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%
% ------------------------------------------------------------------------------
%
\title{The Consistency Strength of
$\ha_\go$ and $\ha_{\go_1}$
being Rowbottom Cardinals without the
Axiom of Choice
% Equiconsistency and Rowbottomness
% in a Choiceless Context
\thanks{2000 Mathematics Subject Classifications:
03E02, 03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Jonsson cardinal, Rowbottom cardinal,
Rowbottom filter,
Prikry forcing, coherent sequence of Ramsey
cardinals, core model.}}
\author{Arthur W.~Apter\thanks{The
first author's research was partially
supported by PSC-CUNY Grant
66489-00-35 and a CUNY Collaborative
Incentive Grant. In addition,
the first author wishes to thank
the members of the set theory
group in Bonn
for all of the hospitality shown him
during his visits to the Mathematisches Institut.}\\
%The City University of New York\\
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
\\
\\
Peter Koepke\\
Mathematisches Institut\\
Rheinische Friedrich-Wilhelms-Universit\"at\\
D-53115 Bonn, Germany\\
http://www.math.uni-bonn.de/people/logic/People/Koepke.html\\
koepke@math.uni-bonn.de}
\date{April 22, 2005\\
(revised March 22, 2006)}
\begin{document}
\maketitle
\begin{abstract}
We show that
for all natural numbers $n$,
the theory ``ZF + ${\rm DC}_{\aleph_n}$ + $\aleph_{\omega}$
is a Rowbottom cardinal carrying a Rowbottom filter''
has the same consistency strength as
the theory ``ZFC + There exists a measurable cardinal''.
In addition, we show that the theory
``{\rm ZF} + $\aleph_{\omega_1}$ is
an $\go_2$-Rowbottom cardinal carrying an
$\go_2$-Rowbottom filter and
$\omega_1$ is regular'' has the same consistency strength as
the theory ``ZFC + There exist $\go_1$ measurable cardinals''.
We also discuss some generalizations of these results.
\end{abstract}
\baselineskip=24pt
%\newpage
\section{Introduction and Preliminaries}\label{s1}
One of the longest standing
open questions in
%the theory of large cardinals,
large cardinals and forcing,
dating back to
Silver's 1965 thesis (see
the published version \cite{Si}),
is whether the theory ``ZFC +
$\ha_\go$ is a Rowbottom cardinal''
is relatively consistent.
%relative to anything short of $0 = 1$.
In spite of numerous attempts
to obtain a solution to this
vexing and intriguing problem,
by Shelah, Foreman, and others,
no solution in either a
positive or negative vein is in sight.
If we are willing to drop
the Axiom of Choice from
our assumptions, i.e.,
if we are willing to settle for
the relative consistency of the theory
``{\rm ZF} + $\neg{\rm AC}$ +
$\ha_\go$ is a Rowbottom cardinal'',
then the situation is quite different.
Everett Bull (unpublished ---
see \cite{Bu}) showed that, relative to
``ZFC + There exists a measurable cardinal'',
the theory
``{\rm ZF} + ${\neg}{\rm AC}_\go$ +
GCH holds below $\ha_\go$ +
$\ha_\go$ is a Rowbottom cardinal carrying
a Rowbottom filter'' is consistent.\footnote{By
GCH holding below $\ha_\go$, we literally
mean, as in the situation when the Axiom
of Choice is true,
that for every $n < \go$,
there is a bijection between
the power set of $\ha_n$
and $\ha_{n + 1}$.}
The first author improved the
amount of choice in Bull's model \cite{A83}
to show that, relative to ``ZFC +
There is an $\go$ sequence of measurable cardinals'',
for an arbitrary $n < \go$, the theory
``{\rm ZF} + ${\rm DC}_{\ha_n}$ + $\ha_\go$ is a
Rowbottom cardinal carrying a Rowbottom filter''
is consistent.
The purpose of this paper is to
obtain equiconsistency results
concerning the theory
``{\rm ZF} + $\neg{\rm AC}$ +
$\ha_\go$ is a Rowbottom cardinal
carrying a Rowbottom filter''
and some generalizations thereof.
Specifically, we prove the
following two theorems.
\begin{theorem}\label{t1}
The theories ``ZFC + There exists
a measurable cardinal'' and
``ZF + $\neg$AC +
${\rm DC}_{\ha_n}$ + $\ha_\go$ is
a Rowbottom cardinal carrying a
Rowbottom filter'' are equiconsistent
for every $n < \go$.
\end{theorem}
\begin{theorem}\label{t2}
The theories ``ZFC + There exist
$\go_1$ measurable cardinals'' and
``ZF + ${\neg}$AC +
$\go_1$ is regular +
$\ha_{\go_1}$
is an $\go_2$-Rowbottom cardinal
carrying an $\go_2$-Rowbottom filter''
are equiconsistent.
\end{theorem}
In showing the forward direction
of Theorem \ref{t2}, we will
indicate how to construct different
models in which various weak forms
of the Axiom of Choice are true.
Also, Theorems \ref{t1} and \ref{t2}
above represent
%the sorts of results which are
our main focus.
We will in addition discuss
at various junctures throughout the course of the paper
generalizations of the theorems
mentioned above, along with proving some
other related results.
%The results are shown by the forcing method and by inner model theory.
We work using forcing and core model theory.
%In particular, a familiarity
We will force to construct the
relevant choiceless inner models in which
$\ha_\go$, $\ha_{\go_1}$, etc$.$ satisfy
the desired properties.
The construction of these choiceless inner models
will be based in large part on the
techniques set forth in \cite{A83}.
As such, we will be assuming some
familiarity with the methods of this paper,
to which we will refer when appropriate.
An overview of the proof of
Theorem \ref{t1} is as follows.
For the forward direction,
if a
measurable cardinal $\kappa$ is made singular
of cofinality $\omega$ by Prikry
forcing, an end segment of the Prikry sequence
$\la \gl_0, \gl_1, \ldots \ra$,
which we denote by
$\la \kappa_0, \kappa_1, \ldots \ra$, is a
{\it{coherent sequence of Ramsey cardinals}}
as defined in \cite{K1}.
(Note that the definition of a
coherent sequence of Ramsey cardinals
can be found in the statement of Theorem \ref{t3}.)
%(see [K1: P. Koepke, The
%consistency strength of the free-subset property for $\omega_{\omega} $,
%Journal Symb. Logic 49 No. 4 (1984), 1198-1204]).
The supremum $\kappa$ of
such a sequence is a Rowbottom cardinal
carrying a Rowbottom filter.
It is then turned into
$\aleph_{\omega}$ by a product of L\' evy collapses
which collapses the Ramsey
cardinals $\kappa_0, \kappa_1, \ldots$ to
$\aleph_i, \aleph_{i + 2}, \ldots$ for
$i < \go$, $i > 0$ a fixed but arbitrary
natural number. We then as in \cite{A83}
define a symmetric submodel
of the generic extension in which {\rm ZF}
holds and in which $\kappa = \aleph_{\omega}$
is still a Rowbottom cardinal
carrying a Rowbottom filter.
For the converse, we use
%arguments from core model theory
the Dodd-Jensen
core model $K$ as presented in the
original articles \cite{DJ1} and \cite{DJ2}
and in the monograph \cite{D}
to get an
inner model with a measurable cardinal from
$\aleph_{\omega}$ being
Jonsson.
The proof of Theorem \ref{t2} is
handled slightly differently, since
when $\go_1$ is regular,
$\ha_{\go_1}$ has uncountable cofinality.
For the forward direction, if
$\la \gk_i \mid i < \go_1 \ra$ is a
sequence of $\go_1$ measurable cardinals
with supremum $\gk$, then $\gk$ is
turned into $\ha_{\go_1}$ as before
by a product of L\' evy collapses which
collapses $\gk_0$, the first measurable cardinal
in the sequence, to some fixed
but arbitrary $\ha_i$ for $i < \go_1$, $i > 0$.
The remaining measurable cardinals
are collapsed in a manner to be
described later, depending on how
much of the Axiom of Choice we wish
to be true in our final symmetric
submodel of the generic extension
in which $\go_1$ is
regular and $\ha_{\go_1}$ is
$\go_2$-Rowbottom and carries an
$\go_2$-Rowbottom filter.
The proofs necessary to
establish the converse
%are based on core model theory, in particular on
%[DJ: A. J. Dodd, R. B. Jensen, The Core Model / The Covering
%Lemma for $K$ / The Covering Lemma for $L [ U ]$, Ann. Math. Logic 20 (1981)
%43-75 and 22 (1982) 127-135]
%[D: A. J. Dodd, The Core
%Model, Lecture Note Series 61 (London Math. Soc. Cambridge, 1982)]
are based on short core models as presented in \cite{K2}.
%[K2: P. Koepke, Ann. Pure Appl.
%Logic 37 (1988) 179-204].
Short core models are constructed from
short sequences
$\bar{\cal U} =
\la {\cal U}_{\kappa} \mid \kappa \in
\dom(\bar{\cal U}) \ra$ of
normal measures ${\cal U}_{\kappa}$
with measurable cardinal $\kappa$.
Note that we say the sequence
$\bar{\cal U}$ is {\it short}
if its order type satisfies
${\rm otp}(\dom(\bar{\cal U})) < \min(\dom(\bar{\cal U}))$.
The structure of this paper is as follows.
Section \ref{s1} contains our
introductory comments and preliminary
remarks concerning notation and terminology.
Section \ref{s2} contains a discussion
of {coherent sequences of Ramsey cardinals}
%a technical device introduced by the second author in \cite{K1}
that will be
critical to the proof of the forward
direction of Theorem \ref{t1}.
Section \ref{s3} contains our proof of Theorem \ref{t1}.
Section \ref{s4} contains our proof of Theorem \ref{t2},
as well as a brief discussion of a generalization
of this theorem.
