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\title{How Many Normal Measures Can
$\alom$ Carry?
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, measurable cardinal,
normal measure, indestructibility, gap forcing,
symmetric inner model.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grant
66489-00-35 and a CUNY Collaborative
Incentive Grant.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{September 5, 2005\\
(revised March 19, 2006)}
\begin{document}
\maketitle
\begin{abstract}
We show that assuming the consistency
of a supercompact cardinal with a
measurable cardinal above it, it is
possible for $\alom$ to be measurable
and to carry exactly
$\gt$ normal measures, where $\gt \ge
\aleph_{\omega + 2}$ is any regular cardinal.
This contrasts with the
%well-known
fact that assuming
AD + DC, $\alom$ is measurable and carries exactly
three normal measures.
Our proof uses the methods of
\cite{AM}, along with a folklore technique
and a new method due to James Cummings.
\end{abstract}
\baselineskip=24pt
%\newpage
\section{Introduction and Preliminaries}\label{s1}
It is a consequence of
AD + DC that $\alom$ is a measurable cardinal
and carries exactly three normal measures.
This follows since assuming AD + DC,
there are only three regular cardinals
(namely $\ha_0$, $\ha_1$, and $\ha_2$)
below $\alom$, AD + DC implies that
$\alom$ satisfies
the strong partition property
$\alom \to {(\alom)}^{\alom}$,
and if a successor
cardinal $\gk$ satisfies the weak partition
property $\forall \gd < \gk
[\gk \to {(\gk)}^\gd]$, then $\gk$ is
measurable and carries exactly the same number of
normal measures as regular cardinals
below $\gk$.
(In fact, if a successor cardinal
$\gk$ satisfies the weak partition property,
then any normal measure $\gk$ carries
must be of the form
$\{x \subseteq \gk \mid x$ contains a set
which is $\gd$ club$\}$, where
$\gd < \gk$ is a regular cardinal.)
The proofs of these last three facts
can be found respectively in
\cite{Kl2} (see also \cite{Kl1}),
\cite{Ja1} (see also \cite{Ja2}),
and \cite{Kl3}.
When the Axiom of Determinacy is not
assumed, however, the situation
concerning the number of normal
measures that $\alom$ can carry if
$\alom$ is measurable
is not so clear.
In fact, in the articles
\cite{A85}, \cite{A90}, \cite{A92},
and \cite{AM}, in which the measurability
of $\alom$ is forced from
supercompactness hypotheses,
the number of normal measures $\alom$
possesses in the relevant models constructed
is completely unclear.
The purpose of this paper is to
shed new light on the situation mentioned in
the preceding paragraph and
construct via forcing models in which
$\alom$ is measurable and carries
%a fixed number of normal measures.
exactly $\gt$ normal measures,
where $\gt \ge \aleph_{\omega + 2}$
is any regular cardinal.
Specifically, we will prove
the following two theorems.
%two theorems, the first of which is as follows.
\begin{theorem}\label{t1}
Let
$V^* \models ``$ZFC + GCH +
$\gk < \gl$
are such that $\gk$ is supercompact
and $\gl$ is the least measurable
cardinal above $\gk$ +
$\gt > \gl^+$ is a fixed but
arbitrary regular cardinal''.
There is then a
generic extension $V$ of $V^*$, a
partial ordering $\FP \in V$, and a
symmetric submodel $N \subseteq V^\FP$
such that
$N \models ``$ZF + ${DC}_{\ha_\go}$ +
$\gl = \alom$ is a measurable cardinal''.
In $N$,
the cardinal and cofinality structure
at and above $\gl$ is the same
as in $V$ (which has the same
cardinal and cofinality structure
at and above $\gl$ as $V^*$),
and $\alom$ carries
exactly $\gt$ normal measures.
\end{theorem}
\begin{theorem}\label{t2}
Let
$V^* \models ``$ZFC + GCH +
$\gk < \gl$
are such that $\gk$ is supercompact
and $\gl$ is the least measurable
cardinal above $\gk$''.
There is then a
generic extension $V$ of $V^*$, a
partial ordering $\FP \in V$, and a
symmetric submodel $N \subseteq V^\FP$
such that
$N \models ``$ZF + ${DC}_{\ha_\go}$ +
$\gl = \alom$ is a measurable cardinal''.
In $N$, $\ha_{\go + 2}$ is regular, and
$\alom$ carries
exactly $\ha_{\go + 2}$ normal measures.
\end{theorem}
Theorems \ref{t1} and \ref{t2}
provide our desired results.
Taken together, these
theorems show that
relative to the appropriate
assumptions, it is consistent for
$\alom$ to be measurable and
to carry $\gt$ normal
measures, where $\gt \ge
\ha_{\go + 2}$ is any regular cardinal.
We digress now to provide
some preliminary information.
