game in which
at any even (successor) stage $\ga + 2$, player II's
response to the condition $p_{\ga + 1}$ chosen
by player I at stage $\ga + 1$ is to choose a
condition $p_{\ga + 2}$ so that
$\sup(S_{p_{\ga + 1}}) \not\in S_{p_{\ga + 2}}$
and so that
$\sup(S_{p_{\ga + 2}}) > \max(\gg, \sup(S_{p_{\ga + 1}}))$.
At limit stages $\gl \le \gd$, player II plays
$\cup_{\ga < \gl} p_\ga$.
Readers can easily verify that this yields
a winning strategy for II, since for any
limit ordinal $\gl \le \gd < \gg$ of uncountable cofinality,
the limit points of $C_\gl = \{ \sup(S_{p_{\ga}}) : \ga < \gl\}$,
having cofinality $< \gd$,
form a club disjoint from
$S_{p_{\gl}}$.
This completes the proof of Lemma \ref{i1}.
\end{proof}
\begin{lemma}\label{i2}
For any
inaccessible cardinal $\gd < \gk$, there is
a partial ordering $\FP(\gk / \gd)$ dense
in $\FP(\gk)$ so that
$\FP(\gk / \gd)$ is
$\gd$-directed closed.
\end{lemma}
\begin{proof}
If $\gd < \gk$ is inacessible,
let $\FP(\gk / \gd) =
\{p : p \in \FP(\gk)$ is either the characteristic
function of the empty set or
$p \in \FP(\gk)$ is so that
%$\dom(p) > \gd$ and
$S_p$ contains an
ordinal $> \gd\}$, ordered in the same way as
$\FP(\gk)$ is.
We claim that $\FP(\gk / \gd)$ is as desired.
Let $p \in \FP$ be given.
Choose $\gg > \max(\sup(S_p), \gd)$,
$\gg < \gk$ an inaccessible cardinal.
Let $q \ge p$ be so that
$\gg \in S_q$.
Clearly, $q \in \FP(\gk / \gd)$, so
$\FP(\gk / \gd)$ is dense in
$\FP(\gk)$.
To see that $\FP(\gk / \gd)$ is
$\gd$-directed closed,
assume without loss of generality that
$\gl < \gd$ is an uncountable
regular cardinal and
$\{r_\ga : \ga < \gl \}$ is a set of
distinct mutually compatible elements of
$\FP(\gk / \gd)$.
It is clear that
$r = \cup_{\ga < \gl} r_\ga$ is the
characteristic function for a subset of some
$\zeta < \gk$.
Further, for $\gg < \sup(S_r)$,
if $\gg$ is inaccessible, then
$S_r - S_r \rest \gg$ is composed of ordinals of
cofinality at least $\gg$.
Thus, it suffices to show that there are sets
$C_r$ and $C_r \rest \gb$
for $\gb < \gl$ a limit ordinal of
uncountable cofinality so that
$C_r$ is club in $\sup(S_r)$,
$C_r \cap S_r = \emptyset$,
$C_r \rest \gb$ is club in
$\sup(S_{\cup_{\ga < \gb} r_\ga})$, and
$C_r \rest \gb \cap S_{\cup_{\ga < \gb} r_\ga} = \emptyset$.
Such a set $C_r$ can be defined by looking at
the sequence
$\la \sup(S_{r_\ga}) : \ga < \gl \ra$,
taking $C_r$ as the limit points of this sequence,
and taking $C_r \rest \gb$ for $\gb < \gl$
a limit ordinal of uncountable cofinality as
the limit points of
$\la \sup(S_{r_\ga}) : \ga < \gb \ra$.
This works since each element of $C_r$ must
be an ordinal $> \gd$
having cofinality $< \gl$, yet by the definition of
$\FP(\gk)$, each element of
$S_r - S_r \rest \gd$ or
$S_{\cup_{\ga < \gb} r_\ga} -
S_{\cup_{\ga < \gb} r_\ga} \rest \gd$
must have cofinality at least $\gd$.
This completes the proof of Lemma \ref{i2}.
\end{proof}
\begin{lemma}\label{i3}
Let $G$ be $V$-generic over
$\FP(\gk)$. Then
$S=S[G]=\cup\{S_p:p\in G\}
\subseteq \gk$ is stationary in $\gk$.
\end{lemma}
\begin{proof}
Let $p \in \FP(\gk)$ be so that
$p \forces ``\dot C$ is club in $\gk$''.
We show that for some $q \ge p$,
$q \forces ``\dot C \cap \dot S \neq \emptyset$''.
To do this, we define a sequence
$\la q_\ga : \ga < \gk \ra$ satisfying the following
properties.
\begin{enumerate}
\item\label{i1a} $q_0 \ge p$.
\item\label{i2a} If $0 \le \ga < \gb < \gk$, then
$q_\gb \ge q_\ga$.
\item\label{i3a} For each $\ga < \gk$, there is
an ordinal $\rho_\ga < \gk$ so that
$q_\ga \forces ``\rho_\ga \in \dot C$''.
\item\label{i4a} The sequence
$\la \rho_\ga : \ga < \gk \ra$ is strictly increasing.
\item\label{i5a} The sequences
$\la \sup(S_{q_\ga}) : \ga < \gk \ra$ and
$\la \rho_\ga : \ga < \gk \ra$ are
``interweaved'' in the sense that
$\rho_0 < \sup(S_{q_0}) < \rho_1 < \sup(S_{q_1}) <
\cdots < \rho_{\gb + 1} < \sup(S_{q_{\gb + 1}}) <
\cdots < \rho_\gl = \sup(S_{q_\gl}) <
\rho_{\gl + 1} < \sup(S_{q_{\gl + 1}}) < \cdots$,
where there is equality between
$\rho_\gl$ and $\sup(S_{q_\gl})$ at limit ordinals $\gl$.
\end{enumerate}
Using ideas from Lemma \ref{i1},
it is easy to construct the sequences
$\la \sup(S_{q_\ga}) : \ga < \gk \ra$ and
$\la \rho_\ga : \ga < \gk \ra$.
Whenever player I chooses a condition $p_\ga$ so that
$p_\ga \forces ``\rho_\ga \in \dot C$'' and
$\rho_\ga > \sup(\la \rho_\gb : \gb < \ga \ra)$,
player II responds by choosing a condition
$q_\ga \ge p_\ga$ so that
$\sup(S_{q_\ga}) > \max(\sup(S_{p_\ga}), \rho_\ga)$ and
$\sup(S_{p_\ga}) \not\in S_{q_\ga}$.
At limit stages $\gl$, player II simply chooses $q_\gl$ as
$\cup_{\ga < \gl} q_\ga$ and $\rho_\gl$ as
$\sup(\la \rho_\ga : \ga < \gl \ra)$.
Readers can verify for themselves that this construction
yields sequences with the desired properties.
By construction,
$C' = \{\rho_\ga : \ga < \gk\}$ is a club in $\gk$.
Since $\gk$ is a Mahlo cardinal, there is some ordinal
$\gl < \gk$ so that $\rho_\gl$ is inaccessible and
$\rho_\gl \in C'$.
We will then have that
$q = q_{\gl} \cup \{\la \rho_\gl, 1 \ra\}$ is so that
$q \ge p$ and
$q \forces ``\dot C \cap \dot S \neq \emptyset$''.
This completes the proof of Lemma \ref{i3}.
