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%
% ------------------------------------------------------------------------------
%
\title{How Many Normal Measures Can $\ha_{\go_1 + 1}$ Carry?
%Controlling the Number of Normal Measures at
%Successors of Singular Cardinals
%On the Number of Normal Measures Successors
%of Singulars Can Carry
\thanks{2000 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal, supercompact
Radin forcing, Radin sequence of measures, symmetric
inner model, normal measure, measurable cardinal.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.}\\
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 21, 2009\\
(revised July 6, 2009)}
\begin{document}
\maketitle
\begin{abstract}
Relative to the existence of a supercompact
cardinal with a measurable cardinal above it,
we show that it is consistent for
$\ha_1$ to be regular and for
$\ha_{\go_1 + 1}$ to be measurable and
to carry precisely $\gt$ normal measures, where
$\gt \ge \ha_{\go_1 + 2} $ is any regular cardinal.
This extends the work of \cite{A06},
in which the analogous result was obtained
for $\ha_{\go + 1}$ using the same hypotheses.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s1}
In \cite{A06}, models were constructed for the
theory ``ZF + ${\rm DC}_{\ha_\go}$ +
$\ha_{\go + 1}$ is a measurable cardinal''
in which $\ha_{\go + 1}$ carries exactly
$\gt$ normal measures, where $\gt \ge \ha_{\go + 2}$
is an arbitrary regular cardinal.
The proof given easily generalizes to handling
successors of other singular cardinals of
cofinality $\go$, such as $\ha_{\go + \go + 1}$,
$\ha_{\go^2 + 1}$, etc.
No attempt was made in that paper, however,
to handle successors of singular cardinals of
uncountable cofinality, such as $\ha_{\go_1 + 1}$.
The purpose of this paper is to rectify this
situation and establish a similar result for
$\ha_{\go_1 + 1}$. Specifically, in analogy to
\cite{A06}, we prove the following two theorems.
\begin{theorem}\label{t1}
Suppose $V^* \models ``$ZFC + GCH + $\gk < \gl$
are such that $\gk$ is supercompact and
$\gl$ is the least measurable cardinal
greater than $\gk$ + $\gt > \gl^+$
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 + $\neg AC_\go$ +
$\ha_1$ is a regular cardinal +
$\gk = \ha_{\go_1}$ +
For every limit ordinal $\nu < \ha_1$,
$\ha_{\nu + 1}$ is a measurable cardinal +
$\gl = \gk^+ = \ha_{\go_1 + 1}$ is a measurable cardinal''.
In $N$,
every successor cardinal less than $\ha_{\go_1}$
is regular,
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
$\ha_{\go_1 + 1}$ carries exactly $\gt$ normal measures.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V^* \models ``$ZFC + GCH + $\gk < \gl$
are such that $\gk$ is supercompact and
$\gl$ is the least measurable cardinal
greater than $\gk$''.
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 + $\neg AC_\go$ +
$\ha_1$ is a regular cardinal +
$\gk = \ha_{\go_1}$ +
For every limit ordinal $\nu < \ha_1$,
$\ha_{\nu + 1}$ is a measurable cardinal +
$\gl = \gk^+ = \ha_{\go_1 + 1}$ is a measurable cardinal''.
In $N$,
every successor cardinal less than $\ha_{\go_1}$
is regular,
$\ha_{\go_1 + 2}$ is regular,
and $\ha_{\go_1 + 1}$ carries exactly
$\ha_{\go_1 + 2}$ normal measures.
\end{theorem}
As in \cite{A06}, Theorems \ref{t1} and \ref{t2}
provide our desired results. Taken together, they
show that relative to the appropriate assumptions,
it is consistent for $\ha_{\go_1 + 1}$ to be
measurable and to carry precisely $\gt$ normal measures,
where $\gt \ge \ha_{\go_1 + 2}$ is any regular cardinal.
\begin{pf}
To prove Theorems \ref{t1} and \ref{t2},
let $V^* \models ``$ZFC + GCH + $\gk < \gl$
are such that $\gk$ is supercompact and
$\gl$ is the least measurable cardinal
greater than $\gk$''.
%For Theorem \ref{t1}, we assume in addition that
%$\gt > \gl^+$ is a fixed but arbitrary regular cardinal in $V^*$.
As in \cite{A06},
we give a uniform proof of Theorems \ref{t1} and \ref{t2}.
In particular, as in the proof of
\cite[Theorem 1]{A06}, for Theorem \ref{t1}
of this paper and $\gt$ as in the
statement of this theorem, we may assume in addition that
$V^*$ has been generically extended to a model $V$
with the same cardinal and cofinality structure as $V^*$
at and above $\gl$ such that $V \models ``\gl$ carries
exactly $\gt$ normal measures''. For Theorem \ref{t2}
of this paper, as in the proof of \cite[Theorem 2]{A06},
we may assume in addition that $V^*$ has been
generically extended to a model $V$ such that
$V \models ``\gl$ carries exactly $\gl^+$
normal measures''.
For the exact manner in which $V$ is constructed
from $V^*$, we refer readers to \cite{A06}.
