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% basic set theory constructions
%
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% IndWC.tex
% This is one section of a joint paper with Arthur Apter.
% The results were proved in the spring of 2003, and written here
% during the summer of 2003.
% This version was sent to Arthur Apter on June 6, 2003, to be
% incorporated into the joint version of the paper.
%
%
%
% The following macros are a selection from Joel's general math
% macros used in the document below
%
\def\tlt{\triangleleft}
\def\k{\kappa}
\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\s{\sigma}
\def\t{\theta}
\def\l{\lambda}
\def\m{\mu}
\def\n{\nu}
\def\lted{{{\leq}\d}}
\def\ltk{{{<}\k}}
%
% "Arthur, in the next two macro definitions, use blackboard bold for
% \bm. Latex uses a different name I think," - Joel Hamkins.
%
\def\P{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Qdot{\dot\Q}
\def\Pforces{\forces_{\P}}
\def\of{{\subseteq}}
%\def\cp #1{{ cp \left( #1 \right)}}
\def\card#1{\left|#1\right|}
\def\boolval#1{\mathopen{\lbrack\!\lbrack}\,#1\,\mathclose{\rbrack\!
\rbrack}}
\def\restrict{\mathbin{\mathchoice{\hbox{\am\char'26}}{\hbox{\am\char'
26}}{\hbox{\eightam\char'26}}{\hbox{\sixam\char'26}}}}
\def\st{\mid}
\def\set#1{\{\,{#1}\,\}}
\def\th{{\hbox{\fiverm th}}}
\def\muchgt{>>}
\def\cof{\mathop{\rm cof}\nolimits}
\def\iff{\mathrel{\leftrightarrow}}
\def\intersect{\cap}
\def\minus{\setminus}
\def\and{\mathrel{\kern1pt\&\kern1pt}}
\def\image{\mathbin{\hbox{\tt\char'42}}}
\def\elesub{\prec}
\def\union{\cup}
\def\iso{\cong}
\def\<#1>{\langle\,#1\,\rangle}
\def\ot{\mathop{\rm ot}\nolimits}
\newcommand{\pr}[1]{{\displaystyle \prod_{#1}}}
\newcommand{\Union}[1]{{\displaystyle \bigcup_{#1}}}
% Definitions of the partition relations
\def\part#1#2#3{#1 \rightarrow [#1]^#2_{#3}} %%\kappa -> [\kappa]^{\delta}_{\mu}
\def\partsub#1#2#3#4{#1 \rightarrow [#1]^#2_{#3, < #4} } %%\kappa -> [\kappa]^{\delta}_{\mu < \nu}$
\def\suppart#1#2#3{#1 \rightarrow [#1]^{<#2}_{#3}} %%\kappa -> [\kappa]^{<\delta}_{\mu}
\def\suppartsub#1#2#3#4{#1 \rightarrow [#1]^{<#2}_{#3, < #4}} %%\kappa -> [\kappa]^{<\delta}_{\mu < \nu}$
\def\npart#1#2#3{#1 \nrightarrow [#1]^#2_{#3}} %%\kappa -> [\kappa]^{\delta}_{\mu}
\def\npartsub#1#2#3#4{#1 \nrightarrow [#1]^#2_{#3, < #4}} %%\kappa -> [\kappa]^{\delta}_{\mu < \nu}$
\def\nsuppart#1#2#3{#1 \nrightarrow [#1]^{<#2}_#3} %%\kappa -> [\kappa]^{<\delta}_{\mu}
\def\nsuppartsub#1#2#3#4{#1 \nrightarrow [#1]^{<#2}_{#3, < #4}} %%\kappa -> [\kappa]^{<\delta}_{\mu < \nu}$
%%%%%%%%%%%%%%
\def\partf#1#2#3#4{#1 \rightarrow_{#4} [#1]^{#2}_{#3}} %%\kappa -> [\kappa]^{\delta}_{\mu}
\def\partfsub#1#2#3#4#5{#1 \rightarrow_{#5} [#1]^#2_{#3, < #4} } %%\kappa -> [\kappa]^{\delta}_{\mu < \nu}$
\def\suppartf#1#2#3#4{#1 \rightarrow_{#4} [#1]^{<#2}_{#3}} %%\kappa -> [\kappa]^{<\delta}_{\mu}
\def\suppartfsub#1#2#3#4#5{#1 \rightarrow_{#5} [#1]^{<#2}_{#3, < #4}} %%\kappa -> [\kappa]^{<\delta}_{\mu < \nu}$
\def\npartf#1#2#3#4{#1 \nrightarrow_{#4} [#1]^#2_{#3}} %%\kappa -> [\kappa]^{\delta}_{\mu}
\def\npartfsub#1#2#3#4#5{#1 \nrightarrow_{#5} [#1]^#2_{#3, < #4}} %%\kappa -> [\kappa]^{\delta}_{\mu < \nu}$
\def\nsuppartf#1#2#3#4{#1 \nrightarrow_{#4} [#1]^{<#2}_#3} %%\kappa -> [\kappa]^{<\delta}_{\mu}
\def\nsuppartfsub#1#2#3#4#5{#1 \nrightarrow_{#5} [#1]^{<#2}_{#3, < #4}} %%\kappa -> [\kappa]^{<\delta}_{\mu < \nu}$
%Definition of the partition function
\def \function #1#2#3 {F: [#1]^{#2} \rightarrow #3} % F:[\kappa]^{\delta} -> \lambda$
\def \functionl #1#2#3{F: [#1]^{<#2} \rightarrow #3} % F:[\kappa]^{\delta} -> \lambda$
%
% ------------------------------------------------------------------------------
%
\title{Jonsson-like Partition Relations and $j: V \to V$
\thanks{Both authors wish to thank the CUNY Research
Foundation for having provided partial support
for this research via the first author's
PSC-CUNY Grant 64455-00-33, under which
the second author was a research assistant.
In addition, both authors wish to thank the
referee for a thorough reading of
the first version of the paper and
for helpful comments and suggestions
which have been incorporated into this
version of the paper.}
\thanks{2000 Mathematics Subject Classifications:
03E02, 03E35, 03E55, 03E65}
\thanks{Keywords: Jonsson cardinals, partition relations,
polarized partitions, elementary embeddings.}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu\\
\\
Grigor Sargsyan\\
%\thanks{The second author wishes to thank Dr. Arthur W. Apter
%for introducing him to Set Theory,
%for generously mentoring his undergraduate years,
%for professionally supervising his individual
%studies in Set Theory,
%and for numerous
%enlightening mathematical and
%non-mathematical conversations.
%He also wishes to express his
%gratitude to the administrative
%staff of the CUNY Baccalaureate
%Program for making his undergraduate
%years as pleasant as he could possibly
%hope.}\\
% 1591 East 18th Street\\
% Brooklyn, New York 11230 USA
%\thanks{The second author's address after August 15, 2003 is
%Program in Logic and the Methodology of Science,
%University of California, Berkeley, California 94720 USA.}\\
% grigors04@yahoo.com}
Group in Logic and the Methodology of Science\\
University of California\\
Berkeley, California 94720 USA\\
grigor@math.berkeley.edu}
\date{July 14, 2003\\
(revised August 6, 2004)}
\begin{document}
\maketitle
\begin{abstract}
Working in the theory ``ZF +
There is a nontrivial elementary
embedding $\J$'',
we show that a final segment of
cardinals satisfies certain square bracket
finite and infinite exponent partition relations. As a
corollary to this, we show that this final segment
%of cardinals
is composed of
Jonsson cardinals. We then show how to
force and bring this situation down to small alephs.
%force and obtain models of ${\rm ZF}$ where all cardinals
%greater than, for instance, $\go_{18}$ satisfy infinite exponent
%partitions with superscript, for instance, $\go_{16}$.
A prototypical result is the construction of a model
for {\rm ZF} in which every cardinal
$\mu \ge \ha_2$ satisfies the square bracket
infinite exponent partition relation
$\mu \to {[\mu]}^{\go}_{\ha_2}$.
We conclude with a discussion of some
consistency questions concerning
different versions of the axiom
asserting the existence of a nontrivial
elementary embedding $j : V \to V$.
By virtue
of Kunen's celebrated inconsistency result,
we use only a restricted amount of the
Axiom of Choice.
\end{abstract}
\newpage
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{R}, Reinhardt speculated on
the possibility of an axiom
asserting the existence of a
nontrivial elementary embedding of the universe
into itself, which we will
usually write as
%when the occasion warrants as
$\J$. Such an axiom can
be viewed as the ultimate
generalization of elementary embeddings of the universe into an
inner model. However, Kunen \cite{Ku1} eventually showed that
%such an axiom is not compatible with
%the Axiom of Choice, and consequently
this axiom is incompatible
with ${\rm ZFC}$.
The use of the Axiom of Choice in Kunen's
original proof and in subsequent proofs (see \cite{Ka},
\cite{S} and \cite{Z}) is essential,
and thus it remained
(and in fact, continues to remain) unknown
whether such an axiom
could be compatible with ${\rm ZF}$ or with any
natural strengthening of ${\rm ZF}$.
%, e.g., ZF + AD + DC or the like.
%(${\rm ZF}+AD+DC$ or the like).
In Kunen's original proof, the use of
the Axiom of Choice is implicit
in the sense that first, the incompatibility of
the Axiom of
Choice with a certain square bracket infinite exponent partition
relation is demonstrated,
and then, using the failure of this partition relation, the
falseness of $\J$ is derived.