Essentially, our notation and terminology
are standard, although exceptions
to this will be noted.
For $\ga < \gb$ ordinals,
$[\ga, \gb]$, $[\ga, \gb)$,
$(\ga, \gb]$, and $(\ga, \gb)$
are as in the usual interval notation.
For $x$ a set of ordinals,
$\ov x$ is the order type of $x$.
When forcing, $q \ge p$ means that
$q$ {\rm is stronger than} $p$.
For $\gk$ a regular cardinal,
the partial ordering $\FP$ is
{\rm $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has a
common extension.
For $\gk$ regular and $\gl$
any ordinal,
$\add(\gk, \gl)$ is the standard
partial ordering for adding
$\gl$ Cohen subsets to $\gk$.
We abuse notation somewhat and
use both $V^\FP$ and $V[G]$
to denote the generic extension
by the partial ordering $\FP$.
If $x \in V[G]$, then $\dot x$
will be a term in $V$ for $x$.
We may, from time to time,
confuse terms with the sets
they denote and write $x$ when
we actually mean $\dot x$ or
$\check x$, especially when $x$
is some variant of the generic
set $G$, or $x$ is in the
ground model $V$.
For $\gk < \gl$ regular cardinals,
${\rm Coll}(\gk, \gl)$ is the
standard L\'evy partial ordering for
collapsing $\gl$ to $\gk$.
${\rm Coll}(\gk, {<} \gl)$ is the
standard L\'evy partial ordering for
collapsing every cardinal
$\gd \in (\gk, \gl)$ to $\gk$.
For such a $\gd$ and any
$S \subseteq {\rm Coll}(\gk, {<} \gl)$,
we define $S \rest \gd =
\{p \in S \mid \dom(p) \subseteq
\gk \times \gd\}$.
It is well-known that if $G$ is
$V$-generic over ${\rm Coll}(\gk, {<} \gl)$
and $\gd \in (\gk, \gl)$
is a cardinal, then
$G \rest \gd$ is $V$-generic over
${\rm Coll}(\gk, {<} \gl) \rest \gd$.
Note that we are assuming
familiarity with the large cardinal
notions of measurability
and supercompactness.
Interested readers may consult \cite{K}
%or \cite{SRK}
for further details.
We conclude Section \ref{s1}
by mentioning that there
are two results
critical to the proofs of
Theorems \ref{t1} and \ref{t2}
which will be taken as ``black boxes''.
For the convenience of readers,
we provide a brief discussion
of these facts here.
The first concerns the folklore
result that if
$V \models ``$ZFC + $\gk$ is
a measurable cardinal + $2^\gk
= \gk^+$'', then
any reverse Easton iteration
adding a single Cohen
subset to every element of
an unbounded normal measure $0$ subset
of $\gk$ (such as a set
of successor cardinals) preserves
the measurability of $\gk$
and increases the number of
normal measures $\gk$ carries
to $2^{2^{\gk}} = 2^{\gk^+}$.
This is the essential content of
Lemma 1.1 of \cite{A01}.
The second is Hamkins' Gap
Forcing Theorem of \cite{H2}
and \cite{H3}.
We state the version of
this theorem we will use here, 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
$\card{\FQ} < \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\rm admits a gap at $\gd$}.
Also, as in the terminology of
\cite{H2} and \cite{H3} (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 relevant form of the
Gap Forcing Theorem is then
the following.
\begin{theorem}\label{t2a}
({\bf Hamkins})
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a gap at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ 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 $V$.
\end{theorem}
\noindent It immediately follows from
Theorem \ref{t2a} that any cardinal
$\gk$ measurable in a generic extension
obtained by forcing that admits a gap
below $\gk$ must also be measurable in the
ground model.
\section{The Proofs of
Theorems \ref{t1} and \ref{t2}}\label{s2}
We turn now to the proofs
of Theorems \ref{t1} and \ref{t2}.
The proofs of these theorems
are quite similar to one another,
so we prove them in tandem,
%distinguishing between the proofs when necessary.
making the relevant distinctions when necessary.
\begin{proof}
Let $V^* \models ``$ZFC + GCH +
$\gk < \gl$
are such that $\gk$ is supercompact
and $\gl$ is the least measurable
cardinal above $\gk$''.
%be as in the hypotheses for Theorem \ref{t1}.
By Laver's result of \cite{L},
we assume that $V^*$ has been
generically extended via the
partial ordering $\FL$ to a
model $\ov V$ such that
$\ov V \models ``\gk$ is indestructibly
supercompact'', i.e.,
$\ov V \models ``\gk$ is supercompact
and $\gk$'s supercompactness is
indestructible under $\gk$-directed
closed forcing''.
