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\title{Level by Level Equivalence and the
Number of
Normal Measures over $P_\gk(\gl)$
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, normal measure,
level by level equivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grants
and CUNY
Collaborative Incentive Grants.}
\thanks{The author wishes to
thank the referee for helpful
comments and suggestions which
have been incorporated into the
current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{July 16, 2006\\
(revised December 12, 2006 and February 15, 2007)}
\begin{document}
\maketitle
\begin{abstract}
We construct two models
for the level by
level equivalence
between strong compactness
and supercompactness in
which if $\gk$ is $\gl$
supercompact and $\gl \ge \gk$
is regular, we are able
to determine exactly the
number of normal measures
$P_\gk(\gl)$ carries.
In the first of these models,
$P_\gk(\gl)$ carries
$2^{2^{[\gl]^{< \gk}}}$ many
normal measures,
the maximal number.
In the second of these models,
$P_\gk(\gl)$ carries
$2^{2^{[\gl]^{< \gk}}}$ many
normal measures, except if
$\gk$ is a measurable cardinal
which isn't a limit of measurable
cardinals. In this case, $\gk$
(and hence also $P_\gk(\gk)$) carries
only $\gk^+$ many normal measures.
In both of these models, there are
no restrictions on the structure
of the class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
One of the advantages of an inner
model for a particular type
of measurable cardinal $\gk$
is that it provides canonical structure
for the universe in which $\gk$ resides.
In particular, in the usual sorts of inner
models for measurability
(see, e.g., the models constructed
and analyzed in \cite{Ku}, \cite{Mi},
and \cite{Ba}),
if $\gk$ is a measurable cardinal,
it is possible to determine exactly
the number of normal measures $\gk$ carries.
Because of the limited inner model theory
currently available for supercompactness,
analogous results for $\gk$-additive,
fine, normal measures over $P_\gk(\gl)$
when $\gl \ge \gk$ is regular
have been relatively few. Aside from
the classical results (see \cite{J})
that if $\gk$ is $2^{[\gl]^{< \gk}}$
supercompact, then $P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}}$ many
$\gk$-additive, fine,
normal measures (the maximal number),
and the more recent results of
\cite{ACH} that
when $\gl \ge \gk$ is regular,
it is consistent
relative to the appropriate assumptions
for $\gk$ to be $\gl$ supercompact and
for $P_\gk(\gl)$ to carry fewer than the
maximal number of
$\gk$-additive, fine, normal measures,
not much has been known concerning
models for supercompactness and the
number of normal measures $P_\gk(\gl)$
can carry.
The purpose of this paper is to
rectify this situation by studying
the number of normal measures
$P_\gk(\gl)$ can carry
when $\gl \ge \gk$ is regular
in the context of
the ``inner model like'' property of level by
level equivalence between strong compactness
and supercompactness. Specifically, we
prove the following two theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
In $V^\FP$, if $\gk$ is $\gl$ supercompact
and $\gl \ge \gk$ is regular,
then $P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
In $V^\FP$, if $\gk$ is a measurable
cardinal which isn't a limit of measurable
cardinals, then $\gk$ carries exactly
$\gk^+$ many normal measures (and hence
$P_\gk(\gk)$ carries exactly $\gk^+$ many
$\gk$-additive, fine, normal measures).
If this isn't the case, i.e., if in
$V^\FP$, $\gk$ is a measurable cardinal
which is a limit
of measurable cardinals, then for any regular
$\gl \ge \gk$ such that $\gk$ is
$\gl$ supercompact,
$P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures.
\end{theorem}
In other words, there is a model
for level by level equivalence
between strong compactness and
supercompactness (the one provided
by Theorem \ref{t1}) in which if
$\gk$ is $\gl$ supercompact
(and not necessarily more) and
$\gl \ge \gk$ is regular, then $P_\gk(\gl)$
always carries the maximal number of
$\gk$-additive, fine, normal measures.
On the other hand, there is also a
model for level by level equivalence
between strong compactness and supercompactness
(the one provided by Theorem \ref{t2})
in which under most circumstances,
if $\gk$ is $\gl$ supercompact and
$\gl \ge \gk$ is regular,
then $P_\gk(\gl)$
carries the maximal number of
$\gk$-additive, fine, normal measures.
However, in this model, this
is not always the case.
%that $P_\gk(\gl)$ carries the maximal number of
%$\gk$-additive, fine, normal measures.
In particular, if $\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals, then both $\gk$
and $P_\gk(\gk)$ carry fewer than the
maximal number of normal measures.
We now very briefly give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult \cite{ACH}.
When forcing, $q \ge p$ means that
$q$ is stronger than $p$.
%and $p \decides \varphi$ means that $p$ decides $\varphi$.
For $\gk$ a regular cardinal
and $\gl \ge \gk$ any cardinal,
$\add(\gk, 1)$ is the
standard partial ordering for adding
a single Cohen subset of $\gk$, and
${\rm Coll}(\gk, \gl)$ is the standard
%L\'evy
collapse partial ordering
(originally used by Cohen)
for collapsing $\gl$ to $\gk$.
%For $\ga < \gb$ ordinals,
%$(\ga, \gb]$ and $[\ga, \gb]$
%are as in standard interval notation.
%is the usual half-open
%interval of ordinals which doesn't
%include $\ga$ but includes $\gb$.
For $\gk$ a cardinal, the
partial ordering $\FP$ is
$\gk$-directed closed if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$, especially
if $x$ is in the ground model $V$,
or $x$ is a variant of the generic
set $G$.
Suppose $V$ is a model of ZFC
%containing supercompact cardinals
in which for all 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.
Such a model will be said to
witness level by level
equivalence between strong
compactness and supercompactness.
We will also say that $\gk$ is a witness
to level by level equivalence between
strong compactness and supercompactness
iff for every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact.