We note only that for Theorem \ref{t1},
$V \models ``2^\gl = \gl^+$ + $2^{\gl^+} = \gt$'',
and for Theorem \ref{t2},
$V \models ``2^\gl = \gl^+$ + $2^{\gl^+} = \gl^{++}$''.
Our proof combines Gitik's techniques of \cite{G85}
with the methods of \cite{A06}.
Our presentation of Gitik's
techniques is based on the one given in
\cite{A92}, but also follows the ones given in
\cite{A85}, \cite{A96}, and \cite{AK}.
All of these rely heavily on \cite{G85}.
As the necessary facts about Radin
forcing are distributed throughout
the literature, our
bibliographical citations will reflect
this.
Our witnessing model $N$ for Theorems \ref{t1} and \ref{t2}
is the specific version of the model
$N_A$ of \cite{A92} described
at the end of that paper, only
constructed using a Radin sequence of
measures of length $\ha_1$ instead of $\gk^+$
and not truncating the universe at $\gk$.
We explicitly give the construction below.
Let $j : V \to M$ be an elementary
embedding witnessing the
$2^\gl$ supercompactness of $\gk$.
Our first step is to define a
{{Radin}} sequence of measures $\mu_{< {{\ha_1}}} = \la \mu_\ga \mid
\ga < {{\ha_1}} \ra$
appropriate for supercompact Radin forcing over $P_{\gk}(\gl)$.
Specifically, if
$\ga = 0$, $\mu_\ga$ is defined by $X \in \mu_\ga$ iff
$\la j(\gb) \mid \gb < \gl \ra \in j(X)$,
and if $\ga > 0$, $\ga < \ha_1$, $\mu_\ga$ is
defined by $X \in \mu_\ga$ iff $\la \mu_\gb \mid \gb < \ga \ra
=_{\hbox{\rm df}} \mu_{< \ga} \in j(X)$.
%Next, using $\mu_{< {{\ha_1}}}$, we let $\FR_{< {{\ha_1}}}$ be
%supercompact {{Radin}} forcing defined over $V_{\gk} \times
%P_{\gk}(\gl)$.
Next, we let $\FR_{< {{\ha_1}}}$ be supercompact Radin
forcing over $P_\gk(\gl)$ defined using $\mu_{< {{\ha_1}}}$.
The \break particulars of the definition
are virtually identical
to the ones found in \cite{A85}, \cite{A92},
\cite{A96}, and \cite{AK},
but for clarity, we repeat them here.
%(which may be taken as our standard references for what follows),
$\FR_{< {{\ha_1}}}$ is composed of all
finite sequences of the form $\la \la p_0, u_0, C_0, \ra, \ldots,
\la p_n, u_n, C_n \ra , \la \mu_{< {{\ha_1}}}, C \ra \ra$ such
that the following conditions hold.
\begin{enumerate}
\item For $0 \le i < j \le n$, $p_i \smag p_j$, where for
$p, q \in P_{\gk}(\gl)$, $\ p \smag q$ means $p \subseteq q$ and
${\rm otp}(p) < q \cap \gk$.
%($\overline{p}$ is the order type of $p$.)
\item For $0 \le i \le n$, $p_i \cap \gk$ is a
measurable cardinal.
\item ${\rm otp}(p_i)$ is the least measurable
cardinal greater than $p_i \cap \gk$.
%which is a measurable cardinal carrying exactly
%$({\rm otp}(p_i))^+$ many normal measures.
In analogy to the notation of \cite{G85},
\cite{A85}, \cite{A92}, \cite{A96}, and \cite{AK},
we write ${\rm otp}(p_i) = {(p_i \cap \gk)}^*$.
By extension of this notation, $\gl = \gk^*$.
\item For $0 \le i \le n$, $u_i$ is a
{{Radin}} sequence of measures
%over $V_{p_i \cap \gk} \times P_{p_i \cap \gk}({\rm otp}(p_i))$
appropriate for supercompact Radin forcing over
$P_{p_i \cap \gk}({\rm otp}(p_i))$
with ${(u_i)}_0$, the
$0$th coordinate of $u_i$, a supercompact measure over
$P_{p_i \cap \gk}({\rm otp}(p_i))$.
\item $C_i$ is a sequence of measure 1 sets for $u_i$.
\item $C$ is a sequence of measure 1
sets for $\mu_{< {{\ha_1}}}$.
\item For each $p \in {(C)}_0$, where ${(C)}_0$ is the
coordinate of $C$ such that ${(C)}_0 \in \mu_0$, $
\bigcup_{i \in \{0, \ldots, n\}} p_i \smag p$.
\item For each $p \in {(C)}_0$, ${\rm otp}(p) =
{(p \cap \gk)}^*$ and $p \cap \gk$ is a
measurable cardinal.
\end{enumerate}
Conditions (5) and (6) are both standard
to any definition of {{Radin}} forcing.
Conditions (1), (2), (4), and (7)
are all standard to any definition of
{\em supercompact} Radin forcing.
Conditions (3) and (8) are
used because of our ultimate
aim of constructing a model in which
$\ha_{\go_1 + 1}$ is measurable
and carries the desired
number of normal measures.
That they may be included and have
the Radin forcing attain its desired goals
follows by the fact that
$V \models ``\gk$ is supercompact and $\gl$
is the least measurable cardinal
greater than $\gk$''.
