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\date{February 11, 1999\\
(revised February 29, 2000)}
\title{Some Remarks on Normal Measures and Measurable Cardinals
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Measurable cardinal, normal measure, GCH}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank the referee for many
helpful comments and suggestions, which were
incorporated into this version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\}
\begin{document}
\maketitle
\begin{abstract}
We prove two theorems which in a certain sense show
that the number of normal measures a measurable
cardinal $\gk$ can carry is independent of a given fixed
behavior of the continuum function on any set having
measure 1 with respect to every normal measure over $\gk$.
First, starting with a model $V \models ``$ZFC + GCH +
$o(\gk) = \gd^*$'' for $\gd^* \le \gk^+$ any finite or infinite
cardinal, we force and construct an inner model
$N \subseteq V[G]$ so that
$N \models ``$ZF + $\forall \gd < \gk[{\rm DC}_\gd]$ +
$\neg {\rm AC}_\gk$ +
$\gk$ carries exactly $\gd^*$ normal measures +
$2^\gd = \gd^{++}$ on a set having measure 1 with respect to
every normal measure over $\gk$''.
There is nothing special about $2^\gd = \gd^{++}$ here, and
other stated values for the continuum function will be
possible as well.
Then, starting with a model
$V \models ``$ZFC + GCH + $\gk$ is supercompact'',
we force and construct models of AC in which,
roughly speaking, regardless of the specified behavior
of the continuum function below $\gk$
on any set having measure 1 with respect to
every normal measure over $\gk$,
$\gk$ can in essence carry any number of normal
measures $\gd^* \ge \gk^{++}$.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
One of the most interesting questions concerning
measurable cardinals is the number of normal measures
they can carry.
Kunen showed in \cite{K} that if $\mu$ is a
normal measure over $\gk$, then for
$\mu^* = \mu \cap L[\mu]$,
$L[\mu] \models ``\mu^*$ is the unique
normal measure over $\gk$''.
The result of Kunen and Paris
\cite{KP} shows the consistency, relative to a
measurable cardinal, of a measurable cardinal $\gk$
carrying exactly $2^{2^\gk}$ normal measures,
the maximal number of normal measures a
measurable cardinal can carry.
If $\gk$ is at least $2^\gk$ supercompact, then
one can prove outright in ZFC that $\gk$ must
possess $2^{2^\gk}$ normal measures.
(See \cite{J}, Lemma 33.11, pages 410-411 for a proof.
The argument given here, combined with an extender
argument, also shows that if $\gk$ is
$\gk + 2$ strong, i.e., there is an elementary embedding
$j : V \to M$ with critical point $\gk$ so that
$M^\gk \subseteq M$,
$j(\gk) > |V_{\gk + 2}|$, and
$V_{\gk + 2} \subseteq M$, then $\gk$ carries
$2^{2^\gk}$ normal measures as well.)
Mitchell in \cite{M} constructed, starting from a
measurable cardinal $\gk$ so that
$o(\gk) = \gd^*$ for $\gd^* \le \gk^{++}$ any
cardinal (finite or infinite), an inner model of
ZFC in which $\gk$ has exactly $\gd^*$ normal
measures.
Beyond the results just mentioned and the results of
\cite{B}, \cite{C1}, and \cite{C2},
not much is known about how many normal measures
a measurable cardinal can carry. In particular, in
$L[\mu]$ and Mitchell's inner models, GCH holds,
and any known ``reasonable '' forcing preserving
measurability acting on an unbounded
measure 0 subset of
a measurable cardinal $\gk$ in any model of ZFC
in which $2^\gk = \gk^+$,
such as a forcing making GCH fail on an unbounded
measure 0 subset of $\gk$
or a forcing adding a single Cohen
subset to every element of an
unbounded measure 0 subset
of $\gk$, automatically increases the
number of normal measures $\gk$ carries to
$2^{2^\gk}$.
Although this result is folklore, and
\cite{C1}, Lemma 6 essentially contains a proof,
for concreteness and specificity, we include
a proof here as well.
First, though, we
mention very briefly some preliminary information.
We presume readers are familiar with the basic
properties of measurable and supercompact cardinals.
Information on this subject can be found in \cite{J}.
Also, when forcing, if $\FP$ is our partial ordering
and $G \subseteq \FP$ is our $V$-generic set,
$V^\FP$ and $V[G]$ will be used interchangeably.
If $p, q \in \FP$, $q \ge p$ means that $q$ is
stronger than $p$.
For $\varphi$ a formula in the forcing language
with respect to $\FP$, $p \decides \varphi$ means
that $p$ decides $\varphi$.
Finally, from time to time, we may confuse terms
with the objects they denote, particularly when
the terms are for ground model sets.
\begin{lemma}\label{l0}
Let $V \models ``$ZFC +
$\gk$ is measurable + $2^\gk = \gk^+$''.
Let $j : V \to M$ be an elementary
embedding witnessing $\gk$'s measurability
generated by a normal measure
over $\gk$. Assume $\FP \in V$ is so that
the following hold.
\begin{enumerate}
\item\label{c1} $\FP \subseteq V_\gk$.
\item\label{c2} $|\FP| = \gk$.
\item\label{c2a} Every $p \in \FP$ has at
least two incompatible extensions.
\item\label{c3} $j(\FP)$ either has the form
$\FP \times \FQ$, where $\FQ$ is $\gk$-closed
in $V$, or $\FP \ast \dot \FQ$, where in $M$
(or $V$, since $M^\kappa \subseteq M$),
$\forces_\FP ``\dot \FQ$ is $\gk$-closed''.