\end{proof}
We remark that the construction given in Lemma \ref{i3}
for the sequences
$\la \sup(S_{q_\ga}) : \ga < \gk \ra$ and
$\la \rho_\ga : \ga < \gk \ra$
essentially shows that either
$\FP(\gk)$ or
$\FP(\gk / \gd)$ for $\gd < \gk$
inaccessible is
${\prec} \gk$-strategically closed.
Also, we note that for the rest of the
paper, if $\FP$ is either a product,
iteration, or some combination of the
two which, for some cardinal $\gk$,
adds a non-reflecting stationary set of
ordinals to $\gk$, then we will say that
$\gk$ is in the support of $\FP$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V$ and ${\mathfrak K}$ be as in the hypotheses of
Theorem \ref{t1}.
%If ${\mathfrak K}$ is a set, then by ``cutting off''
%the universe if necessary at the least supercompact
%cardinal above $\sup({\mathfrak K})$, we assume
%without loss of generality that
%$V \models ``$No cardinal $\gk > \sup({\mathfrak K})$
%is supercompact''.
Let $\FP^0 \subseteq V$ be a (possibly
proper class) partial ordering, defined
as in \cite{A98}, so that
$V^{\FP^0} \models ``$ZFC + Every
$V$-supercompact cardinal $\gk$ is supercompact +
Every $V$-supercompact cardinal is Laver
indestructible \cite{L} + The supercompact and
strongly compact cardinals coincide precisely,
except at measurable limit points''.
Since we may assume without loss of generality
that $\FP^0$ is defined as a reverse Easton
iteration which begins by adding a Cohen
subset to $\omega$,
$\FP^0$ can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \omega$ and
$\forces_\FQ ``\dot \FR$ is
$\ha_1$-strategically closed''.
Thus, in Hamkins' terminology of
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP^0$ ``admits a gap at $\ha_1$'',
so by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
$V^{\FP^0} \models ``$The only supercompact
cardinals are those that were supercompact
in $V$'', i.e.,
$V^{\FP^0} \models ``$Every supercompact cardinal
is Laver indestructible''.
Work now in $V_0 = V^{\FP^0}$. Let
${\mathfrak D} = \seq{\gd_\ga : \ga < \Omega}$,
where $\Omega$ is the appropriate
ordinal if ${\mathfrak K}$ is a set but is
the class of ordinals otherwise,
enumerate in increasing order
$\{\gd : \gd$
is either a
measurable limit of supercompact
cardinals or the cardinal
successor of a
non-measurable limit point of the
measurable limits
of supercompact cardinals$\}
\cup \{\omega\}$.
For each $\ga < \Omega$, let
$\FP_\ga$ be the Easton support
iteration of the partial orderings
which add, for each supercompact
cardinal in the interval
$(\gd_\ga, \gd_{\ga + 1})$,
a non-reflecting stationary set of
ordinals of cofinality $\gd_\ga$
(which by definition is a regular
cardinal).
Let $\FP^1$ be the Easton support
product $\prod_{\ga < \Omega} \FP_\ga$,
and in $V$, let
$\FP = \FP^0 \ast \dot \FP^1$.
Standard arguments concerning class forcing
combined with
the definition of $\FP^1$
yield that
$V^\FP = V^{\FP^1}_0 \models {\rm ZFC}$.
\begin{lemma}\label{l1}
$V^\FP = V^{\FP^1}_0 \models ``$If
$\gk \in {\mathfrak K}$, then $\gk$ is
supercompact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} uses ideas found
in the proof of Lemma 9 of \cite{A97}.
Fix $\gk \in {\mathfrak K}$.
Working in $V_0$, write
$\FP^1$ = $\FQ_\gk \times \FQ^\gk$, where
$\FQ^\gk = \prod_{\gd \ge \k} \FP_\gd$, and
$\FQ_\gk$ is the rest of $\FP^1$.
By the definition of $\FQ^\gk$,
$V_0 \models ``\FQ^\gk$ is
$\gk$-directed closed''.
Thus, by the construction of $V_0$,
$V_1 = V^{\FQ^\gk}_0 \models ``\gk$ is
supercompact''.
It therefore suffices to show that
$V^{\FQ_\gk}_1 \models ``\gk$ is supercompact''.
To do this, let $\gl \ge \gk$ be arbitrary, and let
$\gg = |2^{{[\gl]}^{< \gk}}|$. Let
$j : V_1 \to M$ be an elementary embedding witnessing
the $\gg$ supercompactness of $\gk$ so that
$M \models ``\gk$ isn't supercompact''.
We observe that any $\gd \in (\gk, \gg]$ must be so that
$M \models ``\gd$ isn't supercompact'', for if this
were not the case, then the fact
$M^\gg \subseteq M$ allows us to infer that
$M \models ``\gk$ is ${<}\gd$ supercompact and
$\gd$ is supercompact''.
Hence, a theorem of Magidor found in
\cite{Ma2} tells us that
$M \models ``\gk$ is supercompact'', a
contradiction. Writing
$j(\FQ_\gk) = \FQ_\gk \times \FQ^*$,
the preceding says that in $M$,
the least cardinal $\gg_0$ in the
support of $\FQ^*$ must be so that
$\gg_0 > \gg$.
Let $G$ be $V_1$-generic over $\FQ_\gk$ and
$H$ be $V_1[G]$-generic over $\FQ^*$. In
$V_1[G \times H]$,
$j : V_1 \to M$ extends to
$\overline j : V_1[G] \to M[G \times H]$ via
the definition
$\overline j(i_G(\tau)) =
i_{G \times H}(j(\tau))$. Since
$M \models ``\FQ^*$ is ${<} \gg_0$-strategically
closed'' and $\gg < \gg_0$, the fact
$M^\gg \subseteq M$ implies
$V_1 \models ``\FQ^*$ is $\gg$-strategically closed''
yields that for any cardinal
$\gd \le \gg$, $V_1[G]$ and $V_1[G \times H] =
V_1[H \times G]$ contain the same subsets of $\gd$.
This means the supercompact ultrafilter $\cal U$ over
${(P_\gk(\gl))}^{V_1[G]}$ in $V_1[G \times H]$ given by
$x \in {\cal U}$ iff
$\seq{j(\ga) : \ga < \gl} \in \overline j(x)$
is so that
${\cal U} \in V_1[G]$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP = V^{\FP^1}_0 \models
``$The only supercompact cardinals are
the elements of ${\mathfrak K}$''.
\end{lemma}
\begin{proof}
In analogy to what was done
in the first paragraph of the proof
of Theorem \ref{t1}, write
$\FP = \FQ \ast \dot \FR$, where
$\card{\FQ} = \omega$ and
$\forces_{\FQ} ``\dot \FR$ is $\ha_1$-strategically
closed''.
Thus, as before, $\FP$
``admits a gap at $\ha_1$'', so
by the results of \cite{H1}, \cite{H2}, and
\cite{H3}, any supercompact cardinal in
$V^\FP$ had to have been supercompact in $V$.
Hence, the proof of Lemma \ref{l2} will be
complete once we have shown that any
$V$-supercompact cardinal not in ${\mathfrak K}$
is no longer supercompact in $V^\FP$.
Fix $\gk$ a $V$-supercompact
cardinal not an element of ${\mathfrak K}$.
Work in $V_0$.