Thus, by closure,
$M \models ``\gk$ is measurable and
$\gl$ is the least measurable cardinal
greater than $\gk$''.
This means that by reflection,
$\{ p \in P_{\gk}(\gl) \mid p \cap \gk$ is a measurable
cardinal and ${\rm otp}(p)$ is the
least measurable cardinal greater than $p \cap \gk\} \in \mu_0$.
This will ensure that the Radin
sequence of cardinals eventually produced
can be used in our final symmetric inner model $N$.
%We therefore will be able to
For completeness of exposition,
we recall now the definition of the ordering on
$\FR_{< {{\ha_1}}}$. If $\pi_0 =
%\break
\la \la p_0, u_0, C_0 \ra ,
\ldots , \la p_n, u_n, C_n \ra ,
\la \mu_{< {{\ha_1}}},
C \ra \ra$ and $\pi_1 = \la \la q_0, v_0, D_0 \ra , \ldots ,%\break
\la q_m, v_m, D_m \ra , \la \mu_{< {{\ha_1}}} , D \ra \ra$, then
$\pi_1$ extends $\pi_0$ iff the following conditions hold.
\begin{enumerate}
\item For each $\la p_j, u_j, C_j \ra$
which appears in $\pi_0$, there is a $\la q_i, v_i, D_i \ra$ which
appears in $\pi_1$ such that $\la q_i, v_i \ra = \la p_j, u_j \ra$
and $D_i \subseteq C_j$, i.e., for each coordinate ${(D_i)}_\ga$
and ${(C_j)}_\ga$, ${(D_i)}_\ga \subseteq {(C_j)}_\ga$.
\item $D \subseteq C$.
\item $n \le m$.
\item If $\la q_i, v_i, D_i \ra$ does not appear in $\pi_0$, let
$\la p_j, u_j, C_j \ra$ (or $\la \mu_{< {{\ha_1}}}, C \ra$) be
the first element of $\pi_0$ such that $p_j \cap \gk >
q_i \cap \gk$. Then
\begin{enumerate}
\item $q_i$ is
order isomorphic to some $q \in {(C_j)}_0$.
\item There exists
an $\ga < \ga_0$, where $\ga_0$ is the length
of $u_j$, such that $v_i$ is isomorphic ``in a natural way'' to an
ultrafilter sequence $v \in {(C_j)}_\ga$.
\item For $\gb_0$ the length of $v_i$, there is a function
$f: \gb_0 \to \ga_0$ such that for $\gb < \gb_0$, ${(D_i)}_\gb$ is a
set of ultrafilter sequences such that for some subset
${(D_i)}'_\gb $ of ${(C_j)}_{f(\gb)}$, each ultrafilter sequence
in ${(D_i)}_\gb$ is isomorphic ``in a natural way'' to an ultrafilter
sequence in ${(D_i)}'_\gb $.
\end{enumerate}
\end{enumerate}
\noindent For further information on the definition of the ordering
on $\FR_{< {{\ha_1}}}$ (including the meaning of ``in a natural
way'') and more facts about {{Radin}} forcing in general,
readers are referred to \cite{A85}, \cite{A92},
\cite{A96}, \cite{AK}, \cite{CW}, \cite{FW}, \cite{G85},
\cite{G07}, and \cite{R}.
We are now ready to define the partial
ordering $\FP$ used in the proof
of Theorem \ref{t1}. It is given by
the finite support product
ordered componentwise
$$\prod_{\{\la \ga, \gb \ra \mid
\ha_1 \le \ga < \gb < \gk \ {\rm are} \
{\rm regular} \ {\rm cardinals}\}}
{\rm Coll}(\ga, {<} \gb) \times
\FR_{< {{\ha_1}}},$$
where ${\rm Coll}(\ga, {<} \gb)$ is the
L\'evy collapse of all cardinals of size
less than $\gb$ to $\ga$.
Let $G$ be $V$-generic over $\FP$, and let
$G_0$ be the projection of $G$ onto $\FR_{< {{\ha_1}}}$.
For any condition $\pi \in \FR_{< {{\ha_1}}}$, call
$\la p_0, \ldots p_n \ra$ {\em the
$p$-part of $\pi$}. Let $R = \{p \mid
\exists \pi \in G_0[p \in
{\rm p-part}(\pi)]\}$, and let
$R_\ell = \{p \mid p \in R$ and $p$
is a limit point of $R\}$. Define three sets
$E_0$, $E_1$, and $E_2$ by
$E_0 = \{\ga \mid$ For some
$\pi \in G_0$ and some
$p \in {\rm p-part}(\pi)$, $p \cap \gk = \ga\}$,
$E_1 = \{\ga \le \gk \mid \ga$ is a limit point of $E_0\}$, and
$E_2 = E_1 \cup \{(\ha_1)^V\} \cup \{\gb \mid
\exists \ga \in E_1[\gb = \ga^*]\}$.
By a simple density argument,
forcing with $\FR_{< {{\ha_1}}}$
changes the cofinality of $\gk$ to $(\ha_1)^V$.