\item\label{c4} If $p \in \FP$, then either
$j(p) = \la p, 0 \ra$ or
$j(p) = \la p, \dot 0 \ra$.
\end{enumerate}
Then $V^\FP \models ``\gk$ carries
$2^{2^\gk} = 2^{\gk^+}$ normal measures''.
\end{lemma}
\begin{proof}
Let $G$ be $V$-generic over $\FP$.
We first consider the case where
$j(\FP) = \FP \times \FQ$.
Note that since $|\FP| = \gk$,
by elementarity, we also have that
$|\FQ| = j(\gk)$.
Thus,
$M \models ``$The number of dense open subsets
of $\FQ$ is $2^{j(\gk)} = j(\gk^+)$''.
In addition, since $2^\gk = \gk^+$,
$|j(\gk^+)| = |\{f : f : \gk \to \gk^+$
is a function$\}| = |{[\gk^+]}^\gk| =
2^\gk = \gk^+$. This means we can let
$\la {\cal D}_\ga : \ga < \gk^+ \ra$ be
an enumeration in $V$ of the
dense open subsets of $\FQ$ present in $M$.
Since $M^\gk \subseteq M$,
we can build in $V$ a tree
$\cal T$ of height $\gk^+$ so that:
\begin{enumerate}
\item\label{k1} The root of $\cal T$ is the
empty condition.
\item\label{k2} If $p$ is an element at level
$\ga < \gk^+$ of $\cal T$, then the
successors of $p$ at level $\ga + 1$ are
a maximal incompatible subset of ${\cal D}_\ga$
extending $p$. By condition \ref{c2a} of the
hypotheses of Lemma \ref{l0}, there will be
at least two successors of $p$ at level
$\ga + 1$.
\item\label{k3} If $\gl < \gk^+$ is a limit
ordinal, then the elements of $\cal T$ at
height $\gl$ are upper bounds to any
path through $\cal T$ of height $\gl$.
\end{enumerate}
By conditions \ref{c2a} and \ref{c3} of
the hypotheses of Lemma \ref{l0},
$\cal T$ has height $\gk^+$.
And, any path $H$
of height $\gk^+$ through $\cal T$ is
an $M[G]$-generic object over $\FQ$.
This is since by construction, $H$ is
$M$-generic over $\FQ$, so since
$G$ is $V$-generic over $\FP$,
$M \subseteq V$, and $H \in V$,
$G$ is $M[H]$-generic over $\FP$.
Hence, by the Product Lemma,
$H$ is $M[G]$-generic over $\FQ$.
Therefore, since there are
$2^{\gk^+}$ paths of height
$\gk^+$ through $\cal T$, there are
$2^{\gk^+} = 2^{2^\gk}$ different
$M[G]$-generic objects over
$\FQ$.
By condition \ref{c4} of the hypotheses
of Lemma \ref{l0}, for $H$ an
$M[G]$-generic object over
$\FQ$,
since $j''G \subseteq G \ast H$,
$j : V \to M$ extends to
$j : V[G] \to M[G][H]$. This means that if
$j_1$ is the extension generated by $H_1$ and
$j_2$ is the extension generated by $H_2$,
$j_1(G) = \la G, H_1 \ra$ and
$j_2(G) = \la G, H_2 \ra$, i.e., there are
$2^{2^\gk}$ different extensions of $j$
after forcing with $\FP$.
Since Lemma 1 of \cite{C1} tells us that any
$k : V[G] \to M[G][H]$ is generated by the
normal measure over $\gk$ given by
${\cal U} = \{x \subseteq \gk :
\gk \in k(x)\}$, there are
$2^{2^\gk}$ different normal measures over
$\gk$ in $V[G]$.
Assume now that
$j(\FP) = \FP \ast \dot \FQ$.
In $M$,
$\forces_\FP ``|\dot \FQ| = j(\gk)$''.
Therefore, in $M$,
$\forces_\FP ``$The number of dense open subsets
of $\dot \FQ$ is $2^{j(\gk)} = j(\gk^+)$''.
Hence, as
$|j(\gk^+)| = \gk^+$, we can let
$\la {\cal D}_\ga : \ga < \gk^+ \ra$ be
an enumeration in $V[G]$ of the dense opens
subsets of $i_G(\dot \FQ)$ present in
$M[G]$.
Since by the $\gk^+$-c.c$.$ of $\FP$
(which follows by condition \ref{c2}
of the hypotheses of Lemma \ref{l0}),
${M[G]}^\gk \subseteq M[G]$,
we can build in $V[G]$ as we did above in $V$
a tree ${\cal T}$ of height $\gk^+$.
The construction of ${\cal T}$ ensures that
any path $H$ of height $\gk^+$ through
$\cal T$ is an
$M[G]$-generic object over $i_G(\dot \FQ)$.
We can then argue as we did earlier to show
that there are
$2^{2^\gk}$ distinct normal measures over
$\gk$ in $V[G]$.
This proves Lemma \ref{l0}.
\end{proof}
We note that the conditions of the
hypotheses of Lemma \ref{l0}
basically cover any partial ordering
defined as an Easton support product
or iteration whose support is composed
of a measure 0 subset of $\gk$. Lemma \ref{l0}
therefore means that if we start with a model
of GCH in which $\gk$ is measurable
and $2^{\gk^+}$ is first made
to be any regular cardinal $\gd^* \ge \gk^{++}$
via Cohen forcing,
since this preserves the measurability of
$\gk$, forcing a violation of GCH on
an unbounded measure 0 subset of $\gk$
via either an Easton support product or iteration
will make the number of
normal measures $\gk$ carries
$\gd^*$ as well.