Note that the exception
is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
%When this situation occurs, the
%terminology we will henceforth
%use is that $\gk$ is a witness
%to the Menas exception at $\gl$.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
Note that this property is considered
to be ``inner model like'' in the
sense that, like GCH and a
combinatorial property such as $\diamondsuit$,
it is the sort of regularity phenomenon
one might expect in a ``nice inner model''
for supercompactness.
We assume familiarity with the
large cardinal notions of
measurability, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} for further details.
We do wish to mention, however,
that we will use
``supercompact ultrafilter over $P_\gk(\gl)$'' and
``$\gk$-additive, fine, normal measure
over $P_\gk(\gl)$'' synonymously.
In addition, we state explicitly that
$\gk$ is $\gk$ supercompact iff
$\gk$ is $\gk$ strongly compact
iff $\gk$ is measurable, and
if $\gk$ is measurable,
there is a canonical correspondence
between normal measures over $\gk$
and $\gk$-additive, fine,
normal measures over $P_\gk(\gk)$.
%which is obtained using the observation
%that if $A$ has measure 1 with respect
%to some normal measure over $P_\gk(\gk)$, then
%$\{p \in A \mid p \cap \gk$ is an ordinal$\}$
%has measure 1 with respect to the same
%normal measure over $P_\gk(\gk)$.
This has as an immediate consequence that
if $\gk$ is measurable,
the number of normal measures over
$\gk$ and the number of
$\gk$-additive, fine, normal measures
over $P_\gk(\gk)$ is the same.
We conclude Section \ref{s1}
by mentioning that there
is one result
critical to the proofs of
Theorems \ref{t1} and \ref{t2}
which will be taken as a ``black box''.
For the convenience of readers,
we provide a brief discussion
of this fact here. The result is
a corollary of Theorems 3 and 31
and Corollary 14 of
Hamkins' paper \cite{H5}.
This theorem is a generalization of
Hamkins' Gap Forcing Theorem and
Corollary 16 of
\cite{H2} and \cite{H3}
(and we refer readers to \cite{H2},
\cite{H3}, and \cite{H5} for
further details).
We therefore state the theorem
we will be using now, along
with some associated terminology.
%quoting freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| \le \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H5},
$\FP$ {\rm admits a closure point at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{H3},
$\FP$ is {\rm mild}
with respect to a cardinal $\gk$
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2}, \cite{H3}, \cite{H5},
and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\rm amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Theorems 3 and 31
and Corollary 14 of
\cite{H5} we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a closure point
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$
for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M = V \cap M[j(G)]$
(so $M \subseteq V$). If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\noindent It immediately follows from
Theorem \ref{t3} that any cardinal
$\gk$ measurable in a generic extension
obtained by forcing that admits a closure
point below $\gk$ must also be measurable in the
ground model $V$.
In addition, Theorem \ref{t3} implies that if
$V^\FP \models ``\gk$ is $\gl$ strongly compact'',
$\FP$ is mild with respect to $\gk$, and
$\FP$ admits a closure point below $\gk$, then
$V \models ``\gk$ is $\gg$ strongly compact''
for any ordinal $\gg$ such that
$V^\FP \models ``\card{\gg} = \gl$''.
Similarly, Theorem \ref{t3} implies that if
$V^\FP \models ``\gk$ is $\gl$ supercompact''
%$\FP$ is mild with respect to $\gk$,
and $\FP$ admits a closure point below $\gk$, then
$V \models ``\gk$ is $\gg$ supercompact''
for any ordinal $\gg$ such that
$V^\FP \models ``\card{\gg} = \gl$''.
\section{The Proofs of Theorems \ref{t1}
and \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
The partial ordering $\FP$ which will be used
to prove Theorem \ref{t1} is the same one
used in the proof of Theorem 3 of \cite{A05}.
More specifically, $\FP$ is the proper class
reverse Easton iteration which does nontrivial
forcing only at those stages $\gd$ which
are $V$-regular cardinals. At such a stage,
we force with $\add(\gd, 1)$.
%adds a single Cohen subset to each regular cardinal $\gd$.
By Theorem 3 of \cite{A05},
$V^\FP \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
We must therefore show that
$V^\FP \models ``$If $\gk$ is $\gl$ supercompact
and $\gl \ge \gk$ is regular,
then $P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
Towards this end, assume that
$V^\FP \models ``\gk$ is $\gl$ supercompact
and $\gl \ge \gk$ is regular''.
Clearly, $V \models ``\gl$ is regular''.
Since if $\gl$ is regular and we write
$\FP = \FP_{\gl + 1} \ast \dot \FP^{\gl + 1}$,
$\forces_{\FP_{\gl + 1}} ``\dot \FP^{\gl + 1}$ is
$\gl^+$-directed closed'', we know that
$V^\FP \models ``\gk$ is $\gl$ supercompact'' iff
$V^{\FP_{\gl + 1}} \models ``\gk$ is $\gl$ supercompact''.
Further, any normal measure over
$P_\gk(\gl)$ in $V^{\FP_{\gl + 1}}$ is
also a normal measure over
$P_\gk(\gl)$ in $V^\FP$.
It thus suffices to show that
$V^{\FP_{\gl + 1}} \models ``$If $\gk$ is $\gl$ supercompact
and $\gl \ge \gk$ is regular,
then $P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
To show this last fact,
we begin by noting that
as in Theorem 3 of \cite{A05},
by Theorem \ref{t3} of this paper,
it must be the case that
$V \models ``\gk$ is $\gl$ supercompact''.
We now use a standard analysis
found, e.g., in
Lemma 1.1 of \cite{A01} or Lemma 6 of \cite{C93}
in tandem
with the usual argument
(originally due to Silver)
for lifting a
supercompactness embedding after a reverse
Easton iteration.
