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\begin{document}
\date{August 23, 1997\\(revised December 26, 1998)}
\title {The Calculus of Partition
Sequences,\\Changing Cofinalities,\\and a\\Question of
Woodin\footnote{1991 AMS Subject Classification: 03E15, 03E35, 03E60}}
\author{Arthur W. Apter\footnote{Research partially supported by
PSC--CUNY grant 665337}\\{\footnotesize
Dept. of Mathematics}\\{\footnotesize Baruch College}
\\{\footnotesize New York, NY}\\{\footnotesize
awabb@@cunyvm.cuny.edu}\\James M. Henle\\{\footnotesize
Dept. of Mathematics}\\{\footnotesize Smith College}
\\{\footnotesize Northampton, MA}\\{\footnotesize
jhenle@@smith.edu}\\Stephen C. Jackson \\{\footnotesize
Dept. of Mathematics}\\{\footnotesize University of
North Texas}
\\{\footnotesize Denton, TX}\\{\footnotesize jackson@@jove.acs.unt.edu}}
\maketitle
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\begin{abstract} We study in this paper polarized infinite exponent
partition relations. We apply our results to constructing a model for
the theory ``ZF$+$DC$+\A $ is the only regular, uncountable cardinal
$\le\o_{\A+1}$.'' This gives a partial answer to a question of
Woodin.\end{abstract}
\newpage
In 1994, J. Steel proved what had long been
suspected: that assuming AD, the regular cardinals of $L(\R)$ below
$\T$ are all measurable, where $\T$ is the
least ordinal onto which the real numbers $\R$ cannot be mapped \cite{ST}. This
remarkable theorem motivated
the work of \cite{A}, which showed essentially that the cofinality of any
regular cardinal below $\T$ can be changed to $\o$ without perturbing
cardinal structure. Specifically, the following theorem was proved.
\thm Assume $\V'\sat${\em AD}, and let $\V=L(\R)^{\V'}$. Then for any subset
$A$
of the regular, uncountable cardinals below $\T^\V$, there is a partial
ordering $\P\in\V$ and a symmetric inner model $\N$ of {\em ZF} such that \ret
(1) $\V\subseteq\N\subseteq\V^\P $,\ret
(2) $\N$ and $\V$ contain the same
cardinals,\ret
(3) $\T^\N=\T^\V$, \ret
(4) for all $\k$, $\cof^{\N}(\k)=\o$ if $\cof^\V(\k )\in A$ and
$\cof^\N(\k )=\cof^\V(\k )$ otherwise, and \ret
(5) the measurable cardinals not in $A$ remain measurable in $\N$.
\ethm
What was left unanswered is whether similar facts can
be proven if the cofinality of $\k<\T$ is to be made uncountable.
Consideration of this question was the genesis of this paper. The
principal tool, infinite--exponent partition sequences, proved as
interesting as its application.
We begin, in the first three sections, with background material on
infinite--exponent partition properties of cardinals, their
abundance in models of AD, and their use in ``Magidor--like'' forcing
to change cofinalities.
The second three sections deal with
infinite--exponent partition properties of cardinal sequences. With
appropriate notation, the basic theory of partition sequences and the
associated Magidor--like forcing exactly parallels the cardinal theory.
In ``Finite Support,'' we apply the theory to show the extent to which
possibly uncountable cofinal sequences
can be added if we sacrifice DC. In ``Countable Support,'' we preserve
DC.
In general, uncountable cofinalities present difficulties. In \cite{G}, M.
Gitik
established the consistency relative to an almost huge cardinal of the
theory, ``ZF $+$ $\A$ is the only regular, uncountable cardinal.'' In
his model, all cardinals above $\A$ are given cofinality $\o$ and
consequently DC fails (in fact AC$_\o$ fails). DC is most desirable in
analysis. A. Kechris proved that DC holds in $L(\R )$ assuming AD \cite{ke2}.
W. H. Woodin has asked
if the theory, ``ZF $+$ DC $+$ $\A$ is the only regular, uncountable
cardinal'' is consistent.
``Including Ultrapowers'' and ``Including Ultraproducts'' discuss how
ultrapowers and ultraproducts of cardinals in the most powerful
partition sequences
can be included in
new but less powerful partition sequences.
\newpage
The final section applies our results to obtaining a partial answer to
Woodin's question. In particular, relative to Con(AD), we construct a
model for the theory ``ZF$+$DC$+\A$ is the only regular,
uncountable cardinal $\le\omega_{\A +1}$.''
\vskip.2in
\centerline{\begin{tabular}{lcl}
Partition Cardinals&..........&\pageref{one}\\
AD and Partition Cardinals&..........&\pageref{two}\\
Magidor--like Forcing&..........&\pageref{three}\\
Partition Sequences&..........&\pageref{four}\\
AD and Partition Sequences&..........&\pageref{five}\\
Polarized Magidor--like Forcing&..........&\pageref{six}\\
Finite Support&..........&\pageref{seven}\\
Countable Support&..........&\pageref{eight}\\
Including Ultrapowers&..........&\pageref{nine}\\
Including Ultraproducts&..........&\pageref{ten}\\
Applications&..........&\pageref{eleven}\end{tabular}}
\vskip.3in
Some remarks on notation: When forcing, we will use $\V[G]$ and $\V^\P$
interchangeably to
indicate the generic extension obtained when $\P$ is our partial
ordering, $G$, the generic set. If $p\in\P$ is a condition and $\phi$ is a
formula in the
forcing language with respect to $\P$, then $p\Vert\phi$ will mean
that $p$ decides $\phi$. We adopt the NESTS convention for the
forcing partial order, that is, if $p,q\in\P$ and $q$ extends $p$, we
will write $q\for p$.
We will refer to and use certain weak versions of the Axiom of Choice
throughout the course of this paper. AC$_\o$, the Principle of
Countable Choice, says that if $\< X_n:n<\o\> $ is an $\o$--sequence of
non--empty sets then $\prod_{n<\o}X_n$ is non--empty. DC, the Principle of
Dependent Choices, says that if $R\subseteq X\times X$
satisfies $\forall x \exists y [\< x,y\> \in R]$, $X\neq\emptyset$,
then there is a sequence $\{ x_n\}_{n<\o} $ of elements of $X$ with $\<
x_n,x_{n+1}\> \in R$,
for each $n<\o$.
If
$s$ is a set of ordinals, we use $\overline{s}$ to indicate the order--type of
$s$. If $x$ and $y$ are sequences, $x\cat y$ denotes their concatenation.
General references for results in AD include \cite{Mo2} and \cite{St1}.
\sec {Partition Cardinals}\label{one}
We assume the reader is familiar with infinite--exponent partition
cardinals, but we include the definitions in
this section for convenience. \cite{H1}, \cite{H2}, \cite{H3},
\cite{Kl1}, \cite{Kl2}, and \cite{Kl3} can be
consulted
for matters left unexplained here.
For cardinals $\k$ and $\g$, $[\k]^\g$ is the set of all subsets of
$\k$ of order--type $\g$. We warn readers that we will identify elements of
$[\k]^\g$ with
their increasing enumerations. In practice, this will cause no
confusion; the meaning will follow from the context. We define $[p]^\g$ as
$\{q\subseteq p:\overline{q}=\g\}$, for
$p\in[\k]^\g$, for example,
and it
is clear we are treating $p$ as a set. Similarly, we define
$\po p$ as
\[\{\bu{n<\o}p(\a+n):\a<\g\},\]
i.e., the set of
successive $\o$--sups of elements of $p$, and it is clear we are treating $p$ as
a sequence.
\ef For ordinals $\k,\g,\d$,
$$\k\rightarrow(\k)^\g_\d$$
means that for all partitions $f:[\k]^\g\rightarrow\d$, there is a set
$X\in[\k]^\k$ such that $f$ is constant on $[X]^\g$. $X$ is called
{\em homogeneous} for $f$. If $\d=2$, the subscript is omitted,
and we write simply $\k\rightarrow(\k)^\g$.
For ordinals $\k,\g,\d$, the relation,
$\k\rightarrow(\k)^{<\g}_\d$
means that for all partitions \ret$f:\bu{\b<\g}[\k]^\b\rightarrow\d$, there is a
set
$X\in[\k]^\k$ such that $f$ is constant on $[X]^\b$, for all $\b<\g$.\eef
A few elementary results:
\fact \label{lowerexp} If $\b<\g$, then $\k\rightarrow(\k)^\g_\d$ implies
$\k\rightarrow(\k)^\b_\d$.\efact
\fact \label{ \underset{\beta'<\beta}{\sup}F(\beta')$.
``$\kappa$ strong'' is
equivalent to the statement: for every partition $\cP:[\kappa]^\kappa
\rightarrow \{ 0,1\}$, there is an $i \in \{ 0,1\}$ and a c.u.b. $C\subseteq
\kappa$ such that $\cP(F)=i$ for all $F:\kappa\rightarrow C$ of
the correct type. The equivalence of these two definitions is a
straightforward partition argument left to the reader.
In \rdef{reasonable} below, we isolate the ``minimal hypothesis'' on a
cardinal $\k$ from which one can carry out Martin's proof of the strong
partition relation.
\ef \label{reasonable}
A regular cardinal $\kappa$ is said to be {\em reasonable} if there is
a non--selfdual pointclass $\bG$ closed under
$\exists^{\omega^\omega}$ and a map $\phi$ with
domain $\omega^\omega$ satisfying:
\begin{enumerate}
\item $\forall x \phi(x) \subseteq \kappa \times \kappa$.
\item For every function $F:\kappa\rightarrow\kappa$ $\exists x
\ \phi(x)=F$.
\item $\forall \beta,\gamma <\kappa \ R_{\beta,\gamma} \in
\bD \doteq \bG \cap \bGd$, where
$x \in R_{\beta,\gamma} \leftrightarrow \phi(x)(\beta,\gamma)
\wedge \forall \gamma'<\kappa\ (\phi(x)(\beta,\gamma')\rightarrow
\gamma'=\gamma)$.
\item Suppose $\beta<\kappa$,
$A \in \exists^{\omega^\omega}\bD$,
and $A \subseteq R_{\beta} \doteq \{ x: \exists \gamma <\kappa
\ R_{\beta,\gamma}(x) \}$. Then $\exists \gamma_0 <\kappa$ $\forall x \in A$
$\exists \gamma <\gamma_0$ $R_{\beta,\gamma}(x)$.
\end{enumerate}
\eef
For $\kappa$ reasonable, we will use the notation $R_\beta$,
$R_{\beta,\gamma}$ as above (of course, this depends on the coding function
$\phi$). Also, if $x \in R_\beta$, we let $\phi(x)(\beta)$ denote the
unique $\gamma$ such that $\phi(x)(\beta,\gamma)$.
We present a sketch of Martin's proof, assuming AD$+$DC, that such cardinals
have the
strong partition property. We will show later that $\bG$
is necessarily closed under finite (in fact countable) unions
and intersections, and $\bD$ is closed under
$<\kappa$ unions and intersections; we borrow these facts for the
following proof and \rprop{thmpol}. One more useful general fact
which we use below, and later, is that if $\bG$ is a non--selfdual
pointclass with $\bG=\exists^{\omega^\omega} \bD$ (where
$\bD=\bG \cap \bGd$), then $\bG$ is closed under finite unions and
intersections. To see this, note that if $\bG=\exists^{\omega^\omega}
\bL$ where $\bL$ is non--selfdual and closed under
$\forall^{\omega^\omega}$, then it is easy to see directly that $\bG$
is closed under countable unions and intersections. The only remaining
case (c.f. \cite{St1}) is when $\bG$ is the base of a type I hierarchy.
That is, $\bG=\bigcup_\omega \bL$ for some selfdual $\bL$
closed under quantifiers (and where $\cof(o(\bL))=\omega$).
In this case, easily $\bG$ is closed under finite intersections.
\prop [Martin] \label{marthm}
Every reasonable $\kappa$ has the strong partition property.
\eprop
Proof:
Fix a partition $\cP:[\kappa]^\kappa \rightarrow \{ 0,1 \}$.
Play the integer game where I plays out $x \in \omega^\omega$, II plays
out $y \in \omega^\omega$. If there is a least ordinal $\beta<\kappa$
such that $x \notin R_\beta$ or $y \notin R_\beta$, then II wins provided
$x \notin R_\beta$. Otherwise, let $f_x,f_y:\kappa\rightarrow\kappa$
be the functions they determine. Define $f_{x,y}:\kappa\rightarrow\kappa$
by $f_{x,y}(\beta)=\underset{\beta'<\omega \cdot (\beta+1)} {\sup}
\max(f_x(\beta'),f_y(\beta'))$. II then wins iff $\cP(f_{x,y})=1$.
Assume without loss of generality that II has a winning strategy
$\tau$. Define $x \in S_{\beta,\gamma} \leftrightarrow
\forall \beta' \leq \beta$ $\exists \gamma' \leq \gamma \
x \in R_{\beta,\gamma}$. Thus, $S_{\beta,\gamma} \in \bD$.
An easy computation shows that $\forall \beta,\gamma <
\kappa\ \tau[S_{\beta,\gamma}] \in \exists^{\omega^\omega}
\bD$ (take cases as to
whether $\exists^{\omega^\omega} \bD=
{\bG}$, and use the closure of ${\bG}$
under $\wedge$ in this case). Now, $\forall y \in \tau[S_{\beta,\gamma}]$
$y \in R_\beta$. Thus, $\theta(\beta,\gamma) \doteq
{\sup} \{ \phi(x)(\beta):x \in \tau[S_{\beta,\gamma}] \} < \kappa$.
Let $C\subseteq\kappa$ be the set of points closed under $\theta$, and
$C' \subseteq C$ the set of limit points of C.
Suppose $F:\kappa\rightarrow C'$ is of the correct type. We show that
$\cP(F)=1$. Let $x$ be such that $\phi(x)$ determines a function $f_x:
\kappa\rightarrow C$ such that $F(\beta)=\underset{\beta'<\omega \cdot
(\beta+1)}{\sup} f_x(\beta')$.
We may assume $f_x(\beta)\geq \beta$ for all $\beta$.
Let $y=\tau(x)$. Easily $\phi(y)$
determines a function $f_y:\kappa\rightarrow\kappa$ and
$f_y(\beta')\leq f_x(\beta+1)$ for all $\beta$. Thus,
$F=f_{x,y}$, so $\cP(F)=1$.
\ppf
We now introduce a slight strengthening of the notion of reasonableness.
\ef \label{vreasonable}
A regular cardinal $\kappa$ is {\em very reasonable} if there are
${\bG}$, $\phi$ witnessing $\kappa$ is reasonable,
and for all $\beta < \kappa$ there is an
$\exists^{\omega^\omega} \bD$
relation $B \subseteq \omega^\omega \times \omega^\omega$ such that for
$x,y \in R_{\beta}$, $\phi(x)(\beta) < \phi(y)(\beta)
\leftrightarrow B(x,y)$.