This suggests that if we drop the Axiom
of Choice and assume $\J$, then certain infinite exponent
partition relations will hold.
What we intend to show is that
this is indeed the case. We will prove that
these sorts of partition properties
are satisfied by a final segment of the
cardinals, and that this situation can be
brought down to smaller
alephs. More specifically,
we prove the following
two theorems.
%\footnote{Readers are forewarned
%that until we present our method of
%formalization, the statements of
%our theorems can, strictly speaking,
%only be interpreted informally.
%This will be amended immediatley
%following the statement of Theorem \ref{t2a}.}
%which have several generalizations
%(see Sections \ref{s2} and \ref{s3}).
%(see Section \ref{s3}).
\begin{theorem}\label{t1}
%Let $V \models {\rm ZF} + $\J$'',
%and let $\cp(j) = \k$.
Let $V \models$ ZF, and
suppose there is
a nontrivial elementary embedding
%of the universe into itself having
$\J$ having critical point $\gk$.
There is then a cardinal
$\l > \k$ such that for every
cardinal $\mu \geq \l$ and every $\d <
\l$, $\part \mu \d \l $.
\end{theorem}
\begin{theorem}\label{t2}
%Let $V \models ``ZF + DC + \J$''.
Let $V \models ``$ZF + DC'', and suppose
there is a nontrivial elementary
embedding $j : V \to V$.
%of the universe into itself''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$
such that $N \models ``$ZF +
For every cardinal $\mu \geq
\ha_2$, $\part \mu \omega {\ha_2}$''.
\end{theorem}
In this paper,
we consider three
%types of elementary embeddings
different versions of the axiom
asserting the existence of a
nontrivial elementary embedding
$\J$. Here are the statements
of each of them,
taking as a notational convention that for
$n \in \go$ and $\gk$ the critical point of $j$,
$j^n(\gk)$ denotes the embedding $j$ applied
$n$ times to $\gk$, $j(j(...j(\k)...))$.
\begin{definition}\label{d1}
\item $\bf J_1:$
There is a cardinal $\kappa$ such that
for every $\alpha$,
there is an elementary embedding $\J$ with
$j(\kappa) > \alpha$ and $\cp(j) = \kappa$.
\item $\bf J_2:$
For some cardinal $\kappa$, there is an
elementary embedding $\J$ such that $\cp(j)
=\kappa$ and $DC_\l$
holds in $V$, where $\lambda = \sup(\{j^n(\gk) : n \in \go\})$.
%(\{ j^n(\kappa)=j(j(...j(\k))) : n \in \omega\})$.
\item $\bf J_3:$
For some cardinal $\kappa$, there is an
elementary embedding $\J$ such that
$\cp(j)=\kappa$.
\end{definition}
By $J_1(\k)$ ($ J_2(\k)\ , J_3(\k)$),
we mean that there is a $J_1$
($J_2\ , J_3$) elementary embedding with critical point $\k$. We
will refer to these axioms as the $J$ hierarchy.
We note that this hierarchy has been studied by Woodin,
and we will discuss in Section \ref{s4}
why we put these axioms in this order.
From now
on, by $\k_n$ we mean $j^n(\k)$,
where $j$ is the
elementary embedding under consideration, and
$\k_0 =\k$.
By elementarity, each $\gk_n$ is a regular
(in fact, measurable) cardinal.
The sequence $\la \k_n : n \in \go \ra$,
which is clearly strictly increasing,
is called the critical
sequence, and following \cite{Ku1}, we set
$\k_\go = \sup( \la \k_n :
n\in \go \ra)$.
We will also when necessary write ${(\gk_\go)}_j$ for the
$\gk_\go$ associated with the embedding $j$.
Using the above definitions, our theorems can be
formulated in the following way. \bigskip
\begin{theorem}\label{t1a}
Let $V \models$ ZF, and suppose
$J_3(\gk)$ holds.
Then for every cardinal
$\l \geq
\k_\go$ and every $\d < \k_\go$, $\part \l \d {\k_\go}$.
\end{theorem}
\begin{theorem}\label{t2a}
Let $V \models$ ``ZF + DC'', and
suppose $J_3(\gk)$ holds.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$
such that $N \models ``$ZF +
For every cardinal $\mu \geq
\ha_2$, $\part \mu \omega {\ha_2}$''.
\end{theorem}
In this paper, we will always
start with a model of ${\rm ZF}$,
some or no amount of choice, and one of the three forms of the
above elementary embeddings. However, a moment's thought reveals
that none of the above elementary embeddings is formalizable in
${\rm ZF}$ alone. We can proceed in
a number of different ways to avoid this
difficulty
%(for a third way of formalizing $\J$ see \cite{Su}),
and take our theorems as being stated using
one of these formalizations.
One way is to follow Corazza's treatment of
\cite{C} by enhancing the language of set
theory with a symbol $j$ for the elementary embedding
and then adding additional axiom schema.
For a detailed explanation, we refer readers to \cite{C}.
Another way is to use the formalization
found on page 1593 of Suzuki's paper \cite{Su}.
Our method of formalization, however,
will be to assume
that we have two structures $V \in V'$,
both models of ZF, such that the
elementary embedding $j : V \to V$
is an element of $V'$, in analogy, e.g., to
the situation of having a measurable
cardinal $\gk$ with normal measure
$\mu$, letting $\gd_1$ and
$\gd_2$ be two inaccessible
cardinals such that
$\gd_1 < \gk < \gd_2$,
and considering
$L_{\gd_1}$ and $L_{\gd_2}[\mu]$.
We hasten to add that this is an
imperfect analogy, since in our
circumstances, we will of course have
that for each ordinal $\ga$,
the elementary embedding
%$j_\ga$
defined as the restriction embedding
$j \rest V_\ga : V_\ga \to V_{j(\ga)}$
will be an element of
$V_{j(\ga) + 2}$, which is certainly
not the case
when we are considering fragments
of $L$ and $L[\mu]$.
%in what we mentioned earlier.
%Further, for any $\ga$ above
%$\cp(j)$, $j_\ga$ is not definable,
%as is shown in \cite{H} and \cite{Su}.
We warn readers that
for convenience, we will always be
fairly informal when using
the axioms $J_1$, $J_2$, and $J_3$
as our assumptions.
As a consequence, we will severely
abuse notation and write, e.g.,
%This will result in an abuse of notation
%when we write, e.g.,
$V \models ``$ZF + $J_3$''.
%Con(ZF + $J_3$).
The following definition
will be useful in Sections
\ref{s2} and \ref{s3}.
\begin{definition}\label{d3}
Let $\J$ be an elementary embedding with critical point $\k$,
and let $\l \geq \k_\go$ be a fixed cardinal.
$\la \a^\l_n : n \in \omega \ra$
is the sequence given by taking
$\a^\l_0$ as the least fixed point of $j$ above
$\l^{+\gk}$, and letting
$\ga^\gl_{n + 1}$ be the least
fixed point of $j$ above
${(\ga^\gl_n)}^{+ \gk}$.
\end{definition}
The following definition will also
be useful in Sections \ref{s2} and \ref{s3}.
\begin{definition}\label{d3a}
Let $\J$ be an elementary embedding with critical point $\k$,
and let $\l \geq \k_\go$ be a given cardinal.
$j^\l_n = j \rest V_{\a^\l_n}$ and
$i^\l_n=j^\l_n(j^\l_{n-1}(...(j^\l_0)...))$.
\end{definition}
It is routine to verify the following.
\begin{fact}\label{f0}
$i^\l_n$ is an elementary embedding of $V_{\a^\l_0}$ into itself,
and the critical point of this embedding is $\k_n$. Also, for all
$m \geq n$, $i^\l_n(\k_m)=\k_{m+1}$.
\end{fact}
Once again,
let $\gl$ be a cardinal.
Among various weakenings of AC,
we will use ${\rm DC}_\l$
and ${\rm AC}_\l$.
For convenience,
we recall these definitions here,
and we refer readers to \cite{J2}
for further information.
%The following definition is taken from \cite{J2}.
% Definition of DC_\kappa and AC_\kappa.
\begin{definition}\label{d4}
\item $\bf DC_\l:$ Let $S$ be any set,
with $\FR$ a binary relation on
$S$ such that for every $\a < \l$
and every sequence $ \la x_\b : \b <
\a \ra$ of elements of $S$,
there is $x_\a \in S$ such that for
every $\b < \a$, $x_\b \FR x_\a$.
Then there is a function $f: \l
\to S$ such that for all $\a < \l $ and all $\b < \a$,
$f(\b) \FR f(\a)$.
\item $\bf AC_\l:$
If $\la X_\ga : \ga < \gl \ra$ is a family
of sets such that for every $\ga < \gl$,
$X_\ga \neq \emptyset$, then the cartesian product
$\prod_{\ga < \gl} X_\ga \neq \emptyset$.
%Every family $\FF$ of nonempty sets such that
%$\card \FF \leq \l$ has a choice function.
\end{definition}
We note that ${\rm DC}$ is ${\rm DC}_\go$.
The following facts,
which explain
the connections among ${\rm DC}_\l$, ${\rm AC}_\l$,
and ${\rm AC}$, are proven in
\cite{J2}.
%\begin{fact}
%If $\l < \k$ then ${\rm DC}_\k \implies {\rm DC}_\l$
%and ${\rm AC}_\k \implies
%{\rm AC}_\l$.