Section \ref{s5} contains a further generalization
of our work, along with our final comments.
We conclude Section \ref{s1} with a few
brief words concerning the conventions
we will be following. Basically, our
notation and terminology are standard.
Exceptions to this will be duly noted.
We do wish, however, to state explicitly that
for $\ga < \gb$ ordinals,
$[\ga, \gb]$, $[\ga, \gb)$, $(\ga, \gb]$,
and $(\ga, \gb)$ are as in standard
interval notation.
For $\gk < \gl$ cardinals with $\gk$ regular,
${\rm Coll}(\gk, {<} \gl)$ is the
standard L\'evy partial ordering for
collapsing every $\gd \in (\gk, \gl)$ to $\gk$.
For such a $\gd$ and any
$S \subseteq {\rm Coll}(\gk, {<} \gl)$,
we define $S \rest \gd =
\{p \in S \mid \dom(p) \subseteq
\gk \times \gd\}$.
We also wish to recall for
the benefit of readers the definitions
of what it means for a cardinal to
be Jonsson or Rowbottom.
The cardinal $\gk$ is said to be
{\it Jonsson} if it satisfies the
partition relation
$\gk \to {[\gk]}^{< \go}_\gk$, i.e.,
given a partition
$f : {[\gk]}^{< \go} \to \gk$,
there is a homogeneous set $X \subseteq \gk$
such that $f''{[X]}^{< \go} \neq \gk$.
The filter ${\cal F}$
is called a {\it Jonsson filter} if
some homogeneous set $X$ for $f$
may always be chosen so that
$X \in {\cal F}$.
%Assuming the Axiom of Choice,
The definition of Jonsson cardinal is
equivalent in ZF to saying that any structure
in a countable language whose domain
has cardinality $\gk$ has a proper
elementary substructure of cardinality $\gk$.
The cardinal $\gk$ is said to be
{\it Rowbottom} if for every
cardinal $\gl < \gk$, it satisfies
the partition relation
$\gk \to {[\gk]}^{< \go}_{\gl, \go}$, i.e.,
given a partition
$f : {[\gk]}^{< \go} \to \gl$,
there is a homogeneous set $X \subseteq \gk$
such that $\card{f''{[X]}^{< \go}} \le \go$.
The filter ${\cal F}$
is called a {\it Rowbottom filter} if
some homogeneous set $X$ for $f$
may always be chosen so that
$X \in {\cal F}$.
%Assuming the Axiom of Choice,
The definition of Rowbottom cardinal
is equivalent in ZF to saying that for any structure
$\la A, R, \ldots \ra$ in a countable language
such that $\card{A} = \gk$ and $R$ is
a unary relation having cardinality $\gl < \gk$,
there is a (proper) elementary substructure
$\la A', R', \ldots \ra$ such that
$\card{A'} = \gk$ and $\card{R'} \le \go$.
The notion
of what it means for a cardinal $\gk$ to
be {\it $\gd$-Rowbottom} for
some uncountable cardinal $\gd < \gk$
is a generalization of the definition
of Rowbottom cardinal given in
the preceding paragraph.
This will hold if for every
$\gl$ with $\gd \le \gl < \gk$,
$\gk$ satisfies the partition relation
$\gk \to {[\gk]}^{< \go}_{\gl, {<} \gd}$, i.e.,
given a partition
$f : {[\gk]}^{< \go} \to \gl$,
there is a homogeneous set $X \subseteq \gk$
such that $\card{f''{[X]}^{< \go}} < \gd$.
The definition of $\gd$-Rowbottom cardinal
is equivalent in ZF to saying that for any structure
$\la A, R, \ldots \ra$ in a countable language
such that $\card{A} = \gk$ and $R$ is
a unary relation having cardinality $\gl < \gk$,
%$\gl > \gd$,
there is a (proper) elementary substructure
$\la A', R', \ldots \ra$ such that
$\card{A'} = \gk$ and $\card{R'} < \gd$.
By this definition, a Rowbottom cardinal is
$\go_1$-Rowbottom.
Also, the notion of {\it $\gd$-Rowbottom filter}
is defined as in the preceding two paragraphs.
Further, note that for an uncountable
cardinal $\gd < \gk$,
we have the chain of implications
$\gk$ is Rowbottom $\implies$ $\gk$
is $\gd$-Rowbottom $\implies$ $\gk$ is Jonsson.
%In both Theorems \ref{t1} and \ref{t2}
%and our related work, since we will be
%constructing models in which the
%Axiom of Choice is false,
%we will always assume the partition
%theoretic definitions of Jonsson
%and Rowbottom cardinal.
%Even in a choiceless situation,
%these partition theoretic definitions
%are essentially equivalent to the model theoretic ones,
%assuming we are working with structures
%whose domains can be well-ordered
%(such as structures whose domains are ordinals).
To prove lower bounds on consistency
strength, we employ the theory of
short core models \cite{K2}, which extends
the theory of the Dodd-Jensen
core model \cite{D}, \cite{DJ1}, and \cite{DJ2}.
The theory of short core models is developed
under the assumption that a certain object
$0^{{\rm long}}$, which transcends short
core models in the same way $0^\sharp$
transcends the constructible universe,
does {\it not} exist.
$0^{{\rm long}}$ will be described further
before the proof of Theorem \ref{t8}.
Under the assumption $0^{{\rm long}}$
does not exist, which is denoted by
$\neg 0^{{\rm long}}$,
the core model has the form
$K = L[E]$, where $E$ is some canonical
sequence of total and partial measures.
One also writes
$K = K[\bar{\cal U}_{{\rm can}}]$, where
$\bar{\cal U}_{{\rm can}}$ is the sequence
consisting of the total measures in $E$.
For each ordinal $\ga$, we define the
$\ga^{\rm th}$ level of the core model by
$K_\ga = K_\ga[\bar{\cal U}_{{\rm can}}] = L_\ga[E]$.
Core model theory is usually developed
assuming the Axiom of Choice. For our
study of choiceless combinatorics,
we employ some workarounds which are based
on building core models within the inner model
${{\rm HOD}}$ of
{\it{hereditarily ordinal definable sets}}
or some variants.
Such methods were used, e.g., by Schindler in
\cite{Sc}.
If $a \subseteq
{{\rm HOD}}$ is a set, let
${{\rm HOD}} [ a ]$ be the smallest inner model
such that ${{\rm HOD}} \cup \{ a \} \subseteq {{\rm HOD}} [ a ]$.
We then have the following (see also Lemmas 3 and
4 of \cite{Sc}).
\begin{proposition}\label{p0}
(ZF) Let $a \subseteq {{\rm HOD}}$ be a set. Then
\begin{enumerate}
\item\label{p0i1}
${{\rm HOD}} [ a ]$ is a set-generic extension of ${{\rm HOD}}$, so
${{\rm HOD}} [ a ] \models {\rm ZFC}$.
\item\label{p0i2}
If $\neg 0^{{\rm long}}$, and if
$K$ is (the canonical term for) the
core model, then
$K^{{{\rm HOD}} [ a ]} = K^{{{\rm HOD}}}$.
This equality holds
for every level of the $K$-hierarchy, i.e.,
$K_{\alpha}^{{{\rm HOD}} [ a ]} =
K_{\alpha}^{{{\rm HOD}}}$ for every $\alpha \in {\rm Ord}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Clause (\ref{p0i1})
follows from Vop\v enka's genericity theorem
(see page 142 of \cite{Sc}).
Clause (\ref{p0i2})
follows from the absoluteness of (small) core
model constructions with respect to set-generic extensions.
\end{proof}
\section{Coherent Sequences of Ramsey Cardinals}\label{s2}
Assume $\kappa$ is a measurable cardinal
with normal measure ${\cal U}$. Let
\[ \FP = \{ \la a, X \ra \mid a \in
[ \kappa ]^{< \omega}, X \in {\cal U}, \max(a) < \min(X)
\} \]
be the set of Prikry conditions for
$\kappa$ and ${\cal U}$ with the usual order. Let $G$
be $\FP$-generic over $V$, with
$\la \gl_i \mid i < \omega \ra$ the Prikry
sequence induced by $G$.
In \cite{K1}, the following was proved
as Theorem 3.2.
\begin{theorem}\label{t3}
In $V [ G ]$, there is an ascending sequence
$\la \gk_i \mid i < \omega \ra$ of regular
cardinals cofinal in $\kappa$ which forms a
{{coherent sequence of Ramsey cardinals}},
i.e., for every regressive $f : [ \kappa ]^{< \omega}
\to \kappa$, there is
$\la A_i \mid i < \omega \ra$ such that:
\begin{enumerate}
\item $A_i \subseteq \gk_i \setminus \gk_{i - 1}$
is cofinal in $\gk_i$, where for convenience,
we set $\gk_{-1} = 0$.
\item If $x, y \in [ \kappa ]^{< \omega}$,
$x, y \subseteq \bigcup \{ A_i : i <
\omega \}$, and $\card{x \cap A_i} =
\card{y \cap A_i }$ for
$i < \omega$, then $f ( x ) = f ( y )$.
\end{enumerate}
\end{theorem}
Note that
$f : {[\gk]}^{< \go} \to \gk$ is
{\it regressive} if
$f(x) = 0$ if $0$ is the
minimal member of $x$, or
$f(x) < \min(x)$ for every
$x \in {[\gk]}^{< \go}$ for
which $0$ is not the minimal
member of $x$.
Also, Lemma 3.1 of \cite{K1}
tells us that
$\la \gk_i \mid i < \go \ra$
%(which is definable in $V[G]$)
is actually of the form
$\la \gl_i \mid j \le i < \go \ra$,
i.e., the coherent sequence of
Ramsey cardinals is an end segment of
the Prikry sequence induced by $G$.