\eef
We note again that in all cases where the strong partition relation on
$\kappa$ has been shown to hold, $\kappa$ has been shown to be
very reasonable.
\sec {AD and Partition Cardinals}\label{two}
Our focus in this paper is on models of AD plus
$\V=L(\R)$. As noted, this implies DC as well. Under these
assumptions, there are many partition
cardinals. Conceivably, all regular cardinals below $\T$ possess some
sort of infinite--exponent partition property.
Martin showed that the first projective ordinal, $\de 11= \omega_1$, has the
strong partition property. Martin and Solovay also computed
$\de 13=\omega_{\omega +1}$ and showed that the only regular cardinal
strictly between $\de 11$ and $\de 13$ is $\omega_2$, which is the
ultrapower of $\de 11$ by the unique normal measure on $\de 11$ (given
by the c.u.b. filter).
Jackson (\cite{jad15}, \cite{veryfine}) later proved the strong partition
relation
on $\de 13$, computed $\de 15=\omega_{\omega^{\omega^\omega}+1}$,
and showed that the only regular cardinals strictly between
$\de 13$ and $\de 15$ are the ultrapowers of $\de 13$ by the three
normal measures on $\de 13$ (generated by the c.u.b. filter and
points of cofinality $\omega$, $\omega_1$, or $\omega_2$ respectively).
These three regular cardinals turn out to be $\de 14=\omega_{\omega+2}$,
$\omega_{\omega \cdot 2 +1}$, and $\omega_{\omega^\omega+1}$. (In
\cite{jakh} some additional results along these lines are obtained. For
example, the cofinality of all the successor cardinals between
$\de 13$ and $\de 15$ is determined.)
In \cite{jacab} it is shown how to generalize the main ideas of
\cite{jad15} (the theory of ``descriptions'') to the general projective
level. One thus gets the strong partition property for all the
$\de 1{2n+1}$, and likewise every regular cardinal strictly between
$\de 1{2n+1}$ and $\de 1{2n+3}$ is an ultrapower of $\de 1{2n+1}$
by one of the ($2^{n+1}-1$ many) normal measures on $\de 1{2n+1}$.
Although the full details of the general projective case
have not yet appeared, they are basically the proofs of \cite{jad15}
augmented with the general theory of descriptions of \cite{jacab}.
In fact, the projective ordinal analysis extends with essentially no
modifications to the transfinite $\de 1{\alpha}$, for $\alpha < \omega_1$.
Thus, all of the ``odd'' $\de 1{\alpha}$ (as usual, limit ordinals
are regarded as even) have the strong partition
property, and every regular $\kappa < \sup \{\de 1{\alpha} \colon
\alpha < \omega_1 \}= \omega_{\omega_1}$ is the ultrapower of an odd
$\de 1{\alpha}$ by a normal measure on it.
Past $\omega_{\omega_1}$, the arguments of \cite{jad15}, \cite{jacab}
no longer suffice for the $\de 1{\alpha}$ analysis. Although somewhat
tentative (and not written up), unpublished work appears to show that
this generalized projective analysis (the ``very fine structure of
$\lr$'') can be extended to first inaccessible cardinal in $\lr$.
Up to this point, it appears that every regular Suslin cardinal has the
strong partition property, and that every regular cardinal is an
ultraproduct of regular Suslin cardinals by a normal measure.
Although these results must be regarded as tentative, they would
nevertheless, combined with Corollary~\ref{corinac}, allow us to
extend the results of the final section up to the first inacessible.
Some of the problems arising in attempting to extend the $\lr$ analysis
further are discussed in \cite{jahand}.
We conjecture (assuming AD$+\V=\lr$), that every regular Suslin cardinal
$\kappa<\Theta$ has the strong partition property, and every regular
$\lambda <\Theta$ is an ultraproduct of regular Suslin cardinals by
a normal measure. Granting this, the results of the final section can be
extended to $\Theta$,
although Theorem~\ref{mainthm}
falls short of showing that every (increasing, discontinuous) sequence
of very reasonable cardinals (see section 5) has the strong polarized partition
property.
\sec {Magidor--like Forcing}\label{three}
Assuming that $\k \rightarrow (\k)^{<\gamma}$
and that $\gamma$ is a regular, uncountable cardinal, ``Magidor--like'' forcing
changes the cofinality
of $\k$ to $\gamma$ without adding any bounded subsets to $\k$
(thereby preserving the fact that $\k$ is a cardinal). No use is made
here of
AD$+$DC. The conditions
are defined as follows.
\ef We define the set $\P_{\g,\k}$ by
$$\P_{\g,\k}=\{\< s,x\> : s\in [\k ]^{<\g},x\in
[\k ]^\k, \cup s<\cap x\}.$$
The partial ordering for $\P_{\g,\k}$ is: $\< s',x'\> \for \< s,x\>
$ iff $s\subseteq s'$, $\< x'\> \subseteq \< x\> $, and $s'\minus
s=\po t$ for some $t\in [x]^{<\g}$. We use $\< x\> $ to denote
$\{\po q:q\in[x]^\k\}$. For $p\in\P_{\g,\k}$, we
denote the coordinates of $p$ by $p_0$ and $p_1$, i.e., $p=\<
p_0,p_1\> $. \eef
The sets $\< x\> $ generate a filter on $[\k]^\k$. With the strong
partition property, this is a $\k$--complete
ultrafilter (\cite{H3}). Note that any two conditions $p,p'$ are compatible if
$p_0=p'_0$.
Given $\k \rightarrow (\k)^{<\gamma}$,
and $\gamma$, a regular, uncountable cardinal, Propositions 1.1, 1.3, and 1.5 of
\cite{H2} show the following:
\fact \label{mshrink} If $\phi$ is a formula in the forcing language with
respect to $\P_{\g,\k}$ and $p\in\P_{\g,\k}$, then there is an
$x\in[\k]^\k$ such that $\< p_0,x\> \Vert\phi$.
\efact
\fact \label{mnonewsub} $\V^{\P_{\g,\k}}$ and $\V$ have the same bounded subsets
of $\k$. \efact
\fact \label{mcofinal} The cofinalities
of cardinals in $\V^{\P_{\g,\k}}$ are the same as in $\V$, except that a
cardinal having cofinality $\k$ in $\V$ has cofinality $\g$ in
$\V^{\P_{\g,\k}}$. \efact
We will sometimes be interested in changing the cofinality of $\k$ to
$\o$ instead of to an uncountable regular cardinal.
$\k\rightarrow(\k)^{<\o\cdot\o}$ is sufficient to show that $\P_{\o\cdot\o ,\k}$
changes the cofinality of
$\k$ to $\o$ and also satisfies Propositions 1.1, 1.3, and 1.5 of \cite{H2}.
We can improve on \cite{H2}. First we add some notation,
in light of \rfact{mshrink}. For $s\in[\k]^{<\g}$, we will write
$s\efor\phi$ iff for some
$x\in[\k]^\k$, $\< s,x\> \for \phi$. Read this as ``$s$ insinuates
$\phi$''. Note that by compatibility, it
is impossible for $s$ to insinuate contradictory statements.
\prop \label{kpreserve} If $\k$ satisfies $\k\rightarrow(\k)^\k$, then forcing
with ${\mathcal
P}_{\gamma,\kappa}$, $\g<\k$, preserves all cardinals.
\eprop
Proof: By \rfact{lowerexp} and \rfact{ \underset{\beta<\alpha}
{\sup} k(\beta)$ for all $\alpha<\mu$. If $\lambda=
\underset{\alpha<\mu}{\sup}k(\alpha)$, and $F:\lambda\rightarrow
\lambda$, we say $F$ is a {\em block} function if $\forall
\alpha<\mu$ $\forall \beta \in [\underset{\alpha'<\alpha}{\sup}
k(\alpha'),k(\alpha))$ $F(\beta)\in [\underset{\alpha'<\alpha}{\sup}
k(\alpha'),k(\alpha))$.
We say $C \subseteq \lambda$ is {\em block c.u.b.} if $\forall \alpha
<\mu$ $C \cap k(\alpha)$ is c.u.b. in $k(\alpha)$.
Such a sequence
has the {\em strong polarized partition property} if for every
partition $\cP$ of the block functions on $\lambda$ into
$\{ 0,1\}$, there is an $S \subseteq \lambda$ such that
$\forall \alpha<\lambda \ |S \cap k(\alpha)|=k(\alpha)$ which
is homogeneous for $\cP$, that is, $\exists i \in \{ 0,1\}
\forall$ block functions $F:\lambda\rightarrow S$ $\cP(F)=i$. \eef
Easily, this property on $k$ is
equivalent to the c.u.b version: for every
partition $\cP$ of the block functions on $\lambda$ into
$\{ 0,1\}$, there is a block c.u.b $C\subseteq \lambda$ and an $i \in
\{ 0,1\}$ such that $\cP(F)=i$ for all block functions
$F:\lambda\rightarrow C$ of the correct type.
\sec {AD and Partition Sequences}\label{five}
Throughout this section we assume AD$+$DC. We will indicate when we are
assuming $\V=L(\R)$ as well.
\ef \label{vvreasonable}
A regular cardinal $\kappa$ is {\em very reasonable} if there are
${\bG}$, $\phi$ witnessing $\kappa$ is reasonable,
and for all $\beta < \kappa$ there is an
$\exists^{\omega^\omega} \bD$
relation $B \subseteq \omega^\omega \times \omega^\omega$ such that for
$x,y \in R_{\beta}$, $\phi(x)(\beta) < \phi(y)(\beta)
\leftrightarrow B(x,y)$.
\eef
We note that in all cases where the strong partition relation on
$\kappa$ has been shown to hold, $\kappa$ has been shown to be
very reasonable.
The main result of this section, Theorem~\ref{mainthm}, implies
(c.f. Corollary~\ref{corproj}) that any
sequence of very reasonable cardinals of length less than
$\underset{n}{\sup} \de 1n$ has the desired strong polarized partition
property (in fact Corollary~\ref{corinac} extends this
to longer length sequences).
This has the effect of ``decoupling'' the arguments of this section
with those of the (generalized) projective hierarchy analysis.
Thus, the arguments of this section do not require knowledge of
the projective hierarchy analysis of \cite{jad15} or \cite{jacab}.
For the purposes of this paper, the reader may ``abstractly'' assume that
all of the odd $\de 1{\alpha}$, $\alpha < \omega_1$, are very reasonable, and
that all of the regular cardinals below $\omega_{\omega_1}$ are ultrapowers
of these by normal measures.
One of the main ideas in the arguments of this section
involves the Kechris--Woodin
theory of generic codes (see \cite{kewo}) as well as a category
argument involving the generic codes. Similar ideas were
used in \cite{jasquare} to establish the $\bd^2_1$--
supercompactness of the projective ordinals.
We present now a result giving sufficient conditions for a
sequence $k$ to have the strong polarized
partition property.
It is similar in spirit to the ``abstract form'' of Martin's
Proposition \ref{marthm} earlier.
\ef \label{reasonableseq}
Let $k$ be an increasing, discontinuous $\mu$--sequence
of regular cardinals. We say the sequence is {\em reasonable} if there is a
sequence of non--selfdual pointclasses
$\{ \bG_\alpha\}_{\alpha<\mu}$ each closed under
$\exists^{\omega^\omega}$, and a function
$\phi$ with domain $\omega^\omega$ satisfying:
\begin{enumerate}
\item For all $x \in \omega^\omega$, $\phi(x)\subseteq \lambda \times \lambda
$, where $\lambda = \underset{\alpha<\mu}{\sup}k(\alpha)$. \label{h1}
\item For every block function $F:\lambda\rightarrow\lambda$, $\exists
x \in \omega^\omega\ \phi(x)=F$. \label{h2}
\item $\forall \alpha<\mu$ $R_\alpha \in \bD_{\alpha+1}$,
where $x \in R_\alpha \leftrightarrow \forall \alpha' \leq \alpha$
$\forall \beta \in [\underset{\alpha''<\alpha'}{\sup}k(\alpha''),
k(\alpha'))$ $\exists \gamma \in
[\underset{\alpha''<\alpha'}{\sup}k(\alpha''),
k(\alpha'))$ $[\phi(x)(\beta,\gamma) \wedge \forall \gamma'<\lambda\
\phi(x)(\beta,\gamma')\rightarrow \gamma'=\gamma ]$. Thus, $x \in
R_\alpha$ iff $\phi(x)$ is a block function through $k(\alpha)$. \label{h3}
\item $\forall \alpha <\mu$ $\forall \beta,\gamma \in
[\underset{\alpha'<\alpha}{\sup}k(\alpha'),k(\alpha))$
$R_{\alpha,\beta,\gamma} \in \bD_{\alpha}$, where
$x \in R_{\alpha,\beta,\gamma}\leftrightarrow
\phi(x)(\beta,\gamma) \wedge \forall \gamma' from Lemma~2.4.1 of \cite{kesost}. This includes the case where
${\bG}=\exists^{\omega^\omega}{\bG}'$, for
some non--selfdual ${\bG}'$ closed under $\forall^{\omega^
\omega}$. The only remaining case (c.f. the analysis at the end of
\cite{St1}) is when ${\bG}$ is at the base of a type I
hierarchy. That is, $\bD={\bG}
\cap \bGd$ is closed under
$\exists^{\omega^\omega},\forall^{\omega^\omega}$, and
${\bG}$ is the collection of countable unions of sets in
$\bD$.
We sketch the proof in this case.
Towards a contradiction,
let $\kappa$ be the least
cardinal so that $\bigcup_\kappa {\bG}
\nsubseteq {\bG}$. Thus, $\kappa$ is regular and
by the Coding Lemma $\kappa\geq o(\bD)$, the Wadge
ordinal of $\bD$ (which is also the supremum of the
$\bD$ pre--well--orderings, see \cite{kesost}). Also. $\kappa \neq o(\bD)$
as $\cof(o(\bD))=\omega$. Thus, $\kappa > o(\bD)$.
Let ${\bG}_1=\exists^{\omega^\omega}
\bGd$. By Wadge,
$\bGd \subseteq \bigcup_\kappa {\bG}$, and so $\bG_1 \subseteq
\bigcup_\kappa {\bG}$. Using the regularity of $\kappa$,
let $\< A_\alpha: \alpha<\kappa\> $ be a strictly increasing $\kappa$ sequence
of sets in ${\bG}$ whose union $A$ is in
$\bGd$. Let
$B=\{ x: S_x \subseteq A\}$, where $S$ is universal
${\boldsymbol{\Sigma} }^1_1$. $B \in \bGd$ as
$\bGd$ is closed under $\forall^
{\omega^\omega}$ and $\vee$. Let $B=\bigcup_{\alpha<\kappa}
B_\alpha$, where $B_\alpha \in {\bG}$. If we replace
$A_\alpha$ by $\{y: \exists x \in B_\alpha \ (y \in S_x) \}$,
then the $A_\alpha$ form a ${\boldsymbol{\Sigma} }^1_1$--bounded union of
$\bG$ sets, that is, every ${\boldsymbol{\Sigma} }^1_1$ subset of $A$ is
contained in some
$A_\alpha$. Let $C$ be a universal ${\bG}$ set. Play the
game where I plays $x$, II plays $y,z$, and II wins iff
$x \in A \rightarrow \exists \alpha > |x| (C_y=A_\alpha \wedge
z \in A_\alpha - \cup_{\beta<\alpha}A_\beta)$, where $|x|$ is the least
$\alpha<\kappa$ such that $x \in A_\alpha$. By
${\boldsymbol{\Sigma} }^1_1$--boundedness, II wins, say by $\tau$. Define
$x \prec y$ iff $x,y \in A \wedge \tau(y)_1 \notin C_{\tau(x)_0}$.