%\end{fact}
\begin{fact}\label{f1}
${\rm DC}_\gl$ implies ${\rm AC}_\gl$.
\end{fact}
\begin{fact}\label{f2}
${\rm DC}_\gl$ implies that
for every set $X$,
either $\card X \leq \gl$ or $\card X \geq \gl$,
i.e., either there is an injection from
$X$ into $\gl$, or there is an injection
from $\gl$ into $X$.
\end{fact}
%\begin{fact}
%$(\forall \k){\rm DC}_k \implies {\rm AC}$.
%\end{fact}
\begin{fact}\label{f3}
If $\gl$ is singular,
then $(\forall \gk < \gl[{\rm DC}_\gk]) \implies {\rm DC}_\gl$,
and $(\forall \gk < \gl[{\rm AC}_\gk]) \implies {\rm AC}_\gl$.
\end{fact}
We note that if ${\rm AC}_\l$ holds,
then the cofinality of every
successor cardinal is greater than $\l$.
In particular, if ${\rm AC}_\l$
holds, then $\l^+$ is a regular cardinal.
It is also easily the case that
${\rm AC}_\gl$ (${\rm DC}_\gl$) implies
${\rm AC}_\gd$ (${\rm DC}_\gd$) for
every cardinal $\gd < \gl$.
Now we turn to defining square bracket partition relations.
For any set of ordinals $X$,
we treat $[X]^\d$ in two ways, both as the
set of all subsets of $X$ having
order type $\d$,
and as the set of
all strictly increasing functions from $\d$ into $X$.
As usual,
${[X]}^{< \gd} =
\bigcup_{\ga < \gd} {[X]}^\ga$.
In addition, in the following definition,
$\mu$ is a cardinal.
\begin{definition}[Definition of Square Bracket Partition
Relations]\label{d5}
\item 1. $ \part \k \d \l :$ For every function
$F : {[\k]}^\d \to \l$,
there exists
$X \in [\k]^\k$ such that $F''[X]^{\d} \not = \l$.
%$\npart \k \d \l$ is the negation of $\part \k \d \l$.
\smallskip
\item 2. $ \partsub \k \d \l \mu:$
For every function
$F : {[\k]}^\d \to \l$,
there exists
$X \in [\k]^\k$ such that $\card {F''[X]^{\d}} < \mu$.
%$\npartsub \k \d \l \m$ is the negation of $\partsub \k \d \l \mu$.
\smallskip
\item 3. $ \suppart \k \d \l :$
For every function
$F : {[\k]}^{< \d} \to \l$,
there exists
$X \in [\k]^\k$ such that $F''[X]^{\d} \not = \gl$.
%$\nsuppart \k \d \l$ is the negation of $\suppart \k \d \l $.
\smallskip
\item 4. $\suppartsub \k \d \l \m :$
For every function
$F : {[\k]}^{< \d} \to \l$,
there exists
$X \in [\k]^\k$ such that $\card {F''[X]^{\d}} < \mu$.
%$\nsuppartsub \k \d \l \mu$ is the negation of $\suppartsub \k \d \l \mu$.
\smallskip
\end{definition}
We recall that $\k$ is called a Jonsson cardinal if $\suppart \k
\go \k$. The set $X$ in Definition \ref{d5} is as usual
called a homogeneous set.
The following are the basic properties of square bracket partition
relations.
\begin{fact}\label{f4}
If $\l \leq \mu \leq \k$ and $\d \leq \theta \leq \k$, then $\part
\k \theta \l \implies \part \k \d \mu$.
\end{fact}
\begin{fact}\label{f5}
If $\l \leq \mu \leq \k$ and $\d \leq \theta \leq \k$, then
$\suppart \k \theta \l \implies \suppart \k \d \mu$.
\end{fact}
\begin{fact}\label{f6}
If $\theta \leq \l \leq \k$
and $\theta$ is a cardinal,
then for all $\d \leq \k$, $\partsub
\k \d \theta \theta \implies \part \k \d \l$.
\end{fact}
\begin{fact}\label{f7}
If $\theta \leq \l \leq \k$
and $\theta$ is a cardinal,
then for all $\d \leq \k$,
$\suppartsub \k \d \theta \theta \implies \suppart \k \d \l$.
\end{fact}
\begin{fact} [Erd\"os-Hajnal, ZFC, \cite{EH}]\label{f8}
$\forall \l \geq \go[\npart \l \go \l]$.
\end{fact}
\begin{fact}[{ZF}]\label{f9}
$(\forall \delta \geq \go)(\forall \l \geq \go) [\nsuppart \l \d
\d]$.
\end{fact}
By Fact \ref{f8},
assuming AC, there is a function $\function \k \go \k \ $ such
that for all $X \subseteq \k$ of cardinality $\k$,
it is the case that
$F''[X]^{\go}=\k$. This function is called an
$\go$-Jonsson function
and plays a key role in Kunen's argument that $\J$ is incompatible
with ${\rm ZFC}$. Getting this function is the only place where ${\rm AC}$ is
used, and so if we have ${\rm ZF} + \J $, then for some uncountable
cardinal $\l$,
it is plausible that $\part \l \go \l$. Theorem \ref{t1} is
a generalization of this observation.
We now define square bracket
polarized partition relations.
As in Definition \ref{d5},
$\mu$ is a cardinal.
\begin{definition}
[Definition of Polarized Square Bracket Partition Relations]
Suppose $k$ is an
$\a$ sequence of cardinals and $\d$ is an ordinal.
\item 1. $\part k \d \l:$ For every function $F:\prod_{\k\in
k}\left[\k\right]^{\d}\to \l$, there is a sequence $C\in
\prod_{\k\in k}\left[\k\right]^{\gk}$ such that $F''\prod_{\b <
\a}\left[C(\b)\right]^{\d} \not = \l$.
%$\npart k \d \l$ is the negation of $\part k \d \l$.
\smallskip
\item 2. $\partsub k \d \l \mu:$ For every function $F:\prod_{\k\in
k}\left[\k\right]^{\d}\to \l$, there is a sequence $C\in
\prod_{\k\in k}\left[\k\right]^{\gk}$ such that $\card {F''\prod_{\b <
\a}\left[C(\b)\right]^{\d} } < \mu$.
%$\npartsub k \d \l \mu$ is the negation of $\partsub k \d \l \mu$.
\smallskip
\item 3. $ \suppart k \d \l:$ For every function $F:\prod_{\k\in
k}\left[\k\right]^{<\d}\to \l$, there is a sequence $C\in
\prod_{\k\in k}\left[\k\right]^{\gk}$ such that $F''\prod_{\b <
\a}\left[C(\b)\right]^{<\d} \not = \l$.
%$\nsuppart k \d \l$ is the negation of $\suppart k \d \l $.
\smallskip
\item 4. $\suppartsub k \d \l \mu:$ For every function $F:\prod_{\k\in
k}\left[\k\right]^{<\d}\to \l$, there is a sequence $C\in
\prod_{\k\in k}\left[\k\right]^{\gk}$ such that $\card {F''\prod_{\b <
\a}\left[C(\b)\right]^{<\d} } < \mu$.
%$\nsuppartsub k \d \l \mu$ is the negation of $\suppartsub k \d \l \mu $.
\end{definition}
For more on infinite exponent partition relations, we refer
readers to \cite{AHJ97},
\cite{GP}, \cite{Ka} and \cite{Kl}.
%Our forcing terminology is reasonably
%standard. If $p$ and $q$ are two
%conditions then by $q \leq p$ we mean $p$ extends (is stronger
%than) $q$. We do not distinguish $V^\FP$ from $V[G]$.
We conclude Section \ref{s1} with a
discussion of forcing conventions,
notation, and terminology.
Our ground model $V$ will always be
a model for ZF and either a restricted
or no amount of the Axiom of Choice.
When forcing, $q \ge p$ will mean that
$q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$,
we will abuse notation somewhat and
use both $V[G]$ and $V^\FP$ to
indicate the universe obtained by
forcing with $\FP$.
$\FP$ is $\gk$-closed if
for every sequence
$\la p_\ga : \ga < \gd \le \gk \ra$
of elements of $\FP$ such that
$\gb < \gg < \gd$ implies
$p_\gb \le p_\gg$ (an increasing
chain of length $\gd$), there is
some $p \in \FP$ (an upper bound
to this chain) such that
$p_\ga \le p$ for all $\ga < \gd$.
We will, from time to time,
confuse terms with the sets
they denote and write $x$
when we actually mean $\dot x$
or $\check x$,
especially when $x$ is some
variant of the generic set $G$,
or $x$ is in the ground model $V$.
%$Col(\lambda, \kappa)$ is the standard partial ordering for
%turning inaccessible into a successor; $Col(\lambda, \kappa) =\{ p
%: p : \k \times \l \to \k \wedge \card p < \lambda \wedge
%p(\a, \b) < \a \}$ ordered by inclusion.
For $\gl$ a regular cardinal and
$\gk > \gl$ an arbitrary cardinal,
${\rm Coll}(\gl, {<} \gk)$ is the usual
L\'evy collapse of all cardinals in the open interval
$(\gl, \gk)$ to $\gl$, i.e.,
${\rm Coll}(\gl, {<} \gk) = \{f :
f : \gl \times \gk \to \gk$ is a
function such that $\card{\dom(f)} < \gl$
and $f(\la \ga, \gb \ra) < \gb\}$,
ordered by inclusion.