Lemma 3.1 of \cite{K1} further tells us that
$\la A_i \mid i < \go \ra$ can be taken
so that the $A_i$'s are mutually disjoint and
$A = \bigcup_{i < \go} A_i
\cup \{\gk_i \mid i < \go\}
\in {\cal U}$.
%Thus, the homogeneous set for
%$f$ may be taken to have
%${\cal U}$-measure 1.
We shall use a slightly different technical characterization of coherent
Ramseyness.
\begin{proposition}\label{p1}
$\la \kappa_i \mid i < \omega \ra$
is a coherent sequence of Ramsey cardinals
with supremum $\kappa$ if for all regressive
$f : [ \kappa ]^{< \omega} \to
\kappa$, there is
$\la A_i \mid i < \omega \ra$ such that:
\begin{enumerate}
\item\label{p1i1}
$A_i \subseteq \kappa_i \setminus \kappa_{i - 1}$ is cofinal in
$\kappa_i$, where for convenience,
we set $\kappa_{- 1} = 0$.
\item\label{p1i2}
For every $m < \omega$, if
$x, y \in {[\gk]}^{< \go}$,
$x, y \subseteq \bigcup_{i < m} A_i$, and $\forall
i < m[\card{x \cap A_i} = \card{y \cap A_i} = m]$,
%we have
%the homogeneity property that
%if $\beta < \kappa$, $r \cap \beta = s
%\cap \beta$, and $f ( r ) < \beta$,
then $f(x) = f(y)$.
\end{enumerate}
\end{proposition}
\begin{proof}
Property (2) of Proposition \ref{p1}
is a special case of property (2)
%in the original definition
in Theorem \ref{t3} in which
only arguments with $m$ elements in the
first $m$ sets $A_i$ are considered.
Conversely it implies the original definition.
To see this, let
$f : [ \kappa ]^{< \omega} \to \kappa$
be a regressive function.
%with $m$ the smallest natural number
%such that $c_i = 0$ for all $i \ge m$.
%Note that any sequence of this form
%has the property that there is some $m < \omega$,
%$\max (\la c_i \mid i < \go \ra ) < m$, $c_i \neq 0 \implies i < m$.
%Suppose $\la c_i \mid
%i < \omega \ra$ is a sequence of natural numbers
%which takes the value $0$ on a final segment, and
%$x \in [ \kappa ]^{< \omega}$ has the property that
%$\forall i < \go [\card {x
%\cap [ \kappa_{i - 1}, \kappa_i ] } = c_i]$.
Given $x \in {[\gk]}^{< \go}$,
define $x$'s {\it type} to be
the countable sequence of integers
whose $i^{\rm th}$ member is given by
$\card{x \cap [ \kappa_{i - 1}, \kappa_i ]}$.
Since for any $x \in {[\gk]}^{< \go}$,
there is some $m < \go$ such that
$\forall i \ge m [x \cap [ \kappa_{i - 1}, \kappa_i]
= \emptyset]$, we may infer
that there are only countably
many types.
For any $x \in {[\gk]}^{< \go}$
of some fixed type,
let $m < \go$ and
$y \subseteq \bigcup_{i < m} [
\kappa_{i - 1}, \kappa_i ]$ be such that
$\forall i < m [\card{y \cap
A_i } = m]$,
$\forall i < m [x \cap [ \kappa_{i - 1},
\kappa_i ]$ is an initial segment of
$y \cap [ \kappa_{i - 1}, \kappa_i] ]$, and
$\forall i \ge m [x \cap
[ \kappa_{i - 1}, \kappa_i] = \emptyset ]$.
Note that the finiteness of $x$
ensures that $m$ and $y$ as just stipulated exist.
In addition, the fact there
are only a countable number of different
types implies that a unique $m$ may
be chosen for each distinct type.
One can find $g : [ \kappa]^{< \omega} \to \kappa$
regressively such that for
%for each such sequence of natural numbers and
each $x \in {[\gk]}^{< \go}$ of some fixed type, there are
$m$ and $y$ as just described with
$g(y) = f(x)$.
A homogeneous sequence for $g$ in the sense of
property (2) of Proposition \ref{p1}
will also be fully homogeneous for $f$.
\end{proof}
\begin{proposition}\label{p2}
If $\la \gk_i \mid i < \go \ra$ is
a coherent sequence of Ramsey cardinals
with supremum $\gk$, then $\gk$ is
a Rowbottom cardinal.
\end{proposition}
\begin{proof}
Suppose $\gl < \gk$ and
$f : {[\gk]}^{< \go} \to \gl$.
Without loss of generality,
replace $\gk$ with $B = \gk \setminus \gl$.
$f : {[B]}^{< \go} \to \gl$ is
regressive, so let
$\la A_i \mid i < \go \ra$
be homogeneous
for $f$ in the sense of Theorem \ref{t3}.
Define $A = \bigcup_{i < \go} A_i$.
Since by homogeneity,
$f '' {[A]}^{< \go}$ depends only upon
the number of distinct sequences
$\la \card{x \cap A_i} \mid i < \go \ra$ for
$x \in {[\gk]}^{< \go}$, and since
there are only countably many such sequences,
$\card{f '' {[A]}^{< \go}} \le \go$.
Thus, $A$ is homogeneous for $f$ in the
sense of Rowbottomness.
\end{proof}
The following preservation result for
coherent sequences of Ramsey cardinals
will be essential for the construction
to be given in Section \ref{s3}.
\begin{theorem}\label{t4}
Let $\la \kappa_i \mid i < \omega \ra$
be a coherent sequence of Ramsey
cardinals with supremum $\kappa$.
Let $\la \delta_i \mid i < \omega \ra$ be a
sequence of inaccessible cardinals such that
$\forall i < \go[\delta_i \in ( \kappa^+_{i - 1},
\kappa_i )]$, where $\kappa_{- 1} =
\omega_\ell$ for some $\ell < \go$.
%Let $( P, \leq )$ be the following set of conditions: $P =
Let $\FP =
\{\la p_i \mid i < \omega \ra \mid
p_i \in {\rm Coll}( \kappa^+_{i - 1}, {<} \delta_i)$
for $i < \omega \}$, ordered componentwise.
%where ${Col} ( \sigma, p )$ are the Levy
%conditions for collapsing the
%inaccessible $p$ to $\sigma^+$; they are
%partially ordered by $( y_i \mid i < \omega )
%\leq ( p_i \mid i < \omega )$
%iff $\forall i \hspace{0.25em} y_i \subseteq p_i$.\\
Let $G$ be $\FP$-generic over $V$.
Then in $V [ G ]$, $\la \kappa_i \mid i <
\omega \ra$ is a coherent sequence of Ramsey cardinals.
\end{theorem}
\begin{proof}
Let $p = \la p_i \mid i < \omega \ra \in \FP$
and $p \forces$ ``$\dot{g} :
[ \kappa ]^{< \omega} \to \kappa$ is regressive''.
It suffices to show that some
extension of $p$ forces the existence of a
homogeneous sequence for
$\dot{g}$ in the sense of the characterization of coherent
Ramseyness given in Proposition \ref{p1}.
For $m < \go$, let
$R_m = \{ r \in [ \kappa ]^{m \cdot m}
\mid \forall i < m
[\card{r \cap ( \kappa_i \setminus \delta_i )} = m] \}$.
$R = \bigcup_{m < \go} R_m$ is then the set of
all arguments relevant for the characterization
of coherent Ramseyness given in Proposition \ref{p1}.
Well-order $R$ by $r <' s$
iff either (a) $\card{r} < \card{s}$, or
(b) $\card{r} = \card{s}$ and $\exists \beta [r \setminus
\beta = s \setminus \beta$ and
$\beta \not\in r$ and $\beta \in s]$.
Part (b) corresponds to the usual
well-ordering of $[ {\rm Ord} ]^{< \omega}$
by largest difference.
$\la R, <' \ra$ has order type $\kappa$.
We construct by
recursion on $<'$ a sequence
$\la p ( r ) \mid r \in R \ra$, $p ( r ) = \la p_i ( r
) \mid i < \omega \ra \in \FP$,
and a sequence $\la \omega ( r ) \mid r \in R \ra$
such that the following ``growth condition'' holds: \bigskip
\noindent
(1) If $s <' r$ and $\forall j \geq i
[s \cap [ \delta_j, \kappa_j ]$ is
an initial segment of $r \cap [ \delta_j, \kappa_j ]]$,
then $p_i \subseteq
p_i ( s ) \subseteq p_i ( r )$. \bigskip
\noindent The growth condition will be essential
for the compatibility requirements
of the subsequent construction.
Assume that $r \in R$ and that
for $s <' r$, the condition $p ( s )$ is
constructed so that (1) holds.
Define $\bar{p_i} ( r ) = p_i \cup
\bigcup \{ p_i ( s ) \mid s <' r$
and for $j \geq i$, $s \cap [ \delta_j,
\kappa_j ]$ is an initial segment of
$r \cap [ \delta_j, \kappa_j ] \}$.
$\bar{p_i} ( r )$ is a condition,
since ${\rm Coll} ( \kappa^+_{i - 1}, {<}
\delta_i)$ is closed under unions of
$\subseteq$-increasing sequences of
length $\kappa_{i - 1}$,
and we are taking a union of
%a collection of compatible conditions
a $\subseteq$-increasing chain of conditions
having size at most $\gk_{i - 1}$.
Choose $\omega ( r ) \in \kappa$ and $p ( r ) = \la
p_i ( r ) \mid i < \omega \ra \leq \bar{p} ( r )$
such that $p ( r ) \forces
``\dot{g} ( r ) = \omega ( r )$''.
The definition of $p ( r )$ is consistent
with property (1), and so the recursion works.