Both Theorems \ref{t1} and
\ref{t2} will require that $\ov V$
first be generically extended
to a model $V$ in which
$\gl$ remains the
least measurable cardinal above $\gk$
and carries the appropriate
number of normal measures.
For Theorem \ref{t1},
let $\gt > \gl^+$ be a fixed
but arbitrary regular cardinal
in $V^*$.
We show that $\ov V$ may be generically
extended further to a model $V$ such that
$V \models ``\gk$ is supercompact +
$\gl$ is the least measurable
cardinal above $\gk$ +
$\gl$ carries $\gt$
normal measures''. To do this,
since $\FL$ may
be defined so that $\card{\FL} = \gk$,
standard arguments in tandem with
the L\'evy-Solovay
results \cite{LS} allow us
to assume in addition that
$\ov V \models ``$GCH holds at and above
$\gk$ + $\gl$ is the least
measurable cardinal above $\gk$ +
The cardinal and cofinality structure
at and above $\gk$ is the same as in $V^*$''.
In particular, this means we may infer
that $\ov V \models ``\gt > \gl^+$
is a regular cardinal''.
Let $V$ be the generic extension
obtained by forcing over $\ov V$
with the partial ordering
$\add(\gl^+, \gt) \ast \dot \FR$, where
$\dot \FR$ is a term for the reverse
Easton iteration of length $\gl$
which begins by adding
a Cohen subset to $\gk^+$
and then adds a Cohen subset to
the successor of each
inaccessible cardinal in the
open interval $(\gk^+, \gl)$.
By its definition,
$\add(\gl^+, \gt) \ast \dot \FR$
is $\gk$-directed closed,
which means by indestructibility that
$V \models ``\gk$ is supercompact''.
Further, since $\add(\gl^+, \gt)$
is $\gl^+$-directed closed
and GCH holds at and above $\gk$
in $\ov V$
(GCH holding at and above $\gl$
in $\ov V$ is sufficient for what follows),
$\gl$ remains the least
measurable cardinal above
$\gk$ in
$\ov V^{\add(\gl^+, \gt)}$, and
$\ov V^{\add(\gl^+, \gt)} \models
``2^\gl = \gl^+$ and $2^{\gl^+} = \gt$''.
Therefore, by Lemma 1.1 of \cite{A01},
$V \models ``\gl$ is measurable and carries
$\gt$ normal measures''.
However, by Theorem \ref{t2a},
any cardinal in the open interval
$(\gk^+, \gl)$ measurable in $V$
had to have been measurable in
$\ov V^{\add(\gl^+, \gt)}$.
Since
$\ov V^{\add(\gl^+, \gt)} \models ``\gl$ is the
least measurable cardinal above $\gk$'',
$V \models ``\gl$ is the
least measurable cardinal above $\gk$''
as well.
In addition, by our GCH assumptions,
forcing over $\ov V$ with
$\add(\gl^+, \gt) \ast \dot \FR$
preserves the cardinality and
cofinality structures at and above $\gl$.
For Theorem \ref{t2}, we need
to show that $\ov V$ can be generically
extended further to a model $V$ such that
$V \models ``\gk$ is supercompact +
$\gl$ is the least measurable
cardinal above $\gk$ +
$\gl$ carries $\gl^+$ normal measures''.
To do this, we use a new method
due to James Cummings, which
appears in \cite{ACH} in a
broader context. We isolate
Cummings' techniques in the following lemma,
which we state in a slightly generalized form.
\begin{lemma}\label{lcu}
Suppose
$M \models ``$ZFC + $\gd$ is measurable +
GCH holds at and above $\gd$''.
Then for any $\gg < \gd$, there is a
$\gg$-directed closed partial ordering
$\FP$ such that
$M^\FP \models ``$ZFC + $\gd$ is measurable +
$\gd$ carries $\gd^+$ normal measures''.
\end{lemma}
\begin{proof}
Let $M$ be as in the hypotheses of
Lemma \ref{lcu}.
As above, if we first force over $M$ with
$\add(\gd^+, \gd^{++}) \ast \dot \FR$, where
$\dot \FR$ is a term for the reverse
Easton iteration of length $\gd$
which begins by adding
a Cohen subset to $\gg^+$
and then adds a Cohen subset to
the successor of each
inaccessible cardinal in the
open interval $(\gg^+, \gd)$,
we obtain a model in which
$\gd$ carries
$2^{2^\gd} = 2^{\gd^+} = \gd^{++}$ normal measures.
By its definition, this forcing is
$\gg$-directed closed.
With a slight abuse of notation, we
denote for the rest of Lemma \ref{lcu}
the model which results after the forcing
also as $M$.
Working in $M$,
let $\FQ = \FQ_0 \ast \dot \FQ_1$, where
$\FQ_0 = \add(\gg^+, 1)$, and $\dot \FQ_1$
is a term for ${\rm Coll}(\gd^+, \gd^{++})$.