We can therefore
let $\la \ga_\nu \mid \nu < (\ha_1)^V \ra$ be the
continuous, increasing enumeration of $E_2 -
\{\gk, \gl\}$, and also let
$\ga_{(\ha_1)^V} = \gk$ and $\ga_{(\ha_1)^V + 1} = \gl$.
Note that the sequence $\la \ga_\nu \mid \nu < (\ha_1)^V \ra$
is cofinal in $\gk$. Let
$\nu = \nu' + n$ for some $n \in \go$,
where $\nu' \le (\ha_1)^V$ is either a limit ordinal
or $0$.
For $\gb$ where $\nu < (\ha_1)^V$ and
$\gb \in [\ga_\nu, \ga_{\nu + 1})$ in the
first case, $\nu = (\ha_1)^V$ and
$\gb \in [\gk, \gl)$ in the second case,
and $\gb = \ga_{\nu + 1}$ and $\nu < (\ha_1)^V$ in
the last two cases, define sets $C_i(\ga_\nu, \gb)$
for $i = 1, \ldots, 4$
according to specific conditions on $\nu$ and $\nu'$
in the following manner:
\begin{enumerate}
\item $\nu = \nu' \neq 0$,
$\nu < (\ha_1)^V$, and $n = 0$, i.e.,
$\nu < (\ha_1)^V$ is a limit ordinal. Let
$p(\ga_\nu)$ be the element $p$ of $R$ such that
$p \cap \gk = \ga_\nu$, and let
$h_{p(\ga_\nu)} : p(\ga_\nu) \to {\rm otp}(p(\ga_\nu))$ be
the order isomorphism between $p(\ga_\nu)$ and
${\rm otp}(p(\ga_\nu))$. Then
$C_1(\ga_\nu, \gb) = \{{h_{p(\ga_\nu)}} '' p \cap \gb \mid
p \in R_\ell$, $p \subseteq p(\ga_\nu)$, and
$h^{-1}_{p(\ga_\nu)}(\gb) \in p\}$.
\item $\nu = (\ha_1)^V$. Then $C_2(\gk, \gb) =
C_2(\ga_{(\ha_1)^V}, \gb) =
\{p \cap \gb \mid p \in R_\ell$ and $\gb \in p\}$.
\item $(\nu = \nu' + n$, $\nu' > 0$,
$\nu' < (\ha_1)^V$, and $n \ge 2)$ or
$(\nu' = 0$ and $n \in \go)$, i.e., $\gn$ is neither a
limit ordinal nor the successor of a limit ordinal
less than $(\ha_1)^V$.
Let
$H(\ga_\nu, \ga_{\nu + 1})$ be the projection of $G$ onto
${\rm Coll}(\ga_\nu, {<} \ga_{\nu + 1})$. Then
$C_3(\ga_\nu, \ga_{\nu + 1}) = H(\ga_\nu, \ga_{\nu + 1})$.
\item $\nu = \nu' + 1$ for $\nu' > 0$, $\nu' < (\ha_1)^V$,
i.e., $\nu$ is the successor of a limit ordinal
less than $(\ha_1)^V$. Let
$H(\ga^+_\nu, \ga_{\nu + 1})$ be the projection of $G$ onto
${\rm Coll}(\ga^+_\nu, {<} \ga_{\nu + 1})$. Then
$C_4(\ga^+_\nu, \ga_{\nu + 1}) = H(\ga^+_\nu, \ga_{\nu + 1})$.
\end{enumerate}
$C_1(\ga_\nu, \gb)$ and
$C_2(\gk, \gb)$ % = C_2(\ga_{(\ha_1)^V}, \gb)$
are used to collapse
$\gb$ to $\ga_\nu$ when $\nu \le (\ha_1)^V$ is a
limit ordinal, and are also used to
generate the closed, cofinal sequence
$\la \ga_\gg \mid \gg < \nu \ra$.
$C_3(\ga_\nu, \ga_{\nu + 1})$ is used
to collapse $\ga_{\nu + 1}$ to be the
successor of $\ga_\nu$ when $\nu < (\ha_1)^V$ is
neither a limit ordinal nor the
successor of a limit ordinal, and
$C_4(\ga^+_\nu, \ga_{\nu + 1})$
is used to collapse $\ga_{\nu + 1}$ to be
the successor of $\ga^+_\nu$ when
$\nu < (\ha_1)^V$ is the successor of a limit ordinal.
The $C_i$ have been chosen so as to ensure that
all successor cardinals less than
$\ha_{\go_1}$ are regular and that the
successor of every limit cardinal
less than or equal to $\ha_{\go_1}$ is measurable.
Intuitively, the symmetric inner model $N \subseteq V[G]$
witnessing the conclusions of Theorem \ref{t1}
is the least model of ZF
extending $V$ which contains
$C_1(\ga_\nu, \gb)$ if $\nu < (\ha_1)^V$
is a limit ordinal and
$\gb \in [\ga_\nu, \ga_{\nu + 1})$,
$C_2(\gk, \gb)$ if $\gb \in [\gk, \gl)$,
$C_3(\ga_\nu, \ga_{\nu + 1})$ if
$\nu < (\ha_1)^V$ is neither a limit ordinal
nor the successor of a limit ordinal, and
$C_4(\ga^+_\nu, \ga_{\nu + 1})$ if $\nu < (\ha_1)^V$
is the successor of a limit ordinal.