Note that regardless if $\gk$ is
regular or singular,
${\rm Coll}(\gl, {<} \gk)$ is
$\gd$-closed for every $\gd < \gl$.
Consequently, if ${\rm DC}_\l$ holds, then
forcing with ${\rm Coll}(\gl, {<} \gk)$
adds no new bounded subsets to $\l$, and
hence, preserves all cardinals less than $\l$ and $\l$ itself.
If
${\rm Coll}(\gl, {<} \gk)$ is well-ordered,
then after forcing with it,
%${\rm Coll}(\gl, {<} \gk)$,
we preserve a final segment of cardinals.
Without {\rm AC}, it is
difficult to say what happens with cardinals after forcing with
${\rm Coll}(\gl, {<} \gk)$.
However, even without ${\rm AC}$,
it can be
proved that since
${\rm Coll}(\go, {<} \k)$ has a
well-ordering of order type $\gk$,
forcing with this partial ordering
preserves all cardinals above $\k$.
Finally, we define what we
mean by the notion of
inaccessible cardinal in a
choiceless context.
Under these circumstances, the
cardinal $\gk$ will be said
to be inaccessible iff
$\gk$ is a regular limit
cardinal, and for every
set $x$ of rank less than $\gk$,
there is no cofinal map from
$x$ into $\gk$.
With this definition, it is
easy to show that
$V_\gk \models {\rm ZF}$.
\section{Some Consequences of ZF +
${\bf j : V \to V}$}\label{s2}
We begin this section by proving the following generalization of
Theorem \ref{t1}.
\begin{theorem}\label{t3}
({\bf Sargsyan}) Assume $J_3(\k)$.
\item 1. For all cardinals $\l \geq \k_\go$
and all
$\d < \k_\go$,
there is a cardinal $\mu ( \d, \l)<
\k_\go$ such that for every cardinal $\theta$,
if $\mu(\d, \l) \leq \theta <
\k_\go$, then $\partfsub \l \d \theta \theta {{}}$.
\item 2. For all cardinals $\l \geq \k_\go$
and all
$\d < \k_\go$,
there is a cardinal $\mu ( \d, \l)<
\k_\go$ such that for every cardinal $\theta$,
if $\mu(\d, \l) \leq \theta <
\k_\go$, then
$\suppartfsub \l \d \theta \theta {{}}$.
\end{theorem}
\begin{proof}
Since the proofs of $(1)$ and
$(2)$ are very close
to one another, with minor but obvious
differences, we only explicitly prove $(1)$.
Suppose $\J$ witnesses $J_3(\k)$. We
first prove the theorem in the case where $\d < \k$.
Fix an arbitrary $\d < \k$.
Since $\cp(j) = \gk$,
$j(\gd) = \gd$.
Suppose towards a contradiction that for some
cardinal $\l
\geq \k_\go$, we have that $\partfsub \l \d \theta \theta
{{}}$ fails for unboundedly
many cardinals $\theta < \k_\go$.
Let $\gl$ be the least witness to this fact.
By elementarity, we have that in $V$,
$j(\l)\geq \l$ is the least cardinal
greater than or equal to
$j(\k_\go)=\k_\go$ such
that $\partfsub
{j(\l)} \d \theta \theta {{}}$ fails for
unboundedly many cardinals $\theta < \k_\go$.
Since $\l$ is also the least
cardinal in $V$
that has the same property as $j(\l)$, we conclude that
$j(\l)=\l$.
Now, we claim that for all cardinals $\theta$,
if $\k \leq \theta < \k_\go$, then
$\partfsub \l \d \theta \theta {{}}$.
This is clearly a
contradiction, because we assumed that this
partition property fails for
unboundedly many cardinals $\theta < \k_\go$.
Therefore, towards a contradiction, assume
that for some cardinal $\theta$ with
$\k \leq \theta < \k_\go$, $\npartfsub \l
\d \theta \theta {{}}$.
There is then a function
$\function \l \d \theta \ $ such that for every
$X \in [\l]^\l$, we
have that $\card {F''[X]^{\d}}=\theta$. Since $j(\l)=\l$ and
$j(\d)=\d$, we have that $j(F) : [\l]^\d \to j(\theta)$,
and for all $X \in [\l]^\l$,
$\card {j(F)''[X]^\d} = j(\theta)$.
We next consider $j''\l \in [\l]^\l$.
We claim that $j(F)''[j''\l]^\d \subseteq j''\theta$.
This is a
contradiction, because by virtue of our assumption,
it should be the case that
$\card { j(F)''[j''\l]^\d } =j(\theta)$.
However, since for some $n \in \go$,
$\gk_n \le \theta < \gk_{n + 1}$,
by elementarity,
$\gk_{n + 1} \le j(\theta) < \gk_{n + 2}$,
i.e., as $j(\gk_\ell) =
\gk_{\ell + 1} > \gk_\ell$ for all
$\ell \in \go$, $j(\theta) > \theta$.
This means that $\card {j''\theta}=\theta \l^{+\gk}$ such that
$\gd$ is below the critical point
of $i$,
then the above argument would go through
to give us a contradiction. To get such an elementary embedding,
let $n \in \go$ be such that $\k_n \leq \d < \k_{n+1}$. By Fact
\ref{f0}, $i^\l_{n+1}: V_{\a^\l_0} \to V_{\a^\l_0}$ is an
embedding with $\a^\l_0 > \l^{+\gk}$
such that $\gd$ is below the
critical point of $i^\l_{n+1}$.
The proof is now finished as in the
proof of clause $(1)$ when $\gd < \gk$.
This completes the proof of Theorem \ref{t3}.
\end{proof}
\begin{corollary} Assume $J_3(\k)$.
\item 1.
For all cardinals $\l \geq \k_\go$,
and all $\d < \k_\go$,
$\partf \l \d {\k_\go}
{{}}$.
\item 2.
For all cardinals $\l \geq \k_\go$,
and all $\d < \k_\go$,
$\suppartf \l \d {\k_\go}
{{}}$.
\end{corollary}
\begin{corollary}\label{cjonn}
Assume $J_3(\k)$. Then every cardinal
$\l \geq \k_\go$ is a Jonsson cardinal.
\end{corollary}
\begin{corollary}
Let $A$ be a set theoretic axiom. If ${\rm ZF}+A \vdash$
``There are
unboundedly many non-Jonsson cardinals'', then ${\rm ZF}+A \vdash \neg
J_3$.
\end{corollary}
It is a basic fact that for any $n \in \go$,
the class of all sequences of ordinals
having length $n$ is canonically well-ordered
via the lexicographic ordering,
which we call $\le_{\ell, n}$.
Using this observation, and replacing
$j''\gl$ in the proof of Theorem \ref{t3} with
$\prod_{1 \le i \le n} {[j''\gl_i]}^\gd$ for
$\la \gl_1, \ldots, \gl_n \ra$ the appropriately
chosen $\le_{\ell, n}$ least counterexample, we can
mimic the proof of Theorem \ref{t3}
virtually unchanged to prove the following.
%The next theorem is a generalization of Theorem \ref{t3} for
%polarized partition properties.
\begin{theorem}\label{t4}
Assume $J_3(\k)$.
\item 1. For every finite sequence $k$
consisting of cardinals greater than or
equal to $\k_\go$, and
for all $\d < \k_\go$, there is
a cardinal $\mu ( \d, k)<\k_\go$ such that for
all cardinals $\theta$ satisfying
$\mu(\d, k) \leq \theta < \k_\go$,
$\partsub k \d
\theta \theta $.
\item 2. For every finite sequence $k$
consisting of cardinals greater than or
equal to $\k_\go$, and
for all $\d < \k_\go$, there is a cardinal
$\mu(\gd, k)<\k_\go$ such that for
all cardinals $\theta$ satisfying
$\mu(\gd, k) \leq \theta < \k_\go$,
$\suppartsub k \d
\theta \theta $.
\end{theorem}
\begin{corollary} Assume $J_3(\k)$.
\item 1. For every finite sequence $k$
consisting of cardinals greater than or equal to
$\k_\go$, and for all $\d < \k_\go$,
$\part k \d {\k_\go}$.
\item 2. For every finite sequence $k$
consisting of cardinals greater than or equal to
$\k_\go$, and for all $\d < \k_\go$,
$\suppart k \d {\k_\go}$.
\end{corollary}
\section{Consistency Results from ZF +
${\bf j : V \to V}$}\label{s3}
In this section,
our main goal is to prove Theorem \ref{t2}, which is a
corollary to the following more general theorem.
\begin{theorem}\label{t5}
Assume that $V \models ``$ZF +
$J_3(\k)$ + For some cardinal
$\gd < \gk$, $DC_\gd$ holds''.
%where $\gd$ is either a successor cardinal
%or a regular limit cardinal.
\item 1. There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
$\gd$ and
${(\gd^+)}^V$ are cardinals +
For all cardinals $\l \geq \gd^{++}$,
$\gl \to {[\gl]}^\gd_{\gd^{++}}$''.
\item 2. There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
$\gd$ and
${(\gd^+)}^V$ are cardinals +
For all cardinals $\l \geq \gd^{++}$,
$\gl \to {[\gl]}^{< \gd^+}_{\gd^{++}}$''.
\end{theorem}
In order to prove Theorem \ref{t5},
we will need the following lemmas.
\begin{lemma}\label{l1}
%Assume $J_3(\k)$, and suppose $j : V \to V$ is $J_3(\k)$
%embedding.