Thus, the following question naturally arises:
What can one say about the relationship between
the failure of GCH on a set having measure 1
with respect to any normal measure over a
measurable cardinal $\gk$ and the number
of normal measures $\gk$ can carry?
Note that for $A \subseteq \gk$
a bounded subset of $\gk$ (which of course
automatically has measure 0 for any measure
over $\gk$), for $\gd^*$ the number of
normal measures $\gk$ carries,
by the remarks immediately following
Corollary 4.2 of \cite{B},
it is possible to
force a violation of GCH on $A$
while preserving the
fact that $\gk$ possesses exactly
$\gd^*$ normal measures, regardless
of the value of $\gd^*$.
The purpose of this paper is to provide,
both in the context of the Axiom of
Choice and its negation, an answer to
the above question which in some sense
shows that if $\gk$ is measurable, regardless of
the specified behavior of the continuum function on
sets having measure 1 with respect to any
normal measure over $\gk$, $\gk$ can
carry an arbitrary number of normal measures.
We will prove the
following two theorems for fixed partial
orderings and a fixed measure 1 set.
It will then be clear from the proofs given that
other, more general definitions are also possible.
\begin{theorem}\label{t1}
Let $V \models ``$ZFC + $o(\gk) = \gd^*$ for
$\gd^* \le \gk^+$ any finite or infinite cardinal''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N \subseteq V^\FP$ so that
$N \models ``$ZF + $\forall \gd < \gk[{\rm DC}_\gd]$ +
${\rm AC}_\gk$ fails +
$\gk$ carries exactly $\gd^*$ normal measures
$\la \mu^*_\ga : \ga < \gd^* \ra$ +
$2^\gd = \gd^{++}$
on a set which has measure 1 with respect to
any of the measures $\mu^*_\ga$''.
\end{theorem}
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ so that:
\begin{enumerate}
\item\label{i1} $V^\FP \models ``$ZFC + $\gk$ is
measurable + $2^\gk > \gk^+$ +
GCH fails on a set having measure 1
with respect to any normal measure over $\gk$
in $V^\FP$ or any further generic extension''.
\item\label{i2} For any $V^\FP$-regular cardinal
$\gd^* \ge {({(2^\gk)}^+)}^{V^\FP}$,
there is a partial ordering
$\FQ_{\gd^*} \in V^\FP$ so that
$V^{\FP \ast \dot \FQ_{\gd^*}} \models
``$ZFC +
Any $V^\FP$-cardinal
$\gd \ge {({(2^\gk)}^+)}^{V^\FP} = \gk^{++}$
is a cardinal +
The continuum function below $\gk$
is the same as in $V^\FP$ + $\gk$ carries
exactly $\gd^*$ normal measures''.
\end{enumerate}
\end{theorem}
We remark that in Theorem \ref{t2},
the behavior of the continuum function in
$V^\FP$ will depend on the exact
definition of $\FP$.
Also, as we will make explicit following the
proof of Theorem \ref{t2}, it will be
possible to weaken the hypothesis that
$\gk$ is supercompact.
Finally, in Theorem \ref{t2},
$V$ and $V^\FP$ will contain the same
cardinals and cofinalities, but although
$V^\FP$ and $V^{\FP \ast \dot \FQ_{\gd^*}}$
will contain the same cardinals and cofinalities
$\le \gk^+$ and
$\ge {({(2^\gk)}^+)}^{V^\FP}$, cardinals in the interval
$[\gk^{++}, {(2^\gk)}^{V^\FP}]$
will be collapsed to $\gk^+$ in
$V^{\FP \ast \dot \FQ_{\gd^*}}$.
The structure of this paper is as follows.
Section \ref{s1} contains our introductory
comments and preliminary remarks.
Section \ref{s2} contains the proof of
Theorem \ref{t1}.
Section \ref{s3} contains the proof of
Theorem \ref{t2}.
Section \ref{s4} contains our concluding
remarks.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V \models ``$ZFC + $o(\gk) = \gd^*$'', where
$\gd^* \le \gk^+$ is any fixed but arbitrary
finite or infinite cardinal.
By passing to the inner model of \cite{M},
we can also assume without loss of generality that
$V \models ``$GCH + $\la \mu_\ga : \ga < \gd^* \ra$
enumerates all of the normal measures over $\gk$''.
Let $\la \gd_\ga : \ga < \gk \ra$ enumerate in $V$ the
$V$-inaccessible cardinals below $\gk$.
For each $\ga < \gk$, let $\FP_\ga$ be the usual
partial ordering for adding
$\gd^{++}_\ga$ Cohen subsets to $\gd_\ga$, i.e.,
$\FP_\ga = \{f : \gd_\ga \times \gd^{++}_\ga
\to \{0,1\} : f$ is a function so that
$|\dom(f)| < \gd_\ga\}$, ordered by inclusion.
The partial ordering with which we force is
then the Easton support product
$\prod_{\ga < \gk} \FP_\ga$.
Let $G$ be $V$-generic over $\FP$.
The full generic extension is not our
desired model $N$. In order to define
$N$, we first let $G_\ga$
for $\ga < \gk$ be the
projection of $G$ onto
$\prod_{\gb \le \ga} \FP_\gb = \FQ_\ga$.
By the Product Lemma, $G_\ga$ is
$V$-generic over $\FQ_\ga$.
