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\title{Indestructibility and Measurable Cardinals with Few and
Many Measures
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal,
measurable cardinal, normal measure,
indestructibility.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.
In addition, the author wishes to
thank the referee, for helpful comments,
corrections,
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{January 28, 2007\\
(revised March 19, 2008)}
\begin{document}
\maketitle
\begin{abstract}
If $\gk < \gl$
are such that $\gk$ is
indestructibly supercompact
and $\gl$ is measurable,
then we show that both
$A = \{\gd < \gk \mid \gd$
is a measurable cardinal
which is not a limit of
measurable cardinals and
$\gd$ carries the
maximal number of normal measures$\}$ and
$B = \{\gd < \gk \mid \gd$
is a measurable cardinal
which is not a limit of
measurable cardinals and
$\gd$ carries fewer than the
maximal number of normal measures$\}$
are unbounded in $\gk$.
%must be unbounded in $\gk$.
The two aforementioned
phenomena, however, need not
occur in a universe with
an indestructibly supercompact
cardinal and sufficiently
few large cardinals.
In particular,
we show how to construct a model
with an indestructibly
supercompact cardinal $\gk$ in which
if $\gd < \gk$ is a measurable
cardinal which is not a limit of
measurable cardinals, then
$\gd$ must carry fewer than
the maximal number of normal measures.
We also, however,
show how to construct a model
with an indestructibly
supercompact cardinal $\gk$ in which
if $\gd < \gk$ is a measurable
cardinal which is not a limit of
measurable cardinals, then
$\gd$ must carry the maximal number
of normal measures.
If we weaken the requirements on
indestructibility, then this last
result can be improved to obtain a
model with an indestructibly supercompact
cardinal $\gk$ in which {\em every}
measurable cardinal $\gd < \gk$ carries
the maximal number of normal measures.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
It is an amazing and very interesting fact that
the structure of the universe above an
indestructibly supercompact cardinal $\gk$
can affect what happens
at large cardinals below $\gk$, assuming
the universe is sufficiently rich.
For instance, if $\gk < \gl$ are such
that $\gk$ is indestructibly supercompact
and $\gl$ is $2^\gl$ supercompact, then
the results of \cite{AH4} and \cite{A07} show that
$\{\gd < \gk \mid \gd$ is a measurable
cardinal which is not a limit of
measurable cardinals and $\gd$ violates level
by level equivalence between strong
compactness and supercompactness$\}$
must be unbounded in $\gk$.
On the other hand, it is shown in \cite{A07}
that if $\gk < \gl$ are such that
$\gk$ is indestructibly supercompact and
$\gl$ is measurable, then
$\{\gd < \gk \mid \gd$ is a measurable
cardinal which is not a limit of
measurable cardinals and $\gd$ satisfies level
by level equivalence between strong
compactness and supercompactness$\}$
must also be unbounded in $\gk$.
These results are optimal, since as the
work of \cite{A07} indicates,
in a universe with an indestructibly
supercompact cardinal $\gk$ and sufficiently
few large cardinals, neither of these
sets has to be unbounded in $\gk$.
(For the definition of the concept of {\em
level by level equivalence between strong
compactness and supercompactness}, readers
are urged to consult \cite{AS97a}, \cite{AH4},
or \cite{A07}.)
The purpose of this paper is to
carry on the spirit of the work of \cite{A07},
but in the context of investigating the
number of normal measures a measurable
cardinal which is not a limit of measurable
cardinals can carry in a universe containing
an indestructibly supercompact cardinal.
Specifically, we will begin by proving
the following three theorems.
\begin{theorem}\label{t1}
If $\gk$ is indestructibly supercompact
and there is a measurable cardinal
above $\gk$, then both
$A = \{\gd < \gk \mid \gd$ is a
measurable cardinal which is not a
limit of measurable cardinals
and $\gd$ carries the maximal number
of normal measures$\}$
and
$B = \{\gd < \gk \mid \gd$ is a
measurable cardinal which is not a
limit of measurable cardinals
and $\gd$ carries fewer than the maximal number
of normal measures$\}$
are unbounded in $\gk$.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal is supercompact up to a
measurable cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk$ is
indestructibly supercompact +
No cardinal is supercompact up to a
measurable cardinal''.