Thus, $\prec$ is a $\bGd$
pre--well--ordering of length $\kappa$. By the Coding Lemma, then,
$\bigcup_\kappa {\bG} \subseteq {\bG_1}$,
and hence $\bigcup_\kappa {\bG}=
{\bG_1}$. Now, ${\boldsymbol{\Delta}_1}=
{\bG_1}\cap
\bGd_1$ is clearly also not
closed under $\kappa$ unions, and any $\kappa$ union of sets in
${\boldsymbol{\Delta}_1}$ is in ${\bG}_1$. Thus,
${\bG_1}=\bigcup_{\kappa'}{\boldsymbol{\Delta}_1}$,
where $\kappa' \leq \kappa$ is least such that ${\boldsymbol{\Delta}_1}$
is not closed under $\kappa'$ unions. This, however, shows
$\pwo({\bG_1})$, a contradiction, since by periodicity
$\pwo({\bGd}_1)$ (this last part is an argument of Martin).
The next several lemmas develop the abstract theory of reasonable
pointclasses.
\lem \label{techlem}
Suppose $\kappa_1<\kappa_2$ are reasonable with corresponding pointclasses
${\bG_1},{\bG_2}$. Then
$\exists^{\omega^\omega}\bGd_1 \subseteq \bD_2$ $={\bG_2} \cap \bGd_2$.
\elem
Proof:
If not, then ${\bG_2}=\exists^{\omega^\omega}
\bGd_1$. Then either
${\bG}_1$ or ${\bG}_2$ has the pre--well--ordering property. Assume first
$\pwo({\bG_1})$. Thus, ${\bG}_1$
is closed under well--ordered unions. For $\gamma<\kappa_1$, consider
$R_{0,\gamma} =\{ x: \phi(x)(0)$ is defined and $\leq \gamma \}$. $\phi$
here refers to the coding for $\kappa_1$.
By reasonableness, each $R_{0,\gamma} \in \bD_1$, and
so $R=\cup_{\gamma<\kappa_1}R_{0,\gamma} \in {\bG}_1$.
This contradicts reasonableness if ${\bG}_1$
$=\exists^{\omega^\omega} \bD_1$. So assume
$\bD_1$ is closed under real quantifiers.
Note that ${\bG}_1$ is not closed under $\forall^
{\omega^\omega}$ by assumption. By $\pwo({\bG}_1)$,
an inspection of the hierarchy analysis (c.f. \cite{St1}) shows that
${\bG}_1$ is the base of a type I hierarchy, that is,
${\bG}_1= \bigcup_\omega \bD_1$.
Since $\cof (o(\bD))=\omega$, we must have
$o(\bD_1) \neq \kappa_1$. We can not have $\kappa_1<
o(\bD_1)$, as then (by the Coding Lemma and the fact that
cofinally many of the $R_{0,\gamma}$ lie in some fixed $\bL
\subsetneq \bD_1$)
$R \in \bD_1$, a contradiction to reasonableness.
So $o(\bD_1)<\kappa$. Since $\kappa$ is regular, there is a non--selfdual
$\bL \subseteq \bD_1$ closed under
$\exists^{\omega^\omega}$ with $\pwo(\bL)$, and such that $\kappa$
many of the $R_{0,\gamma}$ are in $\bL$. Since $\bL$ is closed
under well--ordered unions, $R\in \bL$ (note: the $R_{0,\gamma}$ form an
increasing sequence), a contradiction to reasonableness.
Assume now $\pwo({\bG}_2)$.
We may assume $\bD_2$ is closed
under real quantification, arguing as above. This, however, contradicts
${\bG}_2=\exists^{\omega^\omega} \bGd_1$.
\lpf
\lem \label{techlem2}
Let $\kappa$ be reasonable, with corresponding pointclass
${\bG}$, and map $\phi$. Then $\kappa$ is the
supremum of the $\bD$ pre--well--orderings of the
reals. Furthermore, ${\bG}$ is uniquely determined from
$\kappa$. Finally, $\pwo(\bGd)$ if
${\bG}$ is not closed under real quantification.
\elem
Proof:
Let $\bD={\bG}\cap \bGd$, and first assume $\bD$ is closed under
real quantification. Thus, $o(\bD)$ is the
supremum of the $\bD$ pre--well--orderings (c.f. \cite{kesost}).
We can not have $\kappa
o(\bD)$, then by regularity of $\kappa$,
$\kappa$ many of the $R_{0,\gamma}$ lie in some non--selfdual pointclass
$\bL \subseteq \bD$ closed under
$\exists^{\omega^\omega}$ with $\pwo(\bL)$. Then $R_0 \in
\bL \subseteq \bD$, violating reasonableness.
Thus, $\kappa=o(\bD)$. In particular, $o(
\bD)$ is regular, so ${\bG}$ can not
be the base of a type I or II hierarchy. If ${\bG}$
is not closed under real quantification, then from \cite{St1} we
have $\pwo(\bGd)$, as
${\bG}$ is closed under $\exists^{\omega^\omega}$ (\cite{St1} shows that
$\pwo$ falls on the side closed under
$\forall^{\omega^\omega}$ at this level).
Assume now ${\bG}=\exists^{\omega^\omega}
\bD$. We can not have $\pwo({\bG})$,
as then ${\bG}$ is closed under well--ordered unions,
so $R_0 \in \exists^{\omega^\omega}\bD$, violating
reasonableness. Thus, $\pwo(\bGd)$.
Let $\delta$ be the supremum of the lengths of the $\bD$
pre--well--orderings.
Suppose first that ${\bG}=\exists^{\omega^\omega}
{\bG}_{-}$ for some non-selfdual
${\bG}_{-}$ closed under $\forall^{\omega^\omega}$ (this
covers all cases except for ${\bG}= {\bG}
\vee \bGd_{-}$ where
${\bG}_{-}$ is closed under real quantification).
The Coding Lemma again shows $\delta \leq \kappa$.
Suppose towards a contradiction that $\delta<\kappa$. Let
${\bG}_1=\exists^{\omega^\omega}
{\bGd}$, so
${\bG}_1$ is closed under well--ordered unions.
By $\pwo(\bGd)$, $\bigcup_\delta {\bG} \neq {\bG}$, and
hence $\bigcup_\delta {\bG}={\bG}_1$.
In particular, $R_0= \cup_{\alpha<\delta} A_\alpha$, where each
$A_{\alpha} \in {\bG}$. One of the $A_\alpha$ must be
unbounded in $k(\alpha)$ (with respect to the sequence $R_{0,\gamma}$),
contradicting reasonableness.
In the remaining case, ${\bG}=
{\bG}_{-} \vee
{\bGd}_{-}$ where
${\bG}_{-}$ is closed under real quantification.
We will show this case can not occur. Without loss of
generality assume $\pwo({\bG}_{-})$.
First, the Coding Lemma easily shows that $\kappa \geq
\delta_{-}=$ the supremum of the $\bD_{-}$
pre--well--orderings. Also, we can not have $\kappa=\delta_{-}$. If so,
then by the Coding Lemma we could get a ${\bG}_{-}$
set $A \subseteq \cup_{\gamma<\kappa}R_{0,\gamma}$ which is
unbounded, that is, not contained in any $R_{0,\gamma}$.
This contradicts reasonableness. So assume $\kappa>\delta_{-}$.
Let ${\bG}_1=\exists^{\omega^\omega}
({\bG}_{-} \wedge \bGd_{-})$. From \cite{St1}
we have $\pwo(\bGd)$ and thus
$\pwo({\bG}_1)$. Hence, ${\bG}_1$ is
closed under well--ordered unions. In particular,
$R_0=\cup_{\gamma<\kappa}R_{0,\gamma}\in {\bG}_1$.
Also, we easily have ${\bG}_1=\bigcup_{\delta_{-}}
\bGd_{-}$. Write $R_0=\cup_{\alpha<\delta_{-}}A_\alpha$ where $A_\alpha \in
\bGd_{-}$. By the regularity
of $\kappa$, some $A_\alpha$ is unbounded in the $R_{0,\gamma}$ union.
Since $A_\alpha \in \bGd_{-} \subseteq \bD$, this contradicts reasonableness.
It remains to show that $\kappa$ determines ${\bG}$.
Suppose ${\bG}_1$, ${\bG}_2$ both
witness the reasonableness of $\kappa$, and suppose ${\bG}_1
\subseteq {\bG}_2$.
If ${\bG}_1$ is closed under countable intersections but not real
quantification,
then by standard arguments using $\pwo(\bGd)$ there is a
$\bGd_1$ pre--well--ordering of length
$\kappa=$ the supremum of the lengths of the $\bD_1$
pre--well--orderings. Since $\bGd_1 \subseteq
\bD_2$, we contradict $\kappa$ being the supremum of the
$\bD_2$ pre--well--orderings. In the remaining cases,
$\bD_1$ is closed under real quantification.
If $\bG_1$ is not closed under real quantifiers, then
by Lemma~\ref{techlem} (the proof still holds when $\kappa_1=\kappa_2$)
$\exists^{\omega^\omega} \bGd_1 \subseteq
\bD_2$. However, there is an
$\exists^{\omega^\omega} \bGd_1$
pre--well--ordering of length $\kappa$, a contradiction. Finally, if $\bG_1$ is
closed under real quantifiers, then there is a $\bG_1 \wedge
\bGd_1$ pre--well--ordering of length $\kappa$. Since
$\pwo(\exists^{\omega^\omega}(\bG_1 \wedge \bGd_1))$, we must have
$\exists^{\omega^\omega}(\bG_1 \wedge \bGd_1) \subseteq \bD_2$, a
contradiction.
\lpf
\lem \label{techlem3}
Let $\kappa$ be reasonable with corresponding pointclass
${\bG}$ and map $\phi$. Then ${\bG}$ is
closed under countable unions and intersections. Furthermore,
$\bD$ is closed under $<\kappa$ length unions
and intersections.
\elem
Proof:
If ${\bG}=\exists^{\omega^\omega}{\bG}_0$
for some ${\bG}_0$ closed under $\forall^{\omega^\omega}$,
easily ${\bG}$ is closed under countable intersections.
As observed in the previous lemma, for the remaining cases we
have $\bD$ closed under real quantification.
We must show $\bGd$ is closed under
countable unions. From Theorem 2.2 of \cite{St1}, it suffices to show
$\bGd$ is closed under finite
unions. Let ${\bG}'$ be the collection of
${\boldsymbol{\Sigma}}^1_1$--bounded unions of $\bD$
sets of length $\kappa=o(\bD)$.
Let ${\bG}''$
likewise be the collection of $\bD$--bounded unions.
Clearly ${\bG}'' \subseteq {\bG}'$.
The proof of Theorem 3.1 of \cite{St1} shows that ${\bG}'
=\bGd$. By reasonableness,
the $R_{0,\gamma}$, $\gamma<\kappa$, form a $\bD$--bounded
union of $\bD$ sets whose union $R_0$ is not in
$\bD$. Thus, ${\bG}''=
{\bG}'$. We now follow the argument of Theorem 3.2
of \cite{St1}. Let $A,B \in \bGd$,
and by $\red(\bGd)$ we may
assume $A \cap B=\emptyset$. Write $A=\cup_{\alpha<\kappa}A_\alpha$,
$B=\cup_{\alpha<\kappa}B_\alpha$ as $\bD$--bounded unions.
It suffices to show the union $A\cup B= \bigcup_{\alpha<\kappa}
(A_\alpha \cup B_\alpha)$ is ${\boldsymbol{\Sigma}}^1_1$--bounded.
Let $S \subseteq A \cup B$ be ${\boldsymbol{\Sigma}}^1_1$. $S \cap
A \in \bGd$ as
$\bGd$ is closed under conjunction.
Also, $S\cap A=S \cap B^c \in {\bG}$, as
${\bG}$ is closed under conjuction with
${\boldsymbol{\Sigma}}^1_1$ sets as Steel points out (the union of a
$\bGd$ set and a
${\boldsymbol{\Pi}}^1_1$ set can be written in the form
$\forall z (U \vee \bGd)$ where
$U$ is open, and $\bGd$ is
easily seen to be closed under unions with open sets).
Thus, $S \cap A \in \bD$, and likewise $S \cap B \in
\bD$. By $\bD$--boundedness,
$S \subseteq A_\alpha \cup B_\alpha$ for some $\alpha<\kappa$.
The closure of $\bD$ under $< \kappa$ unions follows
easily from the Coding Lemma and the regularity of $\kappa$ if
$\bD$ is closed under quantifiers. Otherwise,
Martin's argument shows $\pwo({\bG})$, a contradiction
to Lemma~\ref{techlem2}.
\lpf
The next is a special case of Theorem~\ref{mainthm} (with reasonable
replaced by very reasonable), but is simple enough to present separately.
\prop \label{thmcount}
Let $k$ be an $\omega$--sequence of reasonable
cardinals with corresponding pointclasses ${\bG}_i$.
Then $k$ has the strong polarized partition
property.
\eprop
Proof:
By countable choice, let $\phi_i$, along with the ${\bG}_i$,
witness the reasonableness
of $k(i)$. We may identify $k(i)$ with
$[\underset{i'*From the Coding Lemma, it follows that $\bigcup_{k(i)}
\bGd{}_i^\ast \subseteq
\exists^{\omega^\omega}\bGd{}_i^\ast$.
If equality holds here, then easily $\pwo(\exists^{\omega^\omega}
\bGd{}_i^\ast)$,
a contradiction. This gives the desired result unless
${\bG}_{i+1}={\bG}_i^\ast$.
This, however, contradicts Lemma~\ref{techlem}.
\ppf
It is natural to ask how far Proposition~\ref{thmcount} can be extended.
That is, for which $\alpha \leq \Theta$ can we show that an increasing,
discontinuous $\alpha$ sequence of reasonable cardinals has the strong
polarized partition property?