We can now intuitively describe $N$ as
the least model of ZF extending $V$
which contains, for each $\ga < \gk$,
the set $G_\ga$.
More formally, let ${\cal L}_1$
be the sublanguage of the forcing language
$\cal L$ with respect to $\FP$ which
contains symbols $\dot v$ for each
$v \in V$, a unary predicate symbol
$\dot V$ (to be interpreted
$\dot V(\dot v)$ iff $v \in V$), and symbols
$\dot G_\ga$ for each $\ga < \gk$.
$N$ can then be defined inside
$V[G]$ as follows.
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_{\ga + 1} = \{x \subseteq N_\ga : x$ is
definable by a term $\tau \in {\cal L}_1$
of set theoretic rank $\le \ga$ over the model
${\la N_\ga, \in, c \ra}_{c \in N_\ga}\}$.
$N_\gl = \cup_{\ga < \gl} N_\ga$ for $\gl$
a limit ordinal.
$N = \cup_{\ga \in {\rm Ordinals}^V} N_\ga$.
\setlength{\parindent}{0in}
The standard arguments show $N \models {\rm ZF}$.
\setlength{\parindent}{1.5em}
\begin{lemma}\label{l1}
Let $y \in V$. If
$x \subseteq y$, $x \in N$, then
$x \in V[G_\ga]$ for some
$\ga < \gk$.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} is very similar to
the proof of Lemma 1 of \cite{A3}.
Let $\tau$ be a term for $x$ so that
$p \forces ``\tau \subseteq y$''.
Without loss of generality, by coding if necessary,
we can assume $\tau$ mentions only one term of the form
$\dot G_\ga$ for some $\ga < \gk$.
For $q \in \FP$, $q = \la q_\gb : \gb < \gk \ra$,
define
$q \rest (\ga + 1) = \la q^*_\gb : \gb < \gk \ra$ by
$q^*_\gb = q_\gb$ if $\gb \le \ga$ and
$q^*_\gb = 0$ (the trivial condition) otherwise.
We can now define a term $\gs$ by
$q \forces ``z \in \gs$'' iff $q \ge p$,
$q \forces ``z \in \tau$'', and
$q \rest (\ga + 1) \forces ``z \in \tau$''.
It is clear that
$p \forces ``\gs \subseteq \tau$''.
We show that
$p \forces ``\tau \subseteq \gs$''.
To see that this is true, let $q \ge p$,
$q = \la q_\gb : \gb < \gk \ra$ be so that
$q \forces ``z \in \tau$'', and assume
towards a contradiction that
$q \rest (\ga + 1) \not\forces ``z \in \tau$''.
Let $r \ge q \rest (\ga + 1)$,
$r = \la r_\gb : \gb < \gk \ra$ be so that
$r \forces ``z \not\in \tau$''. If we define
$s = \la s_\gb : \gb < \gk \ra$ by
$s_\gb = r_\gb$ for $\gb \le \ga$ and
$s_\gb = q_\gb$ otherwise, then by definition,
$s \ge q$ and
$s \forces ``z \in \tau$''.
Let $r_\gb$ and $s_\gb$ be so that
$r_\gb$ and $s_\gb$ are incompatible.
Since $r_\gb, s_\gb \in \FP_\gb$ and
$\FP_\gb$ is the partial ordering for adding
$\gd^{++}_\gb$ Cohen subsets of $\gd_\gb$,
we make as a
\setlength{\parindent}{0in}
{\bf Claim:} There is an automorphism
$\psi_\gb : \FP_\gb \to \FP_\gb$
generated by a permutation of $\gd^{++}_\gb$ so that
$\psi_\gb(r_\gb)$ is compatible with $s_\gb$.
\begin{proof}
Let $\eta < \gd^{++}_\gb$ be an ordinal
$> \sup(\{\rho : \exists \zeta < \gd_\gb
[\la \zeta, \rho \ra \in \dom(r_\gb)$ or
$\la \zeta, \rho \ra \in \dom(s_\gb)]\})$.
$\eta$ exists since for any condition
$t \in \FP_\gb$, $|t| < \gd_\gb$. If
$\la \rho_i : i < \zeta_0 \ra$ enumerates
$\{\rho : \exists \zeta < \gd_\gb
[\la \zeta, \rho \ra \in \dom(r_\gb)]\}$ and
$\la \rho_i' : i < \zeta_0 \ra$ enumerates
the first $\zeta_0$ ordinals $> \eta$, then
$\psi^*_\gb : \gd^{++}_\gb \to \gd^{++}_\gb$
given by
$\psi^*_\gb(\rho_i) = \rho_i'$,
$\psi^*_\gb(\rho_i') = \rho_i$,
and $\psi^*_\gb$ is the identity otherwise
is the desired permutation, with
$\psi_\gb$ defined by applying
$\psi^*_\gb$ to each second coordinate of
a condition's domain.
\end{proof}
\setlength{\parindent}{1.5em}
Thus, if
$\pi = \la \pi_\gb : \gb < \gk \ra$ is defined by
$\pi_\gb = \psi_\gb$ if $r_\gb$ and $s_\gb$ are
incompatible and $\psi_\gb$ is as just described and
$\pi_\gb$ is the identity otherwise,
$\pi$ generates an automorphism of $\FP$ so that
$\pi(r)$ is compatible with $s$.
Note now that $\pi_\gb$ is the identity for
$\gb \le \ga$. Since terms for ground model
sets and terms mentioning only $\dot G_\ga$
can be assumed to be invariant under
automorphisms of $\FP$ not changing the
meaning of $G_\ga$,
$\pi(r) \forces ``z \not\in \tau$'',
$\pi(r)$ is compatible with $s$, and
$s \forces ``z \in \tau$''.