In $V^\FP$, if $\gl$ is a measurable
cardinal which is not a limit of measurable
cardinals, then $\gl$ carries fewer than the
maximal number of normal measures.
In particular, in $V^\FP$, $\gl$ carries exactly
$\gl^+$ many normal measures.
\end{theorem}
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal is supercompact up to a
measurable cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk$ is
indestructibly supercompact +
No cardinal is supercompact up to a
measurable cardinal''.
In $V^\FP$, if $\gl$ is a measurable
cardinal which is not a limit of measurable
cardinals, then $\gl$ carries the
maximal number of normal measures.
In particular, in $V^\FP$, $\gl$ carries exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures.
\end{theorem}
We will also show that if we weaken
the amount of indestructibility
for our supercompact cardinal $\gk$, then
it is possible to extend Theorem \ref{t3}
to the set of all measurable cardinals.
Specifically, the following is true.
\begin{theorem}\label{t4}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal is supercompact up to a
measurable cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\gk$'s
supercompactness is indestructible under
$\gk$-directed closed forcing preserving
$\gk^+$, $\gk^{++}$, and the value of
$2^{2^\gk}$ +
No cardinal is supercompact up to a
measurable cardinal''.
In $V^\FP$, if $\gl$ is a measurable
cardinal, then $\gl$ carries the
maximal number of normal measures.
In particular, in $V^\FP$, $\gl$ carries exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures.
\end{theorem}
Note that the hypotheses of
Theorems \ref{t2} -- \ref{t4}
are consistent relative to the
existence of a supercompact cardinal.
Also, by a well-known theorem of Solovay
(see \cite{J}), if $\gl$ is
$2^\gl$ supercompact, then $\gl$
carries the maximal number of normal measures.
It is thus not possible to get an analogue of
Theorem \ref{t4} extending Theorem \ref{t2}.
We conclude Section \ref{s1}
with a discussion of
some preliminary material.
We presume a basic knowledge
of large cardinals and forcing.
A good reference in this
regard is \cite{J}.
We also mention that the partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D$ of
conditions of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
We abuse notation slightly and
take both $V^\FP$ and $V[G]$ for
$G$ which is $V$-generic over $\FP$
as being the generic
extension of $V$ by $\FP$.
For $\gk$ a regular cardinal and
$\gl > \gk$ a cardinal,
${\rm Coll}(\gk, \gl)$ is the
standard collapse of
$\gl$ to $\gk$.
For $\gk$ a regular cardinal and
$\gl$ an ordinal, $\add(\gk, \gl)$
is the usual partial ordering for
adding $\gl$ many Cohen subsets of $\gk$.
A normal measure $\U$ over a
measurable cardinal $\gk$ has
{\em trivial Mitchell rank}
if for $j : V \to M$ the
elementary embedding generated by $\U$,
$M \models ``\gk$ is not measurable''.
It is a standard fact (see \cite{J})
that every measurable cardinal
carries a normal measure having
trivial Mitchell rank.
An {\em indestructibly supercompact
cardinal} is one as first given
by Laver in \cite{L}, i.e.,
$\gk$ is indestructibly supercompact
if $\gk$'s supercompactness is
preserved in any generic extension
via a $\gk$-directed closed
partial ordering (including
of course trivial forcing).
The cardinal $\gk$ is {\em supercompact
up to the cardinal $\gl$} if
$\gk$ is $\gd$ supercompact
for every $\gd < \gl$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2} and \cite{H3}
will be employed in the
proofs of Theorems \ref{t1} -- \ref{t4}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2} and \cite{H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\em admits a gap at $\gd$}.
%In Hamkins' terminology of \cite{H2}
%and \cite{H3},
%$\FP$ is {\em 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}, and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2} and \cite{H3}
we will be using
is then the following.
\begin{theorem}\label{t5}
%(\bf Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
%Finally, if $\FP$ is mild with
%respect to $\gk$ and $\gk$ is
%$\gl$ strongly compact in $V[G]$
%for any $\gl \ge \gk$, then
%$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
%\noindent
It immediately follows from
Theorem \ref{t5} that if $\gk$ is
measurable in a generic extension
obtained by forcing admitting
a gap below $\gk$, then $\gk$
is measurable in the ground model $V$.