For $i < \go$ and $t \subseteq
\bigcup_{i \leq j < \go} [ \delta_j, \kappa_j ]$,
%$t$ finite,
we take the union of
all the $p_i ( r )$
%which are consistent with $t$
where $t = r \setminus \gd_i$
and define $p'_i ( t ) =
\bigcup \{ p_i ( r ) \mid r \in R$
and $t = r \setminus \delta_i \}$.
$p'_i (t) \in
{\rm Coll} ( \kappa^+_{i - 1}, {<} \delta_i )$,
since ${\rm Coll} (\kappa^+_{i - 1}, {<} \delta_i )$
is $\kappa_{i - 1}$-closed,
and we are taking a union of
at most $\gk_{i - 1}$ many
compatible forcing conditions.
Note that $p'_i$ can be
viewed as a regressive function.
Since $\la \kappa_i \mid i < \omega \ra$ is a
coherent sequence of Ramsey cardinals,
by coding $\go$ partitions into
one in (3) below and then applying
Theorem \ref{t3} twice,
there is $\la A_i \mid i < \omega \ra$,
each $A_i$ cofinal in $\kappa_i$,
such that for
$A = \bigcup_{i < \go} A_i$,
the following homogeneity properties
hold: \bigskip
\noindent (2) If $r, s \in R$,
%$r, s \subseteq \bigcup_{i < \go} A_i$,
$r, s \subseteq A$,
$\forall j < \omega [\card{r \cap
A_j} = \card{s \cap A_j}]$,
then $\omega ( r ) = \omega ( s )$.
\noindent (3) For $i < \go$,
if $r, s \in R$,
%$r, s \subseteq \bigcup_{i < \go} A_i$,
$r, s \subseteq A$,
$r, s \subseteq \bigcup_{i \le j < \go}
[\gd_j, \gk_j]$,
$\forall j < \omega[\card{r \cap
A_j}= \card{s \cap A_j }]$, then
$p'_i(r) =
p'_i ( r \setminus \delta_i ) =
p'_i(s) =
p'_i ( s \setminus \delta_i )$. \bigskip
\noindent Now, for $i < \go$, define
$\bar{p}_i = \bigcup \{ p'_i ( r \setminus \delta_i )
\mid r \in R$,
%$r \subseteq \bigcup_{i < \go} A_i \}$.
$r \subseteq A \}$.
By (3), this is just a countable union.
Consequently, $\bar{p}_i
\in {\rm Coll} ( \kappa^+_{i - 1}, {<} \delta_i )$.
Set $\bar{p} = \la \bar{p}_i
\mid i < \omega \ra$. It is then the case that
$\bar{p} \in \FP$ and $\bar{p} \leq p.$
We show that $\bar{p}$ forces that
$\la A_i \mid i < \omega \ra$ is homogeneous
for $\dot g$ in the sense of the characterization
of coherent Ramseyness given in Proposition \ref{p1}.
To do this,
let $r \in R$, $r \subseteq \bigcup_{i < m} A_i$, where
$\forall i < m [\card{r \cap A_i} = m]$.
It is then the case that
for every $i < \go$,
$p_i(r) \subseteq p'_i(r \setminus \gd_i)
\subseteq \bar{p}_i$.
Hence, $\bar{p} =
\la \bar{p}_i \mid i < \omega \ra \leq
\la p_i(r) \mid i < \omega \ra
= p ( r )$, and $\bar{p}
\forces ``\dot{g} ( r ) = \omega ( r )$''.
If $s \in \bigcup_{i < m} A_i$,
where $\forall i < m[\card{s \cap A_i}
= m]$, then the same calculation yields
$\bar{p} \forces ``\dot{g} ( s ) = \omega(s)$''.
Thus, $\bar{p} \forces ``\dot{g} ( r ) = \dot{g} ( s )$''.
\end{proof}
Take $\gl < \gk$ and
$g : {[\gk]}^{< \go} \to \gl$.
Consider the first-order structure
%\break
$\mathfrak B = \la \gk, R, g,
\gk_0, \gk_1, \ldots \ra$,
where we have distinguished as
constants each member of a coherent
sequence of Ramsey cardinals
$\la \gk_i \mid i < \go \ra$
generated via a Prikry sequence
with respect to the normal measure
${\cal U}$ over $\gk$, and
$R$ is the unary relation
composed of $g''{[\gk]}^{< \go}$.
By Proposition \ref{p2}, let
${\mathfrak A} \prec {\mathfrak B}$
be a Rowbottom elementary substructure.
By the proof of Proposition \ref{p2} and
the remarks made in the
paragraph immediately following
the statement of Theorem \ref{t3},
we may take $A' = \dom({\mathfrak A})$
to be such that
$A = A' \cup \{\gk_i \mid i < \go \} \in {\cal U}$.
Since each member of the set
$\{\gk_i \mid i < \go \}$ was a distinguished
constant in $\mathfrak B$, it is the case that
$\card{g''{[A]}^{< \go}} \le \go$,
i.e., $A \in {\cal U}$ is Rowbottom
homogeneous for $g$.
The proof of Theorem \ref{t4}
therefore yields that if we
first do Prikry forcing and follow
this by the forcing indicated
by Theorem \ref{t4}, then
not only is $\gk$ a Rowbottom
cardinal, but ${\cal U}$ generates a
Rowbottom filter over $\gk$.
This will be critical in the
proof of Theorem \ref{t5}
to be given in the next section.
\section{The Proof of Theorem \ref{t1}}\label{s3}
\begin{pf}
In this section, we will prove Theorem \ref{t1}.
We break the proof up into both its
forward and reverse directions, which
we will establish separately.
We begin with the forward direction,
which we state as a separate theorem.
\begin{theorem}\label{t5}
Let $V_0 \models ``$ZFC + $\gk$ is
a measurable cardinal''.
Let $n < \omega$ be
fixed but arbitrary.
There is then a generic extension
$V$ of $V_0$, a notion of forcing $\FP$, and
a symmetric inner model $N \subseteq
V^\FP$ such that
$N \models ``$ZF + $DC_{\aleph_n}$
+ $\aleph_{\omega}$ is a Rowbottom
cardinal carrying a Rowbottom filter''.
\end{theorem}
We note that Theorem \ref{t5}
represents a significant
reduction in consistency strength
of the hypotheses used for
the main result
(Theorem 1) of \cite{A83}.
As was mentioned in Section \ref{s1},
in that paper, a model witnessing
the same conclusions as in Theorem \ref{t5}
was constructed, but assuming the
consistency of the theory
``ZFC + There is an $\go$ sequence of
measurable cardinals''.
We turn now to the proof of Theorem \ref{t5}.
\begin{proof}
Let
$V_0 \models ``$ZFC + $\gk$ is
a measurable cardinal''.
%Without loss of generality, but with a slight abuse of notation,
We assume that
$V_0$ has been extended generically
via Prikry forcing using a
normal measure ${\cal U}$
over $\gk$ to a model,
which we denote by $V$, containing
a Prikry sequence $\la \gk_i \mid
i < \go \ra$ through $\gk$.
Take $V$ as our ground model.
Let $n < \go$ be fixed but arbitrary. Let
$\FP_0 = {\rm Coll} ( \aleph_{n + 1}, {<} \kappa_0 )$,
and for $1 \le i < \go$, let
$\FP_i = {\rm Coll} ( \kappa^+_{i - 1}, {<} \kappa_i)$.
We then define
$\FP = \prod_{i < \go} \FP_i$ with full support.
Let $G$ be $\FP$-generic over $V$.
$V[G]$, being a model of AC, is not
our desired model $N$. In order to
define $N$, we first note that by
the Product Lemma, $G_i$,
the projection of $G$ onto $\FP_i$,
is $V$-generic over $\FP_i$.
Next, working in $V$, let
$\mathcal{F} = ( \aleph_{n + 1}, \kappa_0 )
\times ( \kappa^+_0, \kappa_1 )
\times ( \kappa^+_1, \kappa_2 ) \times \cdots$.
For each $f \in \mathcal{F}$,
$f = \la f(0), f(1), \ldots \ra$,
define $G \rest f = G_0 \rest f ( 0 ) \times G_1
\rest f ( 1 ) \times \cdots$.
By the Product Lemma and the properties
of the L\'evy collapse, $G \rest f$ is
$\prod_{i < \go} (\FP_i \rest f(i))$-generic over $V$.
%where $\FP_i \rest f(i) =
%\{p \in \FP_i \mid \dom(p) \subseteq
%\gk_i \times f(i)\}$.
$N$ can now intuitively be described as the
least model of ZF extending $V$ which
contains, for each $f \in {\cal F}$,
the set $G \rest f$.
%The
%generic set can be written canonically as $G = \prod G_i$.
%We have to define the desired submodel $N \subseteq V [ G ]$.
%Let
%Then let $N$ be the
%$\subseteq$-least inner model of ${\rm ZF}$ such that
%for any $f \in \mathcal{F} :
%V [ G \rest f ] \subseteq N$.\\
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified 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 symbols $\dot G \rest f$ for each $f \in {\cal F}$. $N$ is
then defined as follows.\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\ga$ 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}\bigskip
The standard arguments show $N \models {\rm ZF}$.
Further, by Lemmas 1.1, 1.2, and 1.4 of
\cite{A83}, which remain valid in the context
of this paper,
$N \models ``{\rm DC}_{\ha_n}$ + $\gk = \ha_\go$''.
The proof of Theorem \ref{t5} will thus
be complete once we have shown that
$N \models ``\ha_\go$ is a Rowbottom cardinal
carrying a Rowbottom filter''.
To do this, fix $\gl < \gk$,
and suppose $g \in N$ is such that
$g : {[\gk]}^{< \go} \to \gl$.
By Lemma 1.1 of \cite{A83},
there is $f \in {\cal F}$ such that
$g \in V[G \rest f]$.