This contradiction means that
$q \rest (\ga + 1) \forces ``z \in \tau$'', i.e.,
$p \forces ``\tau \subseteq \gs$'', i.e.,
$p \forces ``\tau = \sigma$''.
Since $\gs$ can clearly be realized in
$V[G_\ga]$, $x \in V[G_\ga]$.
This proves Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$N \models ``\forall \gd < \gk [{\rm DC}_\gd]$''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} is essentially the
same as the proof of Lemma 2 of \cite{A3}.
Fix $\gd < \gk$ a cardinal in $N$. Assume
$p \forces ``\dot X \in \dot N$ is a set,
$\dot R \in \dot N$,
$\dot R \subseteq {[\dot X]}^{< \gd} \times \dot X$
is a relation so that
$\forall \dot x \in {[\dot X]}^{< \gd}
\exists \dot y \in \dot X [\dot x \dot R \dot y]$,
$\la \tau_\ga : \ga < \gb < \gd \ra \in \dot N$
is a sequence of elements of $\dot X$, and for
all $\gg$ so that $\ga < \gg < \gb$,
$\la \tau_\ga : \ga < \gg \ra \dot R \tau_\gg$''.
We show how to define $\tau_\gb$. Work in $V$.
Note first that inductively, each
$\tau_\gg$ for $\gg < \gb$ can be assumed to
be an element of ${\cal L}_1$.
Let
$\eta = \sup(\{\ga : \exists \gg < \gb
[\dot G_\ga$ occurs in $\tau_\gg]\})$.
Since
$\gk$ is a regular limit cardinal,
$\eta < \gk$, so
$\la \tau_\ga : \ga < \gb \ra$ can be defined
using only $\dot G_\eta$ and hence is an
element of ${\cal L}_1$.
Since $\FP$ is an Easton support product
of the appropriate Cohen partial orderings,
$\FP$ is $\gk$-c.c. Thus, let
${\cal B} \subseteq \FP \times {\cal L}_1$,
$|\cal B| < \gk$,
${\cal B} = \{\la p_\rho, \sigma_\rho \ra :
\rho < \gg^* < \gk\}$ be so that
${\cal A} = \{p_\rho : \rho < \gg^*\}$
forms a maximal antichain of conditions extending
$p$ and
$p_\rho \forces ``\la \tau_\ga : \ga < \gb \ra
\dot R \sigma_\rho$''. As before,
$\eta^* = \sup(\ga : \exists \gg < \gg^*
[\dot G_\ga$ occurs in $\sigma_\gg]\})$ is so that
$\eta^* < \gk$, meaning $\cal B$ can be used to
define a term
$\tau_\gb \in {\cal L}_1$ such that
$p \forces ``\la \tau_\ga : \ga < \gb \ra
\dot R \tau_\gb$''. Since
$\gd < \gk$, as earlier,
$\la \tau_\ga : \ga < \gd \ra \in {\cal L}_1$.
By the fact
$\la \tau_\ga : \ga < \gd \ra$ can be realized in
$N$, $\la \tau_\ga : \ga < \gd \ra$ will denote in
$N$ a ${\rm DC}_\gd$ sequence for $\dot R$ and
$\dot X$.
This proves Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$N \models ``$If $\gd < \gk$ is
inaccessible in $V$, then $\gd$
is inaccessible and
$2^\gd = \gd^{++}$''.
\end{lemma}
\begin{proof}
Let $\ga < \gk$ be so that $\gd = \gd_\ga$. Write
$\FP$ as $\FQ_\ga \times \FQ^\ga$, where
$\FQ^\ga = \prod_{\gb > \ga} \FP_\gb$.
By the definition of $\FP$, $\FQ_\ga$
must be the Easton support product of
partial orderings which add, for every
$\gb \le \ga$, $\gd^{++}_\gb$ subsets of $\gd_\gb$.
Thus, the standard Easton arguments
(see \cite{J}, pages 192-194) show
$V^{\FQ_\ga} \models ``2^{\gd_\gb} =
\gd^{++}_\gb$ for $\gb \le \ga$, and for
$\gb \le \ga$, $\gd_\gb$ is inaccessible''.
Further, by the closure properties of
$\FP_\gb$ for $\gb > \ga$, both
$V^{\FQ^\ga \times \FQ_\ga} = V^\FP$ and
$V^{\FQ_\ga}$ contain the same subsets of
$\gd_\gb$ for $\gb \le \ga$. Thus, since
$V^{\FQ_\ga} \subseteq N \subseteq V^\FP$,
$N \models ``2^{\gd_\gb} = \gd^{++}_\gb$
for $\gb \le \ga$, and for $\gb \le \ga$,
$\gd_\gb$ is inaccessible''.
This proves Lemma \ref{l3}.
\end{proof}
We remark that
Lemma \ref{l3} shows that
$N \models ``\gk$ is inaccessible''. Also,
Lemmas \ref{l1} - \ref{l3}
show that
$N \models \neg {\rm AC}_\gk$.
To see this, define in $N$ for each
$\ga < \gk$ the set
$X_\ga = \{x : x = \la x_\gb : \gb <
\gd^{++}_\ga \ra$ is a sequence of
$\gd^{++}_\ga$ distinct subsets of
$\gd_\ga\}$. Although
$\la X_\ga : \ga < \gk \ra \in N$,
${(\prod_{\ga < \gk} X_\ga)}^N = \emptyset$.