Hence, if $\gk$ is a
measurable cardinal which is not a limit
of measurable cardinals in
$V$, then $\gk$ is not a limit
of measurable cardinals in
a generic extension obtained by forcing
admitting a gap below $\gk$.
In particular, if $\gk$ is the least
measurable cardinal above some ordinal
$\gd$ in $V$
and $\gk$ remains measurable in a generic
extension $V^\FP$ for $\FP$ a partial
ordering admitting a gap at $\gd$,
then $\gk$ is the least measurable
cardinal above $\gd$ in $V^\FP$.
\section{The Proofs of Theorems
\ref{t1} -- \ref{t4}}\label{s2}
Before beginning the proof of Theorem \ref{t1},
we first prove two general lemmas which
show the existence of sufficiently
directed closed partial orderings
which affect the number of normal
measures a measurable cardinal carries.
\begin{lemma}\label{l1}
Suppose $V \models ``$ZFC +
$\gd < \gl$ are such that
$\gd$ is a regular cardinal and
$\gl$ is a measurable cardinal''.
There is then a $\gd$-directed
closed partial ordering $\FP \in V$
admitting a gap below the least
inaccessible cardinal above $\gd$
such that
$V^\FP \models ``$ZFC + $\gl$ is a
measurable cardinal which carries the
maximal number of normal measures
(namely $ \gl^{++} = 2^{2^\gl}$)''.
\end{lemma}
\begin{proof}
Without loss of generality, by first forcing
with a $\gl^+$-directed closed partial
ordering $\FP^*$ which
forces GCH at $\gl$ and $\gl^+$,
we may assume in addition with a
slight abuse of notation that
$V \models ``2^\gl = \gl^+ +
2^{\gl^+} = \gl^{++}$''.
(If $V \models ``2^\gl = \gl^+ +
2^{\gl^+} = \gl^{++}$'' initially, then
$\FP^*$ is taken as being trivial forcing.)
Observe that $\FP^*$
both preserves $\gl$'s measurability and
creates no new measurable
cardinals below $\gl$.
We then force over $V$ with the reverse Easton
iteration $\FP$ having length $\gl$ which
begins by adding a Cohen subset of $\gd$ and
then adds a Cohen subset to each non-Mahlo
inaccessible cardinal in the open interval
$(\gd, \gl)$.
Note that by its definition, regardless if
$\FP^*$ is trivial or nontrivial,
$\FP$ is $\gd$-directed closed and admits
a gap at $({(\gd)}^{< \gd})^+$
(which is of course well below the least
inaccessible cardinal above $\gd$).
By a standard analysis found, e.g.,
in the proof of Lemma 1.1 of \cite{A01}
or Lemma 6 of \cite{C93},
$V^\FP \models ``\gl$ is a measurable
cardinal which carries
$2^{2^\gl} = \gl^{++}$ many normal measures''.
For the convenience of readers of
this paper, we provide an outline
of the proof here, and refer to
\cite{A01} or \cite{C93} for any missing details.
Let $j : V \to M$ be an elementary
embedding witnessing the measurability
of $\gl$ generated by a normal measure
over $\gl$.
By the definition of $\FP$,
$j(\FP) = \FP \ast \dot \FQ$,
where only trivial forcing is done in $M$
at stage $\gl$.
For $G$ which is $V$-generic over $\FP$,
we may lift $j$ in $V[G]$ to
$j : V[G] \to M[G][H]$, where $H$ is
constructed in $V[G]$ by meeting
in turn each member of an
enumeration $\la D_\ga \mid \ga < \gl^+ \ra \in
V[G]$ of dense open subsets of $\FQ$.
Such an $H$, however, may be built in $V[G]$
by constructing a tree $\cal T$ of height
$2^\gl = \gl^+$, each of whose
branches generates an $M[G]$-generic
object for $\FQ$. Since there are
$2^{\gl^+} = 2^{2^\gl} = \gl^{++}$ many distinct
branches through $\cal T$, there are
$2^{\gl^+} = 2^{2^\gl} = \gl^{++}$ many distinct
values of $H$. Each of these values for $H$
generates a distinct normal measure
over $\gl$ in $V[G]$.