Specifically, let $j : V \to M$ be an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk$ generated
by a supercompact ultrafilter over
$P_\gk(\gl)$. Write
$\FP_{\gl + 1} = \FP_\gk \ast \dot \FQ$,
where $\dot \FQ$ is a term for the
partial ordering which adds Cohen subsets
of each regular cardinal in the
closed interval $[\gk, \gl]$.
Then
$j(\FP_{\gl + 1}) = \FP_\gk \ast \dot \FQ
\ast \dot \FR \ast j(\dot \FQ)$, where
$\dot \FR$ is a term for the partial
ordering which adds Cohen subsets of
each regular cardinal in $M$ in the
open interval $(\gl, j(\gk))$.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be $V[G]$-generic
over $\FQ$. Silver's standard arguments,
as given, e.g., in the proof of Lemma 1.2
of \cite{A05} show that
$j$ lifts in $V[G][H]$ to
$j^* : V[G][H] \to M[G][H][H'][H'']$, where
$H'$ and $H''$ are built in $V[G][H]$,
$H'$ is $M[G][H]$-generic over $\FR$,
and $H''$ is $M[G][H][H']$-generic over
$j^*(\FQ)$ and contains a master condition
for ${j^*}''H$.
It is, though, the construction
of $H'$ here which is critical for our purposes.
We therefore examine this more carefully.
Since $\FR$ has
cardinality $j(\gk)$ in $M[G][H]$,
by GCH in both $V$ and $M$, there
are $2^{j(\gk)} = j(\gk^+)$ many
dense open subsets of $\FR$ present
in $M[G][H]$. Again
by GCH in both $V$ and $M$,
%standard arguments show that $j(\gk^+)$
%has cardinality $\gl^+$ in $V[G][H]$.
$V[G][H] \models ``\card{j(\gk^+)} \le
\card{\{f \mid f : P_\gk(\gl) \to \gk^+\}} =
\card{\{f \mid f : \gl \to \gk^+\}} =
2^\gl = \gl^+$''.
This means
since $M[G][H]$ remains $\gl$
closed with respect to $V[G][H]$
that $H'$ is constructed in $V[G][H]$ by
meeting each member of an enumeration
$\la D_\ga \mid \ga < \gl^+ \ra
\in V[G][H]$ of the dense open subsets of
$\FR$ present in $M[G][H]$.
However, it is possible to build a tree
${\cal T}$ of height $\gl^+$ which gives
$2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many distinct
possible values for $H'$. More explicitly,
the root of ${\cal T}$ is the empty condition.
If $p$ is an element at level
$\ga < \gl^+$ of $\cal T$, then the
successors of $p$ at level $\ga + 1$ are
a maximal incompatible subset of $D_\ga$
extending $p$. Note that by the definition
of $\FP$, there will be
at least two successors of $p$ at level
$\ga + 1$. Finally,
if $\gd < \gl^+$ is a limit
ordinal, then the elements of $\cal T$ at
height $\gd$ are upper bounds to any
path through $\cal T$ of height $\gd$.
Since $M[G][H]$ remains $\gl$ closed
with respect to $V[G][H]$ and
$M[G][H] \models ``\FR$ is
$\gl^+$-directed closed'', $\cal T$
is well-defined. In addition, by the
fact that each node of $\cal T$ splits,
there are $2^{\gl^+} = \gl^{++} =
2^{2^\gl} = 2^{2^{[\gl]^{< \gk}}}$
many distinct paths through $\cal T$.
As each path through $\cal T$ generates
an $M[G][H]$-generic object over $\FR$,
there are $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many distinct
possible values for $H'$.
Hence, since for any lift of $j : V \to M$ to
$j^* : V[G][H] \to M[G][H][H'][H'']$,
$j^*(G \ast H) = G \ast H \ast H' \ast H''$,
each distinct value of $H'$ generates
a different value of $j^*$ and consequently,
a distinct normal ultrafilter $\cal U$
over $P_\gk(\gl)$ given by
$X \in {\cal U}$ iff $\la j(\ga) \mid
\ga < \gl \ra \in j^*(X)$.
Thus,
there are $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
%$2^{\gl^+} = \gl^{++} = 2^{2^\gl} = 2^{2^{[\gl]^{< \gk}}}$ many
different normal ultrafilters over $P_\gk(\gl)$
in $V[G][H]$.
This completes the proof of Theorem \ref{t1}.
\end{proof}
Having finished the proof of Theorem \ref{t1},
we turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose
$V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals +
Level by level equivalence between
strong compactness and supercompactness holds''.
Without loss of generality, by Theorem \ref{t1},
we assume in addition that
$V \models ``$If $\gk$ is $\gl$ supercompact
and $\gl \ge \gk$ is regular,
then $P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
For $\gd$ any ordinal, define
$\gg_\gd = \go$ if $\gd$ is less
than or equal to
the least measurable cardinal, and $\gg_\gd$
as the least inaccessible cardinal
above the supremum of all of the
measurable cardinals below $\gd$ otherwise.
The partial ordering $\FP$ used in the
proof of Theorem \ref{t2}
will be defined as $\FP^0 \ast \dot \FP^1$,
where $\FP^0$ and $\FP^1$ are two
reverse Easton iterations.
%$\FP^0$ is the reverse Easton iteration which
$\FP^0$ does nontrivial
forcing only at those stages $\gd$
which are $V$-measurable cardinals which
aren't in $V$ limits of measurable cardinals.
At such a stage, we force with
$\add(\gg_\gd, 1) \ast
\dot {\rm Coll}(\gd^+, \gd^{++})$.
If there are only set many measurable
cardinals in $V$, we let $\Omega$
be their supremum, and conclude
the definition of ${\FP^0}$ by forcing with
$\add(\gg_\Omega, 1)$ (if there are
any inaccessibles above $\Omega$).
Terminology we will use later
at a nontrivial stage of forcing $\gd$ is
that ${\FP^0}$ (or some portion thereof)
acts nontrivially on the ordinals
$\gg_\gd$, $\gd^+$, and $\gd^{++}$.