This follows since an element $y$ of
${(\prod_{\ga < \gk} X_\ga)}^N$ codes a
set of ordinals, so by Lemma \ref{l1},
$y \in V[G_{\ga_0}]$ for some $\ga_0 < \gk$.
This, however, is impossible, as
$|\FQ_{\ga_0}| < \gk$, so there is some
$\gb_0 < \gk$ so that for every
$\gb \ge \gb_0$,
$2^{\gd_\gb} = \gd^+_\gb$ in
$V[G_{\ga_0}]$.
\begin{lemma}\label{l4}
For any $\ga < \gd$,
$N \models ``\mu^*_\ga = \{x \subseteq \gk :
x$ contains a $\mu_\ga$ measure 1 set$\}$ is
a normal measure over $\gk$''.
\end{lemma}
\begin{proof}
Fix $\ga < \gd$. Suppose $x \subseteq \gk$,
$x \in N$. By Lemma \ref{l1},
$x \in V[G_{\gb_0}]$ for some $\gb_0 < \gk$.
Since $|\FQ_{\gb_0}| < \gk$, the
L\'evy-Solovay results \cite{LS} show that
$V[G_{\gb_0}] \models ``$Either $x$ or
$\gk - x$ contains a $\mu_\ga$ measure 1 set''. Since
$V[G_{\gb_0}] \subseteq N$, this same fact is true
in $N$ as well. Further, if
$N \models ``\la x_\gb : \gb < \gd < \gk \ra$ is a
sequence of subsets of $\gk$ so that each
$x_\gb$ contains a $\mu_\ga$ measure 1 set'', then since
$\la x_\gb : \gb < \gd < \gk \ra$ can be coded by a
set of ordinals, by Lemma \ref{l1},
$\la x_\gb : \gb < \gd < \gk \ra \in V[G_{\gb_1}]$ for some
$\gb_1 < \gk$. Since $|\FQ_{\gb_1}| < \gk$, the results of
\cite{LS} yield that
$V[G_{\gb_1}] \models ``\cap_{\gb < \gd} x_\gb$
contains a $\mu_\ga$ measure 1 set''. Again, since
$V[G_{\gb_1}] \subseteq N$, this same fact is true in $N$
as well. Finally, if
$N \models ``f : \gk \to \gk$ is regressive'',
then since $f$ can be coded by a set of ordinals,
Lemma \ref{l1} yields that for some $\gb_2 < \gk$,
$f \in V[G_{\gb_2}]$. Since
$|\FQ_{\gb_2}| < \gk$, the results of \cite{LS} then
show that
$V[G_{\gb_2}] \models ``f$ is constant on a $\mu_\ga$
measure 1 set''. Since
$V[G_{\gb_2}] \subseteq N$, this same fact remains
true in $N$ also.
This proves Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$N \models ``$If $\mu$ is a normal measure over $\gk$,
then $\mu = \mu^*_\gb$ for some $\gb < \gd$''.
\end{lemma}
\begin{proof}
Fix $p \in \FP$ and $\dot \mu$ a term for $\mu$ so that
$p \forces ``\dot \mu \in \dot N$ is a normal measure
over $\gk$''. By coding if necessary, let
$\ga < \gk$ be so that $\dot \mu$ mentions only one
term of the form $\dot G_\ga$. By Lemma \ref{l1},
if $\dot x$ is a term for a subset of $\gk$
mentioning only $\dot G_\ga$, then
$q \decides ``\dot x \in \dot \mu$'' iff
$q \rest (\ga + 1) \decides \dot x \in \dot \mu$''.
Thus, if we let $\dot \mu \rest (\ga + 1)$ be the term
definable in $V[G_\ga]$ as
$q \forces ``\dot x \in \dot \mu \rest (\ga + 1)$'' iff
$q \ge p$, $q \rest (\ga + 1) \in G_\ga$,
$x \in V[G_\ga]$, $x \subseteq \gk$, and
$q \rest (\ga + 1) \forces ``\dot x \in \dot \mu$'',
then standard arguments show that since $\dot \mu$
is a term for a normal measure over $\gk$ in $N$,
$\mu \rest (\ga + 1)$ is a normal
measure over $\gk$ in $V[G_\ga]$.
Therefore, by the results of \cite{LS}, since
$|\FQ_\ga| < \gk$, $\mu \rest (\ga + 1)$ must extend
some ground model normal measure $\nu$ over $\gk$,
i.e., for some $\gb < \gd$ and some $p' \ge p$,
$p' \forces ``\dot \mu \rest (\ga + 1)$ extends $\mu_\gb$''.
(See also the last part of the
proof of Lemma 8 of
\cite{A2} and the proof of
Lemma 0.1 of \cite{A1},
which essentially show that if
$\FQ \times \FQ$ is $\gl$-c.c., then any normal
measure over $\gl$ in $V^\FQ$ must extend a normal
measure over $\gl$ in $V$.)
Since the results of \cite{LS} also imply that
$\mu^*_\gb \rest (\ga + 1) = \{x \subseteq \gk :
x \in V[G_\ga]$ and $x$ contains a
$\mu_\gb$ measure 1 set$\}$ is a normal measure
over $\gk$ in $V[G_\ga]$, and since any
normal measure over $\gk$ in $V[G_\ga]$
extending $\mu_\gb$ must
contain $\mu^*_\gb \rest (\ga + 1)$,
$p' \forces ``\dot \mu \rest (\ga + 1) =
\dot \mu^*_\gb \rest (\ga + 1)$''.