Since the forcing $\FP$ preserves
cardinals, cofinalities, and the value
of the power set function,
this completes the
proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
Suppose $V \models ``$ZFC +
$\gd < \gl$ are such that
$\gd$ is a regular cardinal and
$\gl$ is a measurable cardinal''.
There is then a $\gd$-directed
closed partial ordering $\FQ \in V$
such that
$V^\FQ \models ``$ZFC + $\gl$ is a
measurable cardinal which carries fewer than
the maximal number of normal measures''.
In particular, $V^\FQ \models ``\gl$ carries
$\gl^+$ many normal measures''.
\end{lemma}
\begin{proof}
Without loss of generality,
we assume that $V$ has been generically
extended to the model produced in
Lemma \ref{l1}. With a harmless abuse
of notation, we denote for the duration
of this lemma this model also as $V$.
We now use an argument due to
Cummings, which also appears in
the proof of the Main Theorem of
\cite{ACH}, the proof of Lemma 2.1
of \cite{A06}, and the proof of
Lemma 2.2 of \cite{A07a}.
First, note that by our
assumptions on $V$,
$V \models ``\gl$ carries exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures''.
Define $\FQ$ as $\add(\gd, 1)
\ast \dot {\rm Coll}(\gl^+, \gl^{++})$.
Suppose
$G_0$ is $V$-generic over $\add(\gd, 1)$ and
$G_1$ is $V[G_0]$-generic over
${\rm Coll}(\gl^+, \gl^{++})$.
By the L\'evy-Solovay results \cite{LS},
since $\card{\add(\gd, 1)} < \gl$,
$V[G_0] \models ``\gl$ is a measurable
cardinal carrying
exactly $\gl^{++}$ many normal measures''.
%every normal measure over $\gl$ in $V$
%generates a unique normal measure over $\gl$ in $V[G_0]$.
These remain normal measures over $\gl$
in $V[G_0][G_1]$, since no additional subsets of $\gl$ are
added by the collapse forcing.
Thus, since ${(\gl^+)}^V$
is preserved to $V[G_0][G_1]$,
there are at least
$\gl^+$ many normal measures over $\gl$ in $V[G_0][G_1]$.
Conversely, suppose that ${\cal U}$ is a normal measure over
$\gl$ in $V[G_0][G_1]$, with the associated ultrapower
embedding $j: V[G_0][G_1]\to M[G_0][j(G_1)]$. In particular,
$X\in{\cal U}$ iff $\gl\in j(X)$ for all $X
\subseteq \gl$ in $V[G_0][G_1]$. By Theorem
\ref{t5}, it follows that the restriction
$j\rest V: V\to M$ is a definable class in $V$.
Once more by the results of \cite{LS},
since $\card{\add(\gd, 1)} < \gl$,
$j\rest V$ lifts
uniquely to $V[G_0]$,
and so $j\rest V[G_0]:V[G_0]\to M[G_0]$
is a definable class in $V[G_0]$. The key observation is now
that because $V[G_0]$ and $V[G_0][G_1]$ have the same subsets of
$\gl$, one can reconstruct ${\cal U}$ inside $V[G_0]$ by
observing $X\in{\cal U}$ iff $\gl\in j(X)$,
using only $j\rest V[G_0]$.
Thus, ${\cal U}\in V[G_0]$. Consequently,
every normal measure over $\gl$ in $V[G_0][G_1]$ is actually
in $V[G_0]$. The number of such normal measures, therefore,
is at most ${(\gl^{++})}^{V[G_0]}$, which is $\gl^+$ in
$V[G_0][G_1]$, because ${(\gl^{++})}^{V[G_0]}$ was collapsed by
$G_1$. Hence, in $V[G_0][G_1]$, there are exactly $\gl^+$ many
normal measures over $\gl$, as desired.
Since all partial orderings used are
$\gd$-directed closed,
this completes the proof of Lemma \ref{l2}.
\end{proof}
Having completed the proofs of
Lemmas \ref{l1} and \ref{l2},
we turn now to the proof of
Theorem \ref{t1}.
\begin{proof}
Suppose
$V \models ``\gk < \gl$ are such
that $\gk$ is indestructibly supercompact and
$\gl$ is the least measurable
cardinal above $\gk$''.
%Assume also that $\gl$ is the least measurable cardinal above $\gk$.