We assume $\V=\lr$ for the remainder of this section,
though AD$+$DC$+$ ``within scales'' would suffice for many of the
results (e.g., Corollaries~\ref{corinac} and \ref{corind}). Recall
$\de 21$ is the largest Suslin cardinal in $\lr$. If $\alpha <
\de 21$, let $s(\alpha)$ be the least Suslin cardinal $>\alpha$ such that
the class of $s(\alpha)$--Suslin sets is of the form
${\boldsymbol{\Sigma}}_{1}(J_\beta({\mathbb R}))$ for some $\beta$
beginning a ${\boldsymbol{\Sigma}}_1$--gap. Thus, we ``skip over'' the
Suslin cardinals ocurring at the end of a gap. Let
$S_{s(\alpha)}$ be the pointclass of $s(\alpha)$--Suslin sets.
The point here is that the construction of scales for such
pointclasses is uniform, according to the analysis of \cite{stsc}. That is,
there is a function which assigns to each $\alpha<\de 21$ a
universal $S_{s(\alpha)}$ set $P_{s(\alpha)}$ and an $S_{s(\alpha)}$--scale
$\phi^{s(\alpha)}_n$ on $P_{s(\alpha)}$ with $\phi_0^{s(\alpha)}$
onto $s(\alpha)$. For $x \in P_{s(\alpha)}$, we let $|x|_{s(\alpha)}$
abbreviate $\phi^{s(\alpha)}_0(x)$, and similarly for other scales.
\ef \label{proper}
Let $k$ be an increasing, discontinuous $\mu$--sequence
of reasonable cardinals with corresponding pointclasses
${\bG}_\alpha$ for each $k(\alpha)$. We say that $k$ is {\em proper} if
$\forall \alpha<\mu
\ (\forall^{\omega^\omega} S_{s(\alpha)} \subseteq
{\bG}_\alpha)$.
\eef
In general, properness requires the $k(\alpha)$ to be sufficiently
spread out, although the requirement is trivial for short sequences.
For example, an $\omega_1$ sequence is proper provided only
$k(0) \geq {\boldsymbol{\delta}}^1_3$.
We recall some facts from the Kechris--Woodin theory of generic codes that
we will need for the proof of our main theorem.
Fix for the moment $\alpha_0<\de 21$, and thus
$P_{s(\alpha_0)},\phi_n=\phi^{s(\alpha_0)}_n$.
The main lemma of \cite{kewo} says
there is a Lipschitz continuous function $G:s(\alpha_0)^\omega
\rightarrow \omega^\omega$ such that $\forall
\vec t=(\alpha,\beta_0,\beta_1,\ldots) \in
s(\alpha_0)^\omega \ \forall n \in \omega$ $G(\vec t)_n \in P_{s(\alpha_0)}$
(where $y\rightarrow y_0,y_1,\ldots$ is a recursive bijection).
Furthermore, if $\vec t$ enumerates an {\em honest} set $T$, then
$\forall n \ \phi_0(G(\vec t)_n)= \beta_n$. Recall $T \in
\cP_{\omega_1}(s(\alpha_0))$ is honest if $\forall \beta \in T$
$\exists x \in P_{s(\alpha_0)}$ $[\phi_0(x)=\beta \wedge
\forall n \ \phi_n(x) \in T]$.
Recall also that $\omega_1$ is $\de 21$-- supercompact (Solovay),
that is, there is a fine normal measure on $\cP_{\omega_1}(\de 21)$.
By Woodin \cite{woun}, this measure is in fact unique. We write
$\forall^\ast T \in \cP_{\omega_1}(s(\alpha_0))$ to mean
almost all with respect to this measure. By normality,
almost all $T$ are honest.
Fix now $\mu \leq \de 21$ and a proper $\mu$--sequence $k$.
The proof of the generic Coding Lemma of \cite{kewo} shows that
the coding function $G$ is obtained uniformly in the set $P_{s(\alpha_0)}$
and the scale $\phi^{s(\alpha_0)}_n$ on $P_{s(\alpha_0)}$ for $\alpha_0
<\mu$. $G$ is obtained as a winning strategy in a Suslin, co--Suslin
ordinal game on $s(\alpha_0)$. The trees witnessing that the game is Suslin,
co--Suslin are obtained uniformly from $P_{s(\alpha_0)}$,
$\phi_n^{s(\alpha_0)}$, and the third periodicity argument of
\cite{moog} shows $G$ is obtained uniformly from these trees.
Thus, we have a
Lipschitz continuous function $G$ satisfying:
\begin{enumerate}
\item $\forall \alpha<\mu$ $\forall (\beta_1,\beta_2,\ldots)
\in s(\alpha)^\omega$ $G(\vec t) \in P_{s(\alpha)}$, where
$\vec t=(\alpha,\beta_1,\beta_2,\ldots)$.
\item If $(\alpha,\beta_1,\beta_2,\ldots) \in s(\alpha)^\omega$
enumerates an honest set (with respect to $P_{s(\alpha)}$,
$\phi^{s(\alpha)}_n$), then $\forall n \ \phi^{s(\alpha)}_0(G(
\vec t)_n)=\beta_{n-1}$ (with $\beta_{0}=\alpha$).
\end{enumerate}
We fix this function $G$ for the remainder of this section.
\lem \label{coding}
There is a Lipschitz continuous function
$F:\mu^\omega \rightarrow \omega^\omega$ such that $\forall \alpha<\mu$
$\forall \vec \beta \in s(\alpha)^\omega$, if $\vec t= (\alpha,\vec \beta)$
enumerates an honest set then $u=F(\vec t)$ codes the following:
\begin{enumerate}
\item A ${\bG}_\alpha$ universal set $A_u$. \label{w1}
\item A map $\phi_u$ such that $(k(\alpha),{\bG}_\alpha,
\phi_u)$ is reasonable. We let $R^u_{\alpha}$, $R^u_{\alpha,\beta}$,
$R^u_{\alpha,\beta,\gamma}$ denote the corresponding sets.
\label{w2}
\end{enumerate}
\elem
The exact manner in which $u$ codes these objects is not important, say by
the Coding Lemma relative to a fixed set of high Wadge degree.
Proof:
Consider the game where I plays $\alpha < \mu$,
$\beta_1,\beta_3,\ldots \in s(\alpha)$, and II plays
$\beta_2,\beta_4,\ldots \in s(\alpha)$ and also $u(0),u(1),\ldots$ building
$u \in \omega^\omega$. II wins in case \ref{w1},\ref{w2} above hold,
where in place of $\alpha$ we use $\phi^{s(\alpha)}_0 (G(\vec t)_0)$.
First note that the game is determined by a variation of the usual reflection
argument. The payoff set for II is of the form
$(\alpha,\vec \beta)\in A$ iff $(\alpha, G(\vec t)) \in B$, where
$B \subseteq \mu \times \omega^\omega$, and $G:\mu^\omega \rightarrow
\omega^\omega$ is continuous. If the game is undetermined, then there is a
$\delta< \de 21 $ and $A',B',G',\mu' \in J_\delta({\mathbb R})$ such that
$J_\delta({\mathbb R}) \models ($enough of $ZF)+ (A'$ is not determined$)$.
However, if $B'_\alpha=\{ x: (\alpha,x) \in B' \}$, then the $B'_\alpha$ are
uniformly $\boldsymbol{\Sigma}_{1}(J_{\delta'}({\mathbb R}))$ for some
$\delta' < \de 21$ (that is, there is a function sending $\alpha$ to a
$\boldsymbol{\Sigma}_{1}(J_{\delta'}({\mathbb R}))$ definition of $B_\alpha$).
This shows that $A' \subseteq {\mu'}^\omega$ is Suslin, co--Suslin in
$\lr$, and thus determined. By the Coding Lemma, a winning strategy
for $A'$ lies in $J_{\delta}({\mathbb R})$, a contradiction.
It is easy to see that I can not win this game, for II just enumerates an
honest set containing I's first move $\alpha$ (relative to
$\phi^{s(\alpha)}_n$) and plays an appropriate $u$.
A winning strategy for II gives $F$.
\lpf
We fix this $F$ for the proof of the theorem below.
We shall deal below with Lipschitz continuous functions $\sigma:
\mu^\omega\rightarrow \omega^\omega$ such that for $\alpha<\mu$,
$\vec \beta \in s(\alpha)^\omega$ and $\vec t=(\alpha,\vec \beta)$
enumerating an honest set $T$, $y=\sigma(\vec t)$ codes a comeager
set $A_y \subseteq T^\omega$ and a continuous function
$f_y:A_y \rightarrow \omega^\omega$. By this we mean precisely that
$\sigma$ is a strategy for II in a game where I plays $\alpha<\mu$,
$\beta_1,\beta_3,\ldots *~~from the integer moves of $\sigma$ against $\vec t$.
\item $\forall^\ast T\in \cP_{\omega_1}(s(\alpha))$
$\forall^\ast \vec t=(\alpha,\vec \beta)\in T^\omega$
$\sigma(\vec t)$ codes a comeager set $A_{\sigma(\vec t)}
\subseteq T^\omega$ and a continuous function
$\sigma(\vec t,-):A_{\sigma(\vec t)}\rightarrow\omega^\omega$
such that $\forall^\ast \vec t_1 \in T^\omega$,
if $x=\sigma(\vec t,\vec t_1)$, $u=F(\vec t_1)$, then
$x \in R^u_{\alpha,\beta,\gamma}$.
\end{enumerate}
To finish the proof of the theorem, it suffices to show properties
(1)--(5) of Definition~\ref{reasonableseq} for the coding $\phi$.
Property (1) is trivial.
To verify (2), let $H:\lambda\rightarrow\lambda$
be a block function, where $\lambda =\underset{\alpha<\mu}{\sup}
k(\alpha)$. Play the game where I plays $\alpha< \mu$, then
I and II alternate playing out $\beta_1,\beta_2,\ldots ~~~~s(\alpha)$ and
$G:\cP_{\omega_1}(s(\alpha))\rightarrow \kappa$ then
$\exists \delta <\kappa$ $\forall^\ast T \in \cP_{\omega_1}(s(\alpha))$
$G(\vec t)<\delta$.
Using this and the additivity of category it follows that $\exists \delta
$ the least very reasonable cardinal greater than
$\underset{\alpha'<\alpha}{\sup}k(\alpha')$. Then
$k$ has the strong polarized partition
property.
\ecor
Corollary~\ref{corinac} shows that any sequence of very reasonable
cardinals of not too great a length has the strong polarized partition
property. Corollary~\ref{corind} allows greater length sequences, but
imposes a mild ``spreading out'' condition on the sequence. The next result
shows that we may obtain sequences of any length $< \Theta$ with the
strong polarized partition property, provided we spread the $k(\alpha)$
sufficiently. The proof is easier than that of Theorem~\ref{mainthm},
and uses only the Coding Lemma.
\thm \label{thmtheta}
Let $\mu< \Theta$.
Then there is an increasing, discontinuous $\mu$--sequence
$k$
of regular cardinals having the strong polarized partition property.
\ethm
Proof:
We construct a $\delta \geq \mu$ sequence $k$
of cardinals and corresponding pointclasses
${\bG}_\alpha$ satisfying:
\begin{enumerate}
\item There is a pre--well--ordering $\prec$ of length $\delta$ such that
$\forall \alpha<\delta$ $\prec \restriction \alpha +1\in
\bD_\alpha$. Here, $\prec \restriction \alpha$
denotes the pre--well--ordering restricted to reals of rank $< \alpha$.
\item ${\bG}_\alpha$ is closed under quantifiers
and has the pre--well--ordering
property uniformly in $\alpha$. That is, there is a function which assigns
to $\alpha<\delta$ a ${\bG}_\alpha$--universal set
$A_\alpha$ and a ${\bG}_\alpha$--pre--well--ordering
$\psi_\alpha$ on $A_\alpha$. Also, $k(\alpha)$ is the
supremum of the lengths of the $\bD_\alpha$ pre--well--orderings.
\end{enumerate}
Granting this, we define the desired coding function $\phi$ as
follows. Via the Uniform Coding Lemma and the pre--well--ordering $\prec$,
every real $x$ codes a relation $R_x\subseteq \dom(\prec) \times
\omega^\omega$. Likewise, for all $\alpha<\delta$, every real $z$
codes a relation $R^\alpha_z$ with $\dom(A_\alpha)$
via the Uniform Coding Lemma
and the pre--well--ordering $\psi_\alpha$. For $\alpha<\delta$ and
$\beta,\gamma \in [\underset{\alpha'<\alpha} {\sup}k(\alpha'),
k(\alpha))$ we define
\[\begin{array}{rcl}\phi(x)(\beta,\gamma) & \Leftrightarrow &
\hbox{\rm \parbox[c]{2in}{$\exists y,z,w_1,w_2
[y \in \dom(\prec)$ $\wedge$ $|y|_{\prec}=\alpha$ $\wedge$ $R_x(y,z)$ $\wedge$
$w_1,w_2 \in A_\alpha$ $\wedge$
$|w_1|_{\psi_\alpha}=\beta$ $\wedge$
$|w_2|_{\psi_\alpha}=\gamma$ $\wedge$ $R^\alpha_z(w_1,w_2)$
$\wedge$ $\forall y',z',w'_1,w'_2 [|y'|_{\prec}=|y|_{\prec}$ $\wedge$
$R_x(y',z')$ $\wedge$
$|w'_1|_{\psi_\alpha}= |w_1|_{\psi_\alpha}$
$\wedge$ $R^\alpha_z(w'_1,w'_2) \rightarrow
(w'_2 \in A_\alpha$ $\wedge$
$|w'_2|_{\psi_\alpha} =|w_2|_{\psi_\alpha})]]$}}\end{array}\]
It is straightforward to check properties (1)--(5) of
Definition~\ref{reasonableseq}
for the coding $\phi$. To check (4), for example, note that for
$\alpha,\beta,\gamma$ as above, $R_{\alpha,\beta,\gamma}=
\{x: \phi(x)(\beta,\gamma) \}
\in \bD_\alpha$, as $\bD_\alpha$
is closed under quantifiers.
(5) follows easily from the fact that every $\bD_\alpha$
subset of $A_\alpha$ is bounded below $k(\alpha)$ with respect to
$\psi_\alpha$.
The $k(\alpha),{\bG}_\alpha$ may be constructed in
several ways. For example, let
$\bG$ $=$ $\boldsymbol{\Sigma}^2_1(\mu)$
$=\boldsymbol{\Sigma}_1(\lr;\mu)$
$\overset{def}{=}$ the sets
$\boldsymbol{\Sigma}_1$ definable in $\lr$ from parameters
$\mu$ and reals.
Then $\bG$
is closed under countable unions, intersections, $\exists^{\omega^\omega}$,
$\forall^{\omega^\omega}$. The pointclass $\boldsymbol{\Sigma}^2_1(\mu)$
resembles $\boldsymbol{\Sigma}^2_1$ (except it doesn't have the scale
property). Let $\delta$ be the least ordinal such that
$L_\delta(\mathbb{R}) \prec_1^\mu \lr$, that is, elementary for
$\boldsymbol{\Sigma}_1$ formulas with parameter $\mu$ (and reals).