Let $j : V \to V$ witness $J_3(\gk)$.
Suppose that $X$ is a set of rank less than $\k_n$. Then $\card
X \not \geq \k_n$, i.e.,
there is no injection from
$\gk_n$ into $X$.
\end{lemma}
\begin{proof}
Towards a contradiction,
assume that $\card X \geq \k_n$.
There is then a
one-to-one function $f$
from $\kappa_n$ into $X$. Let
$i^{\k_\go}_n=i_n$,
where $i^{\k_\go}_n = i_n$ is given by
%Fact \ref{f0}.
Definition \ref{d3a}.
Since the critical point of $i_n$ is $\k_n$,
we have that $i_n(X)=X$,
and for every $x \in X$, $i_n(x)=x$. Thus,
it follows that $i_n(f)$ is a
one-to-one function from $i_n(\k_{n})
>\k_n$ into $X$.
Fix $\a$ such that
$\gk_n < \ga < i_n(\gk_n)$, and let
$i_n(f)(\a)=x \in X$.
Note that
$\rge(i_n(f)) = i_n(\rge(f)) =
\rge(f)$ as $\rge(f) \in V_{\gk_n}$,
meaning that
there is $\b < \k_n$ such that $f(\b) = x$.
By elementarity,
$i_n(f)(i_n(\b))=i_n(x)$, i.e.,
since $\gb < \gk_n$ and
$x \in V_{\gk_n}$, $i_n(f)(\gb) = x$.
Since $\ga \neq \gb$, this is
an immediate contradiction to the
fact that $i_n{(f)}$ is one-to-one.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{corollary}\label{c0}
Assume $J_3(\k)$ as witnessed by $j : V \to V$.
%and suppose $j : V \to V$ is $J_3(\k)$ embedding.
Suppose that $X$ is a set of rank less than $\gk_\go$. Then
%${(\gk_\go)}_j$. Then
$\card X \not \geq \gk_\go$.
%{(\gk_\go)}_j$.
\end{corollary}
\begin{corollary}\label{c1}
Assume $J_2(\k)$ as witnessed by $j : V \to V$.
Then
%$V_\a$ for all $\a < \k_\go$ and
$V_{\k_\go}$ is well-orderable.
\end{corollary}
\begin{proof}
By $J_2(\gk)$, we know that
${\rm DC}_{\gk_\go}$ is true in $V$.
By Fact \ref{f2},
we therefore have
that for every set $X$,
for each $n \in \go$,
either $\card X
\geq \k_n$ or $\card X \leq \kappa_n$.
If we now fix
$\a < \k_\go$ and let $\k_n$ for $n \in \go$
be such that
$V_\a$ has rank less than $\kappa_n$, then
by Lemma \ref{l1}, we know that $\card {V_\a } \not \geq
\k_n$.
It thus follows that $\card {V_\a} < \k_n$, and so
$V_\a$ can be well-ordered.
Hence, to well-order $V_{\gk_\go}$,
using ${\rm AC}_\go$, let, for each $n \in \go$,
$\le_n$ well-order
$V_{\gk_n}$, and define a
well-ordering $\le_\go$ on $V_{\gk_\go}$ by
$x \le_\go y$ iff either ${\rm rank}(x) <
{\rm rank}(y)$, or
${\rm rank}(x) = {\rm rank}(y)$ and for the
minimal $n$ such that
$x, y \in V_{\gk_n}$, $x \le_n y$.
It is easily verified that $\le_\go$ is a
well-ordering of $V_{\gk_\go}$.
This completes the proof of Corollary \ref{c1}.
\end{proof}
\begin{lemma}\label{l2}
Assume $V \models ``$ZF + $J_3(\gk)$'',
and let $j : V \to V$ witness $J_3(\gk)$.
%Assume $J_3(\k)$ and let $j : V \to V$ be $J_3(\k)$ embedding.
Let $\FP$ be a partial ordering of
rank less than $\k_n$. Then every
cardinal greater than or equal to
$\k_n$ in $V$ is a cardinal in
$V^\FP$.
\end{lemma}
\begin{proof}
Suppose first
that $\l \geq \k_n$ is a regular cardinal in $V$.
%and $\FP$ is a partial ordering of rank less than $\k_n$.
Suppose towards a contradiction
that $\l$ is not a regular
cardinal in $V^\FP$.
Then for some $\mu < \l$,
there is $f \in V^\FP$
and $p_0 \in \FP$ such that $p_0 \forces ``\dot{f} :
{}{\mu} \to {}{\l}$ is unbounded".
Fix $\a < \mu$, and for
each $\b < \l$, let
$W_{(\a,\ \b)} = \{q \in \FP: q \geq p_0$ and
$q \forces ``\dot{f}({}{\a})={}{\b}$''$\}$.
We have that for $\gb \neq \gg$, either
$W_{(\a, \b)} \neq W_{(\a, \gg)}$, or
$W_{(\a, \b)} =
W_{(\a, \gg)}= \emptyset$.
Let $A=\{\gg: (\exists q \in
\FP)(\exists \a < \mu)[q \geq p_0$ and $q \forces
``\dot{f}({}{\a})={}{\gg}$''$]\}$.
We claim that $ \card A <
\l$.
%which is clearly a contradiction
%to the function $f$ being unbounded in $V^\FP$.
To see this, for each
$\a < \mu$, let
$A_\a =\{\b < \l : \exists q \in \FP [q \geq p_0$ and
$q \forces ``\dot{f}({}{\a})={}{\b}$''$]\}$.
Note that $\card{A_\ga} \le \gl$.
Clearly, $A =\bigcup_{\ga < \mu} A_\ga$, and
the cardinality of $A_\ga$ is the
number of distinct $W_{(\ga, \gb)}$ for $\gb < \gl$.
%$\card {A_\a}="$ the number of different $W_{(\a, \b)}$".
Consider the function $F: A_\a \to 2^\FP$
given by $F(\b)=W_{(\a, \b)}$.
Since ${\rm rank}(2^\FP) < \kappa_n$ and
$\gl \ge \gk_n$,
by Lemma \ref{l1},
if $\card{A_\ga} = \gl$,
$F$ fails to be one-to-one,
and moreover, the set
$X$ on which $F$ is one-to-one
has cardinality less than
$\kappa_n$.
%(such a set $X$ can be defined by induction).
Since $F$ is one-to-one on
$A_\ga$, $\card{A_\ga} < \gk_n$.
Hence, $\card A = \card
{\bigcup_{\ga < \mu} \ A_\a} < \l$,
since otherwise, we would have that the
cardinality of the union of
%fewer than $\l$
$\mu < \gl$ many subsets of $\l$,
each having cardinality less than $\l$
and canonically well-ordered by the
usual ordering on ordinals, is $\l$,
which contradicts
the regularity of $\l$ in $V$.
Thus, again by the regularity of $\gl$
in $V$, $\sup(A) < \gl$. This contradicts
our assumption that $\dot f$ denotes a
function which is unbounded in $\gl$
in $V^\FP$, since
$p_0 \forces ``\rge(\dot f)
\subseteq A$''.
This means that $\gl$ remains
a regular cardinal in $V^\FP$.
Suppose now that $\gl > \gk_n$ is a
singular cardinal in $V$.
Assume that
for some $\mu < \l$,
there is $f \in V^\FP$
and $p_0 \in \FP$ such that $p_0 \forces ``\dot{f} :
{}{\mu} \to {}{\l}$ is onto".
For each $\ga < \mu$ and each $\gb < \gl$,
let $A_\ga$ and $W_{(\ga, \gb)}$ be
defined as above.
We may infer as before that the
cardinality of each $A_\ga$ is
less than $\gk_n$, which means
that we may further infer that
the order type of each $A_\ga$
is less than $\gk_n$.
Let $\gd_\ga$ be this order type.
For each $\gb < \gd_\ga$, let the
ordered pair $\la \ga, \gb \ra$
be sent to the $\gb^{\rm th}$
element of $A_\ga$.
Since $\mu \times \gk_n$
canonically has cardinality
$\max(\mu, \gk_n) < \gl$,
this produces in $V$ a
function from a cardinal
smaller than $\gl$ onto
$\gl$, a contradiction.
Thus, $\gl$ is still a
singular cardinal in $V^\FP$.
%Since all regular cardinals of
%$V$ remain regular in $V^\FP$,
%$\gl$ has the same cofinality
%in $V^\FP$ as it did in $V$.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We note that if $j : V \to V$
witnesses $J_2(\gk)$ instead of just
$J_3(\gk)$, then the proof of
Lemma \ref{l2} becomes much easier.
By Lemma \ref{l1} and
the proof of Corollary \ref{c1},
$\FP$ can be well-ordered via a
well-ordering of order type less than $\gk_n$.
We are then able to employ the standard arguments
used when the Axiom of Choice is true
to show that when forcing with
$\FP$, all
cardinals greater than or
equal to $\gk_n$ are preserved.
%and have the same cofinality as in $V$.
\begin{lemma}\label{l3}
Assume that
$V \models ``$ZF
+ $J_3(\gk)$ +
For some cardinal $\gd < \gk$,
$DC_\gd$ holds'',
and let $j : V \to V$ witness $J_3(\gk)$.
Fix $n \in \go$, and suppose that
$\FP \in V$ is a $\d$-closed partial ordering
of rank less than $\k_n$. Then
$V^\FP \models ``$For all cardinals
$\l \geq \k_\go$,
$\partf \l \d {\k_\go} {{}}$''.