Take $\FP$ as the partial ordering
of Lemma \ref{l1}, defined with
$\gd = \gk$.
By the definition of
$\FP$, Lemma \ref{l1}, and the
remarks in the paragraph immediately
following the statement of
Theorem \ref{t5},
$V^\FP \models ``\gl$ is the
least measurable cardinal above
$\gk$ and $\gl$ carries the
maximal number of normal measures''.
In particular,
$V^\FP \models ``\gl$ is a
measurable cardinal which is not
a limit of measurable cardinals
and $\gl$ carries the
maximal number of normal measures''.
Since $\FP$ is $\gk$-directed closed,
$V^\FP \models ``\gk$ is supercompact'',
so by reflection,
$A = \{\gd < \gk \mid \gd$ is a
measurable cardinal which is not a
limit of measurable cardinals
and $\gd$ carries the maximal number
of normal measures$\}$
is unbounded in $\gk$ in $V^\FP$.
If we now force over $V^\FP$ with
the partial ordering $\FQ$ of Lemma \ref{l2},
then by the definition of
$\FQ$, Lemma \ref{l2}, and the
remarks in the paragraph immediately
following the statement of
Theorem \ref{t5},
$V^{\FP \ast \dot \FQ} \models ``\gl$ is the
least measurable cardinal above
$\gk$ and $\gl$ carries fewer than the
maximal number of normal measures''.
In particular,
$V^{\FP \ast \dot \FQ} \models ``\gl$ is a
measurable cardinal which is not
a limit of measurable cardinals
and $\gl$ carries fewer than the
maximal number of normal measures''.
Once again,
since $\FQ$ is $\gk$-directed closed,
$V^{\FP \ast \dot \FQ} \models ``\gk$ is supercompact'',
so by reflection,
$B = \{\gd < \gk \mid \gd$ is a
measurable cardinal which is not a
limit of measurable cardinals
and $\gd$ carries fewer than the maximal number
of normal measures$\}$
is unbounded in $\gk$ in $V^{\FP \ast \dot \FQ}$.
Since both $\FP$ and $\FP \ast \dot \FQ$
are $\gk$-directed closed, $A$ and $B$
are each unbounded in $\gk$ in $V$.
This completes the proof of Theorem \ref{t1}.
\end{proof}
Having finished with the proof of
Theorem \ref{t1}, we turn now to
the proof of Theorem \ref{t2}.
\begin{proof}
We begin the proof of Theorem \ref{t2}
with a useful definition.
For $\gd < \gk$ 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.
Suppose now $V \models ``$ZFC +
$\gk$ is supercompact +
No cardinal is supercompact up
to a measurable cardinal''.
If necessary, by first doing
the appropriate forcing,
we may assume in addition that
$V \models {\rm GCH}$.
Let $f$ be a Laver function
\cite{L} for $\gk$, i.e.,
$f : \gk \to V_\gk$ is such that
for every $x \in V$ and every
$\gl \ge \max(\gk, \card{{\rm TC}(x)})$, there is
an elementary embedding
$j : V \to M$ generated by a
supercompact ultrafilter over
$P_\gk(\gl)$ such that
$j(f)(\gk) = x$.
The partial ordering $\FP$
which is used to establish
Theorem \ref{t2} is the
reverse Easton iteration of
length $\gk$ which begins by
adding a Cohen subset of
$\go$ and then does nontrivial
forcing only at those
cardinals $\gl < \gk$ which
are measurable in $V$.
At these stages $\gl$, we divide
into two cases.
\bigskip\noindent Case 1: $\gl$ is not
a limit of measurable cardinals in $V$.
In this situation, we force with
$\FP^{0, \gl} \ast \dot \FP^{1, \gl}$. Here,
$\FP^{0, \gl}$ is the partial ordering of
Lemma \ref{l1} defined with
$\gd = \gg_\gl$, and $\dot \FP^{1, \gl}$
is a term for the partial ordering of
Lemma \ref{l2} defined with
$\gd = \gg_\gl$.
%$\add(\gg_\gl, 1) \ast \dot {\rm Coll}(\gl^+, \gl^{++})$.
\bigskip\noindent Case 2: $\gl$ is a
limit of measurable cardinals in $V$.