By the definition of ${\mathfrak D}$, this
means we can find $\ga < \Omega$ so that
$\gk \in (\gd_\ga, \gd_{\ga + 1})$. Writing
$\FP_\ga = \FQ_0 \ast \dot \FQ_1$, where
the least cardinal in the support of
$\dot \FQ_1$ is above $\gk$, we know by
the definition of $\FP_\ga$ that
$V^{\FQ_0}_0 \models ``\gk$ contains a
non-reflecting stationary set of ordinals
of cofinality $\gd_\ga$ and therefore isn't
weakly compact and $\FQ_1$ is
${<} \gd$-strategically closed for
$\gd$ the least $V$-supercompact
cardinal above $\gk$''. Hence,
$V^{\FQ_0 \ast \dot \FQ_1}_0 = V^{\FP_\ga}_0 \models
``\gk$ contains a non-reflecting stationary
set of ordinals of cofinality $\gd_\ga$
and therefore isn't weakly compact''.
Write
$\FP^1 = \FP_{> \ga} \times \FP_\ga \times
\FP_{< \ga}$, where $\FP_{> \ga}$ is the
Easton support product
$\prod_{\gb \in (\ga, \Omega)} \FP_\gb$
and $\FP_{< \ga}$ is the Easton support product
$\prod_{\gb < \ga} \FP_\gb$.
By definition, in $V_0$, $\FP_{> \ga}$ is
$\gd_{\ga + 1}$-strategically closed and
$\card{\FP_{< \ga}} < \gk$. Thus,
$V^{\FP_{> \ga} \times \FP_\ga}_0 \models
``\gk$ contains a non-reflecting stationary
set of ordinals of cofinality $\gd_\ga$ and
therefore isn't weakly compact'', and by the
L\'evy-Solovay results \cite{LS},
$V^{\FP_{> \ga} \times \FP_\ga \times \FP_{< \ga}}_0 =
V^{\FP^1}_0 = V^\FP \models
``\gk$ isn't weakly compact (and hence isn't
supercompact)''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP = V^{\FP^1}_0 \models
``$If $\gk \in {\mathfrak K}$, then
$\gk$ is a limit of (non-supercompact)
strongly compact cardinals''.
\end{lemma}
\begin{proof}
Fix $\gk \in {\mathfrak K}$.
Since $\gk$ is a supercompact limit of
supercompact cardinals in $V_0$ (as well
as in $V$), by Theorem 1 of \cite{A80},
for any $\gg < \gk$, we can find a
non-supercompact measurable limit of
supercompact cardinals $\gd \in (\gg, \gk)$.
For such a $\gd$,
let $\ga$ be so that $\gd = \gd_\ga$.
By Menas' theorem of \cite{Me}, which essentially says
that for any $\gg$, the least measurable limit of
either supercompact or strongly compact cardinals
above $\gg$ is a non-supercompact strongly compact
cardinal, we know that $\gd_{\ga + 1}$ isn't
supercompact, i.e.,
$\gd_{\ga + 1} \in (\gd_\ga, \gk)$.
The proof of Lemma \ref{l3} will thus be complete
once we have shown that
$V^{\FP^1}_0 \models ``\gd_{\ga + 1}$ is
strongly compact''.
As in Lemma \ref{l2}, in $V_0$, write
$\FP^1 = \FP_{> \ga} \times \FP_\ga \times \FP_{< \ga}$.
Since $V_0 \models ``\FP_{> \ga}$ is
$\gd_{\ga + 1}$-directed closed and
$\gd_{\ga + 1}$ is a measurable limit of
indestructible supercompact cardinals'',
$V^{\FP_{> \ga}}_0 \models ``\gd_{\ga + 1}$
is a measurable limit of supercompact cardinals''.
And, by Lemma 2 of \cite{A97}, for any cardinal
$\gg > \gd_{\ga + 1}$, there is a strongly
compact ultrafilter ${\cal U}$ over
$P_{\gd_\ga + 1}(\gg)$ so that for
$j_{\cal U} : V^{\FP_{> \ga}}_0 \to M$ the
associated $\gg$ strongly compact embedding and
$g$ the function representing
$\gd_{\ga + 1}$ in $M$,
$\{p \in P_{\gd_\ga + 1}(\gg) : f'(g(p)) > |p|\} \in
{\cal U}$, where
$f' : \gd_{\ga + 1} \to \gd_{\ga + 1}$,
$f' \in V^{\FP_{> \ga}}_0$ is the
function defined by
$f'(\gb)$ = The least strongly compact cardinal above
$\gb$.
Therefore, if $\gg > \gd_{\ga + 1}$ is an arbitrary
cardinal and
$\gl = |2^{{[\gg]}^{< \gd_{\ga + 1}}}|$,
we can use the arguments given in Lemma 4 of \cite{A97}
to show that for
$j_{\cal U}(\FP_\ga) = \FP_\ga \ast \dot \FS$, if
$G_0$ is $V^{\FP_{> \ga}}_0$-generic over $\FP_\ga$ and
$G_1$ is $V^{\FP_{> \ga}}_0[G_0]$-generic over $\FS$,
$j_{\cal U} : V^{\FP_{> \ga}}_0 \to M$ extends in
$V^{\FP_{> \ga}}_0[G_0][G_1]$ to
$j : V^{\FP_{> \ga}}_0[G_0] \to M[G_0][G_1]$.
Consequently, there is a strongly compact ultrafilter
$\mu$ over
${(P_{\gd_{\ga + 1}}(\gg))}^{V^{\FP_{> \ga}}_0[G_0]}$
present in
$V^{\FP_{> \ga}}_0[G_0][G_1]$, and by
the choice of $\gl$ and arguments found
in Lemma 4 of \cite{A97},
$\mu \in V^{\FP_{> \ga}}_0[G_0]$.
Since $\gg$ was arbitrary, we thus know that
$V^{\FP_{> \ga} \times \FP_\ga}_0 \models
``\gd_{\ga + 1}$ is strongly compact''.
Hence, since
$V_0 \models ``\card{\FP_{< \ga}} < \gd_{\ga + 1}$'',
the results of \cite{LS} imply that
$V^{\FP_{> \ga} \times \FP_\ga \times \FP_{< \ga}}_0 =
V^{\FP^1}_0 = V^\FP \models ``\gd_{\ga + 1}$ is
strongly compact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l1} - \ref{l3} complete the proof of
Theorem \ref{t1}.
\end{proof}
We conclude Section \ref{s2} by observing it is
not true in the model $V^\FP$
constructed above that
every measurable cardinal is
strongly compact.
To see this, by the last paragraph of the
proof of Lemma \ref{l2}, if
$\gd \in (\gd_\ga, \gd_{\ga + 1})$ is measurable, then
$V^{\FP_{> \ga} \times \FP_\ga}_0 \models ``$There
is a $V$-supercompact cardinal $\gk > \gd$
containing a non-reflecting stationary set of
ordinals of cofinality $\gd_\ga$''.
By Theorem 4.8 of \cite{SRK} and the succeeding remarks,
$V^{\FP_{> \ga} \times \FP_\ga}_0 \models
``\gd$ isn't strongly compact'', so
the results of \cite{LS} imply that
$V^{\FP_{> \ga} \times \FP_\ga \times \FP_{< \ga}}_0 =
V^{\FP_1}_0 = V^\FP \models
``\gd$ isn't strongly compact''.
Indeed, the construction of a model from
any hypotheses in which the first $\omega$
measurables are strongly compact is the
main question left open in \cite{KM}.
However, in Section \ref{s3},
the model constructed will be
so that every measurable cardinal has
some non-trivial degree of strong compactness.
\section{The Proof of Theorem \ref{t2}}\label{s3}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V$ and ${\mathfrak K}$
be as in the hypotheses for Theorem \ref{t2}.