At all other ordinals, ${\FP^0}$
acts trivially.
We observe that ${\FP^0}$ is either a set or
a proper class, depending upon whether
the collection of measurable cardinals
in $V$ is a set or a proper class.
Regardless if ${\FP^0}$ is a set or a
proper class, routine arguments show that
$V^{\FP^0} \models ``$ZFC + GCH'' and
that the only cardinals collapsed in
$V^{\FP^0}$ have the form ${(\gd^{++})}^V$,
where $\gd$ is in $V$ a measurable cardinal
which isn't a limit of measurable cardinals.
\begin{lemma}\label{l0}
If
$V^{\FP^0} \models ``\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals'', then
$V \models ``\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals''.
\end{lemma}
\begin{proof}
Suppose
$V^{\FP^0} \models ``\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals''.
Note that it is possible to write ${\FP^0}$ as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is
nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_1$-directed
closed''. Hence, as we observed
at the end of Section \ref{s1}, since any
cardinal which is measurable in $V^{\FP^0}$
had to have been measurable in $V$,
$V \models ``\gk$ is measurable''.
Thus, to prove Lemma \ref{l0},
it suffices to show that
$V \models ``\gk$ isn't a limit of
measurable cardinals''.
To do this, assume to the contrary that
$V \models ``\gk$ is a limit of
measurable cardinals''. In particular,
$V \models ``\gk$ is a limit of
measurable cardinals $\gd$ which themselves
aren't limits of measurable cardinals''.
For any such measurable cardinal $\gd$, write
${\FP^0} = {\FP^0_\gd} \ast \dot \add(\gg_\gd, 1)
\ast \dot {\rm Coll}(\gd^+, \gd^{++})
\ast \dot \FQ'$.
Since
$\card{{\FP^0_\gd} \ast \dot \add(\gg_\gd, 1)} < \gd$,
by the L\'evy-Solovay results \cite{LS},
$V^{{\FP^0_\gd} \ast \dot \add(\gg_\gd, 1)} \models
``\gd$ is measurable''.
Since
$\forces_{{\FP^0_\gd} \ast \dot \add(\gg_\gd, 1)}
``\dot {\rm Coll}(\gd^+, \gd^{++}) \ast \dot \FQ'$ is
$\gd^+$-directed closed'',
$V^{{\FP^0_\gd} \ast \dot \add(\gg_\gd, 1)
\ast \dot {\rm Coll}(\gd^+, \gd^{++})
\ast \dot \FQ'} = V^{\FP^0} \models
``\gd$ is measurable''. Thus,
$V^{\FP^0} \models ``\gk$ is a measurable
cardinal which is a limit of measurable cardinals''.
This contradiction completes
the proof of Lemma \ref{l0}.
\end{proof}
\begin{lemma}\label{l1}
$V^{\FP^0} \models ``$If $\gk$ is a measurable
cardinal which isn't a limit of measurable
cardinals, then $\gk$ carries exactly
$\gk^+$ many normal measures''.
\end{lemma}
\begin{proof}
Suppose
$V^{\FP^0} \models ``\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals''.
By Lemma \ref{l0},
$V \models ``\gk$ is a measurable
cardinal which isn't a limit of
measurable cardinals''.
Therefore, in analogy to
the proof of Lemma \ref{l0}, write
${\FP^0} = {\FP^0_\gk} \ast \dot
\add(\gg_\gk, 1) \ast \dot {\rm Coll}(\gk^+, \gk^{++})
\ast \dot \FQ'$, where
$\card{{\FP^0_\gk}} < \gk$ and
$\forces_{{\FP^0_\gk} \ast \dot
\add(\gg_\gk, 1) \ast \dot {\rm Coll}(\gk^+, \gk^{++})}
``$Forcing with $\dot \FQ'$ adds no bounded subsets of
the least inaccessible cardinal above $\gk$''.
Thus, since by the results of \cite{LS},
$\forces_{{\FP^0_\gk}} ``\gk$ is a measurable
cardinal which isn't a limit of measurable
cardinals'', the proof of Lemma \ref{l1}
will be complete once we have shown that
$\forces_{{\FP^0_\gk} \ast \dot
\add(\gg_\gk, 1) \ast \dot {\rm Coll}(\gk^+, \gk^{++})}
``\gk$ is a measurable cardinal
which isn't a limit of measurable cardinals
carrying exactly $\gk^+$ many normal measures''.
To do this, we use an argument due to
Cummings, which also appears in
the proof of the Main Theorem of
\cite{ACH} and the proof of Lemma 2.1
of \cite{A06}.
First, note that by our
assumptions on $V$,
$V \models ``\gk$ carries exactly
$\gk^{++} = 2^{2^\gk}$ many normal measures''.
Let $\ov V = V^{{\FP^0_\gk}}$.
By the results of \cite{LS},
$\ov V \models ``\gk$ carries exactly
$\gk^{++}$ many normal measures'' as well.
Suppose
$G_0$ is $\ov V$-generic over $\add(\gg_\gk, 1)$ and
$G_1$ is $\ov V[G_0]$-generic over
${\rm Coll}(\gk^+, \gk^{++})$.
Again by the results of \cite{LS},
since $\card{\add(\gg_\gk, 1)} < \gk$,
$\ov V[G_0] \models ``\gk$ is a measurable
cardinal carrying
exactly $\gk^{++}$ many normal measures''.
%every normal measure over $\gk$ in $\ov V$
%generates a unique normal measure over $\gk$ in $\ov V[G_0]$.
These remain normal measures over $\gk$
in $\ov V[G_0][G_1]$, since no additional subsets of $\gk$ are
added by the collapse forcing.