At such a stage $\gl$, if
$f(\gl)$ is a term always forced
to denote a $\gl$-directed
closed partial ordering $\FQ$ having
rank below the least $V$-measurable
cardinal above $\gl$, then we
force with $\FQ$.
If this is not the case, then
we perform trivial forcing.
\begin{lemma}\label{l3}
If
$V^{\FP} \models ``\gl$ is a measurable
cardinal which is not a limit of
measurable cardinals'', then
$V \models ``\gl$ is a measurable
cardinal which is not a limit of
measurable cardinals''.
\end{lemma}
\begin{proof}
Our methods are similar to those
used in the proof of Lemma 2.1 of \cite{A07a}.
Suppose
$V^{\FP} \models ``\gl$ is a measurable
cardinal which is not a limit of
measurable cardinals''.
Note that it is possible to write ${\FP}$ as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is
nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_2$-directed
closed''. Hence, as we observed
at the end of Section \ref{s1}, since any
cardinal which is measurable in $V^{\FP}$
had to have been measurable in $V$,
$V \models ``\gl$ is measurable''.
Thus, to prove Lemma \ref{l3},
it suffices to show that
$V \models ``\gl$ is not a limit of
measurable cardinals''.
To do this, assume to the contrary that
$V \models ``\gl$ is a limit of
measurable cardinals''. In particular,
$V \models ``\gl$ is a limit of
measurable cardinals $\gd$ which themselves
are not limits of measurable cardinals''.
Also, since
$V \models ``$No cardinal is supercompact
up to a measurable cardinal and $\gk$
is supercompact'', we may infer that
$\gl \le \gk$ (and indeed, that any
$V$-measurable cardinal is less than
or equal to $\gk$). Therefore,
for any measurable cardinal $\gd$ which
is not a limit of measurable
cardinals, we may write
${\FP} = {\FP_\gd} \ast \dot \FP^{0, \gd}
\ast \dot \FP^{1, \gd} \ast \dot \FR' =
\FQ' \ast \dot \FR'$.
By Lemma \ref{l2}, $V^{\FQ'} \models ``\gd$ is
a measurable cardinal (and carries exactly
$\gd^+$ many normal measures)''.
Since
$\forces_{\FQ'} ``$Forcing with
$\dot \FR'$ adds no bounded subsets of the
least inaccessible cardinal above $\gd$'',
%$\gd^+$-directed closed'',
$V^{\FQ' \ast \dot \FR'}
= V^{\FP} \models
``\gd$ is a measurable cardinal
(and carries exactly $\gd^+$ many
normal measures)''. Thus,
$V^{\FP} \models ``\gl$ is a measurable
cardinal which is a limit of measurable cardinals''.
This contradiction completes
the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^{\FP} \models ``$If $\gl$ is a measurable
cardinal which is not a limit of measurable
cardinals, then $\gl$ carries exactly
$\gl^+$ many normal measures''.
\end{lemma}
\begin{proof}
%Our methods are similar to the proof of
%Lemma 2.2 of \cite{A07a}.
Suppose
$V^{\FP} \models ``\gl$ is a measurable
cardinal which is not a limit of
measurable cardinals''.
By Lemma \ref{l3},
$V \models ``\gl$ is a measurable
cardinal which is not a limit of
measurable cardinals''.
By the proof of Lemma \ref{l3},
$V^{\FP} \models ``\gl$ carries exactly
$\gl^+$ many normal measures''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is
indestructibly supercompact''.
\end{lemma}
\begin{proof}
Our proof is the same as the
one given in the proof of
Lemma 2.1 of \cite{A07}.
Let $\FQ \in V^\FP$ be such that
$V^\FP \models ``\FQ$ is
$\gk$-directed closed''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{\FP} ``\dot \FQ$ is
$\gk$-directed closed''.
Suppose $\gl \ge
\max(\gk^+, \card{{\rm TC}(\dot \FQ)})$
is an arbitrary cardinal, and let
$\gg = 2^{\card{[\gl]^{< \gk}}}$. Take
$j : V \to M$ as an elementary
embedding witnessing the $\gg$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gg)$ such that
$j(f)(\gk) = \dot \FQ$. Since
$V \models ``$No cardinal is supercompact
up to a measurable cardinal and
$\gk$ is supercompact'',
$V \models ``$No cardinal above
$\gk$ is measurable''.