Without loss of generality, by first doing a
preliminary forcing if necessary, we may assume
that $V \models {\rm GCH}$ as well.
Let ${\mathfrak D} = \la \gd_\ga : \ga < \Omega \ra$
enumerate in increasing order
$\{\gd : \gd$ is a measurable cardinal which
isn't $2^\gd$ supercompact$\}$, where $\Omega$
is the appropriate ordinal if the collection of all
measurable cardinals forms a set
or is the class of all ordinals otherwise.
The partial ordering $\FP$ used in the
proof of Theorem \ref{t2} is the Easton support
iteration which first adds a Cohen subset to
$\omega$ and then
adds, to every
$\gd \in {\mathfrak D}$, a non-reflecting stationary set
of ordinals using the partial ordering
$\FP(\gd)$ described in Section \ref{s1}.
Note that no $\gk \in {\mathfrak K}$
will be an element of the support of $\FP$.
Also, as before, standard arguments concerning
class forcing combined with the definition of $\FP$
yield that $V^\FP \models {\rm ZFC}$.
\begin{lemma}\label{l4}
$V^\FP \models ``$The only supercompact cardinals
are the elements of ${\mathfrak K}$''.
\end{lemma}
\begin{proof}
As in the first paragraphs of the proofs of
Theorem \ref{t1} and Lemma \ref{l2}, write
$\FP = \FP' \ast \dot \FP''$, where
$|\FP'| = \omega$ and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically
closed''.
Hence, again as earlier, $\FP$
``admits a gap at $\ha_1$'', so by the results of
\cite{H1}, \cite{H2}, and \cite{H3}, any
supercompact cardinal in $V^\FP$ had to have been
supercompact in $V$.
This means the proof of Lemma \ref{l4} will be
complete once we have shown that
$V^\FP \models ``$If $\gk \in {\mathfrak K}$, then
$\gk$ is supercompact''.
To do this, fix $\gk \in {\mathfrak K}$, and let
$\gl > \gk$ be an arbitrary successor cardinal. Let
$\gg = |2^{{[\gl]}^{< \gk}}|$, and fix
$j : V \to M$ an elementary embedding witnessing the
$\gg$ supercompactness of $\gk$.
Write
$\FP = \FP^0 \ast \dot \FP^1 \ast \dot \FP^2$, where
$\FP^0$ is the portion of $\FP$ defined through stage
$\gk$, $\dot \FP^1$ is a term for the portion of $\FP$
defined between stages $\gk$ and $\gl$, and
$\dot \FP^2$ is a term for the rest of $\FP$.
By the definition of $\FP$, it will be the case that
$\forces_{\FP^0 \ast \dot \FP^1} ``\dot \FP^2$ is
$\gg$-strategically closed''.
Thus, it will suffice to show that
$V^{\FP^0 \ast \dot \FP^1} \models ``\gk$ is $\gl$
supercompact''.
If this is not the case, then let
$p = \la p_0, \dot p_1 \ra \in \FP^0 \ast \dot \FP^1$
be so that
$p \forces ``\gk$ isn't $\gl$ supercompact''.
By using Lemma \ref{i2} if necessary to find
the necessary terms to extend
coordinatewise, we assume without loss of generality
that each non-trivial coordinate of
$p_1$ is a term for a condition in the appropriate
$\FP(\gd / \gk)$.
Let $G_0$ be $V$-generic over $\FP^0$ so that
$p_0 \in G_0$.
Working in $V[G_0]$ and once again using
Lemma \ref{i2}, let $\FP^3$ be the Easton
support iteration of partial orderings which,
for every measurable
$\gd \in (\gk, \gl)$ which isn't
$2^\gd$ supercompact
(in either $V$ or $V[G_0]$), add non-reflecting
stationary sets of ordinals using
$\FP(\gd / \gk)$.
%Again without loss of generality, we assume that
%every element of $\FP^3$ extends $p_1$.
Note now that if $G_1$ is $V[G_0]$-generic over
$\FP^3$ and $p_1 \in G_1$, then $G_1$ must also
generate a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
To see this, it clearly suffices to show that
$G_1$ meets all dense open subsets of $\FP^1$
above $p_1$.
If $D$ is such a set,
then let $D_1 = \{q \in \FP^3 : q$ extends some
element of $D\}$.
$D_1$ is clearly open.
If $q \in \FP^3$, then $q \in \FP^1$,
so by density, there is $q' \ge q$, $q' \in D$.
By using Lemma \ref{i2} if necessary to
find a term which is forced to extend
each term denoting a
non-trivial coordinate of $q'$ to
a term for an
element of the appropriate $\FP(\gd / \gk)$,
we obtain $q'' \ge q' \ge q$,
$q'' \in D_1$.
Thus, $G_1$ meets $D_1$ and hence meets $D$, so
$G_1$ generates a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
By the definition of
$\FP$ and the closure properties of $M$,
$j(\FP^0 \ast \dot \FP^1) =
\FP^0 \ast \dot \FP^1 \ast \dot \FQ \ast \dot \FR$,
where $\dot \FQ$ is a term for the portion of
$j(\FP^0 \ast \dot \FP^1)$ defined in $M$ between
stages $\gl$ and $j(\gk)$, and
$\dot \FR$ is a term for $j(\dot \FP^1)$, i.e.,
the portion of
$j(\FP^0 \ast \dot \FP^1)$ defined in $M$ between
stages $j(\gk)$ and $j(\gl)$.
If $G_1$ is $V[G_0]$-generic over $\FP^3$ and
$p_1 \in G_1$, then
by the preceding paragraph,
$G_1$ generates a $V[G_0]$-generic
filter $G^*_1$ over $\FP^1$.
We can therefore take $G_2$ as a
$V[G_0][G^*_1]$-generic object over $\FQ$ and
use the usual Easton arguments to infer that
$M[G_0][G^*_1][G_2]$ remains
$\gg$ closed with respect to
$V[G_0][G^*_1][G_2]$ and that $j$ extends
in $V[G_0][G^*_1][G_2]$ to
$j : V[G_0] \to M[G_0][G^*_1][G_2]$.
Further, since
$G_1 \subseteq G^*_1$ and $G_1$ is $V[G_0]$-generic over
a partial ordering ($\FP^3$) that is
$\gk$-directed closed in $V[G_0]$,
$j''G_1$ generates in $V[G_0][G^*_1][G_2]$ a
compatible set of conditions of cardinality
smaller than $\gg < j(\gk)$ in a partial ordering
($j(\FP^3)$) that is $j(\gk)$-directed
closed in $M[G_0][G^*_1][G_2]$.
Therefore, by the fact
$M[G_0][G^*_1][G_2]$ is
$\gg$ closed with respect to
$V[G_0][G^*_1][G_2]$, we can let
$r$ be a master condition for
$j''G_1$ and take $G_3$ to be a
$V[G_0][G^*_1][G_2]$-generic object over
$j(\FP^3)$ containing $r$.
By elementarity, it will be the case that
$G_3$ generates a $V[G_0][G^*_1][G_2]$-generic
object $G^*_3$ over
$\FR = j(\FP^1)$.
As usual, we will then have that in
$V[G_0][G^*_1][G_2][G^*_3]$, $j$ extends to
$j : V[G_0][G^*_1] \to M[G_0][G^*_1][G_2][G^*_3]$, so
$\gk$ is $\gl$ supercompact in $V[G_0][G^*_1][G_2][G^*_3]$.