Consequently, by Theorem \ref{t4}
and the remarks immediately following
its proof, there is a set
$A \in {\cal U} \in V \subseteq N$
which is Rowbottom homogeneous for $g$.
Thus, $N \models ``\gk$ is a Rowbottom
cardinal and ${\cal U}$
generates the Rowbottom filter
${\cal U}^* = \{X \subseteq \gk \mid
\exists Y \in {\cal U}[X \supseteq Y]\}$
for $\gk$''.
\end{proof}
We continue now with the reverse
direction of Theorem \ref{t1}.
We state this as a separate, stronger
theorem which implies our desired result.
\begin{theorem}\label{t6}
Let $\gk$ be a singular Jonsson cardinal
in a model $V$ of ZF.
Then $\gk$ is measurable in some inner model
of ZFC.
\end{theorem}
\begin{proof}
Since ${{\rm HOD}}$ is a
model of the Axiom of Choice,
take $H \in {{\rm HOD}}$
to be a sufficiently elementary submodel of
${{\rm HOD}}$ such that $\gk + 1 \subseteq H$.
Let $X$ be a Jonsson substructure of $H$,
i.e., let $X \prec H$ be
such that $\card{X \cap \gk} = \gk$ and $X \cap \gk
\neq \gk$.
The remainder of the argument will be carried out
in the inner model $M = {{\rm HOD}}
[ X, a ]$, where $a \subseteq \gk$ is cofinal in $\gk$
with order type less than
$\gk$. Note that Proposition \ref{p0}
applies to $M$.
%Also\\ (1) $\gk$ is singular.
Let $\pi : \la \bar{H}, \in \ra
\cong \la X, \in \ra \prec \la H, \in \ra$ be the
Mostowski collapse of $X$, where $\bar{H}$ is
transitive. We then have that
$\pi \rest \gk \neq {\rm id} \rest \gk$
and $\pi ( \gk ) = \gk$.
Let $\bar{K} = K^{\bar{H}}$,
where $K$ is the Dodd-Jensen core
model. Consider the following two cases.\bigskip
\noindent Case 1:
$K_{\gk} \subseteq \bar{K}$. Then $K_{\gk} =
K^{\bar{H}}_{\gk}$, and the elementary map $\pi \rest
K_{\gk} : \la K_{\gk}, \in \ra
\rightarrow \la K_{\gk}, \in \ra$ can be
extended to an elementary map
$\tilde{\pi} : \la K, \in \ra \rightarrow \la K, \in
\ra$. The rigidity theorem for the Dodd-Jensen core model implies
that there is an inner model with
a measurable cardinal less than $\gk$.
Iterating that model, one obtains an inner model with measurable cardinal
$\gk$.\bigskip
\noindent Case 2:
$K_{\gk} \nsubseteq \bar{K}$. Then we can take a mouse
$N$ with $\card{N} < \gk$ and $N \not\in \bar{K}$.
%We assume that $N$ is the minimal such in the
%canonical well-ordering of mice.
$N$ has exactly
one (active) measure, by which it may be iterated.
We therefore
let $\la \kappa_i \mid i \in
{\rm Ord} \ra$ be the sequence of
iteration points in the iteration of $N$.
$\{ \kappa_i \mid i \in {\rm Ord} \}
\cap \bar{K}$ is a set of
order-indiscernibles for $\bar{K}$
which is closed, unbounded in every
sufficiently large cardinal
less than or equal to $\gk$.
Each element $\kappa_i$ of
that set is strongly inaccessible in $\bar{K}$.
In particular, $\gk$ is
a limit of the iteration points,
and hence is regular in $\bar{K}$.
We claim now that
$\gk$ is regular in $K$.
To see this, first note that because
$\gk$ is regular in $\bar{K} = K^{\bar{H}}$ and
$\pi$ is elementary,
$\gk = \pi ( \gk )$ is regular in $K^H$. Since
$H$ is a sufficiently elementary substructure
of ${{\rm HOD}}$ formed in the
original universe, $\gk$ is consequently
regular in the original $K^{{{\rm HOD}}}$. By
clause (\ref{p0i2}) of Proposition \ref{p0},
$K^{{{\rm HOD}}}$ is the
Dodd-Jensen core model $K$
of $M$, the present universe of discourse.
By the facts that
$\gk$ is singular in $M$ but is
regular in $K$,
the covering property fails for $K$ at $\gk$.
By the Dodd-Jensen Covering Theorem,
there exists an inner model with a
measurable cardinal less than or
equal to $\gk$.
By eventually iterating that inner
model, one obtains an inner model with
a measurable cardinal exactly equal to
$\gk$.
\end{proof}
Theorem \ref{t6} clearly implies
the reverse direction of Theorem \ref{t1}.
Thus, the proof of Theorem \ref{t1}
is now complete.
\end{pf}
We remark that Bull's result of
\cite{Bu}, together with Theorem \ref{t6},
yield the following theorem.
\begin{theorem}\label{t6a}
The theories ``ZFC + There exists
a measurable cardinal'' and
``ZF + $\neg AC_\go$ + $\ha_\go$ is
a Rowbottom cardinal carrying a
Rowbottom filter'' are equiconsistent.
\end{theorem}
\section{The Proof of Theorem \ref{t2}}\label{s4}
\begin{pf}
In this section, we will prove Theorem \ref{t2}.
As in Section \ref{s3},
we break the proof up into both its
forward and reverse directions, which
we will establish separately.
We begin with the forward direction,
which we once again state as a separate theorem.
\begin{theorem}\label{t7}
Let $V \models ``$ZFC +
$\la \gk_i \mid i < \go_1 \ra$ is
a sequence of $\go_1$ measurable cardinals
with supremum $\gk$''.
There is then a notion of forcing $\FP$ and
a symmetric inner model $N \subseteq
V^\FP$ such that
$N \models ``$ZF + $\neg AC$ +
$\go_1$ is regular +
$\aleph_{\omega_1}$ is an $\go_2$-Rowbottom
cardinal carrying an
$\go_2$-Rowbottom filter''.
\end{theorem}
\begin{proof}
Suppose $V \models ``$ZFC +
$\la \gk_i \mid i < \go_1 \ra$ is
a sequence of $\go_1$ measurable cardinals''.
Without loss of generality, we
assume in addition that $V \models {\rm GCH}$.
We will give two proofs of Theorem \ref{t7},
one in which the desired model
satisfies ${\rm DC}_{\ha_\ell}$
for a fixed but arbitrary $\ell < \go_1$,
and one in which the desired model
satisfies only DC but also witnesses that
GCH holds below $\ha_{\go_1}$.
Our arguments are slight
generalizations of those given in
\cite{A83} and \cite{Bu}.
For the first of these models,
we proceed in analogy to the proof of
Theorem \ref{t5}. Specifically, let
$\ell < \go_1$ be fixed but arbitrary.
Take $\la \gl_i \mid i < \go_1 \ra$
as the sequence
$\la \gk_i \mid i < \go_1 \ra$, together
with its limit points.
Let $I = \{i < \go_1 \mid i$ is
either a successor ordinal or $0\}$.
%Let $\FP_0 = {\rm Coll}(\ha_{i + 1},
%{<} \gl_0)$,
For $i \in I$, let
$\FP_i = {\rm Coll} ( \gl^+_{i - 1}, {<} \gl_i)$,
where we take $\gl^+_{-1} = \ha_{\ell + 1}$.
We then define
$\FP = \prod_{i \in I} \FP_i$ with full support,
and take
$G$ as being $\FP$-generic over $V$.
By the definition of $\FP$ and the
properties of the L\'evy collapse,
$V[G] \models ``\go_1 = \go^V_1$''.
$V[G]$, being a model of AC, is once again not
our desired model $N$. In order to
define $N$, we first note that by
the Product Lemma,
for $i \in I$, $G_i$,
the projection of $G$ onto $\FP_i$,
is $V$-generic over $\FP_i$.
Next, let
$\mathcal{F} =
\prod_{i \in I} (\gl^+_{i - 1}, \gl_i)$.
For each $f \in \mathcal{F}$,
define $G \rest f =
\prod_{i \in I} (G_i \rest f(i))$.
Once again,
by the Product Lemma and the properties
of the L\'evy collapse, $G \rest f$ is
$\prod_{i \in I} (\FP_i \rest f(i))$-generic over $V$.
As before,
$N$ can now intuitively be described as the
least model of ZF extending $V$ which
contains, for each $f \in {\cal F}$,
the set $G \rest f$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified 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 symbols $\dot G \rest f$ for each $f \in {\cal F}$. $N$ is
then defined in the same way as in
the proof of Theorem \ref{t5}, i.e.,
as follows.\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\ga$ 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}\bigskip
As earlier, $N \models {\rm ZF}$.
Since $V \subseteq N \subseteq V[G]$ and
$V[G] \models ``\go_1 = \go^V_1$'',
$N \models ``\go_1 = \go^V_1$''.
Further, by Lemmas 1.1, 1.2, and 1.4 of
\cite{A83}, which remain valid even when
our sequence $\la \gl_i \mid i < \go_1 \ra$
is uncountable,
$N \models ``{\rm DC}_{\ha_\ell}$ + $\gk = \ha_{\go_1}$''.
In addition, since $N \models {\rm DC}$,
$N \models ``\go_1$ is regular''.
The first proof of Theorem \ref{t7} will thus
be complete once we have shown that
$N \models ``\ha_{\go_1}$ is an
$\go_2$-Rowbottom cardinal
carrying an $\go_2$-Rowbottom filter''.
To do this, let
$\la \mu_i \mid i < \go_1 \ra \in V$
be such that $\mu_i$ is a normal measure
over $\gk_i$. In $N$, define
${\cal F} = \{A \subseteq \gk \mid
\exists i < \go_1
\forall j \in [i, \go_1)[A \cap
\gk_j \in \mu_j]$.