Therefore, because $\gg \ge 2^\gk$
and $M^\gg \subseteq M$,
$M \models ``\gk$ is measurable and
no cardinal in the half-open interval
$(\gk, \gg]$ is measurable''. Hence,
the definition of $\FP$ implies that
$j(\FP \ast \dot \FQ) = \FP \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first nontrivial stage of
forcing in $\dot \FR$ takes
place well above $\gg$.
%$2^{[\gg]^{< \gk}}$.
Laver's original argument from \cite{L} now applies
and shows
$V^{\FP \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be $V$-generic over
$\FP \ast \dot \FQ \ast \dot \FR$,
lift $j$ in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
$V[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ)$ containing $p$, lift $j$
again in $V[G_0][G_1][G_2][G_3]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gg^+$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gl))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0][G_1]$.)
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of
Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$No cardinal
is supercompact up to a
measurable cardinal''.
\end{lemma}
\begin{proof}
Our proof is the same as the one
given in the proof of Lemma 2.2
of \cite{A07}.
We use the factorization of $\FP$
given in Lemma \ref{l3}.
Since $\FP$ admits this factorization,
%by Hamkins' Gap Forcing Theorem of \cite{H2} and \cite{H3},
by Theorem \ref{t5},
for any pair of cardinals
$\gd \le \gl$, if
$V^\FP \models ``\gd$ is $\gl$
supercompact'', then
$V \models ``\gd$ is $\gl$ supercompact''
as well. Consequently, since
$V \models ``$No cardinal is
supercompact up to a measurable cardinal'',
it is also the case that
$V^\FP \models ``$No cardinal is
supercompact up to a measurable cardinal''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l3} -- \ref{l6} complete the
proof of Theorem \ref{t2}.
\end{proof}
\begin{pf}
The proof of Theorem \ref{t3}
is now almost exactly the same as the
proof of Theorem \ref{t2}.
The partial ordering $\FP$ of
Theorem \ref{t3} is the same as
the one used in the proof of Theorem \ref{t2},
with the exception that in Case 1, we force only
with $\FP^{0, \gl}$ and not with $\FP^{1, \gl}$.
More specifically,
suppose once again that $V \models ``$ZFC + GCH +
$\gk$ is supercompact +
No cardinal is supercompact up
to a measurable cardinal''.
Let $f$ be a Laver function
for $\gk$.
The partial ordering $\FP$
which is used to establish
Theorem \ref{t3} is the
reverse Easton iteration of
length $\gk$ which begins by
adding a Cohen subset of
$\go$ and then does nontrivial
forcing only at those
cardinals $\gl < \gk$ which
are measurable in $V$.
At these stages $\gl$, we divide
into two cases.
\bigskip\noindent Case 1: $\gl$ is not
a limit of measurable cardinals in $V$.
In this situation, we force with
$\FP^{0, \gl}$.
\bigskip\noindent Case 2: $\gl$ is a
limit of measurable cardinals in $V$.
At such a stage $\gl$, if
$f(\gl)$ is a term always forced
to denote a $\gl$-directed
closed partial ordering $\FQ$ having
rank below the least $V$-measurable
cardinal above $\gl$, then we
force with $\FQ$.
If this is not the case, then
we perform trivial forcing.
\bigskip
The analogues of Lemmas \ref{l5} and \ref{l6}
then go through exactly as before.
The analogue of Lemma \ref{l3} is then
virtually the same as earlier, with the
exception that since we do not force
with $\FP^{1, \gl}$ for the measurable
cardinal $\gl$ of Lemma \ref{l3},
such measurables carry exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures.
The analogue of Lemma \ref{l4} then
goes through essentially as before, but
this time shows that any measurable cardinal
$\gl$ which is not a limit of measurable cardinals
carries exactly $\gl^{++} = 2^{2^\gl}$ many
normal measures in $V^\FP$.
This completes the proof of Theorem \ref{t3}.
\end{pf}
\begin{pf}
For the proof of Theorem \ref{t4},
we once again assume that
$V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is
supercompact up to a measurable cardinal''.