Since $\card{\FQ_0} < \gd$, by the results of
\cite{LS}, $M^{\FQ_0} \models ``\gd$ is
measurable''. Therefore, as
$M^{\FQ_0} \models ``\FQ_1$ is
$\gd^+$-directed closed''
(which means that $M^{\FQ_0}$ and
$M^{\FQ_0 \ast \dot \FQ_1}$ contain
the same subsets of $\gd$),
$M^{\FQ_0 \ast \dot \FQ_1} \models
``\gd$ is measurable'' as well.
In particular, any normal measure over
$\gd$ in $M^{\FQ_0}$ remains a
normal measure over $\gd$ in
$M^{\FQ_0 \ast \dot \FQ_1}$.
Let $M^* = M^{\FQ_0 \ast \dot \FQ_1}$.
By the preceding paragraph, let
${\cal U}^* \in M^*$ be a normal
measure over $\gd$, with
$j^* : M^* \to N^*$ the associated
ultrapower embedding. Note that
$N^* = N^{j^*(\FQ_0 \ast \dot \FQ_1)}$
for the appropriate model $N$.
In addition, $N^*$
has the properties that
$N^* \subseteq M^*$ and
${(N^*)}^\gd \subseteq N^*$
(so in particular, for any
$\eta < \gd$, ${(N^*)}^\eta \subseteq N^*$).
%and $N^* = N^{j^*(\FQ_0 \ast \dot \FQ_1)}$.
Since $\FQ_0 \ast \dot \FQ_1$ is such that
$\card{\FQ_0} = \card{[\gg^+]^{\gg}}
< \gd$ and
$\forces_{\FQ_0} ``\dot \FQ_1$ is
$\card{\FQ_0}^{++}$-directed closed'',
by Theorem \ref{t2a},
$j^*$ must lift an elementary embedding
$j : M \to N$ such that
$j \rest A \in M$ for any
$A \in M$.
Hence, for ${\cal U} =
\{x \subseteq \gd \mid \gd \in j(x)\}$,
${\cal U} \in M$, ${\cal U}$ is a
normal measure over $\gd$, and
${\cal U} \subseteq {\cal U}^*$.
%for $N \subseteq M$
%Work now in $M^{\FQ_0}$.
By the results of \cite{LS},
${\cal U}' = \{x \subseteq \gd \mid
\exists y \subseteq x[y \in {\cal U}]\}$
is in $M^{\FQ_0}$ a normal measure over
$\gd$.
As was mentioned above,
%Since $M^{\FQ_0}$ and $M^{\FQ_0 \ast \dot \FQ_1}$ contain
%the same subsets of $\gd$,
${\cal U}'$ is a normal measure over
$\gd$ in $M^{\FQ_0 \ast \dot \FQ_1}$
as well. However, by their definitions,
it must be the case that
${\cal U}' = {\cal U}^*$, since otherwise,
if $x \in {\cal U}^*$ but
$x \not\in {\cal U}'$, then
$\gd - x \in {\cal U}'$.
This means that $x$ is disjoint
from a set in ${\cal U}$, which
is absurd since ${\cal U} \subseteq {\cal U}^*$.
Thus, it is actually the case that
${\cal U}^* \in M^{\FQ_0}$, i.e.,
any normal measure over $\gd$ in
$M^{\FQ_0 \ast \dot \FQ_1}$ is actually
an element of $M^{\FQ_0}$.
However, again by the results of
\cite{LS}, there are the same
number of normal measures over $\gd$
in $M^{\FQ_0}$ as there are in $M$, i.e.,
there are ${(\gd^{++})}^M =
{(\gd^{++})}^{M^{\FQ_0}}$ normal measures
over $\gd$ in $M^{\FQ_0}$. Consequently, for
$\gz = {(\gd^+)}^{M} = {(\gd^+)}^{M^{\FQ_0}}
= {(\gd^+)}^{M^{\FQ_0 \ast \dot \FQ_1}}$, as
$M^{\FQ_0 \ast \dot \FQ_1} \models
``\card{{(\gd^{++})}^{M^{\FQ_0}}} = \gz$'',
$\gd$ carries $\gd^+$ normal measures in
$M^{\FQ_0 \ast \dot \FQ_1}$. Since
$\add(\gd^+, \gd^{++}) \ast \dot \FR \ast
\dot \add(\gg^+, 1) \ast \dot
{\rm Coll}(\gd^+, \gd^{++})$ is
$\gg$-directed closed over our ground model,
this completes the proof of Lemma \ref{lcu}.
\end{proof}
Returning to the construction of the
model $V$ used in the proof of Theorem \ref{t2},
let $V^* \models ``$ZFC + GCH +
$\gk < \gl$
are such that $\gk$ is supercompact
and $\gl$ is the least measurable
cardinal above $\gk$''.