To define $N$ more precisely, it is necessary to define
canonical names $\underline{\ga_\nu}$
for the $\ga_\nu$'s when $\nu < (\ha_1)^V$ and canonical
names $\underline{C_i(\nu, \gb)}$ for $i = 1, 2$ and
$\underline{C_i(\nu, \nu + 1)}$ for $i = 3, 4$.
%for the three sets just described.
Recall that when $\nu < (\ha_1)^V$ it is
possible to decide $p(\ga_\nu)$ (and hence
${\rm otp}(p(\ga_\nu))$) by
writing $\omega \cdot \nu = \omega^{\sigma_0} \cdot n_0 +
\omega^{\sigma_1} \cdot n_1 + \cdots + \omega^{\sigma_m} \cdot
n_m$ (where $\sigma_0 > \sigma_1 > \cdots > \sigma_m$ are
ordinals, $n_0, \ldots, n_m > 0$ are integers, and $+$,
$\cdot$, and exponentiation are the
ordinal arithmetical operations), letting
$\pi = \la{\la p_{ij_i}, u_{ij_i}, C_{ij_i} \ra}_{i \le m,
1 \le j_i\le n_i}, \la \mu_{< {{\ha_1}}}, C \ra \ra$ be such that
min($p_{i1} \cap \gk, \omega^{{\hbox{\rm length}}(u_{i1})}) =
\sigma_i$ and length($u_{ij_i}) = {\hbox{\rm min}}(p_{i1} \cap \gk,
{\hbox{\rm length}}(u_{i1}))$ for $1 \le j_i \le n_i$, and letting
$p(\ga_\nu)$ be $p_{mn_m}$.
Further, $D_\nu = \{ r \in \FP \mid r \rest
\FR_{< {{\ha_1}}}$ extends a condition $\pi$ of the above form$\}$
is a dense open subset of $\FP$. $\underline{\ga_\nu}$ is the name of
the $\ga_\nu$ determined by any element of $D_\nu \cap G$; in the
notation of \cite{G85}, \cite{A85}, \cite{A92},
\cite{A96}, and \cite{AK},
$\underline{\ga_\nu} = \{ \la r, \check \ga_\nu(r)
\ra \mid r \in D_\nu \}$, where $\ga_\nu(r)$ is the $\ga_\nu$ determined
by the condition $r$.
The canonical names $\underline{C_i(\nu, \gb)}$
for $i = 1, 2$ and
$\underline{C_i(\nu, \nu + 1)}$ for $i = 3, 4$
are defined in a manner so as to
be invariant under the appropriate group of
automorphisms. Specifically, there are four cases to
consider. We again write $\nu = \nu' + n$,
where $n \in \go$ and $\nu' \le (\ha_1)^V$ is either
a limit ordinal or $0$,
and let $\gb$ be as
before. We also assume without loss of generality that as in
\cite{G85}, \cite{A85}, \cite{A92}, \cite{A96}, and \cite{AK},
$\ga_{\nu + 1}$ is determined by $D_\nu$
when $\nu < (\ha_1)^V$. Further, we adopt
throughout each of the four cases
the notation of \cite{G85}, \cite{A85},
\cite{A92}, \cite{A96}, and \cite{AK}.
\begin{enumerate}
\item $\nu' = \nu \ne 0$,
$\nu < (\ha_1)^V$, and $n=0$, i.e.,
$\nu < (\ha_1)^V$ is a limit ordinal.
$\underline{C_1(\nu, \gb)} = \{
\la r, (\check r \rest \FR_{< {{\ha_1}}}) \rest (\ga_\nu(r), \gb) \ra
\mid r \in
D_\nu \}$, where for $r \in \FP$,
$\pi = r \rest \FR_{< {{\ha_1}}}$, $\pi \rest
(\ga_\nu(r), \gb) = \{ {{h_{p(\ga_\nu)(r)}}}''p\cap \gb \mid p \in$ p-part$
(\pi)$, $ p \subseteq p(\ga_\nu)(r)$, $ p \in R_\ell \rest \pi$, and
$h^{-1}_{p(\ga_\nu)(r)}(\gb) \in p\} $.
\item $\nu = (\ha_1)^V$. $\underline{C_2(\nu, \gb)}
= \{
\la r, (\check r \rest \FR_{< {{\ha_1}}}) \rest (\gk, \gb) \ra
\}$, where for $r \in \FP$,
$\pi = r \rest \FR_{< {{\ha_1}}}$, $\pi \rest
(\gk, \gb) = \{ p\cap \gb \mid p \in$ p-part$
(\pi)$, $ p \in R_\ell \rest \pi$, and
$\gb \in p\} $.
\item $(\nu = \nu' + n$, $\nu' > 0$,
$\nu' < (\ha_1)^V$, and $n \ge 2)$ or
$(\nu' = 0$ and $n \in \go)$, i.e., $\gn$ is neither a
limit ordinal nor the successor of a limit ordinal
less than $(\ha_1)^V$.