Thus, since ${(\gk^+)}^V$
is preserved to $\ov V[G_0][G_1]$,
there are at least
$\gk^+$ many normal measures over $\gk$ in $\ov V[G_0][G_1]$.
Conversely, suppose that ${\cal U}$ is a normal measure over
$\gk$ in $\ov V[G_0][G_1]$, with the associated ultrapower
embedding $j: \ov V[G_0][G_1]\to M[G_0][j(G_1)]$. In particular,
$X\in{\cal U}$ iff $\gk\in j(X)$ for all $X
\subseteq \gk$ in $\ov V[G_0][G_1]$. By Theorem
\ref{t3}, it follows that the restriction
$j\rest \ov V: \ov V\to M$ is a definable class in $\ov V$.
Once more by the results of \cite{LS},
since $\card{\add(\gg_\gk, 1)} < \gk$,
$j\rest \ov V$ lifts
uniquely to $\ov V[G_0]$,
and so $j\rest \ov V[G_0]:\ov V[G_0]\to M[G_0]$
is a definable class in $\ov V[G_0]$. The key observation is now
that because $\ov V[G_0]$ and $\ov V[G_0][G_1]$ have the same subsets of
$\gk$, one can reconstruct ${\cal U}$ inside $\ov V[G_0]$ by
observing $X\in{\cal U}$ iff $\gk\in j(X)$,
using only $j\rest \ov V[G_0]$.
Thus, ${\cal U}\in \ov V[G_0]$. Consequently,
every normal measure over $\gk$ in $\ov V[G_0][G_1]$ is actually
in $\ov V[G_0]$. The number of such normal measures, therefore,
is at most ${(\gk^{++})}^{\ov V[G_0]}$, which is $\gk^+$ in
$\ov V[G_0][G_1]$, because ${(\gk^{++})}^{\ov V[G_0]}$ was collapsed by
$G_1$. Hence, in $\ov V[G_0][G_1]$, there are exactly $\gk^+$ many
normal measures over $\gk$, as desired.
Since forcing with
$\add(\gg_\gk, 1) \ast
\dot {\rm Coll}(\gk^+, \gk^{++})$ doesn't
change the fact that $\gk$ isn't a
limit of measurable cardinals,
this completes the proof of Lemma \ref{l1}.
\end{proof}
For any (measurable) cardinal $\gk$,
define $\gth_\gk$ as the least
cardinal such that $\gk$ isn't
$\gth_\gk$ strongly compact.
\begin{lemma}\label{l2}
Suppose in $V$, $\gk$ is
$\gl^+$ strongly compact and
$\gth_\gk = \gl^{++}$, where
$\gl > \gk$ is a measurable
cardinal which isn't a limit
of measurable cardinals. Then
$V^{\FP^0} \models ``\gk$ isn't
$\gl^+$ strongly compact''.
\end{lemma}
\begin{proof}
By the definition of ${\FP^0}$,
we may write
${\FP^0} = {\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})
\ast \dot \FR$, where
$\forces_{{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})}
``$Forcing with $\dot \FR$ adds no bounded
subsets of the least inaccessible
cardinal above $\gl$''.
It thus suffices to show that
$V^{{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})} \models
``\gk$ isn't $\gl^+$ strongly compact''.
To do this, note that
${{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})}$ may be
written as $\FR_0 \ast \dot \FR_1$, where
%$\FR_0$ is (forcing equivalent to) $\add(\go, 1)$
$\card{\FR_0} = \go$, $\FR_0$ is nontrivial,
and $\forces_{\FR_0} ``\dot \FR_1$ is
$\ha_1$-directed closed''.
In addition, by its definition,
${{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})}$
is easily seen to be mild with
respect to $\gk$. Since
$V^{{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})} \models
%``\card{\gth_\gk} = \gl^+$'',
``\card{{(\gl^{++})}^V} = \gl^+$'' and
$\gth_\gk = {(\gl^{++})}^V$,
by Theorem \ref{t3},
$V^{{\FP^0_\gk} \ast \dot \FQ
\ast \dot {\rm Coll}(\gl^+, \gl^{++})} \models
``\gk$ isn't $\gl^+$ strongly compact''.
This is because otherwise,
as we observed at the end of Section \ref{s1},
$\gk$ would have had to have been
$\gl^{++}$ strongly compact in $V$, which
contradicts the fact that
$\gth_\gk = {(\gl^{++})}^V$.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We remark that the exact same
proof as given in Lemma \ref{l2}
(without the reference to mildness,
which is unnecessary in the
context of supercompactness)
shows that if
$V \models ``\gk$ is $\gl^+$ supercompact
but not $\gl^{++}$ supercompact and
$\gl > \gk$ is a measurable cardinal which
isn't a limit of measurable cardinals'', then
$V^{\FP^0} \models ``\gk$ isn't $\gl^+$ supercompact''.
This observation will be used later.
%in the proof of Lemma \ref{l6}.
\begin{lemma}\label{l3}
Suppose in $V$, $\gk$ is
$\gl$ supercompact for
$\gl \ge \gk$ a regular
cardinal and $\gk$
is a measurable cardinal
which is a limit of measurable cardinals.
%,and $\gk$ isn't a witness to the Menas exception
%at $\gr$ for any $\gr$.
Suppose further that
for any cardinal $\gg > \gk$
which is a measurable cardinal
which isn't a limit of
measurable cardinals,
if $V \models ``\gk$ is $\gg^+$ supercompact'', then
$V \models ``\gk$ is $\gg^{++}$ supercompact'' as well.
Then $V^{\FP^0} \models ``\gk$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
Write
${\FP^0} = {\FP^0_\gk} \ast \dot \FQ \ast \dot \FQ'$,
where $\dot \FQ$ is forced to act
(either trivially or nontrivially)
on ordinals in the closed interval
$[\gk, \gl]$,
and $\dot \FQ'$ is a term for
the rest of ${\FP^0}$.