Then $\bG$ has Wadge degree $\delta$, and
$\boldsymbol{\Delta}= \underset{\alpha<\delta}{\cup} L_\alpha(\mathbb{R}) \cap
{\omega^\omega}$ (c.f. Lemma 1.12 of \cite{stsc}).
Define $k(\alpha),{\bG}_\alpha$ for $\alpha<\delta$
by induction: let $k(\alpha)$ be the least ordinal
$> \underset{\alpha'<\alpha}{\sup}k(\alpha')$ such that
$\bG_\alpha \overset{def}{=}
\boldsymbol{\Sigma}_1(L_{k(\alpha)}(\mathbb{R});\mu) $ is
closed under countable unions, intersections, $\exists^{\omega^\omega}$,
$\forall^{\omega^\omega}$, and is not equal to
$\boldsymbol{\Sigma}_1(L_{\beta}(\mathbb{R});\mu)$ for any $\beta<
k(\alpha)$. The regularity of $\delta$ and a reflection argument
show that $k(\alpha)$ is well--defined.
For $\prec$, we may take the natural pre--well--ordering on reals $x$ viewed
as coding $\Sigma_1(\bar x,\mu)$ statements $\theta_x$
(where $\bar x(n)=x(n+1)$).
Namely, $x \prec y$ iff $\exists \alpha \ L_\alpha(\mathbb{R}) \models
\theta_x \wedge \neg \theta_y$.
Likewise we define $\psi_\alpha$ (so $\psi_\alpha$ is an
initial segment of $\prec$).
\tpf
In view of Corollary~\ref{corinac} and Theorem~\ref{thmtheta}, the
following conjecture seems plausible.
%\begin{con}
Every increasing, discontinuous $\mu$--sequence $k$
of (very) reasonable cardinals of length $\mu<\Theta$ has the
strong polarized partition property.
%\end{con}
A proof of this conjecture may depend on further extensions of the
detailed $\lr$ analysis.
Finally, we present a result which extends Theorem~\ref{mainthm} to
partitions into more than 2 pieces.
\prop
Suppose $k$ is a proper $\mu$--sequence.
Then $k$ has the strong polarized partition property for partitions into
$< k(0)$ many pieces.
\eprop
Proof:
Fix $k$ and the associated pointclasses
${\bG}_\alpha$, and fix a partition $\cP$
of the block functions into $\deltaFrom Lemmas~\ref{techlem2}, \ref{techlem3} it follows that for all $\alpha
<\mu$ there is a pre--well--ordering of length $k(\alpha)$ such that
every $\bD_\alpha$ subset of the field is bounded in
the norm (use $\pwo(\bGd_\alpha)$
and the closure of ${\bG}_\alpha$ under $\wedge$ which
allows the usual boundedness argument to be carried out).
By the usual game argument, there is a strategy $F$ such that
$\forall \alpha<\mu$ $\forall t=(\alpha,\vec \beta)\in s(\alpha)^\omega$
enumerating an honest set closed under $F$, $u=F(\vec t)$ codes
a set $B_u$ and a pre--well--ordering $\prec_u$ of $B_u$ with the
boundedness property for $\bD_\alpha$ sets. Again, the
exact manner in which $u$ codes these objects is not important, say
by continuous pre--images of a fixed set of high Wadge degree.
For $\beta<\delta$, we say a real $\sigma$ is $\beta$--good if
$\sigma$ codes a strategy (also denoted by $\sigma$) such that:
\begin{enumerate}
\item $\forall \alpha< \mu$ $\forall \vec t=(\alpha,\vec \beta)
\in s(\alpha)^\omega$ enumerating an honest set closed under $F$ and
$\sigma$, $\sigma(\vec t)$ codes a comeager set
$A_{\sigma(\vec t)}$ and a continuous function $\sigma(\vec t,-):
A_{\sigma(\vec t)}\rightarrow \omega^\omega$ such that
$\forall \vec t_1 \in A_{\sigma(\vec t)}$ if $w=\sigma(\vec t,\vec t_1)$,
$u=F(\vec t_1)$, then $w$ codes a strategy on $B_u$, that is,
$\forall x \in B_u$ $w(x) \in B_u$ (where we view $w$ as a strategy in
an integer game).
\item For $\alpha<\mu$, define $G^\alpha_\sigma: k(\alpha)
\rightarrow k(\alpha)$ as follows.
If $\gamma $ decides $\phi$. To see this, suppose that $r,r'\for
q$ decide $\phi$ differently. We may assume that these are of uniform
length, i.e., $r_0,r'_0\in[k]^\b$, for some
$\b<\g$. Using AC$_\o$,
choose $t\in[C]^{<\g}$ such that
$r_0\minus p_0=\po t$, and choose $t'$
similarly for $r'$. Then $f(t)\neq f(t')$, contradicting the
homogeneity of $C$. Finally, if $q\for\phi$ then $p\efor\phi$ and if
$q\for\neg\phi$ then
$p\efor\neg\phi$. \ppf
\prop \label{pnonewsub} If $(k)\rightarrow(k)^{<\g}$, then $\V^{\P_{\g,k}}$ and
$\V$ have the same bounded subsets
of $k(0)$. \eprop
Proof: This is a consequence of \rprop{pshrink} together with
\rfact{psubd}. See, for example, the proof of 1.3 in \cite{H2}.
\ppf
We have two preservation theorems involving splitting a polarized
sequence. Let $k_0, k_1$ be consecutive halves of $k$, that is,
$k=k_0\cat k_1$.
\prop \label{forceabove} If $(k)\rightarrow(k)^k$, then $\V^{\P_{\g,k_1}}\sat
(k_0)\rightarrow (k_0)^{k_0}$.\eprop
Proof: Of course, $(k_0)\rightarrow (k_0)^{k_0}$ is true in $\V$ by
\rfact{psubseq}. In $\V^{\P_{\g,k_1}}$, however, there may be new partitions.
Note that since $\P_{\g,k_1}$ adds no new bounded subsets
of $k_1(0)$, the set $[k_0]^{k_0}$ is the same in both $\V$
and $\V^{\P_{\g,k_1}}$, that is, there may be new
partitions, but the domain remains the same. Suppose that
$p\for$``$\dot{f}:[k_o]^{k_0}\rightarrow 2$''.
With the methods used in the proof of \rprop{p $ forces that $[C\r k_0]^{k_0}$ is
homogeneous for $f$. If not, suppose $r,r'\for q$ and $r\for$
``$\dot{f}(x)=0$'' and $r'\for$
``$\dot{f}(x')=1$,'' $x,x'\in [C\r k_0]^{k_0}$. We may assume as before that
the sets $r(\k)_0\minus
p(\k)_0$ and $r'(\k)_0\minus
p(\k)_0$ all have the same length and choose $t,t'\in[C\r k_1]^\b$ so
that $r_0\minus
p_0=\po t$ and $r'_0\minus
p_0=\po t'$. We can then put $x$ and $t$ together to form
$W$ with $g(W)=0$. Similarly, with $x'$ and $t'$ we can form $W'$
such that $g(W')=1$, a contradiction. \ppf
Forcing with the lower half of $k$ is more disruptive to the upper half than
vice--versa. Still, we can prove the following.
\prop \label{forcebelow} If $(k)\rightarrow(k)^{<\A}$, then
$\V^{\P_{\o_1,k_0}}\sat (k_1)\rightarrow (k_1)^{<\A}$.\eprop
Proof: The proof is quite similar to the proof of
\rprop{forceabove}, reversing the roles of $k_0$ and $k_1$, and
replacing the technical fact with:
\[\left(\begin{array}{c}k_0\\k_1\end{array}\right)\rightarrow
\left(\begin{array}{c}k_0\\k_1\end{array}\right)^{<\o_1,<\o_1},\]
by which we mean that for all partitions
$f:\underset{\a,\b<\o_1}{\cup}[k_0]^\a\times [k_1]^\b\rightarrow 2$,
there is $X\in[k_0]^{k_0}$, $Y\in[k_1]^{k_1}$, with $f$ constant on
$\underset{\a,\b<\o_1}{\cup}[X]^\a\times [Y]^\b$ for all
$\a,\b<\o_1$. Any such $f$ can easily be coded as a partition of
$[k]^{<\o_1}$, and so this relation follows from the hypothesis.
The restriction in the statement of the theorem
to exponent ``$^{<\A}$'' is needed because while
$\left([k_1]^{k_1}\right)^\V\neq\left([k_1]^{k_1}\right)^{\V^{\P_{\g,k_0}}}$,
$\left([k_1]^{<\A}\right)^\V$ does equal
$\left([k_1]^{<\A}\right)^{\V^{\P_{\g,k_0}}}$, since $\P_{\g,k_0}$ is
countably closed and we are forcing over a model of DC.
Now if $p\for$``$\dot{f}:[k_o]^{<\o_1}\rightarrow 2$'' we can define a
partition $g$ of $\underset{\a,\b<\o_1}{\cup}[k_0]^\a\times [k_1]^\b$
into 3 as follows. If $W\in[k_0]^{\a}\times[k_1]^{\b}$,
$$\hbox{\rm $g(W)=i$ if $[p_0\cat
\po (W\r k_0)]\efor$``$\dot{f}(W\r k_1)=i$,''}$$
and $g(W)=2$, otherwise.
Let $C$ be homogeneous for $g$. In the manner of \rprop{forceabove}, $q=\<
p_0,C\r k_0\> $ forces that $[C\r k_1]^{k_1}$ is
homogeneous for $f$. \ppf
Using the methods of the previous two propositions, we can prove:
\prop \label{gforceabove} If $(k)\rightarrow(k)^{<\g}$, then
$\V^{\P_{\g,k_1}}\sat (k_0)\rightarrow (k_0)^{<\g}$.\eprop
Simple factoring shows that finite products achieve exactly the
cofinality changes intended. For infinite products, the picture is
not as nice. We can, however, deal with countable products.
\prop \label{cprodcof} Let $k$ be a countable sequence satisfying
$(k)\rightarrow (k)^{<\g}$, $\g \cup k$ and show at the end how to handle cardinals
$k(0)<\cof(\d)< \cup k$.
Suppose that $p\for$``$\dot{F}:\lambda\rightarrow\d$ is unbounded'',
$\l<\cof^\V(\d)$. For $s\in[p_1]^{<\g}$, let $n_s=\{\b<\d:\exists\,\a\,
p_0\cat\po s\efor \dot{F}(\a)=\b\}$. Note that $\Vert
n_s\Vert<\cof^\V(\d)$
since no sequence can insinuate different values for a particular
$\dot{F}(\a)$. Since $n_s$ is defined in $\V$ then, $\cup n_s<\d$.
{\bf Claim:} There is a
$q\in[p_1]^k$ such that for any $\eta<\g$, $\bigcup_{s\in
[q]^\eta} \cup n_s <\d$. This will give us a contradiction, since then
$\< p_0,q\> $ will force that $\dot{F}$ is bounded.
Proof of Claim: We begin by defining
$f:[p_1]^{<\g}\rightarrow 3$
by:
$$f(s\cat t)=\left\{\matll 0&\hbox{\rm if $\cup n_s=\cup n_t$}\cr
1&\hbox{\rm if $\cup n_s<\cup n_t$}\cr
2&\hbox{\rm if $\cup n_s>\cup n_t$}\emat\right.,$$ where $s\cat
t\in[k]^{\eta+\eta}$, $\eta<\g$.
Let $q_1\in[p_1]^k$ be homogeneous for $f$. Clearly, the range of $f$ on
$[q]^{\eta+\eta}$ cannot be $\{2\}$. If the range is $\{0\}$ for a
given $\eta$, then we
have as promised that $\bigcup_{s\in
[q_1]^\eta} \cup n_s<\d$.
Suppose for some fixed $\eta$ $f\r[q_1]^{\eta+\eta}=\{1\}$. Let
$\{\nu_n\}_{n<\o}$ be an enumeration of $k$. For any
$R\subseteq\o$, define the partition $f_R^\eta :[k]^{\eta + \eta}\rightarrow 3$
by:
$$f_R^\eta(s\cat t)=\left\{\matll 0&\hbox{\rm if $\cup n_s=\cup n_{(s,t)_R}$}\cr
1&\hbox{\rm if $\cup n_s<\cup n_{(s,t)_R}$}\cr
2&\hbox{\rm if $\cup n_s>\cup n_{(s,t)_R}$}\emat\right.,$$ where $(s,t)_R\in
[k]^\eta$ is
defined:
$$(s,t)_R(\nu_n)=\left\{\matll s(\nu_n)&\hbox{\rm if $n\notin R$}\cr
t(\nu_n)&\hbox{\rm if $n\in R$.}\emat\right.$$
We don't have the power to find a set homogeneous for all of these
partitions at once [$(k)\rightarrow(k)^{<\g}_{2^\o}$ is sufficient to
produce a set homogeneous for $\o$--many at once, but there are
$2^\o$--many partitions and $(k)\rightarrow(k)^{<\g}_{2^{2^\o}}$ is
generally false] but note two important facts. First, the range of
$f_R^\eta$ on any homogeneous set cannot be $\{2\}$, or we
could easily construct an infinite descending chain of ordinals.
Second, if $q', q''$ are both homogeneous for a certain $f_R^\eta$, then
the ranges of $f_R^\eta$ on the two are the same. This is because the
partition is defined in terms of ``pre--sub--omega'' ($\po$), through the
definition of $n_s$, and it is easy to find $s'\in[q']^\g$,
$s''\in[q'']^\g$ such that $\po s'=\po s''$.
For this reason we can
unambiguously define $m(R)$ for $R\subseteq\o$ as $i$ iff the range of
$f_R^\eta$
on any homogeneous set is $\{i\}$. $m$ is like a measure.
We can prove, for example, that if
$R\subseteq W$, $m(W)=1$, then at least one of $m(R)$, $m(W\minus R)$
is $1$. To see this, choose $s\cat t\cat u$, an $\eta\cdot 3$-sequence
>from any set homogeneous for $f_W^\eta$, $f_R^\eta$, and $f_{W\minus
R}^\eta$. Then
$f_W^\eta(s,t)=1$ implies $\cup n_s<\cup n_{(s,t)_W}$. But
$f_R^\eta(s,t)=0$ implies
$\cup n_s=\cup n_{(s,t)_R}$ and $f_{W\minus R}^\eta((s,t)_R,(t,u)_R)=0$
implies $\cup n_{(s,t)_R}=\cup n_{(s,t)_W}$. Similarly, if $R\subseteq
W$, $m(W)=0$, then both $m(R)$, $m(W\minus R)$
are $0$. This shows that $R\subseteq W\Rightarrow m(R)\le m(W)$. Note
that $m(\omega)=1$.
We claim next that there cannot be an infinite collection of mutually
disjoint sets $\{R_n\}_{n<\o}$ all with measure one, i.e., $m(R_n)=1$.