Since $\FQ \ast \dot \FR$ is $\gg$-directed closed in
$V[G_0][G^*_1]$, it will be the case that
$\gk$ is $\gl$ supercompact in $V[G_0][G^*_1]$.
This, however, contradicts that
$p = \la p_0, p_1 \ra \in G_0 \ast G^*_1$ and
$p \forces ``\gk$ isn't $\gl$ supercompact''.
This contradiction completes the proof of
Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$For every cardinal
$\gd$, $\gd$ is measurable iff
$\gd$ is $2^\gd$ strongly compact''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd$ is measurable''.
By Hamkins' work of \cite{H1}, \cite{H2},
and \cite{H3}, it must be true that
$V \models ``\gd$ is measurable''.
If $V \models ``\gd$ isn't $2^\gd$ supercompact'', write
$\FP = \FP^0 \ast \dot \FP^1$, where
$V^{\FP^0} \models ``\gd$ contains a non-reflecting
stationary set of ordinals
and hence isn't weakly compact''.
Since $\forces_{\FP^0} ``\dot \FP^1$ is
${<}\gg$-strategically closed for $\gg$ the
least $V$ or $V^{\FP^0}$-measurable cardinal
above $\gd$'',
$V^{\FP^0 \ast \dot \FP^1} = V^\FP \models
``\gd$ contains a non-reflecting stationary
set of ordinals
and hence isn't measurable or weakly compact''.
We therefore know that
$V \models ``\gd$ is $2^\gd$ supercompact''.
We can then write
$\FP = \FP^0 \ast \dot \FP^1$, where the support of
$\FP^0$ consists of all measurable cardinals
$\gg < \gd$ which aren't $2^\gg$ supercompact.
Since as in the preceding paragraph,
$\forces_{\FP^0} ``\dot \FP^1$ is ${<}\gg$-strategically
closed for $\gg$ the least $V$ or
$V^{\FP^0}$-measurable cardinal above $\gd$'',
$V^{\FP^0 \ast \dot \FP^1} = V^\FP \models
``\gd$ is $2^\gd$ strongly compact'' iff
$V^{\FP^0} \models ``\gd$ is $2^\gd$
strongly compact''.
To show $V^{\FP^0} \models ``\gd$ is
$2^\gd$ strongly compact'', we use an
unpublished argument of Magidor.
Although the essentials of this argument can
be found in \cite{AC1} and \cite{AC2},
for completeness and for the benefit of
readers, we give the argument here as well.
Let
$\gl = 2^\gd = \gd^+$, and let
$k_1 : V \to M$ be an embedding witnessing the $\gl$
supercompactness of $\gd$ so that
$M \models ``\gd$ isn't $\gl$ supercompact''.
Since $M \models ``\gd$ is measurable'',
we may choose a normal ultrafilter of
Mitchell order $0$ over $\gd$ so that
$k_2 : M \to N$ is an embedding witnessing the
measurability of $\gd$ definable in $M$ with
$N \models ``\gd$ isn't measurable''.
It is the case that if
$k : V \to N$ is an elementary embedding with
critical point $\gd$
and for any $x \subseteq N$ with
$|x| \le \gl$, there is some $y \in N$
so that $x \subseteq y$ and
$N \models ``|y| < k(\gd)$'',
then $k$ witnesses the $\gl$
strong compactness of $\gd$.
Using this fact,
it is easily verifiable that
$j = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl$ strong compactness of $\gd$.
We show that $j$ extends to
$j : V^{\FP^{0}} \to N^{j(\FP^0)}$.
Since this extended embedding witnesses
the $\gl$ strong compactness of $\gd$ in
$V^{\FP^0}$, this proves Lemma \ref{l5}.
To do this, write
$j(\FP^0)$ as
$\FP^0 \ast \dot \FQ^0 \ast \dot \FR^0$, where
$\dot \FQ^0$ is a term for the portion of
$j(\FP^0)$ between $\gd$ and $k_2(\gd)$ and
$\dot \FR^0$ is a term for the rest of
$j(\FP^0)$, i.e., the part above $k_2(\gd)$.
Note that since
$N \models ``\gd$ isn't measurable'',
$\gd \not\in {\rm support}(\dot \FQ^0)$.
Thus, the support of $\dot \FQ^0$
is composed of all $N$-measurable cardinals
$\gg$ in the interval
$(\gd, k_2(\gd)]$
which aren't $2^\gg$ supercompact (so
$k_2(\gd) \in {\rm support}(\dot \FQ^0)$ since
$M \models ``\gd$ is measurable but $\gd$ isn't
$2^\gd$ supercompact''),
and the support of $\dot \FR^0$ is composed of all
$N$-measurable cardinals $\gg$ in the interval
$(k_2(\gd), k_2(k_1(\gd)))$ which aren't
$2^\gg$ supercompact.
Let $G_0$ be $V$-generic over $\FP^0$.
We construct in $V[G_0]$ an
$N[G_0]$-generic object $G_1$ over
$\FQ^0$ and an
$N[G_0][G_1]$-generic object $G_2$ over
$\FR^0$. Since $\FP^0$ is an
Easton support iteration of length $\gd$,
a direct limit is taken at stage $\gd$,
and no forcing is done at stage $\gd$,
the construction of $G_1$ and $G_2$
automatically guarantees that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
This means that
$j : V \to N$ extends to
$j : V[G_0] \to N[G_0][G_1][G_2]$.
To build $G_1$, note that since $k_2$
can be assumed to be generated by an
ultrafilter ${\cal U}$ over $\gd$ and
since in both $V$ and $M$, $2^\gd = \gd^+$,
$|k_2(\gd^+)| = |k_2(2^\gd)| =
|\{ f : f : \gd \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gd| = \gd^+$. Thus, as
$N[G_0] \models ``|\wp(\FQ^0)| = k_2(2^\gd)$'', we can let
$\la D_\ga : \ga < \gd^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of
$\FQ^0$ found in $N[G_0]$.
For the purpose of the construction of
$G_1$ to be given below, we further assume
that for every
dense open subset $D \subseteq \FQ^0$
present in $N[G_0]$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$.
Since the $\gd$ closure of $N$ with respect to either
$M$ or $V$ implies the least element of the support of
$\FQ^0$ is $> \gd^+$, the definition of
$\FQ^0$ as the Easton support iteration which adds
a non-reflecting stationary set of ordinals
to each $N[G_0]$-measurable
cardinal $\gg$ in the interval
$(\gd, k_2(\gd)]$
which isn't $2^\gg$ supercompact implies that
$N[G_0] \models ``\FQ^0$ is
${\prec}\gd^+$-strategically closed''.
By the fact the standard arguments show that
forcing with the $\gd$-c.c$.$ partial ordering
$\FP^0$ preserves that $N[G_0]$ remains
$\gd$-closed with respect to either
$M[G_0]$ or $V[G_0]$,
$\FQ^0$ is ${\prec}\gd^+$-strategically closed
in both $M[G_0]$ and $V[G_0]$.
We can now construct $G_1$ in either
$M[G_0]$ or $V[G_0]$ as follows.
Players I and II play a game of
length $\gd^+$.
The initial pair of moves is generated by
player II choosing the trivial
condition $q_0$ and player I responding by
choosing $q_1 \in D_1$. Then,
at an even stage $\ga + 2$,
%since $\FQ^0$ is
%${\prec} \gd^+$-strategically closed,
player II picks $q_{\ga + 2} \ge
q_{\ga + 1}$ by using some fixed
strategy ${\cal S}$, where
$q_{\ga + 1}$ was chosen by player I
to be so that
$q_{\ga + 1} \in D_{\ga + 1}$ and
$q_{\ga + 1} \ge q_\ga$.