We show now that $\mu$ must be so that
$p' \forces ``\dot \mu = \dot \mu^*_\gb$''.
To see this, assume to the contrary that
$r \ge p'$ is so that $A \in \mu_\gb$ and
$r \forces ``A \subseteq \dot x \subseteq \gk$
is so that $\dot x \in \dot \mu^*_\gb$ but
$\dot x \not\in \dot \mu$'', i.e.,
$r \forces ``A \subseteq \dot x \subseteq \gk$
is so that
$\dot x \in \dot \mu^*_\gb$ and
$\gk - \dot x \in \dot \mu$'', i.e., since
$r \forces ``\gk - A \supseteq \gk - \dot x$'',
$r \forces ``\gk - A \in \dot \mu$''. As
$r \forces
``A \in \mu_\gb \subseteq \dot \mu^*_\gb \rest (\ga + 1) =
\dot \mu \rest (\ga + 1) \subseteq \dot \mu$'', we then have that
$r \forces ``A \in \dot \mu$,
$\gk - A \in \dot \mu$, and
$\dot \mu$ is a normal measure over $\gk$ in $\dot N$'',
an immediate contradiction. Thus,
$p' \forces ``\dot \mu = \dot \mu^*_\gb$''.
This proves Lemma \ref{l5}.
\end{proof}
Since $\{\gd < \gk : \gd$ is inaccessible$\} \in \mu_\ga$
for every $\ga < \gd^*$,
Lemmas \ref{l1} - \ref{l5} complete the proof of
Theorem \ref{t1}.
\end{proof}
We conclude this section by remarking that the
proof just given is equally valid if we replace
$\{\gd < \gk : \gd$ is inaccessible$\}$ with
an arbitrary unbounded subset
$A \subseteq \gk$ composed of regular
cardinals, regardless if $A$ has
measure 1 with respect to any measure
over $\gk$ in $V$.
Also, it is possible to have $\gd^* =
\gk^{++}$ in Theorem \ref{t1},
but since the case of $\gd^* = \gk^{++}$
is explicitly covered in the context of the full
Axiom of Choice in Theorem \ref{t2}, we did not
include a reference to this value of
$\gd^*$ in Theorem \ref{t1} and its proof.
Finally, if
$f : \gk \to \gk$ is so that
$\dom(f) = \{\gd < \gk : \gd$ is inaccessible$\}$
and $f$ is defined as in Easton's theorem
(see \cite{J}, pages 192-194), then we can obtain
by the methods just given
a model for the
conclusions of Theorem \ref{t1} in which
$2^\gd = f(\gd)$ for $\gd < \gk$ inaccessible.
This tells us that in a very real sense, we can
fix some arbitrary behavior of the
continuum function on the set of inaccessible cardinals
below $\gk$ or any unbounded $A \subseteq \gk$
composed of regular cardinals
and still have $\gk$ carry $\gd^*$
normal measures, for $\gd^* \le \gk^+$ any
finite or infinite cardinal.
\section{The Proof of Theorem \ref{t2}}\label{s3}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
The partial ordering $\FP$ to be defined can easily be
modified to handle other, similar definitions,
such as reverse Easton iterations given
using Easton functions as just discussed,
but for concreteness, let $\FP$ be the
reverse Easton iteration which adds, to every inaccessible
$\gd \le \gk^{+5}$, $\gd^{+5}$ Cohen subsets of
$\gd$.
Standard arguments show that
$\FP$ is cardinal and cofinality preserving and
$V^\FP \models ``$ZFC + $\gk$ is $2^\gk$ supercompact +
$2^\gd = \gd^{+5}$ for every inaccessible cardinal
$\gd \le \gk$ + $2^\gd = \gd^+$ for every cardinal
$\gd \ge \gk^{+5}$''. Thus, in $V^\FP$, $\gk$ carries
$2^{2^\gk} = 2^{\gk^{+5}} = \gk^{+6}$ normal measures.
For $\gd^* \ge {({(2^\gk)}^+)}^{V^\FP}
= \gk^{+6}$
a $V$ or $V^\FP$-regular cardinal,
we wish to define
$\FQ_{\gd^*} \in V^\FP$ so that
$V^{\FP \ast \dot \FQ_{\gd^*}} \models ``$ZFC +
Every $V^\FP$-cardinal $\gd \ge
{({(2^\gk)}^+)}^{V^\FP}$ is a cardinal +
The continuum function below $\gk$ is the same
as in $V^\FP$ + $\gk$ carries exactly $\gd^*$
normal measures''.
We consider two cases.
\setlength{\parindent}{0in}
Case 1: $\gd^* = {({(2^\gk)}^+)}^{V^\FP} =
\gk^{+6}$.
In this case, if
$\FQ_{\gd^*}$ is the partial ordering for
adding a Cohen subset of $\gk^+$, then
by the fact
$V^\FP \models ``\FQ_{\gd^*}$ is $\gk$-closed'',
$V^{\FP \ast \dot \FQ_{\gd^*}} \models ``$The
continuum function below $\gk$ is the same
as in $V$ + Every
cardinal $\gd \le \gk^+$ in $V^\FP$ is
a cardinal''. And, since all conditions
$p \in \FQ_{\gd^*}$ have cardinality
$\le \gk$, any subset $x \subseteq \gk$ in
$V^\FP$ can have its characteristic function
$\chi$ coded into an extension $q$ of $p$ by letting
$\la \sigma_\eta : \eta < \gk \ra$ be the first
$\gk$ ordinals not occurring in $\dom(p)$, defining
$\chi'(\sigma_\eta) = \chi(\eta)$, and letting
$q = p \cup \chi'$.
