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\title{On the Number of Normal Measures $\ha_1$
and $\ha_2$ can Carry
\thanks{2000 Mathematics Subject Classifications: 03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, measurable cardinal,
normal measure, symmetric inner model,
supercompact Radin forcing.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY Grants and
CUNY Collaborative
Incentive Grants.
In addition, the author wishes to
thank the referees
for helpful comments and
suggestions which have been
incorporated into the current version
of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{April 12, 2007\\
(revised August 13, 2007 and October 17, 2007)}
\begin{document}
\maketitle
\begin{abstract}
We show that assuming the consistency
of certain large cardinals (namely
a supercompact cardinal with a
measurable cardinal above it), it is
possible to force and
construct choiceless universes
of ZF in which the first two
uncountable cardinals
$\ha_1$ and $\ha_2$ are both measurable
and carry certain fixed numbers of
normal measures.
%The Axiom of Choice fails in these models.
Specifically, in
the models constructed,
$\ha_1$ will carry exactly one normal
measure, namely
$\mu_\go = \{x \subseteq \ha_1 \mid x$
contains a club set$\}$, and $\ha_2$
will carry exactly
$\gt$ normal measures, where $\gt \ge
\ha_3$ is any regular cardinal.
This contrasts with the
well-known
facts that assuming
AD + DC, $\ha_1$ is measurable and carries exactly
one normal measure, and $\ha_2$ is
measurable and carries exactly two normal measures.
\end{abstract}
\baselineskip=24pt
%\newpage
%\section{Introduction and Preliminaries}\label{s1}
We begin with a brief introduction
to our terminology, notation, and conventions.
We will primarily be discussing the
construction of models of Zermelo-Fraenkel
(ZF) set theory without the Axiom of Choice (AC).
These universes, however, will satisfy
the {\em Axiom of
Dependent Choice (DC)}, a weakened form of AC
which says, roughly speaking, that it is
possible to make countably many arbitrary
choices, each dependent on the preceding.
A more precise statement of DC may
be found in \cite[page 50]{J}.
We will also mention an axiom that
contradicts AC, the {\em Axiom of Determinacy (AD)}.
This axiom, roughly speaking, says that
certain infinite two-person games of perfect information
are {\em determined}, i.e., one of the players must
have a winning strategy for the game.
A more precise statement of AD may
be found in \cite[page 627]{J}.
We will be working with different
{\em large cardinal axioms}, i.e., axioms
asserting the existence of cardinal
numbers not provable in either
ZF or ZF + AC alone. Large cardinals,
both with and without AC, have played
a significant role in modern set theory,
as can be seen by consulting \cite{J}.
A large cardinal axiom that will be of
particular importance to us is the
axiom asserting the existence of a
{\em measurable cardinal}.
The cardinal $\gk$ is {\em measurable}
if $\gk$ carries a $\gk$-additive,
nonprincipal ultrafilter $\mu$,
which is frequently referred to as a {\em measure}.
The measure $\mu$ over the measurable cardinal
$\gk$ is {\em normal} if for every function
$f : \gk \to \gk$ such that
$\{ \ga < \gk \mid f(\ga) < \ga \} \in \mu$, there is
some $\ga_0 < \gk$ with
$\{ \ga < \gk \mid f(\ga) = \ga_0 \} \in \mu$.
In addition, there is a generalization
of the notion of measurable cardinal
which is important for the purposes of this
paper. For $\gk < \gl$ two cardinals,
{\em $\gk$ is $\gl$ supercompact} if
the set $P_\gk(\gl) = \{ p \subseteq \gl \mid
\card{p} < \gk \}$ carries a
$\gk$-additive, fine, normal ultrafilter
${\cal U}$. (The ultrafilter ${\cal U}$ over $P_\gk(\gl)$
is {\em fine} if for every ordinal
$\ga < \gl$, $\{ p \mid \ga \in p \} \in {\cal U}$.
${\cal U}$ is {\em normal} if for every function
$f : P_\gk(\gl) \to \gl$, there is an ordinal
$\ga < \gl$ such that
$\{p \in P_\gk(\gl) \mid f(p) =
\ga\} \in {\cal U}$.)
The existence of a $\gl$ supercompact
cardinal for a cardinal $\gl > \gk$ is
much stronger in consistency strength than
the existence of a measurable cardinal.
There are some combinatorial notions
which are also relevant to the forthcoming discussion.