$\underline{C_3(\nu, \nu + 1)} =
\{ \la r, (\check r \rest
{\hbox{\rm Coll}}(\ga_\nu(r), {<} \ga_{\nu + 1}(r)))
%\rest \gb
\ra \mid r \in D_\nu \}$.
\item $\nu = \nu' + 1$ for $\nu' > 0$, $\nu' < (\ha_1)^V$,
i.e., $\nu$ is the successor of a limit ordinal
less than $(\ha_1)^V$.
$\underline{C_4(\nu, \nu + 1)} =
\{ \la r, (\check r \rest
{\hbox{\rm Coll}}(\ga^+_\nu(r), {<} \ga_{\nu + 1}(r)))
%\rest \gb
\ra \mid r \in D_\nu \}$.
\end{enumerate}
\noindent As in \cite{G85}, \cite{A85}, \cite{A92},
\cite{A96}, and \cite{AK}, since for any
$r , r' \in D_\nu \cap G$, $p(\ga_\nu)(r) =
p(\ga_\nu)(r')$, each of the definitions
just given is unambiguous.
Let ${\cal G}$ be the group of automorphisms of \cite{G85},
and let $\underline{C(G)} =
\{\psi(\underline{C_1(\nu,\gb)}) \mid \psi
\in {\cal G}$, $0 \le \nu< (\ha_1)^V$, and $\gb \in [\nu,\gk)$ is a
cardinal$\} \cup
\{\psi(\underline{C_2(\nu,\gb)}) \mid \psi
\in {\cal G}$, $\nu = (\ha_1)^V$, and $\gb \in [\gk,\gl)$ is a
cardinal$\} \cup
\bigcup_{i = 3, 4} \{
\psi(\underline{C_i(\nu,\nu + 1)}) \mid \psi
\in {\cal G}$ and $0 \le \nu< (\ha_1)^V\}$.
$C(G) =
\{i_G(\psi(\underline{C_1(\nu,\gb)})) \mid \psi
\in {\cal G}$, $0 \le \nu< (\ha_1)^V$, and $\gb \in [\nu,\gk)$ is a
cardinal$\} \cup
\{i_G(\psi(\underline{C_2(\nu,\gb)})) \mid \psi
\in {\cal G}$, $\nu = (\ha_1)^V$, and $\gb \in [\gk,\gl)$ is a
cardinal$\} \cup
\bigcup_{i = 3, 4} \{
i_G(\psi(\underline{C_i(\nu,\nu + 1)})) \mid \psi
\in {\cal G}$ and $0 \le \nu< (\ha_1)^V\}
= i_G(\underline{C(G)})$.
$N$ is then the set of all sets
%of rank less than $\gk$ of the model consisting of all sets
which are hereditarily $V$ definable from $C(G)$, i.e.,
%$N = V^{ {\rm HVD} (C(G)) }_{\gk}$.
$N = { {\rm HVD} (C(G)) }$.
Let $\la \gd_\nu \mid \nu \le (\ha_1)^V + 1 \ra$ be the
continuous, increasing enumeration of
$\{\ga_\nu \mid \nu \le (\ha_1)^V + 1\} \cup \{
(\ga^+_\nu)^V \mid \nu = \gamma + 1$ and
$\gamma < (\ha_1)^V$ is a limit ordinal$\}$.
%\ \cup \ \{\gl\}\}$.
The arguments of \cite{G85} and
\cite{A92} allow us to conclude that
$N \models ``$ZF + $\neg {\rm AC}_\go$ +
$\ha_1 = (\ha_1)^V$ is a regular cardinal +
$\la \ha_\nu \mid \nu < \ha_1 \ra =
\la \gd_\nu \mid \nu < \ha_1 \ra$ +
For every limit ordinal $\nu < \ha_1$,
$\ha_{\nu + 1}$ is a measurable cardinal +
$\gk = \ha_{\go_1}$ + Every successor
cardinal less than $\ha_{\go_1}$ is
regular + $\gk^+ = \gl =
\ha_{\go_1 + 1}$ is a measurable cardinal''.
%Every successor cardinal is regular +
%Every limit cardinal is singular + The
%successor of every singular cardinal is measurable''.
In addition, we
know that for any ordinal
$\gamma$ and any set $x \subseteq \gamma$, $x \in N$,
$x = \{\ga<\gamma \mid
V[G] \models
\phi(\ga, i_G(\psi_1(\underline{C_{i_1}(\nu_1,\gb_1)})),\ldots,
i_G(\psi_n(\underline{C_{i_n}(\nu_n,\gb_n)})), C(G))\}$,
where $i_j$ is
an integer, $1 \le j \le n$, $1 \le i_j \le 4$, each $\psi_i
\in {\cal G}$, each $\gb_i$ is an appropriate ordinal for $\nu_i$,
and $\phi(x_0, \ldots, x_{n+1})$ is a formula which may also
contain some parameters from $V$ which we shall suppress.
Let $$\overline \FP =
\prod_{i_j = 3, j \le n}
{\rm Coll}(\ga_{\nu_j}, {<} \ga_{\nu_j + 1})
\times
\prod_{i_j = 4, j \le n}
{\rm Coll}(\ga^+_{\nu_j}, {<} \ga_{\nu_j + 1})
\times \FR_{< {{\ha_1}}}.$$
For $\pi \in \FR_{< {{\ha_1}}}$,
%and $\gamma$ an arbitrary ordinal,
let $\pi \rest \gamma = \{\la q,
u, C \ra \in \pi \mid q \cap \gk \le \gamma \}$.