This is because we could find $x$ homogeneous simultaneously for all
the $f_{R_n}^\eta$, and then for $s\cat t\in[x]^{\eta+\eta}$ we would have
$$\cup n_{(s,t)_\o}>\cup n_{(s,t)_{\o\minus R_0}}>\cup n_{(s,t)_{\o\minus
(R_0\cup R_1)}}>...$$ (If $s,t,u$ are as before, apply $f^\eta_{R_0}$
to $\left((s,t)_{\o\minus R_0},(t,u)_{\o\minus R_0}\right)$ for the
first inequality, then
apply $f^\eta_{R_1}$
to $\left((s,t)_{\o\minus (R_0\cup R_1)},(t,u)_{\o\minus (R_0\cup
R_1)}\right)$ for the second inequality, and so on.)
Let $Q^\eta=\{n:m(\{n\})=1\}$. From the discussion
above, $Q^\eta$ must be finite.
Now we claim that $m(\o\minus Q^\eta)=0$. If not, we would have that for some
infinite
subset $R$ of $\o\minus Q^\eta$, $m(R)=1$ and $W\subseteq R$ implies that
exactly one of the sets $W$, $R\minus W$ is measure one (otherwise we
could construct an infinite disjoint collection of measure one sets).
In effect, this gives us a non--principal ultrafilter on $R$ (additivity
follows from this and the properties proved above), contradicting an hypothesis
of the proposition.
Let $x\in[q_1]^k$ be homogeneous for $f_{\o\minus Q^\eta}$. Take
$$\zeta=\bigcup\left\{\cup
n_s:\left[\matll \hbox{\rm if $n\in Q^\eta$, then $s(\nu_n)$ is an interval of
$x(\nu_n)$}\cr
\hbox{\rm if $n\notin Q^\eta$, then
$s(\nu_n)=x(\nu_n)\r\eta$.}\emat\right]\right\}$$
Since $Q^\eta$ is finite, there are no more than $\nu_n$--many $s$ in the
definition of $\zeta$, where $\nu_n\in k$ is greatest with $n\in Q^\eta$. Since
$\cof(\d)>\nu_n$, we must have $\z<\d$. This then gives us $\bigcup_{s\in
[x]^\eta} \cup n_s <\d$ as follows: Suppose $s\in [x]^\eta$ and for some $y$
$\<
p_0\cat\po s,y\> \for$
``$\dot{F}(\a)=\b$''. Take $t\in[x]^\eta$, $t>s$, such that $t(\nu_n)$ is an
interval of $x(\nu_n)$ for all $n\in Q^\eta$. Homogeneity for $f$ gives us
that $\cup n_t > \cup n_s \ge \b$. Homogeneity for $f_{\o\minus
Q^\eta}^\eta$
gives us that $\cup n_t=\cup n_{s'}$, where $$s'(\nu_n)=\left\{\matll
t(\nu_n)&\hbox{\rm for $n\in Q^\eta$}\cr
x(\nu_n)\r \eta&\hbox{\rm for $n\notin Q^\eta$,}\emat \right.$$
since $t=(s',(t,u)_{Q^\eta})_{\o\minus Q^\eta}$, for any $u>t$. Then we have
$\b\le\cup n_{s'}<\z$.
Finally, use $(k)\rightarrow (k)^{<\g}$ to obtain $q\in[q_1]^k$ homogeneous
simultaneously for all $f_{\o\minus Q^\eta}$, $\eta<\g$. This satisfies
our Claim and finishes the case: $\cof(\d)>\cup k$.
Suppose now that $\cof(\d)$ lies somewhere within the limits of the sequence.
We can factor the partial ordering into a piece below $\cof(\d)$ and a
piece (at and)
above. By \rprop{pnonewsub}, forcing first with the piece (at and)
above either changes the cofinality of $\d$ to $\g$ or leaves it
unchanged (depending on whether $\cof(\d)$ is an
element of the sequence (at and) above). By \rprop{gforceabove}, this forcing
preserves the partition relation necessary to carry out the forcing on
the piece below. All that remains is to show that after this first
forcing, there is still no non-principal ultrafilter on $\o$, so that
the preceding proof applies.
Suppose that $p\for$``$\dot{U}$ is a non--principal ultrafilter on
$\o$.'' We can then define an ultrafilter $U^*$ in $\V$ as follows: For
$x\subseteq\o$, $x\in\V$, we say $x\in U^*$ iff $p_0\efor$``$x\in\dot{U}$.''
$U^*$ is definable in $\V$ and it is straightforward to show that it
is a non--principal
ultrafilter, contradicting our hypothesis.
\ppf
\sec{Finite Support}\label{seven}
Questions of Choice limit us when dealing with products. One solution
is to use inner models based on finite products. We exploit this to
prove the following theorem.
\thm \label{finsup} Suppose that $\V=L(\R)\sat${\em AD} and $A$ is a sequence
of cardinals such
that for all $k\in[A]^{<\o}$, $(k)\rightarrow (k)^k$. Let
$h:A\rightarrow \Theta^\V\minus A$ be any function in $\V$ such that
for
$\k\in A$, $h(\k )<\k$ and $h(\k )$ is a regular cardinal. Then there is
a partial ordering $\P\in\V$ and a symmetric inner
model $\N$ of {\em ZF} such that\ret
(1) $\V\subseteq\N\subseteq \V^\P$, \ret
(2) $\N$ and $\V$
contain the same cardinals, \ret
(3) $\Theta^\N=\Theta^\V$, and \ret
(4) the
cofinality function in $\N$ is given by:
$$\cof(\k)=\left\{\matll h(\k )&\hbox{\rm if }\cof^\V(\k )\in A \cr
\cof^\V(\k )&\hbox{\rm otherwise. }\emat\right.$$\ethm
Proof: For convenience, we will write $\P_\k$ for $\P_{h(\k),\k}$ if
$h(\k)>\o$ and $\P_\k$ for $\P_{\o\cdot\o,\k}$ if $\k=\o$. The partial
ordering $\P$ with which we force to
construct the model is all elements of
$\pr{\k\in A}\P_\k$ having finite support, i.e., those $p$ such that $\{\k\in
A:p(\k)\neq\< \emptyset,\k\> \}$ is finite. We will denote the support
of $p$
by $\supp(p)$. For any set $B \subseteq A$,
we will write $\P_B$ for $\pr{\k\in
B}\P_\k$. The ordering
on $\P_B$ is by component.
Let $G$ be $\V$--generic over $\P_A = \P$.
The full generic extension fails
to satisfy the theorem if $A$ is infinite. To define our model
$\N$, we first define $G_{\k}$ for $\k\in A$ as the $\V$--generic object
over $\P_\k$ generated by $G$, i.e., $q\in G_{\k}$ iff for some $p\in
G$, $q=p(\k)$. By the Product Lemma, each $G_{\k}$
is $V$--generic over $\P_\k$ and for $k\in[A]^{<\o}$,
$\pr{\k\in k}G_{\k}$ (which we will write as $G_k$) is $\V$--generic
over $\P_k$. We
can describe $\N$ as the least model of ZF extending
$\V$ which contains, for each $\k\in A$, the set $G_{\k}$. More
formally, let $\L_1$ be the sublanguage of the forcing language $\L$
with respect to $\P$ which contains symbols $\dot{v}$ for each $v\in
\V$, a unary predicate symbol $\dot{\V}$ (to be interpreted
$\dot{\V}(\dot{v})$ iff $v\in \V$), and symbols $\dot{G}_\k$ for each
$\k\in A$. We can define $\N$ inside $\V[G]$ as follows:
$\matll \N_0&=\emptyset .\cr
\N_{\alpha +1}&=\{x\subseteq \N_\alpha : x \,\,\,\hbox{\rm
is definable by $\tau\in\L_1$ of rank $\le\alpha$
over }
\< \N_\alpha , \in , c\> _{c\in\N_\alpha }\}.\cr
\N_\lambda &={\displaystyle \bu{\alpha<\lambda}}\N_\alpha\,\,\,
\hbox{\rm for $\lambda$ a
limit ordinal.}\cr
\N &= {\displaystyle \bu{\alpha\in\hbox{\rm ORD}^V}}\N_\alpha .\emat$
Standard arguments show $\N\sat$ ZF and if $A$ is finite, $\N=\V
[G]$.
\lem \label{finreduce} Let $x$ and $y$ be sets with $x,y\in\V$, and let
$f:x\rightarrow y$ be a function with $f\in\N$. Then $f\in\V
\left[G_k\right]$, where $k\in[A]^{<\o}$. \elem
Proof: The proof is similar to that
of Lemma 2.1 of \cite{A}. Let $p\in\P$ be so that $p\for\hbox{\rm
``$\dot{f}:x\rightarrow y$ is a function.''}$ Since $f\in\N$, we can
choose $\dot{f}\in\L_1$, and $k\in[A]^{<\o}$, such that $\dot{f}$ mentions only
terms of the form
$\dot{G}_\k$ for $\k\in k$. By
extending $p$ and/or $k$ if necessary, we can assume $\supp(p)=k$. We will
identify $p$ with $p\r \supp(p)$, so we may consider $p$ both as a
condition of $\P$ and as a condition of $\P_k$.
Thinking of $\dot{f}$ as a term for a set, define in
$\V\left[G_k\right]$ the set $$\hbox{\rm $g=\{\< w,z\> \in \V:
\exists q\in G_k$ such that $q\for p$
and $q\for_\P$``$\< w,z\> \in\dot{f}$''$\}$}.$$
We claim
$p\for$``$\dot{f}=\dot{g}$''. Certainly
$p\for$``$\dot{g}\subseteq \dot{f}$''. To show that
$p\for$``$\dot{f}=\dot{g}$'', suppose $q\for p$, $q\in G$ with $q\for$``$\<
w,z\>
\in\dot{f}$''. We will be done if we can show that $q'=q\r\supp(p)\for$``$\<
w,z\>
\in\dot{f}$''.
If $q'\nfor$``$\< w,z\>
\in\dot{f}$'', then choose $r\for q'$, $r\for$ ``$\< w,z\>
\notin\dot{f}$'', with $\supp(q)\subseteq\supp(r)$. Extend $q$ to $s$
with $\supp(s)=\supp(r)$. Note that $s\for$ ``$\< w,z\>
\in\dot{f}$''. We may assume that $s\r \supp(p)=r\r \supp(p)$.
By extending the appropriate
sequence of ordinals as in Proposition 1.1 of \cite{H2}, we can assume
$r(\k)_0$ and $s(\k)_0$ (the first coordinates) have the same length
for all $\k$. Further, if we denote by $\cl(X)$ the $\o$--closure of
$X$, then $\cl(r(\k)_1)\cap\cl(s(\k)_1)\in[\k]^\k$ and $\<
\cl(r(\k)_1)\cap\cl(s(\k)_1)\> \subseteq \< r(\k)_1\> \cap\< s(\k)_1\>
$. Using this, we may assume for all $\k\in \supp(r)$ that
$r(\k)_1=s(\k)_1$. Now for $\k\in\supp(r)$ define $\pi_\k:\k\rightarrow\k$ by
$\pi_\k(r(\k)_0(\xi))=s(\k)_0(\xi)$ for each $\xi<\overline{s(\k)_0}$ and the
identity otherwise. Define $\psi_\k:\P_\k\rightarrow\P_\k$ by $\psi_\k(x)=\<
\pi_\k\circ x_0, \pi_\k''x_1\> $. $\psi_\k$ is an isomorphism of the
conditions in $\P_\k$
extending $r(\k)$ to the conditions in $\P_\k$ extending $s(\k)$. In addition,
$\psi_\k(r(\k))=s(\k)$.
Now define $\psi:\P\rightarrow\P$ by $$\psi(u)(\k)=\left\{\matll
\psi_\k(u(\k))&\hbox{\rm if $\k\in \supp(r)$, and}\cr u(\k)&\hbox{\rm
otherwise.}\emat\right.$$ Note that for $\k\in \supp(p)$, $\psi_\k$ is
the identity, since $r(\k)=s(\k)$, hence $G_{\k}$ and $\dot{f}$ can be
assumed to be invariant under $\psi$. Finally, $w$ and $z$, being (terms for)
ground model sets, can also be
assumed to be invariant under $\psi$ and so $s=\psi(r)\for$``$\< w,z\>
\notin\dot{f}$''.
Since
$s\for$``$\< w,z\> \in\dot{f}$'', this is a
contradiction and so $q'\for$``$\< w,z\> \in\dot{f}$'', i.e.,
$p\for$``$\dot{f}\subseteq \dot{g}$''. \lpf
Our strategy is to reduce the problem to the finite case.
\rprop{kpreserve}, for example, tells us that cardinals are preserved in
$\V^{\P_{k(n-1)}}$ and \rprop{forceabove} tells us that $(k\r
(n-1))\rightarrow(k\r (n-1))^{k\r (n-1)}$ is preserved as well.
Working our way down, we see that all cardinals are preserved in
$\V^{\P_k}$, for $k$ finite. Combining this with the
previous lemma gives us (2) of the theorem. In a similar fashion,
Fact 3.3 gives (4). To complete the proof, we need only the following:
\lem \label{fintheta} For any $\g\in A$, $\Theta^\V=\Theta^{\V^{\P_\g}}$.\elem
Proof: Since $\g>\o$ and $\V$
and $\V^{\P_\g}$ contain the same bounded subsets of $\g$,
$\R^\V=\R^{\V^{\P_\g}}$. Thus ``$\R$'' is unambiguous.
Suppose $\< s,X\> \for$``$\dot{f}:\R\rightarrow\Theta^\V$ is onto.''
For each $r\in\R$ and each $t\in[X]^{ )=\left\{\matll \alpha&\hbox{\rm if $(s\cup\po t)
\efor$``$\dot{f}(r)=\alpha$''}\cr
0&\hbox{\rm otherwise.}\emat \right.$$
By the fact that any two conditions $\< u,Y\> $ and $\< u,Y'\> $ are
compatible (as in the proof of \rlem{finreduce}), $g$ is
well--defined. Further, $g$ can easily be seen to be a surjection onto
$\Theta^\V$. Since any pair $\< r,t\> $ as above codes a subset of
$\g$, $g$ can be used to define in $\V$ a mapping $h$ of $2^\g$ onto
$\Theta^\V$. Since $\g<\Theta^\V$, however, there is a
pre--well--ordering of $\R$ in $\V$ of size $\g$. We can use the Coding Lemma
(see
\cite{Mo2}) to produce a mapping $k$ from $\R$ onto $2^\g$, and then
$h\circ k$ is a mapping in $\V$ from $\R$ onto $\Theta^\V$,
a contradiction. \ltpf
Note
that the only differences between the cofinality function in $\V$ and
$\N$ are those that are forced to occur. Thus, since $\Theta$ is
regular in $\V$ (see \cite{Mo2}), \rlem{fintheta} implies $\Theta$ is
regular in $\N$.