If $\gl$ is a nontrivial stage
of the forcing, i.e., if $\gl$ is a
measurable cardinal which isn't
a limit of measurable cardinals, then
$\forces_{{\FP^0_\gk} \ast \dot \FQ} ``$Forcing
with $\dot \FQ'$ adds no bounded subsets of
$\gl^+$''.
If $\gl$ is a trivial stage of the forcing, then
$\forces_{{\FP^0_\gk} \ast \dot \FQ} ``$Forcing
with $\dot \FQ'$ adds no bounded subsets of the
least inaccessible cardinal above $\gl$''.
Thus, to show
$V^{\FP^0} \models ``\gk$ is $\gl$ supercompact'',
it suffices to show that
$V^{{\FP^0_\gk} \ast \dot \FQ} \models ``\gk$ is
$\gl$ supercompact''.
To do this, let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$.
Note that
by hypothesis, if $\gl = \gg^+$
where $\gg$ is a measurable
cardinal which isn't a limit
of measurable cardinals, then
$\gk$ is actually $\gl^+ = \gg^{++}$
supercompact in $V$
(so under these circumstances,
$M$ may be taken as being
$\gl^+$ closed).
Consequently, regardless if this is the case,
$M$ has enough closure so that
$j({\FP^0_\gk} \ast \dot \FQ) =
{\FP^0_\gk} \ast \dot \FQ \ast \dot \FR
\ast j(\dot \FQ)$, where
the first ordinal at which
$\dot \FR$ is forced to act
nontrivially is above $\gl$.
Silver's standard lifting arguments,
as given, e.g.,
in the proof of Lemma 1.2 of \cite{A05}
(and mentioned in the proof of Theorem \ref{t1})
once again show that if
$G$ is $V$-generic over ${\FP^0_\gk}$ and
$H$ is $V[G]$-generic over $\FQ$, then
$j$ lifts in $V[G][H]$ to
$j^* : V[G][H] \to M[G][H][H'][H'']$
which witnesses the $\gl$ supercompactness
of $\gk$,
where $H'$ and $H''$ are the
generic objects constructed in $V[G][H]$
%for the appropriate partial orderings, and
for $\FR$ and $j^*(\FQ)$, and
$H''$ contains a master condition for
${j^*}''H$. Hence,
$V^{{\FP^0_\gk} \ast \dot \FQ} \models ``\gk$
is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^{\FP^0} \models ``\K$ is the class of
supercompact cardinals''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l0},
we may write
${\FP^0} = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_1$-directed
closed''. Hence, by Theorem \ref{t3},
any cardinal supercompact in $V^{\FP^0}$
had to have been supercompact in $V$.
However, by Lemma \ref{l3}, any cardinal
supercompact in $V$ remains supercompact
in $V^{\FP^0}$.
Thus,
$V^{\FP^0} \models ``\K$ is the class of
supercompact cardinals''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^{\FP^0} \models ``$Level by level equivalence
between strong compactness and supercompactness holds''.
\end{lemma}
\begin{proof}
Suppose
$V^{\FP^0} \models ``\gk < \gl$ are such that
$\gk$ is $\gl$ strongly compact and
$\gl$ is a regular cardinal''.
By its definition, ${\FP^0}$ is mild
with respect to $\gk$.
Therefore, by the factorization of ${\FP^0}$
given in Lemmas \ref{l0} and \ref{l4}
and Theorem \ref{t3}, it must be true that
$V \models ``\gk$ is $\gl$ strongly compact''.
By level by level equivalence between
strong compactness and supercompactness,
$V \models ``$Either $\gk$ is $\gl$
supercompact,
%and witnesses level by level equivalence between
%strong compactness and supercompactness,
or $\gk$ is a measurable
limit of cardinals $\gd$ which
are $\gl$ supercompact''.
%and each $\gd$ witnesses level by level equivalence
%between strong compactness and supercompactness''.
By Lemma \ref{l2} and the paragraph
immediately following, it cannot be the case that
$V \models ``\gl = \gg^+$, $\gg$ is a measurable
cardinal which isn't a limit of measurable
cardinals, and either strong compactness
or supercompactness first fails at
$\gg^{++}$
for any cardinal $\gd$ which is either $\gl$
strongly compact or $\gl$ supercompact''.
%$\gth_\gd = \gg^{++}$
Hence, by Lemma \ref{l3},
$V^{\FP^0} \models ``$Either $\gk$ is $\gl$
supercompact, or $\gk$ is a measurable
limit of cardinals $\gd$ which are
$\gl$ supercompact'', i.e.,
$V^{\FP^0} \models ``$Level by level equivalence
between strong compactness and supercompactness holds''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
Because $\gl^{++}$ is collapsed if
$\gk$ is exactly $\gl$ supercompact
and $\gl$ is in $V$ a measurable
cardinal which isn't a limit of
measurable cardinals, we need to
do an additional forcing to ensure
that if
$\gk$ is $\gl$
supercompact, $\gl \ge \gk$ is regular, and
$\gk$ is a limit of measurable
cardinals, then
$P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures.
To this end, let
$\ov V = V^{\FP^0}$.
The partial ordering $\FP^1 \in \ov V$
we use to complete the proof of
Theorem \ref{t2} is the reverse Easton
iteration which begins by adding a
Cohen subset of $\go$ and then
does nontrivial forcing only
at those stages $\gd$ which are
$\ov V$-measurable cardinals which
aren't in $\ov V$ limits of
measurable cardinals. At such
a stage $\gd$, we force with
$\add(\gd^*, 1)$, where $\gd^*$ is
(in either $\ov V$ or
$\ov V^{\FP^1_\gd}$) the least
inaccessible cardinal above $\gd$.