%and let $\FR_\gamma = \{\pi \rest \gamma \mid \pi \in \FR_{< {{\ha_1}}}\}$.
For
$p \in \overline{\FP}$, $p = \la p_1, \ldots, p_m, \pi \ra$, $m \le n$,
$\pi \in \FR_{< {{\ha_1}}}$, let
$p \rest \gamma = \la q_1, \ldots, q_m, \pi \rest
\gamma \ra$, where $q_j = p_j$ if $\ga_{\nu_j} \le \gamma$
and $q_j = \emptyset$ otherwise. In
other words, $p \rest \gamma$ is the part of p below or at
$\gamma$. Without loss of generality,
we ignore the empty coordinates
and let $\overline{\FP} \rest \gamma
= \{p \rest \gamma \mid p \in \overline{\FP} \}$. Let
$G \rest \gamma$ be the projection of
$G$ onto $\overline{\FP} \rest \gamma$. An
analogous fact to \cite[Theorem 3.2.11]{G85}
holds, using the same
proof as in \cite{G85},
%namely for any $x \subseteq \gamma$,
namely $x \in
V[G \rest \gamma]$. In addition, the elements
of $\overline{\FP} \rest \gamma$
can be partitioned into equivalence
classes (the ``almost
similar'' equivalence classes of \cite{G85}) with respect to
$\underline{C_{i_1}(\nu_1,\gb_1)},
\ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$
via an equivalence relation to be called $\sim$
such that if
$\varphi$ is any formula mentioning
only (terms for ground model sets and)
$\underline{C_{i_1}(\nu_1,\gb_1)},
\ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$, and
$\underline{C(G)}$, $p \decides \varphi$
(i.e., $p$ {\em decides} $\varphi$), and
$q \sim p$, then $q \decides \varphi$ in
the same way that $p$ does.
It thus follows as an immediate
corollary of the work of \cite{G85}
that if we define
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg
= \{ {[p]}_\sim \mid p \in G \rest \gamma \}$,
then $x \in
V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg]
$ and $V[
G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg
] \subseteq N$.
Further, suppose
$\nu \le (\ha_1)^V$ is a limit ordinal.
The work of \cite{G85} also tells us that for
$\gamma = \ga_{\nu + 1} = \gd_{\nu + 1} =
(\ha_{\nu + 1})^N$, since $x$ is a set of ordinals,
we may assume that for
$\gb = \max(\{\gb_i \mid i \le n\})$,
$\gb \in [\ga_\nu, \ga_{\nu + 1})$.
Because $\gb \in [\ga_\nu, \ga_{\nu + 1})$, it follows that
%From this, it follows that
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg$
is $V$-generic
over a partial ordering forcing equivalent to a partial ordering
%$\overline{\FP} \rest \gamma$ which is forcing equivalent
%to a partial ordering
$\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gg$
such that
$\card{\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gg}
< \gamma$.
And, if $\gg \ge \gl$ is a $V$-cardinal,
it is the case that
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg$
is $V$-generic
over a partial ordering forcing equivalent to a partial ordering
$\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gg$
such that
$\card{\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gg}
< \gl$.
The facts in the next to last sentence
and \cite[Lemma 2]{A92}, in tandem with the
way in which $N$ is defined, provide a proof that if
$\U \in V$ is a normal measure over $\gg$, then
$N \models ``\U' = \{x \subseteq \gg \mid
\exists y \subseteq x [y \in \U]\}$ is a
normal measure over $\gg$''.
From this, we will be able to
infer that $\ha_{\go_1 + 1}$
carries the desired number of normal measures in $N$.
In particular, the following two lemmas
complete the proof of Theorem \ref{t1}.
\begin{lemma}\label{l1}
Suppose $\U^* \in N$ is a normal measure over $\gl$.
Then for some normal measure $\U \in V$
over $\gl$, $\U^* = \{x \subseteq \gl \mid
\exists y \subseteq x [y \in \U]\}$.
\end{lemma}
\begin{proof}
Our proof is very similar to the proofs of
\cite[Lemma 2.2]{A06} and \cite[Lemma 1.1]{A08},
both of which use ideas from the proof of
\cite[Theorem 2.3(e)]{BK}.
Let $\gs$ be a term for $\U^*$. Since $\U^* \in N$,
we may choose ordinals $\nu_1, \ldots, \nu_n,
\gb_1, \ldots, \gb_n$ and terms
$\underline{C_{i_1}(\nu_1,\gb_1)},
\ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$
such that $\gs$ mentions only
$\underline{C_{i_1}(\nu_1,\gb_1)},
\ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$,
$\underline{C(G)}$, and canonical terms for sets in $V$.
We may assume (by ``padding'' if necessary) that for
$\gb = \max(\{\gb_i \mid i \le n\})$, $\gb \in [\gk, \gl)$.
This means by our remarks in the last paragraph that
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gl$
is $V$-generic over a partial ordering
forcing equivalent to one
having size less than $\gl$.