Suppose $\gk$ is a regular uncountable cardinal.
The set $C \subseteq \gk$ is called
{\em closed unbounded} or {\em club} if
for every $\ga < \gk$, there is some
$\gb \ge \ga$, $\gb \in C$ (unbounded), and for
every increasing sequence
$\la \gb_\ga \mid \ga < \gg \ra$
of elements of $C$, of any length $\gg < \gk$,
$\sup(\la \gb_\ga \mid \ga < \gg \ra) \in C$ (closed).
If the set $C \subseteq \gk$ is unbounded
and closed under suprema of increasing
$\gg$ sequences for $\gg < \gk$
a regular cardinal, then $C$ is called
{\em $\gg$ closed unbounded} or {\em $\gg$ club}.
Finally, for ordinals $\ga, \gb, \gg$ with
$\gg \le \gb \le \ga$,
the {\em partition property}
$\ga \to {(\gb)}^\gg$ means that
for every $F : {[\ga]}^\gg \to 2$, there is
some $X \subseteq \ga$ having
order type $\gb$ such that
$\card{F '' {[X]}^\gg} = 1$.
The Axiom of Choice contradicts all
such partition properties with $\gg$ infinite.
We continue now with the main narrative.
It is a consequence of
AD + DC that $\ha_1$ and
$\ha_2$ are measurable cardinals,
$\ha_1$ carries exactly one normal measure
(namely
$\mu_\go = \{x \subseteq \ha_1 \mid x$
contains a club set$\}$), and $\ha_2$
carries exactly two normal measures.
This follows since assuming AD + DC,
$\ha_1 \to {(\ha_1)}^{\ha_1}$,
$\forall \gd < \ha_2 [\ha_2 \to
{(\ha_2)}^\gd]$,
and if a successor
cardinal $\gk$ satisfies the {\em 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 $\gk$.
The proofs of these first two facts
(along with a historical discussion)
can be found in \cite[pages 1--7, pages 39--45,
and page 67]{Kl2}, and the proof of
this last fact can be found in
\cite[Section 2,
pages 416--420]{Kl3}.\footnote{In fact,
if a successor cardinal
$\gk$ satisfies the weak partition property,
then any normal measure $\gk$ carries
must be of the form $\mu_\gd =
\{x \subseteq \gk \mid x$ contains a set
which is $\gd$ club$\}$, where
$\gd < \gk$ is a regular cardinal. (When
$\gk = \ha_1$, this definition
of $\mu_\go$ coincides with the one
given earlier.)
From this and the preceding, we may immediately infer
that assuming AD + DC, $\ha_2$ carries
exactly two normal measures, which are
given by $\mu_\go$ and $\mu_{\ha_1}$.}
When the Axiom of Determinacy is not
assumed, however,
%the situation concerning
the number of normal
measures that $\ha_1$ and
$\ha_2$ can carry if both of these cardinals
are measurable is not so clear.
This motivates
the purpose of this note, which is to
shed new light on this situation and
construct, via forcing
over a ground model of ZFC containing large cardinals,
models of ZF + DC in which
both $\ha_1$ and $\ha_2$
are measurable, $\ha_1$ carries
exactly one normal measure
(specifically, $\mu_\go$), and $\ha_2$ carries
%a fixed number of normal measures.
exactly $\gt$ normal measures,
where $\gt \ge \aleph_3$
is any regular cardinal.
More explicitly, 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 are 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 + $\gk = \ha_1$ and
$\gl = \ha_2$ are measurable cardinals''.
In $N$,
the regular cardinals greater than
or equal to $\gl$ are the same
as in $V$ (which has the same
cardinal and cofinality structure
at and above $\gl$ as $V^*$),
$\ha_1$ carries exactly one normal
measure (namely $\mu_\go$),
and $\ha_2$ 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 are 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 + $\gk = \ha_1$ and
$\gl = \ha_2$ are measurable cardinals''.
In $N$,
the regular cardinals greater than
or equal to $\gl$ are the same
as in $V$ (so $\ha_3$ is regular),
$\ha_1$ carries exactly one normal
measure (namely $\mu_\go$),
and $\ha_2$ carries
exactly $\ha_3$ 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
%$\ha_2$ to be measurable and
%to carry $\gt$ normal
%measures, where $\gt \ge
%\ha_3$ is any regular cardinal,
%while $\ha_1$ is also measurable
%and carries exactly one normal
%measure, which is given by $\mu_\go$.