By Lemma 1.3 of \cite{A83}, which
again remains valid working with
an uncountable sequence of cardinals
$\la \gl_i \mid i < \go_1 \ra$,
${\cal F}$ is in $N$ an $\go_2$-Rowbottom
filter over $\gk = \ha_{\go_1}$.
This completes our first proof
of Theorem \ref{t7}.
\end{proof}
\begin{pf}
For our second proof of Theorem \ref{t7},
let $\la \gk_i \mid i < \go_1 \ra$,
$\la \gl_i \mid i < \go_1 \ra$, $\gl_{-1}$,
$I$, $\FP_i$, and $\FP$
be as in the first proof of Theorem \ref{t7}.
(The exact value of $\ell < \go_1$ will
be irrelevant.)
Let $G$ be $\FP$-generic over $V$,
and for $i \in I$, let $G_i$ be
the projection of $G$ onto $\FP_i$.
For $j \in I$, let
$\FQ_j = \prod_{i \le j, i \in I} \FP_i$ and
$H_j = \prod_{i \le j, i \in I} H_i$.
It is again the case, by the
properties of the L\'evy collapse
and the Product Lemma, that
$H_j$ is $\FQ_j$-generic over $V$.
The $N$ for our second proof of
Theorem \ref{t7} can now be intuitively
described as the least model of ZF
extending $V$ which contains, for
every $j \in I$, the set $H_j$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified 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 symbols $\dot H_j$ for every $j \in I$.
$N$ is then defined as follows.\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\ga$ 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}\bigskip
As earlier, $N \models {\rm ZF}$.
Also, in analogy to our first
proof of Theorem \ref{t7},
for $j \in I$,
$\go^V_1 = \go^{V[H_j]}_1 =
\go^N_1 = \go^{V[G]}_1$.
In addition, Lemma 1 of \cite{A00}
(using the homogeneity properties
of the L\'evy collapse instead of
the homogeneity properties of Cohen forcing)
or Lemma 1.4 of \cite{Bu} (for an
uncountable sequence of cardinals)
gives us the fundamental homogeneity property that
if $x \in N$ is a set of ordinals, then
$x \in V[H_j]$ for some $j \in I$.
(The proofs of) Lemmas 1.2 and 1.4 of \cite{A83},
which again remain valid even when our
sequence of cardinals $\la \gl_i \mid i < \go_1 \ra$
is uncountable, then once more tell us that
$N \models ``$DC + $\gk = \ha_{\go_1}$''.
Further, as before, since
$N \models {\rm DC}$,
$N \models ``\go_1$ is regular''.
Our second proof of Theorem \ref{t7} will
thus be complete once we have shown that
$N \models ``$GCH holds below $\ha_{\go_1}$ +
$\ha_{\go_1}$ is an $\go_2$-Rowbottom cardinal
carrying an $\go_2$-Rowbottom filter''.
To see that the first of these facts is true,
%by the fundamental homogeneity property
%just mentioned and
by the properties of
the L\'evy collapse,
since $\FP$ is a full support product,
if $j \in I$ and
$x \subseteq \gl_j$, $x \in V[G]$,
then $x \in V[H_j]$.
Since $V \models {\rm GCH}$, we may
once again use the properties of the
L\'evy collapse and the fact $j < \go_1$
to infer that
$V[H_j] \models ``$There are no
cardinals in any of the open
intervals $(\gl^+_{i - 1}, \gl_i)$
for $0 \le i \le j$ + $\gl^+_{i - 1}$
and $\gl_i$ remain cardinals for
$0 \le i \le j$ + GCH holds for
any cardinal less than or
equal to $\gl_j$ +
$\gl_j < \ha_{\go_1}$''.
Since $V[H_j] \subseteq N \subseteq V[G]$, these
facts remain true in $N$ as well.
Consequently, since
$N \models ``\ha_{\go_1} = \gk =
\sup(\la \gl_i \mid i < \go_1 \ra)$'',
$N \models ``$GCH holds below $\ha_{\go_1}$''.
To see that the second of these facts is true,
%we argue in analogy to our first
%proof of Theorem \ref{t7}. Specifically,
as before, let
$\la \mu_i \mid i < \go_1 \ra \in V$
be such that $\mu_i$ is a normal measure
over $\gk_i$. In $N$, define
${\cal F} = \{A \subseteq \gk \mid
\exists i < \go_1
\forall j \in [i, \go_1)[A \cap
\gk_j \in \mu_j]$.
By the fundamental homogeneity
property mentioned above,
since any
$f : {[\gk]}^{< \go} \to \gl$
for any $\gl < \gk$ can be
coded as a set of ordinals,
for some $j \in I$,
$f \in V[H_j]$.
Since by the definition of
$\FQ_j$, there is some $j' > j$,
$j' \in I$ such that
$\card{\FQ_j} < \gk_{j'}$, by
the L\'evy-Solovay results
\cite{LS}, the sequence
$\la \gk_i \mid j' \le i < \go_1 \ra$
is composed of cardinals which are
measurable in $V[H_{j}]$, and for
any $i$ with $j' \le i < \go_1$,
$\mu^*_i = \{X \subseteq \gk_i \mid
\exists Y \in \mu_i[X \supseteq Y]\}$
is a normal measure over $\gk_i$ in
$V[H_{j}]$.
As in the last
paragraph of the proof of
Lemma 1.3 of \cite{A83}, which
still remains valid working with
the uncountable sequence of cardinals
$\la \gk_i \mid j' \le i < \go_1 \ra$,
we may infer that there is
$A \subseteq \gk$, $A \in V[H_{j}] \subseteq N$
which is $\go_2$-Rowbottom homogeneous for
$f$ such that $\forall i \in [j', \go_1)[A \cap
\gk_i \in \mu^*_i]$.
Without loss of generality, as in \cite{A83},
we may further assume that
$\forall i \in [j', \go_1)[A \cap \gk_i \in \mu_i]$.
%Since for any $i \ge j'$, $\mu^*_i \supseteq \mu_i$,
It then immediately follows that
$A \in {\cal F}$.
Thus, ${\cal F}$ is in $N$ an $\go_2$-Rowbottom
filter over $\gk = \ha_{\go_1}$.
Our second proof of Theorem \ref{t7}
is now complete.
\end{pf}
We continue now with the reverse
direction of Theorem \ref{t2}.
As before,
we state this as a separate, stronger
theorem which implies our desired result.
\begin{theorem}\label{t8}
Let $\gk$ be a singular Jonsson cardinal of
uncountable cofinality in a model $V$
of ZF. Then there is an inner model whose class of
measurable cardinals is cofinal in $\gk$.
\end{theorem}
The proof of Theorem \ref{t8} will
use the theory of short core models,
but in the context of models
{\it not} satisfying the Axiom of Choice.
We therefore briefly mention now
some terminology and notation
attendant to this theory.
The smallest mouse which
is not an element of a short core model
is uniquely determined and is the set
$0^{{\rm long}}$ mentioned in
Section \ref{s1}. It is a countable
iterable structure of the form
$0^{{\rm long}} =
L_{\delta}[ {\bar{\cal U}}^{{\rm long}} ]$,
in which the measure
sequence ${\bar{\cal U}}^{\rm long}$ is not short.
Indeed, it is the case that
otp$({\dom}({\bar{\cal U}}^{{\rm long}})) =
\min({\dom}({\bar{\cal U}}^{{\rm long}}))$.
The existence of $0^{{\rm long}}$ is a
large cardinal axiom.
%The non-existence of $0^{{\rm long}}$ is denoted by
%$\neg 0^{{\rm long}}$.
We begin our discussion of the proof of
Theorem \ref{t8} with
a useful preliminary result.
\begin{proposition}\label{p3}
Assume that $0^{{\rm long}}$ exists.
Then for any singular cardinal $\gk$,
there is an inner model whose class
of measurable cardinals is
cofinal in $\gk$.
\end{proposition}
\begin{proof}
Let $\gl = {\rm cof} ( \gk ) < \gk$.
Choose a strictly monotone
sequence $\la \gk_i \mid i < \gl \ra$
which is cofinal in $\gk$ such that
$\gk_0 = \gl$. Construct a minimal iterate
$L_{\eta} [ {\bar{\cal U}} ]$ of
$0^{{\rm long}}$ such that
for every $i < \gl$,
it is the case that the $i^{\rm th}$
measurable cardinal of ${\bar{\cal U}}$ is
greater than $\gk_i$.
%$> \gk_i $.
Since ${\bar{\cal U}}$ has a $\gl^{\rm th}$
measurable cardinal
greater than or equal to $\gk$
%$\geqslant \gk$
which can be iterated out of the
ordinals, one gets that
$L [ {\bar{\cal U}} \rest \gk ] \models ``{\bar{\cal U}}
\rest \gk$ is a sequence of measures''.
The model $L [ {\bar{\cal U}}
\rest \gk ]$ is as desired.
\end{proof}
We turn now to the proof
of Theorem \ref{t8}.
\begin{proof}
If $0^{{\rm long}}$ exists,
Theorem \ref{t8} holds by the previous proposition.
Consequently, we assume $\neg 0^{{\rm long}}$.
Then the theory of short core models of
\cite{K2} is adequate for models of the form
${{\rm HOD}} [ a ]$ with $a \subseteq
{{\rm HOD}}$.
Let $K [ {\bar{\cal U}}_{{\rm can}} ]$
be the canonical short core model
formed in ${{\rm HOD}}$,
with measure sequence ${\bar{\cal U}}_{{\rm can}} $.
$K [ {\bar{\cal U}}_{{\rm can}} ] \models
``{\bar{\cal U}}_{{\rm can}}$ is a sequence of measures'',
so it suffices to show that
${\dom} ( {\bar{\cal U}}_{{\rm can}} \rest \gk)$
is cofinal in $\gk$.