As in the proof of Theorem \ref{t1},
again using indestructibility,
we may assume that $V^*$ has been
generically extended to a model $\ov V$
such that
$\ov V \models ``\gk$ is indestructibly
supercompact +
GCH holds at and above
$\gk$ + $\gl$ is the least
measurable cardinal above $\gk$ +
The cardinal and cofinality structure
at and above $\gk$ is the same as in $V^*$''.
We then force over $\ov V$ with
$\add(\gl^+, \gl^{++}) \ast \dot \FR \ast
\dot \add(\gk^+, 1) \ast \dot
{\rm Coll}(\gl^+, \gl^{++})$, where
$\dot \FR$ is a term for the reverse
Easton iteration of length $\gl$
which begins by adding
a Cohen subset to $\gk^+$
and then adds a Cohen subset to
the successor of each
inaccessible cardinal in the
open interval $(\gk^+, \gl)$.
Call the resulting model $V$.
Since this partial ordering
by its definition is
$\gk$-directed closed,
$V \models ``\gk$ is supercompact''.
By Lemma \ref{lcu},
$V \models ``\gl$ is measurable
and carries $\gl^+$ normal measures'',
and by the
remarks immediately prior to the
proof of Lemma \ref{lcu},
$\ov V^{\add(\gl^+, \gl^{++}) \ast \dot \FR}
\models ``\gl$ is the least measurable
cardinal above $\gk$''. Since
$\ov V^{\add(\gl^+, \gl^{++}) \ast \dot \FR}
\models ``\card{\add(\gk^+, 1)} < \gl$'',
by the results of \cite{LS},
$\ov V^{\add(\gl^+, \gl^{++}) \ast \dot \FR
\ast \dot \add(\gk^+, 1)} \models
``\gl$ is the least measurable
cardinal above $\gk$''. Therefore, since
$\ov V^{\add(\gl^+, \gl^{++}) \ast \dot \FR
\ast \dot \add(\gk^+, 1)} \models
``{\rm Coll}(\gl^+, \gl^{++})$ is
$\gl^+$-directed closed'',
$\ov V^{\add(\gl^+, \gl^{++}) \ast \dot \FR
\ast \dot \add(\gk^+, 1) \ast \dot
{\rm Coll}(\gl^+, \gl^{++})} = V \models
``\gl$ is the least measurable
cardinal above $\gk$'' as well.
We continue with a unified
proof of Theorems \ref{t1} and \ref{t2}.
We summarize where we are at this point.
For both of these theorems,
we have that
$V \models ``$ZFC +
$\gk < \gl$
are such that $\gk$ is supercompact
and $\gl$ is the least measurable
cardinal above $\gk$''.
For Theorem \ref{t1}, for $\gt$
as in the statement of
that theorem, we have that in addition,
$V \models ``\gl$ carries $\gt$
normal measures''.
For Theorem \ref{t2}, we have that in addition,
$V \models ``\gl$ carries $\gl^+$
normal measures''.
%We will return to this distinction shortly,
%but in the meantime, we once again
%Regardless of the exact nature of $V$,
%we once again
%abuse notation and henceforth
%refer to $V$ as $V$, since we will
%take the appropriate $V$
%as our new ground model.
We outline now the construction of
the model $N$ witnessing the
conclusions of the Theorem of
\cite{AM}, since this model
(built within $V[G]$) will
witness the desired conclusions of
our theorems.
%Theorems \ref{t1} and \ref{t2} as well.
We quote freely from \cite{AM},
using portions verbatim as necessary.
As in \cite{AM}, the fact that $\kappa$ is $2^\l$ supercompact for
$\l>\kappa$ the least
measurable cardinal implies there is a supercompact
ultrafilter $\U$ over $\pkl$ with the Menas partition property
\cite{Me} such that
$C_0 = \{p \in \pkl \mid p \cap \kappa$ is a
measurable cardinal and $\ov p$ is the least measurable cardinal
greater than $p \cap \kappa \} \in \U$.
The forcing conditions $\FP$ used in the proof of
Theorems \ref{t1} and \ref{t2}
are the set of all finite sequences of the form
$\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
satisfying the following properties.
\begin{enumerate}
\item Each $p_i$ for $1 \le i \le n$ is an element of $C_0$, and
for $1 \le i < j \le n$, $p_i \smag p_j$, where as in \cite{AM},
$p_i \smag p_j$ means $p_i \subseteq p_j$ and $\ov{p_i} <
p_j \cap \kappa$.
\item $f_0 \in {\rm Coll}(\omega_1,
{<} \ov{p_1})$, for $1 \le i < n$,
$f_i \in {\rm Coll}({\ov p}^+_i,
{<} \ov{p_{i+1}})$, and
$f_n \in {\rm Coll}({\ov p}^+_n, {<} \l)$.