%$p_\ga \in D_\ga$ extending
%$\sup(\la q_\gb : \gb < \ga \ra)$
%(initially, $q_{-1}$ is the empty condition)
%and player II responds by picking
%$q_\ga \ge p_\ga$ (so $q_\ga \in D_\ga$).
If $\ga$ is a limit ordinal, player II uses
${\cal S}$ to pick
$q_\ga$ extending each $q_\gb$ for $\gb < \ga$.
%$\sup(\la q_\gb : \gb < \ga \ra)$.
By the ${\prec}\gd^+$-strategic closure of
$\FQ^0$ in both $M[G_0]$ and $V[G_0]$,
%player II has a winning strategy for this
%game, so
the sequence $\la q_\ga : \ga < \gd^+ \ra$
as just described exists. By construction,
%can be taken
%as an increasing sequence of conditions with
%$q_\ga \in D_\ga$ for $\ga < \gd^+$. Clearly,
$G_1 = \{p \in \FQ^0 : \exists \ga <
\gd^+ [q_\ga \ge p]\}$ is our
$N[G_0]$-generic object over $\FQ^0$.
It remains to construct in $V[G_0]$ the
desired $N[G_0][G_1]$-generic object
$G_2$ over $\FR^0$.
To do this, we first note that as
$M \models ``\gd$ is measurable but $\gd$
isn't $2^\gd$ supercompact'',
we can write
$k_1(\FP^0)$ as
$\FP^0 \ast \dot \FS^0 \ast \dot \FT^0$, where
$\forces_{\FP^0} ``\dot \FS^0 = \dot \FP(\gd)$'', and
$\dot \FT^0$ is a term for the rest of
$k_1(\FP^0)$.
Note now that
$M \models ``$No cardinal
$\gg \in (\gd, \gl]$ is measurable''.
Thus, the support of
$\dot \FT^0$ is composed of all
$M$-measurable cardinals $\gg$
in the interval $(\gl, k_1(\gd))$
which aren't $2^\gg$ supercompact,
which implies that in $M$,
$\forces_{\FP^0 \ast \dot \FS^0}
``\dot \FT^0$ is ${\prec}\gl^+$-strategically
closed''. Further, since
$V \models {\rm GCH}$ and $\gl$ is regular,
$|{[\gl]}^{< \gd}| = \gl$ and $2^\gl = \gl^+$.
Therefore, as $k_1$ can be assumed to be
generated by an ultrafilter ${\cal U}$ over
$P_\gd(\gl)$,
$|k_1(\gl)| =
|k_1(\gd^+)| = |k_1(2^\gd)| =
|2^{k_1(\gd)}| =
|\{ f : f : P_\gd(\gl) \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gl| = |{[\gl]}^\gl| = \gl^+$.
Work until otherwise specified in $M$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{C}, Section 1.2.5, page 8) associated with
$\dot \FT^0$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^0 \ast \dot \FS^0$ and
$\forces_{\FP^0 \ast \dot \FS^0} ``\tau \in
\dot \FT^0$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP^0 \ast \dot \FS^0} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT^0$ to be a set,
it is possible to restrict the terms
appearing in it to a sufficiently large
set-sized collection, with the additional
crucial property that any term $\tau$
forced to be in $\dot \FT^0$ is also forced
to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M \models ``|\FT^*| = k_1(\gd)$''.
Clearly, $\FT^* \in M$. Also, since
$\forces_{\FP^0 \ast \dot \FS^0} ``\dot \FT^0$ is
${\prec}\gl^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl^+$-strategically closed in $M$ and, since
$M^\gl \subseteq M$, in $V$ as well.
Observe that
$M \models ``k_1(\gd)$ is measurable and
$|\FP^0 \ast \dot \FS^0| < k_1(\gd)$'' and
$\forces_{\FP^0 \ast \dot \FS^0} ``\dot \FT^0$ is
an Easton support iteration of length $k_1(\gd)$ and
$|\dot \FT^0| = k_1(\gd)$''.
We can thus let $\dot f$ be a term so that
$\forces_{\FP^0 \ast \dot \FS^0}
``\dot f : k_1(\gd) \to \dot \FT^0$ is
a bijection''.
Since
$M \models ``|\FP^0 \ast \dot \FS^0| < k_1(\gd)$'',
for each $\ga < k_1(\gd)$, let
$S_\ga = \{ r^\ga_\gb : \gb < \eta^\ga < k_1(\gd) \}$
be a maximal incompatible set of elements of
$\FP^0 \ast \dot \FS^0$ so that for some term
$\tau^\ga_\gb$,
$r^\ga_\gb \forces ``\tau^\ga_\gb = \dot f(\ga)$''.
Define $T_\ga = \{\tau^\ga_\gb : \gb < \eta^\ga \}$ and
$T = \bigcup_{\ga < k_1(\gd)} T_\ga$. Clearly,
$|T| = k_1(\gd)$, so we can let
$\la \tau_\ga : \ga < k_1(\gd) \ra$ enumerate the
members of $T$.
%Each sequence
%$\la \tau^\ga_\gb : \gb < \eta^\ga < k_1(\gd) \ra$
%can be used to define a sequence of terms
$\la \tau_\ga : \ga < k_1(\gd) \ra$ is so that if
$\forces_{\FP^0 \ast \dot \FS^0} ``\tau \in \dot \FT^0$'',
then for some $\ga < k_1(\gd)$,
$\forces_{\FP^0 \ast \dot \FS^0} ``\tau = \tau_\ga$''.
Therefore, we can restrict the set of terms we choose so that
we can assume that in $M$,
$|\FT^*| = k_1(\gd)$. Since
$M \models ``2^{k_1(\gd)} = {(k_1(\gd))}^+ =
k_1(\gd^+) = k_1(\gl)$'',
this means we can let
$\la D_\ga : \ga < \gl^+ \ra$
enumerate in $V$ the dense open subsets of $\FT^*$
found in $M$,
so that as before, for every
dense open subset
$D \subseteq \FT^*$ present in $M$,
for some odd ordinal $\gg + 1$,
$D = D_{\gg + 1}$,
and argue as we did earlier to construct in
$V$ an $M$-generic object $H_2$ over $\FT^*$.
Note now that since $N$ can be assumed to be given
by an ultrapower of $M$ via a normal ultrafilter
${\cal U} \in M$ over $\gd$,
Fact 2 of Section 1.2.2 of \cite{C} tells us that
$k_2 '' H_2$ generates an $N$-generic object
$G^*_2$ over $k_2(\FT^*)$. By elementariness,
$k_2(\FT^*)$ is the term forcing in $N$
defined with respect to
$k_2(k_1(\FP_\gd)_{\gd + 1}) =
\FP^0 \ast \dot \FQ^0$.
Therefore, since
$j(\FP^0) = k_2(k_1(\FP^0)) =
\FP^0 \ast \dot \FQ^0 \ast
\dot \FR^0$,
$G^*_2$ is $N$-generic over
$k_2(\FT^*)$, and $G_0 \ast G_1$ is
$k_2(\FP^0 \ast \dot \FS^0)$-generic over
$N$, Fact 1 of Section 1.2.5 of \cite{C}
tells us that for
$G_2 = \{i_{G_0 \ast G_1}(\tau) : \tau \in
G^*_2\}$, $G_2$ is $N[G_0][G_1]$-generic over
$\FR^0$.