We therefore assume towards a contradiction that
${\dom} ( {\bar{\cal U}}_{{\rm can}}
\rest \gk )$ is bounded below $\gk$.
Set $\gl = {\rm cof}
( \gk )$ and $\theta = \sup ( {\dom} ( {\bar{\cal U}}_{{\rm can}}
\rest \gk ) )$.
We begin by showing there
is a closed, unbounded $D \subseteq \gk$ such that
every element of $D$ is singular in
$K [ {\bar{\cal U}}_{{\rm can}} ]$.
For this, we consider two cases.\bigskip
\noindent Case 1: $\gk$ is a singular successor
cardinal. Then let $\bar{\gk}$
be the cardinal predecessor of $\gk$,
so $\gk = \bar{\gk}^+$.
Choose a strictly monotone sequence
$\la \gk_i \mid i < \gl \ra$
which is closed, unbounded in $\gk$. Let $\gk'
= ( \bar{\gk}^+ )^{{{\rm HOD}}
[\la \gk_i \mid i < \gl \ra]} \le
\bar{\gk}^+ = \gk^{}$. Since $\gk'$ is regular in
${{\rm HOD}} [\la \gk_i \mid i < \gl \ra]$
and $\gk$ is singular in ${{\rm HOD}}
[ \la \gk_i \mid i < \gl \ra]$,
we have $\gk' < \gk$.
Since $\gk' >
\bar{\gk} \ge {\rm cof}^V ( \gk )
= \gl \ge \omega_1^V$, we have
${{\rm HOD}} [ \la \gk_i \mid i < \gl \ra ]
\models ``\gk' \ge \omega_2 $''.
Work in ${{\rm HOD}} [ \la \gk_i \mid i < \gl \ra ]$.
Let $D = \{ \gk_{\omega \cdot i} \mid i < \gl$
and $\gk_{\omega \cdot i} > \gk'$ and
$\gk_{\omega \cdot i} > \theta \}$. Consider $\gk_{\omega
\cdot i} \in D$. It is then the case that
${\rm cof} ( \gk_{\omega \cdot i} ) \le \omega \cdot i < \gl
\le \bar{\gk} \le \card{\gk_{\omega \cdot i}}$,
$\gk_{\omega \cdot i} > \omega_2$,
and $\gk_{\omega \cdot i} >
\sup(\dom( {\bar{\cal U}}_{{\rm can}} \rest
(\gk_{\omega \cdot i} + 1) ))$.
By the Covering Theorem 3.20(i) of \cite{K2},
$\gk_{\omega \cdot i}$ is
singular in $K [ {\bar{\cal U}}_{{\rm can}} ]$.
The claim follows, since
$K [{\bar{\cal U}}_{{\rm can}} ]^{{{\rm HOD}}
[ \la \gk_i \mid i < \gl \ra ]} =
K [{\bar{\cal U}}_{{\rm can}} ]^{{{\rm HOD}}}$.\bigskip
\noindent Case 2: $\gk$ is a
singular limit cardinal. Choose a closed,
unbounded set $D \subseteq \gk$ of order type $\gl$. Since the limit
cardinals are closed, cofinal in $\gk$,
we may assume that, in ${{\rm HOD}}
[ D ]$, every $\tau \in D$ is a limit cardinal
of cofinality less than
or equal to $\gl < \tau$ which is
greater than $\max ( \omega_2, \theta, \gl )$.
Then, by the Covering Theorem
3.20(ii) of \cite{K2},
$\tau$ is singular in
$K [ {\bar{\cal U}}_{{\rm can}} ]^{{{\rm HOD}} [
D ]} = K [ {\bar{\cal U}}_{{\rm can}} ]^{{{\rm HOD}}}$.
$D$ is hence as desired.\bigskip
Continuing with the proof of Theorem \ref{t8},
take $H \in {{\rm HOD}}$ to be a sufficiently
elementary submodel of
${{\rm HOD}}$ such that
$\gk + 1 \subseteq H$.
Let $X$ be a Jonsson
substructure of the first-order structure
$\mathfrak{H}= \la H, \in, D \ra$,
where $D$ is taken as above.
By the choice of $X$,
$X \prec \la H, \in, D \ra$,
$\card{X \cap \gk } = \gk$, and
$X \cap \gk \neq \gk$.
The remainder of the argument will be carried out
inside the structure ${{\rm HOD}}[D, X ]$.
Note that by Proposition \ref{p0},
notions of short core model theory are
absolute between
${{\rm HOD}}$ and ${{\rm HOD}} [ D, X ]$.
Let $\pi : \la \bar{H}, \in, \bar{D} \ra
\cong \la X, \in, X \cap D \ra \prec \la H, \in, D \ra$
be the Mostowski collapse of $X$, where
$\bar{H}$ is transitive.
As in the proof of Theorem \ref{t6},
it is then the case that
$\pi \rest \gk \neq {\rm id} \rest \gk$
and $\pi ( \gk ) = \gk$.
Let $\bar{{\cal U}} = \pi^{- 1}
( {\bar{\cal U}}_{{\rm can}} \rest \gk )$, and let
$\bar{K} = K [ \bar{{\cal U}} ]^{\bar{H}}$
be the short core model over $\bar{{\cal U}}$
as defined in $\bar{H}$.
Since being closed, unbounded can be defined
absolutely in the $\in$-language,
$\bar{D}$ is closed, unbounded in $\gk$.
We note now that
$\forall \gg \in \bar{D}
[\bar{K} \models ``\gg$ is singular''$]$.
To see this, recall that
$\forall \gg \in D[K [ {\bar{\cal U}}_{{\rm can}}]
\models ``\gg$ is singular''$]$,
from which it follows
by the definition of $\pi$ and elementarity that
$\forall \gg \in D[K [ {\bar{\cal U}}_{{\rm can}}
\rest \gk ] \models ``\gg$ is singular''$]$.
Here, the core model
can be considered as being defined
in the model ${{\rm HOD}}$. Since $H$ is a
sufficiently elementary submodel of ${{\rm HOD}}$,
we have $\forall \gg
\in D[( K [ {\bar{\cal U}}_{{\rm can}} \rest \gk ] )^H
\models ``\gg$ is
singular''$]$. This is downwards absolute
to $\bar{H}$, so $\forall \gg \in
\bar{D}[( K [ \bar{{\cal U}} ] )^{\bar{H}}
\models ``\gg$ is singular''$]$.
It is also easily seen that
$K_{\gk} [ \bar{{\cal U}} ] \subseteq \bar{K}$.
This is verified by checking the
Condensation Criterion 3.24(ii) of \cite{K2}.
Consider a closed, unbounded set $C \subseteq \gk$.
Since ${\rm cof}(\gk ) = \gl > \omega$,
there is $\gg \in C \cap \bar{D}$.
By the preceding paragraph,
$\bar{K} \models ``\gg$ is singular'', as required.
In addition,
it is easily shown that
$\bar{{\cal U}}$ is a strong measure sequence, i.e.,
$K [ \bar{{\cal U}} ] \models
``\bar{{\cal U}}$ is a sequence of measures''.
To see this,
by elementarity, $\bar{{\cal U}}$
is a sequence of measures in
$\bar{K}$. If $\xi \in {\dom} ( \bar{{\cal U}} )$,
then $\bar{{\cal U}}_{\xi}$ is a
measure in $\bar{K}$.
%By the preceding paragraph,
By the fact that
$K_{\gk} [ \bar{{\cal U}} ] \subseteq \bar{K}$,
$\bar{{\cal U}}_{\xi}$ is a measure in
$K_{\gk} [\bar{{\cal U}} ]$. Since
$K_{\gk} [ \bar{{\cal U}} ] = H_{\gk}^{K [ \bar{{\cal U}} ]}$,
$\bar{{\cal U}}_{\xi}$ is a measure in $K [ \bar{{\cal U}} ]$.
Further,
since $K_{\gk} [ \bar{{\cal U}} ] =
H_{\gk}^{\bar{K}}$ and $\pi \rest H_{\gk}^{\bar{K}} :
H_{\gk}^{\bar{K}} \to
H_{\gk}^{K [ {\bar{\cal U}}_{{\rm can}} ]^H} =
K_{\gk} [ {\bar{\cal U}}_{{\rm can}} ]$, we
have that
$\pi \rest K_{\gk} [ \bar{{\cal U}} ] : K_{\gk} [ \bar{{\cal U}} ]
\to K_{\gk} [ {\bar{\cal U}}_{{\rm can}} ]$ is elementary.
By Theorem 3.16 of \cite{K2},
there is an iterated ultrapower
$\sigma : K [{\bar{\cal U}}_{{\rm can}} ]
\to K [ {\cal U}' ]$ such that $\bar{{\cal U}}$ is an initial
segment of the measure sequence ${\cal U}'$.
We may assume that $\bar{{\cal U}} = {\cal U}'
\rest \gk$, by possibly further iteration
of measures in ${\cal U}'$
above $\bar{{\cal U}}$. Then
$K_{\gk} [ {\cal U}' ] = K_{\gk} [ \bar{{\cal U}} ]$,
and $\pi \rest K_{\gk} [ {\cal U}' ] :
K_{\gk} [ {\cal U}' ] \to K_{\gk}
[{\bar{\cal U}}_{{\rm can}} ]$
is elementary.
Since ${\rm cof} ( \gk )$ is uncountable,
the upward extension
embeddings techniques known from
the standard proof of the Covering Theorem
(see Theorem 3.25 of \cite{K2})
may be applied to lift the map
$\pi \rest K_{\gk} [ {\cal U}' ]$ up to $K [ {\cal U}' ]$.