The partial ordering $\FP$ is the
same as the one used to establish
Theorem \ref{t3}, but with one
slight modification. This is that
in Case 2, we only do nontrivial forcing
at a stage $\gl$ which is a limit of
measurable cardinals in $V$ when
$f(\gl)$ is a term always forced to
denote a $\gl$-directed closed
partial ordering $\FQ$ preserving
$\gl^+$, $\gl^{++}$, and $2^{2^\gl}$
having rank
below the least $V$-measurable cardinal
above $\gl$.
The analogue of Lemma \ref{l6} is
once again exactly as before.
The analogue of Lemma \ref{l5}
is almost the same as before, with
the exception that the only
nontrivial $\gk$-directed closed
forcing allowed at
stage $\gk$ in $M$ is one which
preserves $\gk^+$, $\gk^{++}$, and $2^{2^\gk}$.
This is the reason the amount of
indestructibility obtained is under
$\gk$-directed closed partial orderings preserving
$\gk^+$, $\gk^{++}$, and $2^{2^\gk}$.
This of course includes the trivial
partial ordering, so that $\gk$ is
supercompact in $V^\FP$.
The proof of Theorem \ref{t4} is
now completed by the following lemma.
\begin{lemma}\label{l7}
$V^\FP \models ``$If $\gl$ is a
measurable cardinal, then $\gl$ carries exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures''.
%the maximal number''.
\end{lemma}
\begin{proof}
Since $V^\FP \models ``$No cardinal
is supercompact up to a measurable cardinal
and $\gk$ is supercompact'',
$V^\FP \models ``$No cardinal above $\gk$ is measurable''.
Hence, if $V^\FP \models ``\gl$ is
measurable'', as before, it follows that $\gl \le \gk$.
Further, since
$\FP$ may be defined so that
$\card{\FP} = \gk$ and
$V \models {\rm GCH}$, we know that
$V^\FP \models ``$GCH holds for all cardinals
at and above $\gk$''. The supercompactness of
$\gk$ in $V^\FP$ then allows us to infer that
$V^\FP \models ``\gk$ carries exactly
$\gk^{++} = 2^{2^\gk}$ many normal measures''.
We may therefore assume
for the rest of the proof of this lemma that $\gl < \gk$.
By the factorization given in
Lemma \ref{l3} (which remains valid
for the current definition of $\FP$) and
Theorem \ref{t5} and the remarks immediately following,
if $V^\FP \models ``\gl$ is measurable'', then
$V \models ``\gl$ is measurable'' as well.
Fix $\U$ a normal measure over
$\gl$ having trivial Mitchell rank, and let
$j : V \to M$ be the elementary
embedding generated by $\U$.
Write $\FP = \FP_\gl \ast \dot \FP^\gl$. Since
$M \models ``\gl$ is not measurable'',
$j(\FP) = \FP \ast \dot \FQ$, where only
trivial forcing is done at stage $\gl$ in $M$.
The arguments of Lemma \ref{l1} then apply
and show that
$V^{\FP_\gl} \models ``\gl$ carries exactly
$\gl^{++} = 2^{2^\gl}$ many normal measures''.
Since regardless if $\gl$ is a limit of
measurable cardinals in $V$, the
current definition of $\FP$ implies that
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
$\gl$-directed closed and
forcing with $\dot \FP^\gl$
preserves $\gl^+$, $\gl^{++}$, and $2^{2^\gl}$'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models
``\gl$ carries exactly $\gl^{++} = 2^{2^\gl}$
many normal measures''.
This completes the proof of both
Lemma \ref{l7} and Theorem \ref{t4}.
\end{proof}
\end{pf}
We note that analogues to
Theorems \ref{t1} -- \ref{t4}
hold if $\gk$ is an indestructibly strong
cardinal in the sense of \cite{GS}.
We leave it to readers to work out
the specific details for themselves.
Of course, the amount of indestructibility for $\gk$
obtained in Theorem \ref{t4} is less than optimal.
We therefore ask if it is possible
to prove an analogue of Theorem \ref{t4}
in which $\gk$'s supercompactness is
fully indestructible under arbitrary
$\gk$-directed closed forcing.
In addition, as pointed out by the referee,
we may ask if the models for Theorems
\ref{t2} and \ref{t3} of this paper
may also witness the (mutually exclusive)
properties of the models of Theorems 3
and 4 of \cite{A07}.
These are the questions with which we conclude.
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\end{document}