Regardless if $\FP^1$ is a set
or a proper class, routine arguments
show that forcing with $\FP^1$
preserves all cardinals and cofinalities and
$\ov V^{\FP^1} \models ``$ZFC + GCH''.
\begin{lemma}\label{l6}
If
$\ov V^{\FP^1} \models
``\gk$ is a measurable cardinal
which isn't a limit of measurable
cardinals'', then
$\ov V \models ``\gk$ is a measurable cardinal
which isn't a limit of measurable
cardinals''.
\end{lemma}
\begin{proof}
We mimic to a certain
extent the proof of Lemma \ref{l0}.
The exact same arguments as in
Lemma \ref{l0} show that
$\ov V \models ``\gk$ is measurable''.
Thus, it once again suffices to show that
$\ov V \models ``\gk$ isn't a limit
of measurable cardinals''.
As before, to do this, we
assume to the contrary that
$\ov V \models ``\gk$ is
a limit of measurable cardinals'',
so that in particular,
$\ov V \models ``\gk$ is
a limit of measurable cardinals
which themselves aren't limits
of measurable cardinals''.
For any such measurable cardinal $\gd$, write
$\FP^1 = \FP^1_\gd \ast \dot \FQ$.
Since $\card{\FP^1_\gd} < \gd$, by the
results of \cite{LS},
$\ov V^{\FP^1_\gd} \models ``\gd$ is measurable''.
Since $\forces_{\FP^1_\gd} ``\dot \FQ$ is
$\gd^+$-directed closed'',
$\ov V^{\FP^1_\gd \ast \dot \FQ} = \ov V^{\FP^1}
\models ``\gd$ is measurable''. Hence,
$\ov V^{\FP^1} \models ``\gk$ is a measurable
cardinal which is a limit of measurable
cardinals'', a contradiction which then
completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$\ov V^{\FP^1} \models ``$If $\gk$ is a
measurable cardinal which isn't a limit of
measurable cardinals, then $\gk$ carries
exactly $\gk^+$ many normal measures''.
\end{lemma}
\begin{proof}
Suppose
$\ov V^{\FP^1} \models ``\gk$ is a
measurable cardinal which isn't a
limit of measurable cardinals''.
By Lemma \ref{l6},
$\ov V \models ``\gk$ is a
measurable cardinal which isn't a
limit of measurable cardinals''.
Therefore, in analogy to the
proof of Lemma \ref{l6}, write
$\FP^1 = \FP^1_\gk \ast \dot \FQ$.
By Lemma \ref{l1},
$\ov V \models
``\gk$ carries exactly
$\gk^+$ many normal measures''. Hence, since
$\card{\FP^1_\gk} < \gk$, by the results of \cite{LS},
$\ov V^{\FP^1_\gk} \models
``\gk$ is a
measurable cardinal which isn't a
limit of measurable cardinals and $\gk$ carries
exactly $\gk^+$ many normal measures''.
Consequently, as
$\forces_{\FP^1_\gk} ``$Forcing with $\dot \FQ$
adds no bounded subsets of the least
inaccessible cardinal above $\gk$'',
$\ov V^{\FP^1_\gk \ast \dot \FQ} =
\ov V^{\FP^1} \models
``\gk$ is a
measurable cardinal which isn't a
limit of measurable cardinals and $\gk$ carries
exactly $\gk^+$ many normal measures''.
This completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
If $\ov V \models ``\gk$ is $\gl$
supercompact and $\gl > \gk$
is regular'', then
$\ov V^{\FP^1} \models ``\gk$
is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
We mimic to a certain extent
the proof of Lemma \ref{l3}.
Write
${\FP^1} = {\FP^1_\gk} \ast \dot \FQ \ast \dot \FQ'$,
where $\dot \FQ$ is forced to act
(either trivially or nontrivially)
on ordinals in the closed interval
$[\gk, \gl]$,
and $\dot \FQ'$ is a term for
the rest of ${\FP^1}$.
%Since forcing with ${\FP^1}$ preserves GCH and
Since $\forces_{\FP^1_\gk \ast \dot \FQ}
``\dot \FQ'$ is $\gl^+$-directed closed'',
to show
$\ov V^{\FP^1} \models ``\gk$ is $\gl$ supercompact'',
it suffices to show that
$\ov V^{\FP^1_\gk \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
To do this, let
$j : \ov V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$.
$M$ has enough closure so that
$j({\FP^1_\gk} \ast \dot \FQ) =
{\FP^1_\gk} \ast \dot \FQ \ast \dot \FR
\ast j(\dot \FQ)$, where
the first ordinal at which
$\dot \FR$ is forced to act
nontrivially is above $\gl$.
As before,
Silver's standard lifting arguments,
as given, e.g.,
in the proof of Lemma 1.2 of \cite{A05}
(and mentioned in the proof of Theorem \ref{t1}
and Lemma \ref{l3})
once again show that if
$G$ is $\ov V$-generic over ${\FP^1_\gk}$ and
$H$ is $\ov V[G]$-generic over $\FQ$, then
$j$ lifts in $\ov V[G][H]$ to
$j^* : \ov V[G][H] \to M[G][H][H'][H'']$
which witnesses the $\gl$ supercompactness
of $\gk$,
where $H'$ and $H''$ are the
generic objects constructed in $\ov V[G][H]$
%for the appropriate partial orderings, and
for $\FR$ and $j^*(\FQ)$, and
$H''$ contains a master condition for
${j^*}''H$. Hence,
$\ov V^{{\FP^1_\gk} \ast \dot \FQ} \models ``\gk$
is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l8}.
\end{proof}
\begin{lemma}\label{l9}
$\ov V^{\FP^1} \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
We mimic to a certain extent the proof of Lemma \ref{l5}.
Suppose
$\ov V^{\FP^1} \models ``\gk < \gl$
are such that $\gk$ is $\gl$ strongly
compact and $\gl$ is a regular cardinal''.
By its definition, $\FP^1$ is mild
with respect to $\gk$. In addition, it
is possible to factor $\FP^1$ as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_1$-directed closed''.