\item $A \subseteq C_0$, $A \in \U$, and for every $q \in A$,
$p_n \smag q$ and the range and domain of $f_n$ are both subsets
of $q$, meaning that if $\la \la \a, \b \ra, \gg \ra \in f_n$,
$\a,\b,\gg \in q$.
\item $F$ is a function defined on $A$
such that for $p \in A$,
$F(p) \in {\rm Coll}({\ov p}^+, {<} \l)$, and if $q \in A$,
$p \smag q$, then the range and domain of $F(p)$ are both subsets
of $q$.
\end{enumerate}
Before we can define the ordering on $\FP$, we need to
define for $p,q \in A$, $p \smag q$ and $f \in
{\rm Coll}({\ov p}^+,
{<} \l)$ such that the range and domain of $f$
are subsets of $q$ the collapse of $f$ in $q$, denoted
$f^*_q$. Let $h : q \to {\ov q}$ be the unique order
isomorphism between $q$ and $\ov q$. Then $f^*_q : {\ov p}^+
\times {\ov q} \to {\ov q}$ is defined as $f^*_q(\la \a,
h^{-1}(\b) \ra) = h(f(\la \a, h^{-1}(\b) \ra))$ if
$h^{-1}(\b) \in q$. In other words, to define $f^*_q$ given
$f$, we transform using $h^{-1}$ the appropriate $\la \a,
\b \ra \in {\ov p}^+ \times {\ov q}$ into an element of
${\ov p}^+ \times \l$, apply $f$ to it, and collapse the result
using $h$. It is easily checked $f^*_q \in
{\rm Coll}({\ov p}^+, {<} {\ov q})$.
We are now able
to define the ordering on $\FP$.
If $\pi_0 =
\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
and $\pi_1 = \la q_1, \ldots, q_m, g_0, \ldots, g_m, B, H \ra$,
then $\pi_1 \ge \pi_0$ iff the following conditions
hold.
\begin{enumerate}
\item $n \le m$, $p_i = q_i$ for $1 \le i \le n$, and
$q_i \in A$ for $n+1 \le i \le m$.
\item $f_i \subseteq g_i$ for $0 \le i < n$, and ${(f_n)}^*_{
q_{n+1}} \subseteq g_n$. If $n=m$, then $f_n \subseteq g_n$.
\item ${(F(q_i))}^*_{q_{i+1}} \subseteq g_i$ for $n+1 \le i < m$,
and $F(q_m) \subseteq g_m$.
\item $B \subseteq A$.
\item For every $p \in B$, $F(p) \subseteq H(p)$.
\end{enumerate}
Let $G$ be $V$-generic over $\FP$.
As in \cite{AM}, we can define
sequences $r = \la p_i \mid i\in\omega-\{0\}\ra$
and $g = \la G_i \mid i <
\omega \ra$, where $p_i \in r$ iff $\exists \pi \in G [ p_i \in
\pi]$ and $G_i = \bigcup \{ f_i \mid \exists \pi \in G [\pi =
\la p_1, \ldots, p_n, f_0, \ldots, f_i, \ldots, f_n, A, F \ra
]\}$. These sequences will be well-defined by the genericity of
$G$.
%Further, as in \cite{AM}, $V[G] \models ``\gk = \ha_\go$''.
We are now in a position to describe the inner model
$N \subseteq V[G]$ which,
when appropriately constructed,
will witness either the conclusions of
Theorem \ref{t1} or the conclusions
of Theorem \ref{t2}.
For $\d \in [\gk, \gl)$, $\d$ inaccessible, let
$r \rest \d = \la p_i \cap \d \mid i \in \omega - \{0\} \ra$, and
let $g \rest \d = \la G^\d_i \mid i < \omega \ra$, where $G^\d_i
= G_i \rest {\ov {p_{i+1} \cap \d}}$.
%(If $f \in {\rm Coll}(
%\omega_1, < {\ov{p_1}})$ or $f \in {\rm Coll}({\ov{p_i}}^+, <
%{\ov{p_{i+1}}})$, then let $f^\d = f \rest {\ov{p_1 \cap \d}}$ or
%$f^\d = f \rest {\ov{p_{i+1} \cap \d}}$.)
Intuitively,
$N$ is the least model of ZF extending $V$ which contains, for
each inaccessible $\d \in [\gk, \gl)$, the sequences $r \rest \d$
and $g\rest\d$. More formally, let ${\cal L}_1$ be the sublanguage of
the forcing language
$\cal L$ with respect to $\FP$ which contains
symbols $\check v$ for each $v \in V$, a
unary predicate symbol $\check V$ (to be interpreted ${\check V}(
\check v)$ iff $v \in V$), and for $\d \in [\gk, \gl)$, $\d$
inaccessible, symbols ${\dot r} \rest \d$ for $r \rest \d$ and
${\dot g} \rest \d$ for $g\rest\d$. $N$ can then be defined inside $V[G]$
as follows.