\end{lemma}
\begin{proof}
Let $\l \geq \k_\go$ be a cardinal in $V^\FP$.
By Theorem \ref{t3}, there
is $m > n$ such that $\partfsub \l \d {\k_m} {\k_m}
{{}} $ holds in $V$.
To prove Lemma \ref{l3},
since $\gk_m < \gk_\go$,
by Fact \ref{f6},
it is enough to show that
$V^\FP \models
``\partfsub \l \d {\k_m} {\k_m} {{}}$''.
To do this, note that
by Lemma \ref{l2}, $\k_m$ is a regular cardinal in $V^\FP$.
Also, since ${\rm DC}_{\d}$ holds in $V$
and $\FP$ is $\d$-closed, all
cardinals less than or equal to $\gd^+$
are preserved, and for every $\eta$, $[\eta]^{\d}$
and $[\eta]^{< \d^+}$
are the same in both $V$ and $V^\FP$.
Now, let $\function \l \d
{\k_m}\ $ be a function in $V^\FP$.
Define $H : [\l ]^\d \to
\k_m$ in $V$ by $H(s) =
\sup(\{\b : \exists p \in \FP[p \forces
``\dot{F}({}{s})={}{\b}$''$]\})$.
We claim that $H$ is
well-defined, i.e., that
$\sup(\{\b : \exists p \in \FP[p \forces
``\dot{F}({}{s})={}{\b}$''$]\}) < \k_m$.
To see this, fix
$s \in [\l]^\d$, and
let $W_{(s,\a)} = \{ p \in \FP : p \forces
``\dot{F}({}{s})={}{\a}$''$\}$.
As in the proof of Lemma \ref{l2},
we have that for $\ga \neq \gb$,
either $W_{(s,\a)} \neq W_{(s,\b)}$,
or $W_{(s,\a)} = W_{(s,\b)} = \emptyset$.
Again in analogy to the
proof of Lemma \ref{l2},
consider the map $f :\k_m \to 2^\FP$
given by $f(\a)= W_{(s,\a)}$.
Since ${\rm rank}(2^\FP) < \k_n$
and $\gk_m > \gk_n$,
by Lemma \ref{l1}, $f$ must
fail to be one-to-one,
and in fact, the set $X \subseteq \k_m$ on
which $f$ is one-to-one
has cardinality less than $\k_m$.
%(such set $X$ can be defined by induction).
Since $\k_m$ is regular, this
means that $\sup(\{\b : \exists p \in \FP[p \forces
``\dot{F}({}{s})={} {\b}$''$]\}) < \k_m$.
Let $C \in {[\gl]}^\gl$ be homogeneous for $H$.
We have that $\card{H''[C]^\d} < \k_m$.
Then clearly, $F''[C]^{\d} \subseteq \sup
(\{H''[C]^{\delta}\}) < \gk_m$,
which in fact means that
$\card {F''[C]^{\d}} < \kappa_m$.
This completes the proof of Lemma \ref{l3}.
\end{proof}
We note that as a corollary to
Theorem \ref{t3} and the
proof of Lemma \ref{l3},
by replacing a superscript of
$\gd$ with a superscript of
${<}\gd^+$, we have the following.
\begin{lemma}\label{l3a}
Assume that
$V \models ``$ZF
+ $J_3(\gk)$ +
For some cardinal $\gd < \gk$,
$DC_\gd$ holds'',
and let $j : V \to V$ witness $J_3(\gk)$.
Fix $n \in \go$, and suppose that
$\FP \in V$ is a $\d$-closed partial ordering
of rank less than $\k_n$. Then
$V^\FP \models ``$For all cardinals
$\l \geq \k_\go$,
$\partf \l {< \d^+} {\k_\go} {{}}$''.
\end{lemma}
We can now prove Theorem \ref{t5}.
\begin{proof}
Since the proofs of clauses $(1)$ and
$(2)$ are very close to one
another, with minor but
obvious differences,
we will only explicitly prove $(1)$.
Suppose $\J$ witnesses
$J_3(\k)$ and $\gd < \gk$ is a
cardinal in $V$ for which
${\rm DC}_\gd$ holds. We will obtain
our inner model $N$
witnessing the conclusions of
Theorem \ref{t5}
as a symmetric submodel of the
forcing extension $V^{\FP}$, where
$\FP={\rm Coll}(\d^+, {<} \k_\go)$.
Let $G$ be $V$-generic over $\FP$,
and for $i \in \go$,
let $\FP_i = {\rm Coll}(\d^+, {<} \k_i)$.
As usual, the set
$G_i = \{p \in G : p \in \FP_i\}$ is
$V$-generic over $\FP_i$.
%In order to define $N$,
$N$ can now intuitively be described as the least model of
${\rm ZF}$ extending $V$ which contains,
for every $i \in \go$, the set $G_i$.
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
$\check V(\check v)
\iff v \in V$, i.e., $\check V$ allows us
to determine members of the ground model),
and symbols $\dot G_i$ for each $G_i$. $N$ is then
defined as follows.
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\gl$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of
set theoretic rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{0em}
The standard arguments show $N \models {\rm {\rm ZF}}$.
\setlength{\parindent}{1.5em}
To see that $N$
is our desired model, we first show that
$N \models ``\k_\go=\d^{++}$''.
Note that as ${\rm DC}_\d$ holds and $\FP$ is
$\d$-closed, as we observed in the
proof of Lemma \ref{l3},
all $V$-cardinals less than or equal to $\gd^+$
remain cardinals in $V[G]$
(and also retain the same cofinalities
as in $V$).
In addition,
for any $\nu \le \gd$
and any ordinal $\eta$,
${([\eta]^\nu)}^V = {([\eta]^\nu)}^{V[G]}$.
Since
$V \subseteq N \subseteq V[G]$,
these facts remain true with $V[G]$
replaced by $N$.
%below $\d$, $\d$ and $\d^+$ are cardinals in $N$.
Because $G_i \in N$ for each $i \in \go$, $N$
contains a bijection from $\d^+$ onto $\k_n$.
This means that $N
\models ``\k_\go \leq \d^{++}$''.
It is therefore enough to show that
$\k_\go$ is a cardinal in $N$.
To demonstrate this, assume otherwise.
It is then the case that
$N \models ``\card {\k_\go }=\d^+$''.
Consequently, let $f \in N$ be a bijection
from $\d^+$ onto $\k_\go$.
By the usual automorphism argument
(see, e.g., Lemma 1.1 of \cite{A83}),
%\cite{A85}, and \cite{AH86}
which, as is noted in the proof of
Lemma 2.1 of \cite{AH86},
remains valid in the absence of the
Axiom of Choice,
we know that if $X$ is a set of ordinals in $N$,
then there is $i\in \go$ such that $X \in V[G_i]$.
The same proof also applies to show that
if $X$ and $Y$ are sets in $V$
and $h : X \to Y$ is
a function in $N$, then for some $i \in \go $,
$h \in V[G_i]$.\footnote{An outline
of the argument is as follows.
Let $\tau$ be a term for $h$
which, besides containing symbols
for sets in $V$, mentions only
a term of the form $\dot G_i$
for some fixed $i \in \go$.
Such a term $\tau$ can easily
be found using the appropriate coding.
Suppose $p \forces ``\tau(x) = y$'',
and let $q = p \rest i =
\{\la \la \ga, \gb \ra, \gg \ra \in p :
\gb < \gk_i\}$. If
$q \not \forces ``\tau(x) = y$'',
then let $r \ge q$ be such that
$r \forces ``\tau(x) \neq y$''.
Let $s = p \cup r \rest i$.
$s$ is well-defined, and since
$s \ge p$, $s \forces ``\tau(x) = y$''.
By the usual properties of the
L\'evy collapse, there is an
automorphism $\pi$ of $\FP$,
which can be defined canonically
and without any use whatsoever of
the Axiom of Choice, such that
$\pi(s)$ is compatible with
$r$, and $\pi$ is
generated by a permutation of $\gk_\go$
which is the identity on $\gk_i$.
Since any term for a ground model
set or any term mentioning only terms
for ground model sets and $\dot G_i$
can be assumed to be invariant under
$\pi$, $\pi(s)$ is compatible with
$r$, $\pi(s) \forces ``\tau(x) = y$'', and
$r \forces ``\tau(x) \neq y$''.
This contradiction means
$q \forces ``\tau(x) = y$'', so
$h$ is definable in $V[G_i]$ as
$\{\la x, y \ra$ : For some
$t \in G_i$, $t \forces ``\tau(x) = y$''$\}$.}
This implies that there is $i \in \go$
such that $f \in V[G_i]$.
This means that $\k_\go$ is not a cardinal in
$V[G_i]$, which contradicts Lemma \ref{l2}.
Thus, $\gk_\go$ is a cardinal in $N$.
We next show that in $N$, for any cardinal
$\gl \ge \gd^{++}$,
the partition relation
$\partf \l \d {\d^{++}} {{}}$ holds.
Let $\l \geq (\d^{++})^N$
be a fixed cardinal of $N$.
Since $(\d^{++})^N
=\k_\go$, we will show that
$N \models ``\partf \l \d {\k_\go}
{{}}$''.
To do this, let
$\function \l \d {\k_\go} \ $ be a function in
$N$. We want to show that there is
$X \subseteq \gl$, $X \in N$ such
that $\card{X} = \gl$ and
$F''[X]^\d \not = \k_\go$.