We note that of necessity,
the models constructed witnessing
the conclusions of Theorems \ref{t1}
and \ref{t2} must violate the
Axiom of Choice.
This is since a successor cardinal
which is measurable can only exist
in a choiceless model of ZF.
Also, as Schindler's results of
\cite{Sc} show, the consistency
strength of two successive
measurable cardinals is quite large,
thereby requiring strong assumptions
for the proofs of Theorems \ref{t1}
and \ref{t2}.
Our method of proof will hinge on a
use of Woodin's technique exposited in
\cite[Theorem 1]{AH} for forcing
both $\ha_1$ and $\ha_2$ to be
simultaneously measurable.
Since this method will ensure that
$\mu_\go$ is a normal measure over
$\ha_1$, $\mu_\go$ is in fact the
{\em unique} normal measure over $\ha_1$.
We encapsulate this in the following
easy proposition.
\setcounter{proposition}{+2}
\begin{proposition}\label{p1}
ZF $\vdash ``$If $\mu_\go$ is a normal
measure over $\ha_1$, then it is
the {\em unique} normal measure
over $\ha_1$''.
\end{proposition}
\begin{proof}
By standard arguments
(see \cite[Exercise 8.8, page 104
and Lemma 8.11, page 96]{J}),
in ZF alone, any
normal measure over a measurable
cardinal must contain all club sets.
Thus, if $\mu_\go$ is a normal
measure, and in particular, an
ultrafilter over $\ha_1$, it is
automatically the case that for
any other normal measure $\mu$
over $\ha_1$, $\mu_\go \subseteq \mu$.
It then immediately follows as usual that
$\mu = \mu_\go$.
%This completes the proof of Proposition \ref{p1}.
\end{proof}
Having completed our introductory comments,
we turn now to the proofs of
Theorems \ref{t1} and \ref{t2}.
We stress that we will be presuming
henceforth a reasonably good understanding
of large cardinals and forcing.
Some knowledge of the proof of \cite[Theorem 1]{AH}
will be helpful as well.
\begin{proof}
Our presentation is similar
in spirit to that given in \cite{A06}.
As in the proofs of
\cite[Theorems 1 and 2]{A06}, we
present a unified proof of the
results in question. We begin by noting that
by \cite[Lemma 2.1]{A06} and the remarks
immediately preceding, we may assume
without loss of generality that
$V^*$ has been generically extended to
a model $V$ having certain additional key properties.
For Theorem \ref{t1}, $V$ has the same
cardinal and cofinality structure as $V^*$, and
$V \models ``$ZFC + $\gk < \gl$ are
such that $\gk$ is $\gl$ supercompact
and $\gl$ is the least measurable
cardinal above $\gl$ + $\gl$ carries
exactly $\gt$ normal measures''.
Here, $\gt$ is as in the statement
of Theorem \ref{t1}. For Theorem \ref{t2},
$V \models ``$ZFC + $\gk < \gl$ are
such that $\gk$ is $\gl$ supercompact
and $\gl$ is the least measurable
cardinal above $\gl$ + $\gl$ carries
exactly $\gl^+$ normal measures''.
We are now able to describe the
symmetric inner model $N$ which
will witness the conclusions of
either Theorem \ref{t1} or Theorem \ref{t2}.
What we are about to present is almost
completely dependent on the discussion of
the proof of \cite[Theorem 1]{AH}.
Since this material is quite complicated,
we will not duplicate it here, but will
refer readers to \cite{AH} for
any missing details.
The forcing conditions $\FP$ to be used are
${\rm SC}(\gk, \gl) \times {\rm Coll}(\go, {<} \gk)$,
where ${\rm SC}(\gk, \gl)$ is supercompact
Radin forcing as described in \cite{AH}, and
${\rm Coll}(\go, {<} \gk)$ is the usual
L\'evy collapse of $\gk$ to $\ha_1$.
Let $G$ be $V$-generic over $\FP$.
Take ${\cal G}$ as the set
of restrictions of $G$ described in \cite[page 595]{AH},
which code collapses
of cardinals in the open interval
$(\go, \gk)$ to $\ha_1$ and collapses of
cardinals in the
open interval $(\gk, \gl)$ to $\gk^+$.