Note also that \rthm{finsup} does not require that $(A)\rightarrow(A)^A$.
Note that if $A$ is infinite, then as in \cite{A},
$\N\nsat \hbox{\rm AC}_\o$. To see this, suppose $k\in[A]^\o$. For all $i<\o$,
let $X(i)=\{G:G$ is $\V$--generic over
$\P_{k(i)}\}$. The collection $\{X(i)\}_{i<\o}$ is in $\N$, but
has no choice function $f$. If $f$ is in the cartesian product, then
since by identifying generic sets with cofinal sequences, $f$ can be
coded by a set of ordinals,
by \rlem{finreduce}, $f$ can be
obtained by forcing over $\V$ using only a finite number of
Magidor--like orderings. Such an extension, however, only destroys the
regularity
of the finitely--many $k(i)$, while $f$ witnesses the singularity of
all the $k(i)$, a contradiction.
On the other hand, if $A$ is finite and each $\g\in A$ has its
cofinality changed to some uncountable cardinal, then $\N\sat$ DC,
since $\N=\V[G]$, $\V\sat$ DC (recall that assuming AD,
$L(\R )\sat$ DC), and $\P$ is countably closed. If, however, some
successor cardinal $\k\in A$ has its cofinality changed to $\o$, then
regardless of the size of $A$, $\N\nsat \hbox{\rm AC}_\o$.
\sec {Countable Support}\label{eight}
In this section, we present a general theorem in which a certain polarized
partition property holding for a set $A$ of
cardinals is used to change the cofinality of each member of $A$ to $\A$ and
preserve the fact that they are cardinals. We then apply
this theorem (in section 11) to derive
the existence of the model mentioned in the Introduction which
provides a partial answer to Woodin's question on the consistency of
the theory ``ZF $+$ DC $+$ $\A$ is the only regular uncountable
cardinal.'' The challenge is to preserve DC and the method is to move
to countable support.
The general theorem we prove is
\thm \label{countsup} Suppose that $\V=L(\R)\sat${\em AD} and
$A\subseteq\Theta^\V$, $A\in\V$, is such
that all $k\in[A]^{<\A}$ satisfy $(k)\rightarrow(k)^{<\A}$.
Then there is a partial ordering $\P\in\V$ and an
inner model $\N$ of {\em ZF} such that \ret
(1) $\V\subseteq\N\subseteq\V^\P$, \ret
(2) $\Theta^\N=\Theta^\V$, \ret
(3) $\N\sat$``{\em ZF} $+$ {\em DC} $+$ $\k$ is a
cardinal of cofinality $\A$ for all $\k\in A$,'' and\ret
(4) for all cardinals $\d$ of $\V$,
$\cof^{\N}(\d)=\cof^\V(\d)$ if $\cof^\V(\d)\notin A$ and $\cof^\N(\d)=\o_1$
otherwise. \ethm
Proof: The proof has similarities with the proof of
\rthm{finsup}. For each $\k\in A$, let $\P_\k=\P_{\A,\k}$. The partial
ordering $\P$ we use now will, as before, be a subset of $\P_A$. This time,
however, $\P$
will be composed of all elements having countable support.
The ordering on $\P$ is once again componentwise.
As before, we may assume our conditions have first coordinates of
equal length in all non--trivial components.
Now let $G$ be $\V$--generic over $\P$. For $\k\in A$, let
$G_\k$ be the $\V$--generic object over $\P_\k$
generated by $G$, i.e., $q\in G_\k$ iff for some $p\in G$, $q=p(\k)$. The
Product Lemma again
implies that for $k\in[A]^{<\A}$, $G_k=\pr{\k\in k}G_\k$ is
$\V$--generic over $\P_k$. This allows us to describe
$\N$ intuitively as the least model of ZF extending $\V$ which
contains, for each $k\in[A]^{<\A}$, the set
$G_k$. More formally, let $\L_1$ be the sublanguage of the forcing
language $\L$ with respect to $\P$ which contains symbols $\dot{v}$
for each $v\in\V$, a unary predicate symbol $\dot{\V}$ (to be
interpreted $\dot{\V}(\dot{v})$ iff $v\in \V$), and symbols $\dot{G}_k$
for each $k\in[A]^{<\A}$,
which are interpreted as $G_k$.
$\N$ can then be defined inside $\V[G]$ as follows:
$\matll \N_0&=\emptyset .\cr
\N_{\alpha +1}&=\{x\subseteq \N_\alpha : \hbox{\rm $x$ is
definable by $\tau\in\L_1$ of rank $\le\alpha$
over }
\< \N_\alpha , \in , c\> _{c\in\N_\alpha }\}.\cr
\N_\lambda &={\displaystyle \bu{\alpha<\lambda}}\N_\alpha\,\,\,
\hbox{\rm for $\lambda$ a
limit ordinal.}\cr
\N &= {\displaystyle \bu{\alpha\in\hbox{\rm ORD}^V}}\N_\alpha .\emat$
Standard arguments show $\N\sat$ ZF and if $A$ is
countable, $\N=\V[G]$.
\lem \label{countreduce} Let $x$ and $y$ be sets with $x,y\in\V$ and let
$f:x\rightarrow
y$ be a function with $f\in\N$. Then
$f\in\V\left[G_{k}\right]$, for some $k\in[A]^{<\o_1}$.\elem
Proof: The proof is virtually identical to the proof of
\rlem{finreduce}. We note that by a coding argument, a term $\dot{f}$
for $f$ can be assumed to mention only one term of the form
$\dot{G}_k$, where $k\in[A]^{<\A}$. Once this
has been done, the only difference in the proofs is that the supports
used now are countable, not finite as in \rlem{finreduce}. However, since we
can extend the first coordinates of conditions
canonically, and since the union of two countable supports is also a
countable support, all aspects of the proof of the current lemma can
be carried out as before. \lpf
\lem \label{countDC} $\N\sat$ {\em DC}. \elem
Proof: Let
$p$ force:$$\hbox{\rm ``$\dot{R}\subseteq\dot{X}\times\dot{X}$,
$\dot{R}$, $\dot{X}\in\dot{\N}$, $\dot{X}\neq\emptyset$, and $\forall
x\in\dot{X}\exists
y\in\dot{X} \< x,y\> \in\dot{R}.$''}$$ Using the fact that
$\V=L(\R)\sat$ DC since $\V=L(\R)\sat$ AD (see [Ke]), we can define
inductively a sequence of conditions $\< p_n:n<\o\> $ and a
sequence of terms $\< \tau_n:n<\o\> $ so that\ret
(1) each $\tau_i\in\L_1$,\ret
(2) $p_0\for p$, \ret
(3) $p_0\for$``$\tau_0\in\dot{X}$'', and \ret
(4) for $n>0$, $p_n\for p_{n-1}$ and $p_n\for$``$\< \tau_{n-1},\tau_n\>
\in\dot{R}$.''\ret
Since $\V\sat$ DC and $\V\sat$``$\P_\k$ for $\k\in A$ is countably
closed,'' the definition of $\P$ ensures that $\V\sat$``$\P$ is
countably closed.'' This means there is a single $q$ extending every
$p_n$, so that for any $n<\o$, $q\for$``$\< \tau_{n-1},\tau_n\>
\in\dot{R}$.'' By AC$_\o$, $\V\sat$``The countable union of
countable sets is countable,'' so $\bu{n<\o}\supp(\tau_n)$ is
countable and can be used as a support to define a term $\tau\in\L_1$
for $\< \tau_n:n<\o\> $. Thus, $q\for$``$\tau\in\dot{\N}$ is a
witness for DC for $\dot{R}$.'' \lpf
\lem \label{countpreserve}$\N\sat$``If $\g\in A$, $\g$ is
a cardinal.'' Further, for all cardinals $\d$ of $\V$,
$\cof^\N(\d)=\cof^\V(\d)$ if $\cof^{\V }(\d)\notin A$ and $\cof^\N(\d)=\o_1$
otherwise. \elem
Proof: Since ZF$+$AD$\vdash$``All sets of reals are
Lebesgue--measurable'' and a non--principal ultrafilter on $\o$ is a
non--Lebesgue--measurable set, $\V\sat$``There is no non--principal
ultrafilter on $\o$.'' Therefore, by \rprop{cprodcof} and \rlem{countreduce},
the cofinality is what it should be.
For the cardinality, fix $\g\in A$. By \rlem{countreduce}, if $\zeta<\g$ and
$f:\zeta\rightarrow\g$ is a function, $f\in \N$, then
$f\in\V\left[G_{k}\right]$ for some $k\in[A]^{<\A}$. Since
$\V\left[G_{k}\right]\subseteq\N$, it suffices to show
that for any $k\in[A]^{<\A}$, $\V\left[G_{k}\right]\sat$``$\g$ is a cardinal.''
Fix $k\in[A]^{<\A}$, $k\in\V$, and assume without
loss of generality that $\g\in k$. Let $k_0$ be the part of $k$ below
$\g$ and $k_1$ the part of $k$ at and above $\g$. By
\rprop{forcebelow}, $\V^{\P_{k_0}}\sat(k_1)\rightarrow(k_1)^{<\A}$,
so by \rfact{seqtocard}, $\V^{\P_{k_0}}\sat\g\rightarrow(\g)^{<\A}$.
Hence, $\g$ is still a cardinal. By
\rprop{pnonewsub}, forcing with $\P_{k_1}$ over $\V^{\P_{k_0}}$ adds
no new bounded subsets of $k_1(0)=\g$. Hence, $\g$ is still a cardinal
in $\left(\V^{\P_{k_0}}\right)^{\P_{k_1}}=\V^{\P_k}$.
\lpf
\lem \label{counttheta} $\T^\N=\T^\V$. \elem
Proof: The proof is similar in spirit and method
to the proof of \rlem{fintheta}. By \rlem{countreduce}, any $x\subseteq \o$
lies in
some $\V\left[G_k\right]$ for some $k\in[A]^{<\A}$. Since all members of $k$
are uncountable and since $\V$ and
$\V\left[G_k\right]$ have the same bounded subsets of
$k(0)$, $x\in\V$. This means that $\R^\N=\R^\V$. We will therefore
once more write $\R$ unambiguously.
Suppose now that $f\in\N$ maps $\R$ onto $\T^\V$. Using
\rlem{countreduce}, choose $k$ so that
$f\in\V\left[G_k\right]$, $k\in[A]^{<\A}$,
and suppose
$p\in\P_k$ forces ``$\dot{f}:\R\rightarrow\T^\V$ is
surjective.'' For each $r\in\R$ and $t\in[p_1]^{<\A}$, define
$$g(\< r,t\> )=\left\{\matll \alpha&\hbox{\rm if $ (p_0\cat\po t)
\efor$``$\dot{f}(r)=\alpha$''}\cr
0&\hbox{\rm otherwise.}\emat \right.,$$
$g$ is well--defined and the range is all
of $\T^\V$. Since $\T$ is regular in $\V$ and $k$ is
countable, $\cup k=\lambda<\T^\V$, so any $t$ as above is coded by a
subset of $\lambda$. The proof now finishes in the manner of the
proof of \rlem{fintheta}, with first a map in $\V$ from $\R$ onto $2^\lambda$
and then a
map onto $\T^\V$, producing a contradiction.
\ltpf
We remark that since $\V\sat$``$\Theta$ is regular'',
\rlem{countpreserve} and \rlem{counttheta} imply $\N\sat$``$\Theta$ is
regular'' as well.
\sec {Including Ultrapowers}\label{nine}
In our chosen universe, not every regular cardinal is a member of a
sequence satisfying $(k)\rightarrow(k)^k$.
If $\mu$ is a normal measure on $\k$, we will use $\k_\mu$ to represent
the ultrapower, $\k^\k/\mu$, of $\k$ via $\mu$. These cardinals satisfy
powerful partition properties but fall short of being strong partition
cardinals. The following is due to E. M. Kleinberg.
\fact \label{ultdo} If $\k$ satisfies $\k\rightarrow(\k)^\k$,
then
$\k_\mu\rightarrow(\k_\mu)^\d$, for all $\d<\A$. Further, if
$\k^\k/\mu=\k^+$, then $\k_\mu\rightarrow(\k_\mu)^\d$, for all
$\d<\k_\mu$. At the same time,
$\k_\mu\rightarrow\!\!\!\!\!\!/\,\,\hskip1pt(\k_\mu)^{\k_\mu}$. {\em (See
\cite{Kl1}, \cite{Kl2}.)}\efact
This implies that if $\mu$ is a
normal measure on $\k\in k$ and $(k)\rightarrow(k)^k$, then $\k_\mu$ cannot be
in any
sequence satisfying the same property. Under AD, many regular
cardinals are of this sort.
Ultrapowers can, however, be in a
sequence of less extravagant power.
In this section, we assume DC throughout.
\prop \label{addingult} Suppose $(k)\rightarrow(k)^k_{\A}$ and $k^+$ is
a countable sequence composed
of members of $k$ and
ultrapowers of members of $k$. Then
$(k^+)\rightarrow(k^+)^{<\A}$.\eprop
Proof: We will need some of the extensive machinery developed by
Kleinberg. Familiarity with his work is useful, but not necessary.
For $\k$ measurable, $\mu$ a normal measure on $\k$, and for
$p\in[\k]^\k$, let $\lb
p\rb^\mu$ be the ordinal below $\k_\mu$ represented by the
increasing sequence $p$. Let $S_\mu^p=\{\lb
q\rb^\mu:q\in[p]^\k\}$. Central to Kleinberg's work
are the
``break'' and ``shuffle'' functions:
$bk_\zeta:[\k]^\k\rightarrow\left[[\k]^\k\right]^\zeta$ and
$sh_\zeta:\left[[\k]^\k\right]^\zeta\rightarrow[\k]^\k$, for all $\zeta
<\k^{^+}$. We list
some
of his results.
\fact \label{bksh} Assume $\mu$ is a normal measure on $\k$. Then\ret
a) For all $\alpha<\zeta <\k^{^+}$, $p\in [\k]^\k$, $\lb bk_\zeta
(p)(\alpha)\rb^\mu =S^p_\mu(\alpha )$.\ret
b) For all $\zeta <\k^{^+}$, $p\in [\k]^\k$, $sh_\zeta(bk_\zeta(p))=p$.\ret
c) For all $\alpha<\zeta <\k^{^+}$, $\{p_\delta\}_{\delta<\zeta}\in
\left[[\k]^\k\right]^\zeta$,
$\lb bk_\zeta(sh_\zeta(\{p_\delta\}_{\delta
<\zeta}))(\alpha)\rb^\mu =\lb p_\alpha\rb^\mu$.