By our earlier remarks,
${({[\gl]}^\gd)}^N = {({[\gl]}^\gd)}^V$.
Therefore, by
what we said in the preceding paragraph,
for some $n \in \go$, $F
\in V[G_n]$. Since $\FP_n$ is
$\d$-closed and has rank less than $\k_{n + 1}$,
by Lemma \ref{l3},
$V[G_n] \models ``\partf \l \d
{\k_\go} {{}}$''.
Now, we can find $X \in
{V[G_n]}$, $X \subseteq \gl$ such that
$\card{X} = \gl$ and $F''[X]^\d \neq \gk_\go$.
Hence, as $V[G_n] \subseteq N$,
$X \in {N}$.
Finally, we finish the proof of
Theorem \ref{t5} by noting that by
Lemma \ref{l3a} and the argument
just given, we have that not only does
$N \models ``\gl \to {[\gl]}^\gd_{\gk_\go}$'', but
$N \models ``\gl \to {[\gl]}^{< \gd^+}_{\gk_\go}$''
as well.
This completes the proof of Theorem \ref{t5}.
\end{proof}
We take this opportunity to make as an
explicit remark that as the above
proof shows, clauses $(1)$ and $(2)$ of
Theorem \ref{t5} are satisfied by the
same symmetric submodel $N$.
In addition, as
$N \models ``\gk_\go = \gd^{++}$'', i.e.,
as there is a successor cardinal in
$N$ having cofinality $\go$,
${\rm AC}_\go$ fails in $N$.
\begin{corollary}
\item 1. Assume that $V \models ``$ZF
+ $DC$ + $J_3(\gk)$''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
For all cardinals $\gl \ge \ha_2$,
%$\l \geq (\go_2)^N$,
$\partf \l \go {\ha_2} {{}}$''.
\item 2. Assume that $V \models ``$ZF
+ $DC$ + $J_3(\gk)$''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
For all cardinals $\gl \ge \ha_2$,
%$\l \geq (\go_2)^N$,
$\partf \l {< \ha_1} {\ha_2} {{}}$''.
\end{corollary}
We note that by Corollary \ref{cjonn},
assuming $J_3(\gk)$, every cardinal
$\gl \ge \gk_\go$ is a Jonsson cardinal.
Further, if the partition property
we wish to preserve
witnesses Jonssonness, then
the proof of Lemma \ref{l3} remains
valid without using any amount
of Dependent Choice or any amount of
closure for the partial ordering
in question.
This means that if
we force with
${\rm Coll}(\go, {<} \gk_\go)$,
then the methods employed when
proving Theorem \ref{t5}
yield the following.
\begin{corollary}\label{cjonn2}
Assume that $V \models ``$ZF
+ $J_3(\gk)$''.
%Assume $J_3(\k)$.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
Every uncountable cardinal
is a Jonsson cardinal''.
\end{corollary}
We remark that
models in which all uncountable
cardinals are Jonsson were constructed
by the first author in \cite{A85},
using an almost huge cardinal
and methods of Gitik from \cite{G2},
and also by the first author in
\cite{A92}, using these same methods,
along with a cardinal which in
consistency strength lies between
a supercompact limit of
supercompact cardinals and an
almost huge cardinal.
Further, although this has not
been verified to this point,
it seems quite reasonable to assume
that in Gitik's model of \cite{G1},
which is constructed from a proper
class of strongly compact cardinals
and in which all uncountable cardinals
are singular, all uncountable
cardinals are Jonsson as well.
\begin{theorem}\label{t6}
\item 1. Assume
that $V \models ``$ZF +
$J_3(\k)$ +
For some cardinal $\d < \k$, $DC_\d$
holds''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
$\gd$ and ${(\gd^+)}^V$ are cardinals +
For every finite sequence
$k$ consisting of cardinals greater than
or equal to $\gd^{++}$,
$\part k \gd {\d^{++}}$''.
\item 2. Assume
that $V \models ``$ZF +
$J_3(\k)$ +
For some cardinal $\d < \k$, $DC_\d$
holds''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
$\gd$ and ${(\gd^+)}^V$ are cardinals +
For every finite sequence
$k$ consisting of cardinals greater than
or equal to
$\d^{++}$, $\suppart k {\gd^+} {\d^{++}}$''.
\end{theorem}
\begin{proof}
The proof of Theorem \ref{t6} is very close
to the proof of Theorem \ref{t5}, but
with one main difference.
In order to prove Theorem \ref{t6}, we need the
following polarized versions of Lemmas \ref{l3}
and \ref{l3a}.
\begin{lemma}\label{l4}
Assume that $V \models ``$ZF +
$J_3(\k)$ +
For some cardinal $\d < \k$, $DC_\d$
holds'',
and let $j : V \to V$ witness $J_3(\gk)$.
Fix $n \in \go$, and
suppose $\FP$ is a
$\delta$-closed partial ordering
having rank less than $\kappa_n$.
Then $V^\FP \models ``$For every
finite sequence $k$ consisting
of cardinals greater than
or equal to $\k_\go$,
$\part k \d {\k_\go}$''.
\end{lemma}
\begin{lemma}\label{l4a}
Assume that $V \models ``$ZF +
$J_3(\k)$ +
For some cardinal $\d < \k$, $DC_\d$
holds'',
and let $j : V \to V$ witness $J_3(\gk)$.
Fix $n \in \go$, and
suppose $\FP$ is a
$\delta$-closed partial ordering
having rank less than $\kappa_n$.
Then $V^\FP \models ``$For every
finite sequence $k$ consisting
of cardinals greater than
or equal to $\k_\go$,
$\suppart k {\d^+} {\k_\go}$''.
\end{lemma}
Since the proofs of Lemmas \ref{l4}
and \ref{l4a} are very close
to the proofs of Lemmas \ref{l3}
and \ref{l3a}, but
with obvious modifications, we leave their
proofs to the readers of this paper.
The proof of Theorem \ref{t6} can then
be carried out in analogy to the proof
of Theorem \ref{t5}, as readers may
verify for themselves.
This completes our discussion of the
proof of Theorem \ref{t6}.
\end{proof}
We note explicitly that the
models $N$ witnessing the
conclusions of Theorems
\ref{t5} and \ref{t6} are the same.
\begin{corollary}
\item 1. Assume that $V \models ``$ZF
+ $DC$ + $J_3(\gk)$''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
For every finite sequence $k$ consisting of
cardinals greater than or
equal to $\ha_2$,
$\part k \go {\ha_2}$''.
\item 2. Assume that $V \models ``$ZF
+ $DC$ + $J_3(\gk)$''.
There is then a partial ordering
$\FP \in V$ and a symmetric submodel
$N \subseteq V^\FP$ such that
$N \models ``$ZF +
For every finite
sequence $k$ consisting of cardinals
greater than or equal to
$\ha_2$,
$\suppart k {\ha_1} {\ha_2}$''.
\end{corollary}
It is important to note that
${\rm DC}_\gd$
in Theorems \ref{t5} and \ref{t6}
allows us to characterize
completely the cardinal
structure of $N$.
As we observed at the beginning
of the proof of Theorem \ref{t5},
all $V$-cardinals less than or
equal to $\gd^+$ remain cardinals
in $N$.
%(and also retain the same cofinalities as in $V$).
In addition,
all $V$-cardinals greater than or equal to
$\k_\go$ remain cardinals in
$N$.
This is since as we observed
during the proof of Theorem \ref{t5},
any function in $N$ between two
ground model sets must actually be
a member of $V^{\FP_i}$ for some
$i \in \go$.
Therefore, by Lemma \ref{l2},
no such function can collapse any
cardinal greater than or equal to
$\gk_\go = {(\gd^{++})}^N$.
\section{Some Remarks on the Consistency Strength of
the ${\bf J}$ Hierarchy}\label{s4}
We conclude our paper by a discussion of the
consistency strength of the $J$ hierarchy
and some of its consequences.
We begin by noting that it is not always true
that ``ZF + $J_3(\k)$ + ${\rm DC}_\d$
for some cardinal
$\d < \k$'' is weaker than
``ZF + $J_2$'', assuming
there is also an inaccessible cardinal
$\ga > \gk_\go$ in the universe.
\begin{fact}\label{f9a}
Let $j : V \to V$ witness $J_3(\gk)$,
and assume that
$V \models ``$ZF +
$\ga$ is the least inaccessible cardinal
above $\gk_\go$''.
There is then a cardinal $\gd < \gk$ such that
if $V \models DC_\gd$,
then some elementary embedding
$i : V_\ga \to V_\ga$ witnesses
$J_2(\gd)$ for the ZF model $V_\ga$.
\end{fact}
\begin{proof}
Let $j$, $V$, $\gk$, and
$\ga$ be as in the
hypotheses of Fact \ref{f9a}.
Observe that since
$\gk_\go$ is fixed by $j$
and $\ga$ is the least
inaccessible cardinal above
$\gk_\go$, by elementarity,
$\ga$ is fixed by $j$ as well.
Also, $k = j \rest V_\ga$ is an
elementary embedding from
$V_\ga$ to $V_{j(\ga)} = V_\ga$.
We now show the existence of a
%$\gd < {(\gk_\go)}_j$
%(which for the rest of the proof of
%Fact \ref{f9a}, we write as just $\gk_\go$)
$\gd < \gk_\go$
and an elementary embedding
$i : V_\ga \to V_\ga$ with
$\cp(i) = \gd$ and
${(\gk_\go)}_i = \gk_\go$
such that if $V \models {\rm DC}_\gd$,
then $V_\ga \models {\rm DC}_{\gk_\go}$.