$N$ is then given by ${\rm HVD}^{V[G]}({\cal G})$,
the class of all sets hereditarily $V$-definable
in $V[G]$ from an element of the set ${\cal G}$.
Standard arguments now show that
$N \models {\rm ZF}$.
By \cite[Lemmas 1.1 - 1.5]{AH} and the
intervening remarks,
$N \models ``{\rm DC} + \gk = \ha_1$ +
$\gl = \gk^+ = \ha_2$ +
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 $\gl$ +
$\ha_1$ is measurable via $\mu_\go$''.
In addition, \cite[Lemmas 1.2 and 1.3]{AH}
and their proofs provide
us with the following proposition.
\begin{proposition}\label{p2}
For every set
$r = \{ r_1, r_2 \}$ for which
$r_1, r_2 \in {\cal G}$,
there is a term $\dot r$ such that
any formula mentioning only
(canonical terms for ground model sets and)
$\dot r$ may be
decided in $V[r]$
the same way as in $V[G]$.
Further, $V[r]$
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
$V[r]$ for the appropriate $r$.
\end{proposition}
Proposition \ref{p2} 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}
Our proof is almost identical
to the proof of \cite[Lemma 2.2]{A06}.
Let $\gt$ be a term for
$\U^*$. Since $\U^* \in N$, we
may choose $r = \{r_1, r_2 \}$ with
$r_1, r_2 \in {\cal G}$
such that
$\gt$ mentions only $\dot r$
and canonical terms for sets in $V$.
By Proposition \ref{p2}, the set
${\cal U}^* \rest r = \U^* \cap
V[r] \in V[r]$, which
immediately implies that
${\cal U}^* \rest r$ is in
$V[r]$ a normal measure over $\gl$.
Again by Proposition \ref{p2}
and the L\'evy-Solovay results \cite{LS},
it must consequently be the case that
for some $\U \in V$ a normal measure
over $\gl$,
${\cal U}^* \rest r$ is definable in
$V[r]$ as
$\{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$.
Therefore, since ${\cal U} \subseteq
{\cal U}^* \rest r \subseteq {\cal U}^*$
and ${\cal U}' =
\{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$
as defined in $N$
is an ultrafilter over $\gl$,
${\cal U}' = {\cal U}^*$.
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 regular cardinals greater than or
equal to $\gl$ are the same as in $V$.
\end{lemma}
\begin{proof}
We mimic the proof of \cite[Lemma 2.3]{A06}.
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 Proposition \ref{p2},
$f \in V[r]$ for some $r = \{ r_1, r_2 \}$
where $r_1, r_2 \in {\cal G}$.
Since $V[r]$
is obtained by forcing with a partial
ordering having size less than $\gl$,
$f$ cannot witness that any $V$-regular cardinal
greater than or equal to
$\gl$ has a different
cardinality or cofinality.
This completes the
proof of Lemma \ref{lcard}.
\end{proof}
By Lemmas \ref{lnorm} and \ref{lcard}
and our earlier remarks,
if $V^*$ and $V$ are as in
the proof of Theorem \ref{t1},
then $N$ witnesses the conclusions
of Theorem \ref{t1}.
Similarly, Lemmas \ref{lnorm} and \ref{lcard}
and our earlier remarks imply that if
$V^*$ and $V$ are as in
the proof of 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}
We conclude by asking the general question
of how many normal measures each of
$\ha_1$ and $\ha_2$ can carry when both
$\ha_1$ and $\ha_2$ are simultaneously measurable.
Because of the present state of
set theoretic technology,
the results under AD + DC and of this paper
seem to be all that can be
currently established.
These theorems paint what appears to be a rather incomplete
picture of what we conjecture the general situation
most likely is, i.e., that
$\ha_1$ and $\ha_2$ can carry any number
of normal measures when both are measurable.
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\end{document}
${\cal U}^*$ is a normal measure
over $\gl$,
for ${\cal U}'$
the normal measure over $\gl$
defined in $N$ as
$\{x \subseteq \gl \mid
\exists y \subseteq x[y \in {\cal U}]\}$,
${\cal U}' = {\cal U}^*$.
This is since otherwise, there is
$x \in {\cal U}^*$ such that
$x \not\in {\cal U}'$, i.e.,
$\gl - x \in {\cal U}'$. This means
that $x$ is disjoint from a set in
${\cal U}$, which is absurd since
${\cal U} \subseteq {\cal U}^*$.