\efact
Now suppose we are given $f:[k^+]^{\d}\rightarrow 2$, $\d<\A$. Let \ret
$g_\d:[k]^k\rightarrow 2$
be defined by $g_\d(W)=f(W^{(\d)})$, where
$$W^{(\d)}(\k)=\left\{\matll W(\k)\r\d&\hbox{\rm if $\k\in k$}\cr
\{\lb bk_{\d}(W(\g))(\a)\rb^\mu\}_{\a<\d}&\hbox{\rm if
$\k=\g_\mu$, $\g\in k$.}\emat\right.$$
Let $X\in[k]^k$ be homogeneous for $g_\d$. Then $X'\in[k^+]^{k^+}$ is
homogeneous for $f$, where
$$X'(\k)=\left\{\matll X(\k)&\hbox{\rm if $\k\in k$}\cr
S^{X(\g)}_\mu&\hbox{\rm if
$\k=\g_\mu$, $\g\in k$,}\emat\right.$$
since if $Y\in[X']^{\d}$, we can construct $W\in[X]^k$
such that $W^{(\d)}=Y$ as follows: There is no difficulty in dealing with a
cardinal $\g$ and its ultrapower at the same time (the value of $\{\lb
bk_{\d}(W(\g))(\a)\rb^\mu\}_{\a<\d}$ is independent of
$W(\g)\r\d$), but there is a difficulty when we have several measures on
a single $\g$. For this we use a lemma.
\lem \label{atonce} Given $\{\mu_n\}_{n\in\o}$, normal measures on
$\k$, $p\in[\k]^\k$, and $q_n\in\left[S^p_{\mu_n}\right]^\d$, $\d<\A$,
there exists $r\in[p]^\k$ such that for each $n$, $q_n=\lb
bk_\d(r)\rb^{\mu_n}$. \elem
Proof: We first find $r_0\in[p]^\k$ such that $q_n(0)=\lb
r_0\rb^{\mu_n}$ for all $n$. Begin by partitioning $\k$ into disjoint
sets, $\k=\underset{n\in\o}{\cup}A_n$ such that for each $n$,
$\mu_n(A_n)=1$. Choose for each $n$ $t_n\in[p]^\k$ such that $\lb
t_n\rb^{\mu_n}=q_n(0)$. Now set
$r_0(\a)=\underset{\b<\a}{\cup}r_0(\b)\cup t_n(\a)$, where $\a\in A_n$.
The set, $\{\alpha\,|\,\underset{\b<\a}{\cup}r_0(\b)=\a\}$, is closed and
unbounded (closed is obvious; for unbounded, note that for any
$\alpha$, the sup of $r_0(\alpha),
r_0(r_0(\alpha)), r_0(r_0(r_0(\alpha))), ...$ is in the set) and consequently
for $\mu_n$--measure one many $\a$,
$r_0(\a)=t_n(\a)$, and so $\lb
r_0\rb^{\mu_n}=\lb
t_n\rb^{\mu_n}=q_n(0)$. To complete the proof, we do this for $r_\xi$
and $q_n(\xi)$ for all $\xi<\d$ and set $r=sh_\d(\{r_\xi\}_{\xi<\d})$. \lpf
Returning to our task, from $(k)\rightarrow(k)^k$ we infer
$(k^+)\rightarrow(k^+)^{\d}$. Given now a partition
$f:[k^+]^{<\A}\rightarrow 2$, we can define an auxiliary partition
$g:[k]^k\rightarrow \A+1$ by $g(W)=\d$ iff $\d$ is least such that $W$
is not homogeneous for $g_\d$. Let $X\in[k]^k$ be homogeneous for
$g$. The range of $g$ on $[X]^k$ cannot be $\d<\A$, since we can
always find $Y\in[X]^k$ homogeneous for a particular $g_\d$. Thus
$X'$, as defined above, is homogeneous for $f$.
\ppf
The extra assumption of the subscript poses no problem for
our applications. The following proposition gives us this, assuming
$\A^{\A}/\mu=\o_2$.
\prop \label{getsubb} If $(\A\cat k)\rightarrow (\A\cat k)^{\A\cat
k}$ and $\o_1^{\o_1}/\mu=\o_2$, $\mu$ normal on $\o_1$, then
$(\o_2\cat k)\rightarrow(\o_2\cat k)^{\o_1\cat k}_{\A}$. \eprop
Proof: Suppose
$f:[\o_2\cat k]^{\o_1\cat k}\rightarrow \A$. For $W\in[\A\cat
k]^{\A\cat k}$, we will write $W^*$ for all but the first component of
$W$, that is, $W=W(0)\cat W^*$. Now define, for all $\d < \A$,
$f_\d:[\o_2\cat k]^{\o_1\cat k}\rightarrow 2$ and
$g:[\A\cat k]^{\A\cat
k}\rightarrow 2$ by $f_\d(X)=0$ iff $f(X)<\d$, and $g(W)=0$ iff $f_\d$ is
constant on
$\left\{S^{W(0)}_{\mu}\r\A\right\}\times [W^*]^{k}$
for all $\d\A$ satisfies
$\k\rightarrow(\k)^\k_{\A}$, or $\k=\A$ satisfies
$\k\rightarrow(\k)^\k$ and $\mu$ is a
normal measure on $\k$ with $\k^\k/\mu = \k^+$, then forcing with ${\mathcal
P}_{\A,\k_\mu}$ preserves all cardinals.\eprop
Proof: By \rprop{ultdo} and \rfact{seqtocard}, if $\k>\A$, we have
$\k_\mu\rightarrow(\k_\mu)^{<\A}$. If $\k=\A$, \rfact{$ extends $p=$ (for example, if $q$ is homogeneous for the
partition: $F(x)=0$ iff there is a member of $p_1$ between $bk_2(x)(0)$
and $bk_2(x)(1)$).
Let $\beta =h(q)$. Choose $s\in [S^{q}_\mu]^{\xi}$ for some
$\xi<\A$ with $\beta\in n_
s$. We wish to proceed as before but there is a
difficulty. We want to take the part of $S^{q}_\mu$ above $\cup s$,
then add $
s$ as in
the previous proof. The problem is that we are working at the
$\k$--level. If we thin $q$ to $r$ so that the least element of $
S^r_\mu$ is above $s$,
it may not be true that $h(r)\le h(q)$, that is, intervals of
$S^r_\mu$ are not necessarily intervals of $S^{q}_\mu$ since the former
is
not a final segment of the latter.
To deal with this, choose, for each $\a<\xi$, $v_\a\in[q]^\k$
such that $\lb v_\a\rb=s(\a)$. Let $w=sh_\xi(\< v_\a:\a<\xi\> )$,
so that $\lb bk_\xi(w)\rb=s$. Now
form $t,r\in [q]^{\k}$ as follows: place the first $\xi$--many
elements of $w$ in $t$. Next place the first
element of $q$ above $t$ in $r$. In general, at stage
$\alpha$, place the first $\xi$ elements of $w$ above $r$ so far defined
in $t$
and then place the
first $\rho$ elements of $q$ above this in $r$, where $\rho$ is the
least indecomposable greater than $\alpha$ (indecomposable means that
the sum of ordinals less that $\rho$ is again less than $\rho$).
Since $q\supseteq t\cup r\supseteq r$, $h(q)\le h(t\cup r)\le
h(r)$. Kleinberg's methods give us that $\lb bk_\xi(t\cup
r)\rb=\lb bk_\xi(t)\rb=\lb bk_\xi(w)\rb=s$ (since for a closed,
unbounded set of ordinals $\alpha$, $t$, $t\cup r$, and $w$ all have the
same $\alpha^{th}$ element and the same $(\alpha+\eta)^{th}$ elements for
all $\eta<\xi$), so that $\beta\in n_{S^{t\cup r}_\mu}$ (and so
$h(t\cup r)\neq\beta$).
Claim: every
interval of $S^{r}_\mu$ of length less than $\A$ is also an interval of
$S^{q}_\mu$. This claim completes the proof, since the definition of
$h$ and $n_t$ will then give us $h(r)\le h(q)$, so $h(t\cup r)=h(q)=\beta$,
a contradiction. Proof of the claim: suppose that $x$ is a
$\tau$--sequence from $S^{r}_\mu$, $\tau<\A$, forming an interval in
$S^{r}_\mu$. Let
$a\in [\k ]$$^{\k}$ be such that
$x(0)$ is represented by $r$ composed with $a$. For all $\sigma <\tau$,
the $\k$--sequence $\left\{r(a(\alpha )+\sigma )\right\}_{\alpha <\k}$
represents $x(\sigma )$. But for each $\alpha>\tau$,
$r(a(\alpha ))$ is
in a sequence of members of $q$ of indecomposable length greater than
$\tau$, hence $r(a(\alpha )+\sigma )$ is an element of $q$ and so
$x=\lb bk_\tau(r\comp a)\rb$ is an interval of $S^q_\mu$. \ppf
\sec{Including Ultraproducts}\label{ten}
Ultraproducts of members of partition
sequences can be included as well. We assume DC throughout this section.
If $\mu$ is a countably additive measure on $\k$ and $k$ is a $\k$--sequence of
cardinals, we will use $k_\mu$ to represent
the ultraproduct, $\left[\pr{\a\in k}\a\right]/\mu$. For $q\in
\pr{\a\in k}\a$, we will write $\lb q\rb^\mu$ for
the ordinal corresponding to $q$ in $\left[\pr{\a\in k}\a\right]/\mu$. If
$X\in[k]^k$, we will abbreviate
$\left[\pr{\a\in k}X(\a)\right]/\mu$ by $\Pi^X_\mu$.
\prop \label{ulpdoes} If $k$ satisfies $(k)\rightarrow(k)^\d$, $\mu$ a
countably additive measure on len($k$), $\d\cup s$ and set $r=\pr{\k\in k}(q(\k)\minus y(\k))$.
Let $t=\pr{\k\in k} \left(r(\k)\cup s'(\k)\right)$. Then as in
\rprop{kpreserve},
\[h\left(\Pi^q_\mu\right)\le h\left(\Pi^t_\mu\right)\le
h\left(\Pi^r_\mu\right)\]
by our construction of $q$, and
\[h\left(\Pi^r_\mu\right)\le h\left(\Pi^q_\mu\right)\]
by the definition of $h$. But
$\b=h\left(\Pi^r_\mu\right)< h\left(\Pi^t_\mu\right)$, a contradiction. \ppf
Since the value $\lb p\rb^\mu$, of a sequence $p$ in an ultrapower is
independent of initial segments, we can manage both ultraproducts and
ultrapowers at once. The following proposition sums this up:
\prop \label{prodsum2} Suppose that $(\A\cat k)\rightarrow (\A\cat
k)^{\A \cat k}$, $k(0)>\A$, $\A ^{\A }/\mu=\o_2$ for some normal measure $\mu$
on $\A $, $k^+$ is
a countable sequence composed
of members of $k$,
ultrapowers of members of $k$, and ultraproducts of members of $k$,
and that there is no
non--principal ultrafilter on $\o$. Then $(k^+)\rightarrow(k^+)^{<\A }$.
Further, ${\mathcal
P}_{\A,\o_2\cat k^+}$ changes the cofinality of
cardinals with cofinalities in $\o_2\cat k^+$ to $\A $ while leaving all other
cofinalities unchanged. \eprop
\sec{Applications}\label{eleven}
\thm Con({\em ZF} + {\em AD}) $\Rightarrow$
Con({\em ZF} + {\em DC} + $\o_1$ is the only regular uncountable
cardinal $\le \o_{\omega_1 + 1}$).\ethm
Proof: Let $\V' \models $ ZF + AD, and let
$\V = {L(\R )}^{\V'}$. By the work of [J1] and [J2], the remarks
of Section 2 and the second paragraph immediately following
Definition 5.1, and Corollary 5.1, the odd
$\bd^1_\alpha$ for $\alpha < \o_1$ form a sequence
satisfying the strong polarized partition property such
that every regular cardinal $\delta < \o_{\omega_1}$
is either so that $\delta = \bd^1_\alpha$
or $\delta = {\bd^1_\alpha}^{\bd^1_\alpha} / \mu$
for some normal measure $\mu$ on $\bd^1_\alpha$. By the
claim on page 152 of [St2] (using $h(\alpha) =
\bd^1_\alpha$ for $\alpha < \A $ as the function of
this claim), $\o_{\omega_1 + 1}$ is the ultraproduct
as in Section 10 via the unique normal measure $\mu_\omega$
on $\A $ (generated by the filter of c.u.b$.$ sets)
of the odd $\bd^1_\alpha$ for $\alpha < \A $. Thus,
since Martin has shown that assuming AD, for $\bd^1_1
= \A $,
${\A }^{\A } / \mu_\omega = \o_2$
(see [Kl2] for a proof of this fact),
by Proposition 10.5,
for any countable subsequence $s$ of the regular cardinals in
the interval $[\o_2, \o_{\omega_1 + 1}]$,
$(s) \rightarrow {(s)}^{< \omega_1}$. This means the model
$\N$ of Theorem 8.1 constructed using the set $A$ of regular
cardinals in the interval $[\o_2, \o_{\omega_1
+ 1}]$ is so that $\N \models ``$ZF + DC +
$\A $ is the only regular uncountable cardinal
$\le \o_{\omega_1 + 1}$''. This proves Theorem 11.1. \tpf
We remark that by Lemma 8.1 and Proposition 10.5,
in $\N$, for $\delta$ a cardinal in $\V$,
${\hbox{\rm cof}}(\delta) = \A $ if ${\hbox{\rm cof}}(\delta) \in A$, and
${\hbox{\rm cof}}(\delta) = {\hbox{\rm cof}}^\V(\delta)$
otherwise. Also, by Lemma 8.1 and Proposition 6.2,
${\A }^{\A } / \mu_\omega = \o_2$
is true in $\N$.
In conclusion, we remark that the desired ultimate goal is to
assume $\V' \models {\hbox{\rm AD}}$, force over
$\V = {L(\R )}^{\V'}$, and obtain a model $\N$ so that
$\Theta^\N = \Theta^\V$ and
$\N \models ``$ZF + DC + $\A $ is the only regular
uncountable cardinal $< \Theta$''. (A model $\N$ for
``ZF + $\A $ is the only regular uncountable
cardinal $< \Theta$'' in which ${\hbox{\rm AC}}_\omega$
is false was constructed in [A].) Theorem 8.1 provides us
with a possible way of constructing such a model. If one
could show that the set $A$ of Theorem 8.1 were so that
$A$ could be composed of all
$\V = {L(\R )}^{\V'}$-regular cardinals in the interval
$[\o_2, \Theta)$, then the model
$\N$ of Theorem 8.1 would be our desired model. In fact, even though the
unpublished analysis mentioned in
Section 2 may be difficult to extend beyond the first inaccessible,
and even if the conjecture given in the last paragraph of
Section 2 is false,
it is still conceivable that the aforementioned structural
conjecture about the regular cardinals in the interval
$[\o_2, \Theta)$ assuming AD + $\V = L(\R )$ is
true. Whether this is indeed the case remains open.
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\end{document}
~~