Thus, in $V[G_0]$,
$j : V \to N$ extends to
$j : V[G_0] \to N[G_0][G_1][G_2]$.
As $V^{\FP^0} \models ``\gd$ is $2^\gd$
strongly compact'', this completes the
proof of Lemma \ref{l5}.
\end{proof}
Lemmas \ref{l4} and \ref{l5} complete the
proof of Theorem \ref{t2}.
\end{proof}
We conclude Section \ref{s3}
and this paper with two remarks.
First, we observe we can guarantee
that in Theorem \ref{t2}, there are measurable
cardinals $\gd$ which aren't strongly compact.
If we force over models such as those given in
\cite{A98}, \cite{AS97a}, or \cite{KM} in which the classes
of strongly compact and supercompact cardinals
coincide precisely except at measurable
limit points, then there will be many measurable
cardinals, such as those below the least
supercompact cardinal, which won't be
strongly compact. If we force
using the partial ordering $\FP$
as just described, then by the work of
\cite{H1}, \cite{H2}, and \cite{H3},
no new strongly compact cardinals are created.
Thus, any cardinal which wasn't strongly compact
in $V$ isn't strongly compact in $V^\FP$.
Second, we note
that in Theorem \ref{t2}, it is possible
to have each measurable cardinal
$\gd$ exhibit a greater degree of
strong compactness than just $2^\gd$
strong compactness.
The proof given in Lemma \ref{l5}
will work for any regular value of
$\gl$ below the least $V$-measurable
cardinal above $\gd$.
Further, the nature of the partial
ordering $\FP$ used in the proof of
Theorem \ref{t2} ensures that the
cardinal structure between any
$V$-measurable cardinal $\gd$ and the
least $V$-measurable cardinal $\gd'$ above it
remains the same in $V^\FP$, regardless if
the measurability of either $\gd$ or
$\gd'$ is destroyed by $\FP$.
Thus, e.g., it is possible to construct
a model in which the ground model and
the generic extension have the same class
${\mathfrak K}$ of supercompact cardinals
and $\gd$ is measurable iff
$\gd$ is $\gl$ strongly compact for
$\gl$ assuming values such as
$2^{2^\gd}$, $\gd^{+ 17}$, the least
inaccessible cardinal above $\gd$,
the least weakly compact cardinal above $\gd$,
etc.
In certain of these models, it may be
necessary to assume additional large
cardinal hypotheses, e.g.,
if there is only one supercompact
cardinal $\gk$ in the universe and we wish
every measurable cardinal $\gd$ to be
$\gl$ strongly compact for $\gl$ the
least inaccessible cardinal above $\gd$,
it will be necessary to assume the
existence of an inaccessible cardinal
above $\gk$.
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Like It'',
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Generalizing the L\'evy-Solovay Theorem'',
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%Academic Press, New York and San
%Francisco, 1978.
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%Higher Infinite}, Springer-Verlag,
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\end{document}
Suppose $V \models ``$ZFC + $\gk$ is
supercompact + No cardinal $\gl > \gk$
is measurable".
There is then a partial ordering
$\FP \in V$, $|\FP| = \gk$ so that
$V^\FP \models ``$ZFC + $\gk$ is
supercompact +
No cardinal $\gl > \gk$ is measurable +
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH + $\gk$ is huge''.
There is then a partial ordering
$\FP \in V$, $|\FP| = \gk$ so that
$V^\FP_\gk \models ``$ZFC + There is a
proper class of supercompact limits of
supercompact cardinals + For every
cardinal $\gd$, $\gd$ is measurable iff
$\gd$ is $2^\gd$ strongly compact''.
\end{theorem}
\setlength{\parindent}{0pt}
In Theorem \ref{t3},
$V^\FP_\gk$ is $V_\gk$ of $V^\FP$.
\setlength{\parindent}{1.5em}
Suppose that
$\gk < \gl$ are regular cardinals.
A partial ordering $\FP$ used
during the course of this paper is the
partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically, $\FP$ is
defined as
$\{ p$ : 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
$\gb > \gk$ 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 $\FP$ (see
\cite{Bu} or \cite{KM}), in $V[G]$,
%if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\cup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced,
and the
bounded subsets of $\gl$ are the same as those in $V$.
%and cardinals, cofinalities, and GCH
%have been preserved.
It is also virtually immediate that $\FP$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu} or \cite{KM}) that
$\FP$
is ${\prec}\gl$-strategically closed.
The proof of Lemma \ref{l4} is similar to the proof of
Lemma \ref{l1}.
Fix $\gl \ge 2^\gk$ a cardinal, and let
$\gg = 2^{[\gl]^{< \gk}}$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gg$ supercompactness of $\gk$.
Since $M \models ``\gk$ is
$2^\gk$ supercompact + No cardinal $\sigma > \gk$ is
measurable'', we can write
$j(\FP) = \FP \ast \dot \FQ$, where the least
ordinal in the support of $\dot \FQ$ is
larger than $\gg$.
Thus, if $G$ is $V$-generic over $\FP$ and
$H$ is $V[G]$-generic over $\FQ$,
since $\FP$ is an Easton support iteration,
$j$ extends in $V[G][H]$ to
$\ov j : V[G] \to V[G \ast H]$ via the definition
$\ov j(i_G(\tau)) = i_{G \ast H}(j(\tau))$.
Also, as $M[G]$ remains $\gg$ closed
with respect to $V[G]$,
the definition of $\FP$ ensures that
$\FQ$ is $\gg$-strategically closed in both
$M[G]$ and $V[G]$.
This means the supercompact ultrafilter
${\cal U}$ over ${(P_\gk(\gl))}^{V[G]}$
given by $x \in {\cal U}$ iff
$\la j(\ga) : \ga < \gl \ra \in \ov j(x)$
is so that ${\cal U} \in V[G]$.
This completes the proof of Lemma \ref{l4}.
\section{The Proof of Theorem \ref{t3}}\label{s4}
We turn now to the proof of Theorem \ref{t3}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is huge''.
Let $j : V \to M$ be an elementary embedding
witnessing the hugeness of $\gk$, with
$j(\gk) = \gl$.
In analogy to the proof of Theorem \ref{t2}, let
${\mathfrak D} = \la \gd_\ga : \ga < \gl \ra$
enumerate in increasing order
$\{\gg < \gl : \gg$ is a measurable cardinal which
isn't $2^\gg$ supercompact$\}$.
The partial ordering $\FP$ used in the proof of
Theorem \ref{t3} is the Easton support iteration
which adds, to every element
${\gg \in \mathfrak D}$, a non-reflecting stationary set
of ordinals of cofinality
$\sigma_\gg = (\sup(\{\gd < \gg : \gd$ is $\gl$
supercompact$\}))^+$, where if $\gg$ is
below the least $\gl$ supercompact cardinal,
then $\sigma_\gg = \omega$.
We note that the proof of Lemma \ref{l5},
which remains valid regardless of the
cofinality of the ordinals in the
non-reflecting stationary sets
added by $\FP$,
shows that
$V^\FP_\gk \models ``\gd$ is measurable iff
$\gd$ is $2^\gd$ strongly compact''.
Further, if $V^\FP \models ``\gk$ is huge'',
then by Exercise 24.12, page 334 of \cite{K},
$V^\FP_\gk \models ``$There is a proper
class of supercompact limits of
supercompact cardinals''.
Thus, the proof of Theorem \ref{t3} will be
complete once we have shown the following.
\begin{lemma}\label{l6}
$V^\FP \models ``\gk$ is huge''.