Therefore, by Theorem \ref{t3},
it must be the case that
$\ov V \models ``\gk$ is $\gl$ strongly compact''.
By level by level equivalence between
strong compactness and supercompactness in $\ov V$,
$\ov V \models ``$Either $\gk$ is $\gl$
supercompact, or $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact''.
Hence, by Lemma \ref{l8},
$\ov V^{\FP^1} \models ``$Either $\gk$ is $\gl$
supercompact, or $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact'', i.e.,
$\ov V^{\FP^1} \models ``$Level by level
equivalence between strong compactness and
supercompactness holds''.
This completes the proof of Lemma \ref{l9}.
\end{proof}
The proof of Theorem \ref{t1} now
applies almost verbatim to show that
$\ov V^{\FP^1} \models ``$If $\gk$ is $\gl$
supercompact, $\gl \ge \gk$ is regular, and
$\gk$ is a limit of measurable
cardinals, then
$P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
The same proof as in Lemma \ref{l4}
(replacing a reference to Lemma \ref{l3}
with a reference to Lemma \ref{l8})
shows that
$\ov V^{\FP^1} \models ``\K$ is the class
of supercompact cardinals''.
Therefore, by letting
$\FP = \FP^0 \ast \dot \FP^1$,
Lemmas \ref{l0} -- \ref{l9} and the
intervening remarks complete the
proof of Theorem \ref{t2}.
\end{proof}
As we remarked at the beginning
of this paper, if $\gk$ exhibits
enough supercompactness, it will
be the case that $P_\gk(\gl)$
carries the maximal number of
$\gk$-additive, fine, normal measures.
However, since this may not always
be the case, we conclude by asking
what the other possibilities are
for the number of normal measures
over $P_\gk(\gl)$ in a universe
containing supercompact cardinals in
which level by level equivalence between
strong compactness and supercompactness holds.
In particular, if $\gk$ is $\gl$
supercompact, $\gl \ge \gk$ is regular,
and $\gk$ is not $\gl^+$ supercompact,
is it possible, in a model
satisfying GCH and level by level
equivalence between strong compactness
and supercompactness,
for $P_\gk(\gl)$ to
carry exactly 1 normal measure?
What about $\gd$ many normal
measures, where $\gd$ is an
arbitrary cardinal less than
$2^{2^{[\gl]^{< \gk}}}$?
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\begin{lemma}\label{l6}
$V^{\FP^0} \models ``$If $\gk$ is $\gl$
supercompact, $\gl \ge \gk$ is regular, and
$\gk$ is a limit of measurable
cardinals, then
$P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
\end{lemma}
\begin{proof}
Suppose
$V^{\FP^0} \models ``\gk$ is $\gl$
supercompact, $\gl \ge \gk$ is regular
in $V$, and
$\gk$ is a measurable cardinal which is a
limit of measurable cardinals''. Again
by the factorization of ${\FP^0}$ given in
Lemmas \ref{l0} and \ref{l4} and
Theorem \ref{t3},
$V \models ``\gk$ is $\gl$
supercompact, and
$\gk$ is a measurable cardinal which is a
limit of measurable cardinals''.
Therefore, by our assumptions on $V$,
we know that
$V \models ``P_\gk(\gl)$ carries
exactly $2^{2^{[\gl]^{< \gk}}} =
2^{2^\gl} = \gl^{++}$ many
$\gk$-additive, fine, normal measures''.
In addition, by the paragraph
immediately following Lemma \ref{l2},
we know that
$V \models ``\gl \neq \gg^+$ where
supercompactness first fails
at $\gg^{++}$ and $\gg > \gk$
is a measurable cardinal which isn't
a limit of measurable cardinals''.
Hence, from the proof of Lemma \ref{l3}
(including the notation therein), if
$j : V \to M$ is an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$, $j$ lifts in
$V^{{\FP^0_\gk} \ast \dot \FQ}$
to an elementary embedding $j^*$ witnessing
the $\gl$ supercompactness of $\gk$.
Since there are ${(2^{2^{[\gl]^{< \gk}}})}^V$
many supercompact ultrafilters over
$P_\gk(\gl)$ in $V$, there are
${(2^{2^{[\gl]^{< \gk}}})}^V$ many such lifts
$j^*$, and as a consequence,
${(2^{2^{[\gl]^{< \gk}}})}^V$ many $\gk$-additive,
fine, normal measures over $P_\gk(\gl)$ in
both $V^{{\FP^0_\gk} \ast \dot \FQ}$ and $V^{\FP^0}$.
Thus, it suffices to show that there are actually
${(2^{2^{[\gl]^{< \gk}}})}^{V^{\FP^0}}$ many
$\gk$-additive, fine, normal measures over
$P_\gk(\gl)$ in $V^{\FP^0}$.
Because the only cardinals collapsed have the form
${(\gd^{++})}^V$ where $\gd$ is in $V$ a
measurable cardinal which isn't a limit
of measurable cardinals, the only
ambiguity occurs when
$V \models ``\gk$ is $\gl^+$ supercompact and
$\gl$ is a measurable cardinal which isn't
a limit of measurable cardinals''.
However, by the paragraph immediately
following Lemma \ref{l2}, in this
situation, it must be the case that
$V \models ``\gk$ is $\gl^{++}$ supercompact''.
As we remarked above,
$V^{\FP^0} \models ``$There are
${(2^{2^{[\gl^{++}]^{< \gk}}})}^V$ many
$\gk$-additive, fine, normal measures over
$P_\gk({(\gl^{++})}^V)$'', which means that
$V^{\FP^0} \models ``$There are
$2^{2^{[\gl^{+}]^{< \gk}}}$ many
$\gk$-additive, fine, normal measures over
$P_\gk(\gl^+)$''.
This completes the proof of Lemma \ref{l6}.
\end{proof}