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\gl$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}
The standard arguments show
$N \models {\rm ZF}$.
By Lemmas 1 - 7 and the
intervening remarks of \cite{AM},
$N \models ``\gk = \ha_\go$ +
$\gl = \gk^+ = \ha_{\go + 1}$ +
For any normal measure $\U \in V$
over $\gl$,
${\cal U}^* = \{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$
is a normal measure over $\gl$ +
${\rm DC}_{\ha_\go}$''.
Further, Lemmas 3 and 4 of
\cite{AM} and their proofs
tell us that
for $\gd < \gk$ inaccessible,
any formula mentioning only
(terms for ground model sets and)
$\dot r \rest \gd$ and
$\dot g \rest \gd$ may be
decided in $V[r \rest \gd, g \rest \gd]$
the same way as in $V[G]$,
and that $V[r \rest \gd, g \rest \gd]$
is obtained by forcing with a partial
ordering having size less than $\gl$.
In particular, any set of ordinals
in $N$ is actually a member of
$V[r \rest \gd, g \rest \gd]$
for the appropriate $\gd < \gk$.
These facts will be critical in
the proof of Theorems \ref{t1}
and \ref{t2} and the following two
lemmas.
\begin{lemma}\label{lnorm}
Suppose $\U^* \in N$ is a
normal measure over $\gl$.
Then for some normal measure
$\U \in V$ over $\gl$,
${\cal U}^* = \{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$.
\end{lemma}
\begin{proof}
We use ideas found in the proof of
Theorem 2.3(e) of \cite{BK}.
Let $\gt$ be a term for
$\U^*$. Since $\U^* \in N$, we
may choose $\gd < \gk$,
$\gd$ inaccessible, such that
$\gt$ mentions only $\dot r \rest \gd$
and $\dot g \rest \gd$.
By our remarks in the paragraph
immediately preceding the
statement of Lemma \ref{lnorm}, the set
${\cal U}^* \rest \gd = \U^* \cap
V[r \rest \gd, g \rest \gd] \in
V[r \rest \gd, g \rest \gd]$, which
immediately implies that
${\cal U}^* \rest \gd$ is in
$V[r \rest \gd, g \rest \gd]$ a
normal measure over $\gl$.
Again by our remarks in the paragraph
immediately preceding the
statement of Lemma \ref{lnorm}
and the results of \cite{LS},
it must consequently be the case that
for some $\U \in V$ a normal measure
over $\gl$,
${\cal U}^* \rest \gd$ is definable in
$V[r \rest \gd, g \rest \gd]$ as
$\{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$.
Therefore, since in $N$,
${\cal U}^*$ is a normal measure
over $\gl$, by the same
argument as found in the last
paragraph of the proof of
Lemma \ref{lcu},
for ${\cal U}'$ defined in $N$ as
$\{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$,
${\cal U}' = {\cal U}^*$.
This completes the proof of Lemma \ref{lnorm}.
\end{proof}
%By Lemma \ref{lnorm},
%the proofs of Theorems
%\ref{t1} and \ref{t2} will be complete
%once we have shown
%that the cardinal structure in $N$ above $\gl$ is
%the same as in $V$. We see that this is so via
%the following lemma.
\begin{lemma}\label{lcard}
In $N$, the cardinal and cofinality structure
above $\gl$ is the same
as in $V$.
\end{lemma}
\begin{proof}
Let $\gb$ and $\gg$
be arbitrary ordinals,
and suppose
$N \models ``f : \gb \to \gg$ is
a function''.
Since $f$ may be coded by
a set of ordinals, by our remarks
in the paragraph immediately
preceding the statement
of Lemma \ref{lnorm},
$f \in V[r \rest \gd, g \rest \gd]$
for some $\gd < \gk$.
Since $V[r \rest \gd, g \rest \gd]$
is obtained by forcing with a partial
ordering having size less than $\gl$,
$f$ cannot witness that any $V$-cardinal
greater than or equal to
$\gl$ has a different
cardinality or cofinality.
This contradiction completes the
proof of Lemma \ref{lcard}.
\end{proof}
By Lemmas \ref{lnorm} and \ref{lcard}
and our earlier work,
if $V^*$ is as in Theorem \ref{t1},
then $N$ witnesses the conclusions
of Theorem \ref{t1}.
Similarly, Lemmas \ref{lnorm} and \ref{lcard}
and our earlier work imply that if
$V^*$ is as in Theorem \ref{t2},
then $N$ witnesses the conclusions of
Theorem \ref{t2}.
This completes the proofs of
Theorems \ref{t1} and \ref{t2}.