Thus, standard density arguments
(see also Exercise 19.7, page 183 of \cite{J} and
Lemma 4 of \cite{A3}) show that
${(2^{\gk})}^{V^\FP} =
\gk^{+5}$
is collapsed to $\gk^+$. Since
$V^\FP \models ``|\dot \FQ_{\gd^*}| = \gk^{+5}$'',
standard chain condition arguments show that every
$V$ or $V^\FP$-cardinal $\gd \ge
\gk^{+6} =
{({(2^\gk)}^+)}^{V^\FP}$ is preserved.
Also, this means
$V^{\FP \ast \dot \FQ_{\gd^*}} \models
``2^{2^\gk} = 2^{\gk^+} = \gd^* = {(\gk^{+6})}^{V^\FP}
={({(2^\gk)}^+)}^{V^\FP}
= \gk^{++}$ + For every cardinal $\gd \ge \gk^{++}$,
$2^\gd = \gd^+$''.
Therefore, as any normal measure over $\gk$ in
$V^\FP$ remains a normal measure over $\gk$ in
$V^{\FP \ast \dot \FQ_{\gd^*}}$ by the
$\gk$-closure of $\FQ_{\gd^*}$, and as
$\gk$ carries
${(\gk^{+6})}^{V^\FP}$ normal measures in
$V^\FP$, $\gk$ carries $\gd^*$
normal measures in
$V^{\FP \ast \dot \FQ_{\gd^*}}$.
Case 2: $\gd^* > {({(2^\gk)}^+)}^{V^\FP}$.
In this case, let
$\FQ_{\gd^*} = \FQ_0 \ast \dot \FQ_1 \ast
\dot \FQ_2$, where $\FQ_0$ is the partial ordering
for adding a Cohen subset of $\gk^+$,
$\dot \FQ_1$ is a term for the usual Cohen ordering
for making
$2^{\gk^+} = \gd^*$, and $\dot \FQ_2$
is a term for the Easton support product
of partial orderings which add, for every
inaccessible cardinal $\gd < \gk$, a Cohen
subset of $\gd^{+6}$.
By the arguments from Case 1 and the
closure properties of $\FQ_0$,
$V^{\FP \ast \dot \FQ_0} \models
``$Every $V^\FP$-cardinal $\gd \ge
{({(2^\gk)}^+)}^{V^\FP} =
\gk^{++}$ is a cardinal +
For every cardinal
$\gd \ge \gk$, $2^\gd = \gd^+$ + $\gk$ is measurable +
The continuum function below $\gk$ is
the same as in $V^\FP$''.
By the closure and cardinal preservation
properties of $\FQ_1$,
the arguments of \cite{J}, pages 192-194,
which show that forcing with $\FQ_2$ over
$V^{\FP \ast \dot \FQ_0 \ast \dot \FQ_1}$
doesn't collapse cardinals or
change the value of the continuum
function below $\gk$, and Lemma \ref{l0}
applied to forcing over
$V^{\FP \ast \dot \FQ_0 \ast \dot \FQ_1}$
with $\FQ_2$,
$V^{\FP \ast \dot \FQ_0 \ast \dot \FQ_1 \ast \dot \FQ_2} =
V^{\FP \ast \dot \FQ_{\gd^*}} \models ``$Every
$V^\FP$-cardinal $\gd \ge
{({(2^\gk)}^+)}^{V^\FP} =
\gk^{++}$ is a cardinal + The
continuum function below $\gk$ is the same as in
$V^\FP$, $V^{\FP \ast \dot \FQ_0}$, or
$V^{\FP \ast \dot \FQ_0 \ast \dot \FQ_1}$ +
$\gk$ carries exactly $\gd^*$ normal measures''.
\setlength{\parindent}{1.5em}
Cases 1 and 2 complete the proof of Theorem \ref{t2}.
\end{proof}
We observe that
if we wish $\gk$ to carry
$\gk^{+6}$ normal measures in our
final model,
$\dot \FQ_{\gd^*}$ can be taken as a term
for the trivial partial ordering.
Also, by unpublished results of Woodin,
it is possible to start with a model
$V \models ``$ZFC + GCH + $\gk$ is strong''
and use a somewhat more complicated forcing
$\FP$ to obtain a model
$V^\FP \models ``$ZFC + $\gk$ is $\gk + 2$
strong +
GCH holds except that
for every inaccessible
cardinal $\gd \le \gk$ and every cardinal $\gg$
so that $\gd \le \gg \le \gd^{+4}$,
$2^\gg = \gd^{+5}$''.
Since
under these circumstances, $\gk$ carries
$2^{2^\gk}$ normal measures, this allows
the hypotheses of Theorem \ref{t2} to be
greatly reduced in consistency strength,
even when using an Easton function to
define $\FP$, regardless of the exact
nature of $\FP$.
\section{Concluding Remarks}\label{s4}
We conclude by remarking that as
observed in Sections \ref{s2} and
\ref{s3}, modulo the nature of the
forcing constructions used, the
continuum function below $\gk$ in
both Theorems \ref{t1} and \ref{t2}
has a great deal of freedom.
However, trying to obtain a model in which the
Axiom of Choice is true, GCH fails on a measure 1
subset of $\gk$ (or more strongly, $2^\gk > \gk^+$),
and $\gk$ carries $\gd^*$ normal measures for
$1 \le \gd^* < 2^{2^\gk}$ remains a tantalizing
open problem.
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\end{document}