Towards this end,
by the definition of $k$,
it is the case that
$V \models ``$There is an elementary
embedding $i'$ of $V_\ga$ into itself
such that ${(\gk_\go)}_{i'} = \gk_\go$''.
If $i$ is such an elementary embedding
with $\cp(i)$ minimal and
${(\gk_\go)}_i = \gk_\go$, then by
elementarity,
$V \models ``j(i) : V_\ga \to V_\ga$
is an elementary embedding with
$\cp(j(i))$ minimal and
${(\gk_\go)}_{j(i)} = \gk_\go$''.
This means that
$\cp(i) = \cp(j(i))$.
%Consider now $j^\gb_0$, where
%$j^\gb_0$ is as given in
%Definition \ref{d3a}.
By its definition,
%Definition \ref{d3a} and
%Fact \ref{f0}, $j_0^\gb$ is
%an elementary embedding of
%$V_\ga$ into itself such that
$\cp(k) = \gk$ and
${(\gk_\go)}_{k} = \gk_\go$.
This means, by our minimality assumption,
that $\cp(i) \le \gk$. However, if
$\cp(i) = \gk$, then $j(\cp(i)) =
\cp(j(i)) = j(\gk)$, which, since
$j(\gk) > \gk$, contradicts
the last sentence of the preceding
paragraph.
Set $\gd = \cp(i) < \gk$.
If $V \models {\rm DC}_\gd$,
then also,
$V_\ga \models {\rm DC}_\gd$,
%since $V_\ga \models {\rm ZF}$.
%since by the fact $\ga$ is the
%least fixed point of $j$ above
%a certain point,
since $\ga > \gd$ is a limit ordinal.
By elementarity, for each $n \in \go$,
$V_\ga \models {\rm DC}_{i^n(\gd)}$,
which immediately implies that
since $\gk_\go =
\sup(\la i^n(\gd) : n \in \go \ra)$,
$V_\ga \models {\rm DC}_{\gk_\go}$.
Since $\ga$ is inaccessible
in $V$, $V_\ga \models {\rm ZF}$.
Thus, $i$ witnesses $J_2(\gd)$
for the ZF model $V_\ga$.
This completes the proof of
Fact \ref{f9a}.
\end{proof}
We continue our discussion
by noting that
Definition \ref{d1} is stated
in a way which suggests that
$J_1$ is stronger than $J_2$
which is stronger than $J_3$. From
Definition \ref{d1}, it is clear that
$J_1$ implies $J_3$ and
$J_2$ implies $J_3$.
However, the following fact holds.
\begin{fact}\label{f10}
$\forall \k[J_1(\k) \implies \neg J_2(\k)]$.
\end{fact}
\begin{proof}
Assume otherwise,
and suppose that $\J$ witnesses $J_2(\k)$. Let
$\l ={(\k_\go)}_j$, and
fix an arbitrary cardinal $\gd$ such that
$\gd > \gl > \gk$.
Using $J_1(\k)$,
let $\I$ be an elementary embedding such that
$i(\k)>\d$. It is then the case that
$i(\gd) > i(\gl) > i(\gk) > \gd > \gl > \gk$.
%Let $\l_1 = \sup \la i ^n(\k) : n \in \go \ra$.
Since $V \models {\rm DC}_\l$,
by elementarity,
we have that $V \models {\rm DC}_{i(\l)}$.
%Iterating this, and using Fact \ref{f3},
%we get that $V \models {\rm DC}_{\l_1}$.
Hence, as $\gd < i(\gl)$,
$V \models {\rm DC}_\gd$.
%Now it is clear that in this way, we can show that
Since $\gd$ was arbitrary,
it follows that we can show that
$V \models \forall \d
[{\rm DC}_\d]$, but this, by virtue of Fact \ref{f2},
is equivalent to ${\rm AC}$.
Thus, we get that $V \models {\rm AC}$,
which is a contradiction.
This completes the proof of Fact \ref{f10}.
\end{proof}
Nevertheless, by an unpublished result
of Woodin (which is quoted in
\cite{C} as Theorem 1.1(2)),
if $V \models {\rm ZF}$ and
$j : V \to V$ witnesses $J_1(\gk)$,
then there is a forcing extension
$V^\FP$ and an elementary embedding
$i : V^\FP \to V^\FP$ witnessing $J_2(\gk)$.
This means that our ordering of
the $J$ hierarchy is justified,
because of Woodin's result, and the fact that
$J_2$ directly implies $J_3$.
Suppose now $V \models {\rm ZF}$, and
$j : V \to V$ is an elementary embedding
witnessing $J_2(\gk)$. If
$\gl = {(\gk_\go)}_j$,
%then as in Theorem 3.13 of \cite{Co},
then by elementarity and the fact that
$V_\gk \models {\rm ZF}$,
$V_\gl \models {\rm ZF}$.
Hence, by Corollary \ref{c1},
since $V_\gl$ is well-orderable,
$V_\gl \models {\rm AC}$.
Therefore, by Theorem 3.13 of
\cite{C}, in $V_\gl$, for
every $n \in \go$, there is
an $n$ huge cardinal (and in
fact, for every $n \in \go$,
there is a super $n$ huge
cardinal, where this is as in
Definition 2.1 of \cite{C}).
This shows that $J_2$ is
quite strong.\footnote{In fact,
the methods of Theorem 3.13 of
\cite{C} tell us that the structure
$\la V_\l, \in,
j\rest V_\l \ra$ is a model for ${\rm ZFC}+ {\rm WA}$,
where WA is the Wholeness Axiom of
\cite{C}. This allows us to infer the
existence of super $n$ huge cardinals
for every $n \in \go$.
For further details,
including the definition of the
Wholeness Axiom and a discussion
of its strength, readers are
urged to consult \cite{C}.}
We conclude this paper with
a discussion of the optimality
of $J_2$ and one of its
consequences. First, we address
the question of
how much choice is consistent with $\J$.
In particular, does $J_2$
admit the optimal amount of
choice? The next fact
shows that indeed the amount of choice in $J_2$
is best possible.
\begin{fact}\label{f12}
$J_3(\gk) \implies \neg AC_{\k_\go^+}$.
\end{fact}
\begin{corollary}
The amount of choice in $J_2$ is best possible.
\end{corollary}
\newenvironment{proof1}{\noindent{\bf
Proof of Fact \ref{f12}:}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\begin{proof1}
It is a celebrated result of ${\rm ZFC}$,
proven by Solovay,
that for every regular cardinal $\gd$,
any stationary subset of $\gd$ can be
partitioned into $\gd$
many disjoint stationary subsets.
We refer readers to \cite{Ku2}, Theorem 6.11,
page 79, for a proof of the
case when $\gd$ is a
successor cardinal.
(The general case is given as
Theorem 16.9 in \cite{Ka}.)
A careful examination of this proof shows that if
$\gd$ is a successor cardinal and
${\rm ZF}+{\rm AC}_{\gd}$ holds,
then any stationary subset of $\gd$ can be
partitioned into $\gd$
many disjoint stationary subsets.
However,
an argument of Woodin
(see page 320 of \cite{Ka}) shows that if
$V \models {\rm ZF}$ and
$j : V \to V$ witnesses $J_3(\gk)$, then
for $\gl = \gk_\go^+$,
there is a stationary subset of $\gl$
which cannot be partitioned into
$\gl$ many disjoint stationary subsets.
(Woodin's argument, which
makes no use of any amount of the
Axiom of Choice other than the
fact implied by
${\rm AC}_{\gk_\go}$ that $\gl$ is regular
and the fact implied by
${\rm AC}_\gk$ that the union of
$\gk$ many nonstationary subsets
of $\gl$ is nonstationary,
actually shows that this stationary
subset of $\gl$ cannot be
partitioned into $\gk$ many
disjoint stationary subsets.)
Thus, $J_3(\gk)$ and ${\rm AC}_\gl$
are incompatible with one another.
This completes the proof of Fact \ref{f12}.
\end{proof1}
Finally, we recall that by
Corollary \ref{c1}, if
$j : V \to V$ witnesses $J_2(\gk)$,
then for $\gl = \gk_\go$,
$V_\gl$ can be well-ordered.
Our last results show that this,
too, is optimal.
Specifically, we have the following.
\begin{fact}\label{f13}
If $j : V \to V$ witnesses
$J_3(\gk)$, then for $\gl = \gk_\go$,
$V_{\gl + 1}$ is not well-orderable.
\end{fact}
\begin{proof}
If $V_{\gl + 1}$ were well-orderable,
then since
$\wp(\gl) \subseteq V_{\gl + 1}$,
$\wp(\gl)$ and any of its
subsets, such as
${[\gl]}^\go$ and ${[\gl]}^\gl$,
would be well-orderable
as well.
The fact that
${[\gl]}^\go$ and ${[\gl]}^\gl$
are well-orderable now
allows us to carry out Kunen's
original proof (see pages 319--320 of
\cite{Ka}) to show that $J_3(\gk)$
is false.
This completes the proof
of Fact \ref{f13}.
\end{proof}
The proof just given
also shows the following.
\begin{fact}\label{f14}
If $j : V \to V$ witnesses
$J_3(\gk)$, then for $\gl = \gk_\go$,
$\wp(\gl)$ is not well-orderable.
\end{fact}
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\end{document}