\end{proof}
Suppose $V$ is an inner model
(e.g., as given in \cite{Mi}) with
$V \models ``\gk < \gl$ are such that
$\gk$ is regular and $\gl$ is measurable
+ For some cardinal $\gt$
which is either less than or equal to
$\gk$ or is one of the cardinals
$\gl$, $\gl^+$, or $\gl^{++}$,
$\gl$ carries $\gt$
normal measures''.
We observe that a simplified version of
the proof of Theorem 3.1 of \cite{BK}
shows the existence of a
partial ordering $\FP$ and a symmetric
inner model $N \subseteq V^\FP$ such that
$N \models ``\gk$ is regular + $\gl = \gk^+$ +
$\gt$ is a cardinal +
$\gl$ is measurable and carries $\gt$
normal measures''.
In addition, suppose
we start with a model
$V^* \models ``$ZFC + GCH holds at and
above $\gl$ + $\gk < \gl$
are such that $\gk$ is regular and
$\gl$ is measurable + $\gt > \gl^+$
is a regular cardinal'' and then force
with the partial ordering
$\add(\gl^+, \gt) \ast \dot \FR$, where
$\dot \FR$ is a term for the reverse
Easton iteration of length $\gl$
which begins by adding
a Cohen subset to $\gk^+$
and then adds a Cohen subset to
the successor of each
inaccessible cardinal in the
open interval $(\gk^+, \gl)$.
If we denote the resulting
generic extension by $V$,
then by standard arguments,
$\gk$ remains regular in $V$.
In addition,
by our earlier remarks,
$V \models ``\gt$ is a regular
cardinal + $\gl$ is measurable
and carries $\gt$ normal measures''.
Once again, a simplified version of
the proof of Theorem 3.1 of \cite{BK}
shows the existence of a
partial ordering $\FP$ and a symmetric
inner model $N \subseteq V^\FP$ such that
$N \models ``\gk$ is regular + $\gl = \gk^+$ +
$\gt$ is a regular cardinal +
$\gl$ is measurable and carries $\gt$
normal measures''.
Note that in both cases mentioned above,
$\FP = {\rm Coll}(\gk, {<} \gl)$, and if
$G$ is $V$-generic over $\FP$,
$N$ may intuitively be described as
the least model of ZF extending $V$
which contains, for each
inaccessible cardinal $\gd$
in the open interval $(\gk, \gl)$,
the set $G \rest \gd$.
It is thus true that because of
the existence of the relevant
inner models, it is relatively
consistent for the successor
of a regular cardinal to be
measurable and to carry essentially
any desired (regular) cardinality
of normal measures.
Due to the current state
of knowledge, however,
%(including the lack of any sort of
%viable inner model theory for
%supercompactness), however,
the existence of a model in
which $\alom$ carries, say,
exactly four normal measures
remains open.
We therefore conclude this
paper by reiterating and
expanding upon the title
question, i.e., by asking
how many normal measures
$\alom$, or indeed, the
successor of any singular
cardinal, can carry.
More specifically, is it
relatively consistent for
$\alom$ to carry exactly
$\gt$ normal measures, where
$\gt$ is a cardinal and
either $\gt = 1$, $\gt = 2$, or
$4 \le \gt \le \alom$?
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\begin{gr}
Theorem \ref{t1} by itself shows that
relative to the appropriate
assumptions, it is consistent for
$\alom$ to carry $\gt$ normal
measures, where $\gt \ge
\ha_{\go + 3}$ is any regular cardinal.
However, it still leaves open the
question as to whether it is
relatively consistent for $\alom$
to carry $\ha_{\go + 2}$ normal measures.
As we have indicated above, this
is certainly possible, as witnessed
by the following theorem.
Let $\gk$ be a regular cardinal
and $\gl > \gk$ be a measurable
cardinal of Mitchell order $\gt$,
for $\gt$ any cardinal which is
either less than or equal to
$\gk$ or is one of the cardinals
$\gl$, $\gl^+$, or $\gl^{++}$.
We note that a simplified version of
the proof of Theorem 3.1 of \cite{BK}
shows the existence of a generic
extension $V^\gt$ and a symmetric
inner model $N^\gt \subseteq V^\gt$
such that
$N^\gt \models ``\gl = \gk^+$ +
$\gt$ is a cardinal +
$\gl$ carries exactly $\gt$
normal measures''.
for any regular cardinal $\gt \ge \gk^{++}$,
it is possible to force and obtain
a model in which $\gk$ remains
measurable and carries exactly
$\gt$ normal measures.
It is well-known (see \cite{Kl2})
that assuming AD + DC, $\alom$
carries exactly three normal measures.
The proof of this fact
uses the regularity properties provided
by determinacy, along with the
theory \cite{Kl1} of infinite
exponent partition relations.
\end{gr}