\end{lemma}
\begin{proof}
We begin by observing that
for $\gd < \gl$,
$V \models ``\gd$ is $\gl$ supercompact'' iff
$M \models ``\gd$ is $\gl$ supercompact''.
For the forward direction, we note that since
$V \models ``\gd$ is $\gl$ supercompact'',
$V \models ``\gd$ is ${<}\gl$ supercompact'',
so by the fact $M^\gl \subseteq M$,
$M \models ``\gd$ is ${<}\gl$ supercompact'' as well.
By elementariness,
$M \models ``\gl$ is measurable'', so
$M \models ``\gd$ is $\gl$ supercompact''.
The reverse direction follows since
$M^\gl \subseteq M$.
Further, by elementariness, since
$M \models ``\gl$ is $j(\gl)$ supercompact'',
if $\gd < \gl$ is so that
$M \models ``\gd$ is $\gl$ supercompact'',
then Magidor's theorem from \cite{Ma2}
implies that
$M \models ``\gd$ is $j(\gl)$ supercompact''.
Also, since $j(\gl) > \gl$, it clearly is
the case that if
$M \models ``\gd$ is $j(\gl)$ supercompact'', then
$M \models ``\gd$ is $\gl$ supercompact''.
This means that $\FP_\gk$, the partial
ordering $\FP$ through stage $\gk$, is
an initial segment of $j(\FP)$, and $\FP$ is an
initial segment of $j(\FP)$.
Let $G$ be $V$-generic over $\FP$,
where $G = G_0 \ast G_1$ is so that
$G_0$ is $V$-generic over $\FP_\gk$,
and $G_1$ is $V$-generic over
$\FP^\gk$, the portion of $\FP$
defined between stages $\gk$ and $\gl$.
Since $\FP_\gk$ is an Easton support
iteration and $\gk$ is measurable,
this means that
$j''G_0 \subseteq G_0 \ast G_1$.
Hence, in $V[G]$, $j$ extends to
$j_0 : V[G_0] \to M[G_0][G_1]$.
Further, as $\gl$ is measurable in both
$V$ and $M$ and $\FP$ is an Easton
support iteration, $\FP$ is $\gl$-c.c.
Thus, standard arguments show that
$M[G]$ remains $\gl$ closed with
respect to $V[G]$. Therefore,
working in $V[G]$, since
$A = \{j''p : p \in G_1\}$ is a compatible
set of conditions of size $\gl$
in $j(\FP^\gk)$,
a partial ordering which is
$\gl^+$-directed closed in both
$M[G]$ and $V[G]$, we can let $q$ be
a master condition for $A$.
Since we may assume that $j$ is generated
by a huge ultrafilter ${\cal U}$ over
$P^\gk(\gl)$, we can infer by GCH in $V$ that
$|j(\gl)| =
|\{f : f : P^\gk(\gl) \to \gl\}| =
|\{f : f : \gl \to \gl\}| = 2^\gl = \gl^+$.
And, since $j(\FP^\gk)$ is $j(\gl)$-c.c$.$
in $M[G]$, in $M[G]$,
$|\{{\cal A} : {\cal A}$ is a maximal antichain of
$j(\FP^\gk)\}| =
|{[j(\gl)]}^{< j(\gl)}| = j(\gl)$.
This means that in
$V[G]$, this set has cardinality $\gl^+$.
Thus, if
$\la {\cal A}_\ga : \ga < \gl^+ \ra$ is
an enumeration in $V[G]$ of these maximal antichains,
since $j(\FP^\gk)$ is $\gl^+$-directed closed in $V[G]$,
we can let $p_0$ be a common extension of $q$ and
the element of ${\cal A}_0$ with which it is
compatible, take $p_\gb$ for $\gb < \gl^+$ a
limit ordinal as an upper bound for
$\la p_\ga : \ga < \gb \ra$, and take
$p_{\ga + 1}$ for $\ga < \gl^+$ as a
common extension of $p_\ga$ and the element of
${\cal A}_{\ga}$ with which it is compatible.
(To avoid redundancy, we let $p_1 = p_0$.)
If we then define
$G_2 = \{p \in j(\FP^\gk) : p$ is extended
by an element of the sequence
$\la p_\ga : \ga < \gl^+ \ra\}$, then
$G_2$ is an $M[G]$-generic object over
$j(\FP^\gk)$ constructed in $V[G]$ so that
$j''G = j''(G_0 \ast G_1) \subseteq
G_0 \ast G_1 \ast G_2$.
Thus, in $V[G]$, $j$ extends to
$\ov j : V[G] \to M[G][G_2]$, i.e.,
$V[G] \models ``\gk$ is huge''.
This proves Lemma \ref{l6}.
\end{proof}
The proof of Lemma \ref{l6} completes the
proof of Theorem \ref{t3}.
\end{proof}
We conclude Section \ref{s4} by remarking
that as in Theorem \ref{t2}, it is
possible for each measurable cardinal
$\gd$ in $V^\FP_\gk$ to exhibit a greater
degree of strong compactness than just
$2^\gd$ strong compactness.
\section{Concluding Remarks}\label{s5}
In conclusion to this paper, we ask if
it is possible to weaken the hypotheses
used to prove Theorem \ref{t3}.
Although given the nature of the desired
model we feel such a weakening is possible,
to this point, we have not been able to
come up with a proof of Theorem \ref{t3}
from weaker hypotheses.
%In conclusion to this paper, we comment on
%some of the problems encountered when
%trying to prove Theorem \ref{t2}
%for more than one supercompact cardinal,
%i.e., when trying to construct a model
%with many supercompact cardinals in
%which every measurable cardinal has a
%certain degree of strong compactness.
The properties of $\FP(\gk)$
given in Lemmas \ref{i1} - \ref{i3} can all
be demonstrated by arguments found on
pages 435--437 of \cite{Bu}.
For completeness and comprehensibility,
we include proofs of each of these lemmas below.
We are now in essentially the same situation as in
Lemma 4 of \cite{A97},
so an application of this lemma yields
By the results of \cite{LS},
since $V \models ``|\FP| = \gk$'',
$V^{\FP} \models ``$No cardinal
$\gl > \gk$ is measurable''.
This observation, together with
such a situation, i.e., where each measurable
cardinal below $\gk$ has a degree of strong
compactness which is a certain large cardinal,
we will need to assume the existence of the
analogous large cardinal above $\gk$.
adds a
non-reflecting stationary set of ordinals of
cofinality $\omega$ to $\gd$'', and
In certain of these models, it may be
necessary to assume additional large
cardinal hypotheses, e.g.,
if there is only one supercompact
cardinal $\gk$ in the universe and we wish
every measurable cardinal $\gd$ to be
$\gl$ strongly compact for $\gl$ the
least inaccessible cardinal above $\gd$,
it will be necessary to assume the
existence of an inaccessible cardinal
above $\gk$.
${\mathfrak K}$ is a set, and there are
unboundedly many measurable cardinals in
the universe, it may be necessary to
assume the existence of unboundedly
many measurable cardinals
in the universe possessing
the appropriate degree of supercompactness.
$S_r \rest \gb$ is not stationary in $\gb$, and
We remark that the proof of Lemma \ref{i1}
just given actually shows that
$\FP(\gk)$ is ${\prec} \gk$-strategically closed.
and for any
$\gb < \gl$ of uncountable cofinality,
If $\FP$ is either a product or an iteration,
we will say that an
ordinal $\ga$ is in the support of $\FP$ if
forcing with $\FP$ adds a non-empty subset of $\ga$.