Again by what was mentioned
in the preceding paragraph, the set
%$\U^* \rest \la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra
$\U^{**} = \U^* \cap
V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gl]
\in
V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gl]$,
which immediately implies that
%$\U^* \rest \la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra$
$\U^{**}$ is in %\break
$V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gl]$
a normal measure over $\gl$.
Once more by what was stated in the last paragraph
and the L\'evy-Solovay results \cite{LS},
it must consequently be the case that for some
$\U \in V$ a normal measure over $\gl$, $\U^{**}$
is definable in
$V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gl]$
as $\{x \subseteq \gl \mid \exists y \subseteq x
[y \in \U]\}$. Therefore, since
$\U \subseteq \U^{**} \subseteq \U^*$ and
$\U' = \{x \subseteq \gl \mid \exists y \subseteq x
[y \in \U]\}$ as defined in $N$ is an
ultrafilter over $\gl$, $\U' = \U^*$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
In $N$, the cardinal and cofinality structure
above $\gl$ is the same as in $V$.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.3]{A06}.
Let $\gb$ and $\gg$ be arbitrary ordinals, and suppose that
$N \models ``f : \gb \to \gg$ 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{l1},
$f$ is a member of a generic extension of $V$
via a partial ordering having cardinality less than $\gl$.
Thus, $f$ cannot witness that any $V$-cardinal
greater than or equal to $\gl$ has a different
cardinality or cofinality.
This completes the proof of Lemma \ref{l2}.
\end{proof}
By Lemmas \ref{l1} and \ref{l2} and
our earlier exposition, if $V^*$ is
as in Theorem \ref{t1}, then $N$ witnesses
the conclusions of Theorem \ref{t1}.
Similarly, Lemmas \ref{l1} and \ref{l2}
and our earlier exposition 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{pf}
As with the proof found in \cite{A06}, the proof that has
just been given easily generalizes to
handling successors of different singular
cardinals of uncountable cofinality, such as
$\ha_{\go_1 + \go_1 + 1}$, $\ha_{\go_2 + 1}$, etc.
It is only necessary to change the length of
the Radin sequence of measures accordingly,
i.e., to $\go_1 + \go_1$, $\go_2$, etc.
As in \cite{A06}, though, it is unknown if
it is possible for $\ha_{\go_1 + 1}$ to carry
precisely $\gt$ normal measures, where
$\gt \le \ha_{\go_1 + 1}$ is an arbitrary
infinite or finite cardinal.
It is natural to wonder about the possibility of
extending Theorems \ref{t1} and \ref{t2}
so as also to control the number of normal measures
over $\ha_{\nu + 1}$, where $\nu < \ha_1$ is a limit ordinal.
The methods of this paper do not seem to allow
this to be done. In particular, the proof of
Lemma \ref{l1} breaks down. To see this, let
$\gg$ be such that
$N \models ``\gg = \ha_{\nu + 1}$, where
$\nu < \ha_1$ is a limit ordinal''.
We will not always be able to infer that
$\U^{**} \in
V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }_\gg]$.
%is a normal measure over $\gg$.
This is since $\U^{**}$ is not a set of ordinals,
so we cannot necessarily assume that
$\gb \in [\ga_\nu, \ga_{\nu + 1})$, where as before,
$\gb = \max(\{\gb_i \mid i \le n\})$.
%It thus won't automatically be the case that
%$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gg$
%is $V$-generic over a partial ordering
%forcing equivalent to one %a partial ordering
%having size less than $\gg$.
We therefore conclude by asking whether it is possible to
extend Theorems \ref{t1} and \ref{t2} so that the
successor of each singular cardinal
less than or equal to $\ha_{\go_1}$,
or even more generally, the successor
of each singular cardinal, is both measurable
and has the number of normal measures it carries
exactly controlled.
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%{{Radin}}
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\end{document}
It thus follows as an immediate
corollary of the work of \cite{G85} that if we define $
G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }
= \{ {[p]}_\sim \mid p \in G \rest \gamma \}$,
then $x \in
V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }]
$ and $V[
G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra }
] \subseteq N$.
$p \decides
\varphi(\ga, i_G(\psi_1(\underline{C_{i_1}(\nu_1,\gb_1)})),\ldots,
i_G(\psi_n(\underline{C_{i_n}(\\nu_n,\gb_n)})), C(G))$
(where $\decides$ means {\em decides}) and $q \sim p$, then
$q \decides
\varphi(\ga, i_G(\psi_1(\underline{C_{i_1}(\nu_1,\gb_1)})),\ldots,
i_G(\psi_n(\underline{C_{i_n}(\\nu_n,\gb_n)})), C(G))$
in the same way that $p$ does.
%$\sigma < \gamma$, $\tau$ is a suitable term for $x$, and $p
%\forces ``\sigma \in \tau$'', then for any $q \sim p$,
%$q \forces ``\sigma \in \tau$''.
(This is true for $\gl$ because in this situation, for
$\gb = \max(\{\gb_i \mid i \le n\})$,
$\gb \in [\gk, \gl)$,
which means that
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}_\gl$
is $V$-generic over a partial ordering
forcing equivalent to one %a partial ordering
having size less than $\gl$.)