In particular, there is a map
$\tilde{\pi}$ and a transitive inner model $W$ such that
$\tilde{\pi} : K [ {\cal U}' ] \to W$
is elementary, $\tilde{\pi}
\supseteq \pi \rest K_{\gk} [ \bar{{\cal U}} ]$,
and $\tilde{\pi} (\gk ) = \gk$.
By Theorem 3.13 of \cite{K2},
$W = K [ \tilde{{\cal U}} ]$, where $\tilde{{\cal U}} =
\tilde{\pi} ( {\cal U}' )$. Hence,
$\tilde{\pi} : K [ {\cal U}' ] \to
K [ \tilde{{\cal U}} ]$ is elementary.
In addition, $\tilde{{\cal U}} \rest \gk
= {\bar{\cal U}}_{{\rm can}} \rest \gk$.
This is since by the choice of ${\cal U}'$,
we have $\bar{{\cal U}} = {\cal U}' \rest
\gk$. Thus,
$\tilde{{\cal U}} \rest \gk = \tilde{\pi} ( {\cal U}' ) \rest
\gk = \tilde{\pi} ( {\cal U}' ) \rest \tilde{\pi} ( \gk ) =
\tilde{\pi} ( {\cal U}' \rest \gk ) = \tilde{\pi} ( \bar{{\cal U}} ) =
{\bar{\cal U}}_{{\rm can}} \rest \gk$.
It then immediately follows that
$\tilde{\pi} \circ \sigma : K [ {\bar{\cal U}}_{{\rm can}} ] \to K [
\tilde{{\cal U}} ]$ is elementary.
By Theorem 3.17 of \cite{K2},
$\tilde{\pi} \circ \sigma$ is a normal iterated
ultrapower of $K [ {\bar{\cal U}}_{{\rm can}} ]$ which,
since $\pi \rest \gk \neq
{\rm id} \rest \gk$, is not the identity on
$\gk$. Let $\alpha < \gk$
be the critical point of $\tilde{\pi} \circ \sigma$.
Then $\alpha$ is measurable in
$K [ {\bar{\cal U}}_{{\rm can}} ]$, and
$\alpha \in {\dom}({\bar{\cal U}}_{{\rm can}})$.
Since ${\bar{\cal U}}_{{\rm can}} ( \alpha )$
is the first measure used in the normal iteration,
$\alpha \not\in {\dom}(\tilde{{\cal U}})$.
This, however, contradicts that
$\tilde{{\cal U}} \rest \gk
= {\bar{\cal U}}_{{\rm can}} \rest \gk$,
thereby completing the proof of Theorem \ref{t8}.
\end{proof}
Theorem \ref{t8} clearly implies the
reverse direction of Theorem \ref{t2}.
Thus, the proof of Theorem \ref{t2}
is now complete.
\end{pf}
We conclude this section
by noting that our methods of
proof for Theorem \ref{t2}
routinely generalize to the
situation where $\ga$ is an
ordinal such that
$\go_\ga > \ga$ and
$\go_\ga$ is regular.
More specifically, the methods
of this section allow us to
prove the following theorem.
\begin{theorem}\label{t8a}
Suppose $\ga$ is a definable
ordinal whose definition is
absolute between transitive
models of ZF.
Suppose further that for
any transitive model $V$ of ZFC,
$V \models ``\go_\ga > \ga$
and $\go_\ga$ is regular''.
The theories
``ZFC +
There exist $\go_\ga$
measurable cardinals'' and
``ZF + $\neg$AC + $\go_\ga$ is regular +
$\ha_{\go_\ga}$ is an $\go_{\ga + 1}$-Rowbottom
cardinal carrying an $\go_{\ga + 1}$-Rowbottom
filter''
are then equiconsistent.
\end{theorem}
\section{Some Generalizations and
Additional Remarks}\label{s5}
We begin this section by
noting that in Theorem \ref{t2},
we require for our equiconsistency
that $\go_1$ be regular.
That this is not a superfluous
requirement is shown by the
following theorem.
\begin{theorem}\label{t9}
Let
$V \models ``$ZFC + $\gk$ is
a measurable cardinal''.
There is then a notion of
forcing $\FP$ and a symmetric
inner model $N \subseteq V^\FP$ such that
$N \models ``$ZF + $\neg$AC${}_\go$ +
$\go_1$ is singular +
$\ha_{\go_1}$ is a Rowbottom cardinal
carrying a Rowbottom filter''.
\end{theorem}
\begin{proof}
As in the proof of
Theorem \ref{t5}, we
assume that the ground
model $V$ for the
hypotheses of Theorem \ref{t9}
has been extended generically
via Prikry forcing using a
normal measure ${\cal U}$
over $\gk$ to a model,
which we also denote by $V$, containing
a Prikry sequence $\la \gk_i \mid
i < \go \ra$ through $\gk$.
Let
$\FP_{-1} = {\rm Coll} (\go, {<} \ha_\go)$,
$\FP_0 = {\rm Coll} ( \aleph_{\go + 1}, {<} \kappa_0 )$,
and for $1 \le i < \go$, let
$\FP_i = {\rm Coll} ( \kappa^{+ \ha_{i-1} + 1}_{i - 1}, {<} \kappa_i)$.
We then define
$\FP = \FP_{-1} \times \prod_{i < \go} \FP_i$ with full support.
Let $G$ be $\FP$-generic over $V$.
To define our desired model $N$
witnessing the conclusions of
Theorem \ref{t9}, we first note that
as before, by
the Product Lemma, for $-1 \le i < \go$, $G_i$,
the projection of $G$ onto $\FP_i$,
is $V$-generic over $\FP_i$.
Next, let
$\mathcal{F} =
(\go, \ha_\go) \times
( \aleph_{\go + 1}, \kappa_0 )
\times ( \kappa^{+\ha_0 + 1}_0, \kappa_1 )
\times ( \kappa^{+\ha_1 + 1}_1, \kappa_2 ) \times \cdots$.
For each $f \in \mathcal{F}$,
$f = \la f(-1), f(0), f(1), \ldots \ra$,
define $G \rest f = G_{-1} \rest f(-1) \times
G_0 \rest f ( 0 ) \times G_1
\rest f ( 1 ) \times \cdots$.
By the Product Lemma and the properties
of the L\'evy collapse, $G \rest f$ is
$(\FP_{-1} \rest f(-1)) \times
\prod_{i < \go} (\FP_i \rest f(i))$-generic over $V$.
$N$ can now intuitively be described as the
least model of ZF extending $V$ which
contains, for each $f \in {\cal F}$,
the set $G \rest f$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified 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 symbols $\dot G \rest f$ for each $f \in {\cal F}$. $N$ is
then defined as follows.\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\ga$ 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}\bigskip
The standard arguments show $N \models {\rm ZF}$.
Further, by Lemmas 1.1 and 1.2 of
\cite{A83}, which remain valid even with
the current definition of $\FP$,
$N \models ``\gk = \ha_{(\ha_\go)^V}$''.
Since by Lemmas 1.1 and 1.2 of \cite{A83},
$N \models ``(\ha_\go)^V = \ha_1$'',
it is actually the case that
$N \models ``\gk = \ha_{\go_1}$ and
${\rm cof}(\gk) = \go$''.
As a consequence of this,
$N \models ``\neg{\rm AC}_\go$''.
Further, the proof of Theorem \ref{t4},
as well as the remarks in the
succeeding paragraph,
remain valid and show that for
any $f \in {\cal F}$ and any
$g \in V[G \rest f]$ with
$g : {[\gk]}^{< \go} \to \gl$
for $\gl < \gk$, there is
$A \in {\cal U}$ which is
Rowbottom homogeneous for $g$.
The proof that
$N \models ``\ha_{\go_1}$ is a Rowbottom
cardinal carrying a Rowbottom filter''
is therefore the same as
the one given in Theorem \ref{t5}. Hence,
the proof of Theorem \ref{t9} is now complete.
\end{proof}
From Theorems \ref{t9} and \ref{t6},
we may consequently now immediately
infer the following theorem.
\begin{theorem}\label{t10}
The theories
``ZFC + There exists a measurable cardinal'' and
``ZF + $\neg$AC + $\ha_{\go_1}$ is a
Rowbottom cardinal carrying a Rowbottom filter''
are equiconsistent.
\end{theorem}
One might also wonder if it is possible to
have additional instances of the
Axiom of Choice holding in
our aforementioned models in
which $\ha_\go$ is Rowbottom or
$\ha_{\go_1}$ is $\go_2$-Rowbottom
and GCH holds below either
$\ha_\go$ or $\ha_{\go_1}$.
In particular, in Bull's model of
\cite{Bu}, ${\rm AC}_\go$ fails,
and in the model constructed for
the second proof of Theorem \ref{t7},
we can only show that DC is true.
In fact, the degree of the Axiom
of Choice in these models
appears to be optimal
%is optimal
for the large cardinal strength
available in the relevant ground models.
If one had models, e.g.,
in which $\ha_\ga$ were Jonsson for
$\ga = \go$ or $\ga = \go_1$,
${\rm AC}_\ga$ were true, and
for each $i < \ga$, it were possible
to well-order $\wp(\ha_i)$, then
it would be possible to well-order
$\bigcup_{i < \ga} \wp(\ha_i)$ as well.
This would then allow us to run the
argument of Theorem 5.2 of \cite{K2}
and infer the existence of $0^{\rm long}$.
In conclusion,
we ask what the consistency
strength is for the theory
``ZF + $\neg$AC + The least
regular cardinal is both a
limit and Jonsson cardinal''.
By the results of \cite{A96},
by forcing over a model of AD,
we may establish an upper bound of
$\go$ Woodin cardinals.
Is it possible to lower this
upper bound, and even establish
an equiconsistency result, in
analogy to what is done in this paper?
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\end{document}