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\title{On a Problem of Foreman and Magidor
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords:
Supercompact cardinal, indestructibility,
almost huge cardinal, mutual stationarity,
symmetric inner model}}
\author{Arthur W.~Apter
\thanks{The author wishes to thank
James Cummings
for helpful correspondence on the
subject matter of this paper.
The author also wishes to thank
the referee and Andreas Blass,
the corresponding editor,
for helpful comments and
suggestions that have been
incorporated into this version
of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{February 22, 2004\\
(revised June 6, 2004)}
\begin{document}
\maketitle
\begin{abstract}
A question of Foreman and Magidor
asks if it is consistent for
every sequence of stationary
subsets of the $\ha_n$'s
for $1 \le n < \go$ to
be mutually stationary.
We get a positive answer to
this question in the context
of the negation of the
Axiom of Choice.
We also indicate how a positive
answer to a generalized
version of this question
in a choiceless context may be obtained.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{FM}, Foreman and Magidor
introduced the concept of mutual
stationarity by giving the following
definition.
\begin{definition}\label{d1}
Let $\cal K$ be a set of regular
cardinals having supremum $\gl$.
Suppose that $S_\gk \subseteq \gk$
for each $\gk \in {\cal K}$. Then
$\la S_\gk : \gk \in {\cal K} \ra$
is mutually stationary iff for all
algebras $\mathfrak A$ on $\gl$,
there is an elementary substructure
$\mathfrak B \prec \mathfrak A$ such that
for all $\gk \in {\mathfrak B} \cap
{\cal K}$, $\sup({\mathfrak B} \cap \gk)
\in S_\gk$.
\end{definition}
Note that in Definition \ref{d1},
$\mathfrak A$ is an
{\it algebra on $\gl$} iff
${\mathfrak A} = \la \gl, \in,
\la f_n : n < \go \ra \ra$,
where each $f_n$ is a function from a
finite cartesian product of $\gl$ to $\gl$.
%itself.
Many interesting properties of mutual
stationarity were proven by Foreman
and Magidor in \cite{FM}.
In particular, Theorem 7 of
\cite{FM} tells us that if
$\la \gk_\ga : \ga < \gb \ra$ is
an increasing sequence of regular cardinals
and $S_\ga \subseteq \gk_\ga$ is stationary
and consists of ordinals of cofinality
$\go$, then $\la S_\ga : \ga < \gb \ra$
is mutually stationary.
Theorem 24 of \cite{FM} indicates,
however, that in $L$,
the analogous property fails if
the sets in the sequence $\la S_\ga : \ga < \gb \ra$
consist of points of uncountable cofinality.
In particular, there is a sequence of
stationary sets $\la S_n : 1 < n < \go \ra$
such that $S_n \subseteq \ha_n$, $S_n$ is
stationary and consists of points having
cofinality $\ha_1$, yet
$\la S_n : 1 < n < \go \ra$ isn't
mutually stationary.
This result was generalized by Schindler
\cite{Sc} to show that a similar fact
%the same fact
is true in higher core models.
It is thus not a theorem of ZFC that
if $\cal K$ consists of an increasing
sequence of regular cardinals
and for each $\gk \in {\cal K}$,
$S_\gk \subseteq \gk$ is stationary in
$\gk$, then $\la S_\gk : \gk \in {\cal K} \ra$
is mutually stationary.
In fact, the results of the preceding
paragraph tell us that this can
fail in the $\ha_n$'s.
However, as Foreman and Magidor implicitly
do in \cite{FM}, page 290,
it is still possible to ask the
following
\bigskip\setlength{\parindent}{0pt}
%\begin{question}\label{q1}
Question: Is it possible to
construct a model of ZFC in which if
$\la S_n : 1 \le n < \go \ra$
is such that $S_n \subseteq \ha_n$
and $S_n$ is stationary, then
$\la S_n : 1 \le n < \go \ra$
is mutually stationary?
%\end{question}
\bigskip\setlength{\parindent}{1.5em}
Note that there is a positive
relative consistency result in this regard.
It is a theorem of
Shelah (see Section 6 of \cite{CFM}) that starting
from an $\go$ sequence of supercompact
cardinals, it is possible to
construct a model of ZFC in which in the
$\ha_n$'s, there are
%mutual stationarity
%sequences not concentrating on a fixed cofinality.
sequences of stationary sets which are
mutually stationary which do not
concentrate on a fixed cofinality.
More explicitly, in this model, suppose
$f : \go \to 2$ is an arbitrary function, and
we define
$S^f_n = \{\ga < \ha_n : {\rm cof}(\ga) =
\ha_{f(n)}\}$ if $n > 1$. We then have that
the sequence
$\la S^f_n : 1 < n < \go \ra$ is mutually stationary.
Unfortunately, though, this theorem doesn't give an
answer to the aforementioned Question.
The purpose of this note is to provide a
complete answer in a choiceless context
to our Question above.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC +
$\la \gk_n : n < \go \ra$ is a
sequence of supercompact cardinals
whose supremum is $\gl$''.
There is then a partial ordering
$\FP \in V$ and a symmetric inner model
$N \subseteq V^\FP$ such that
$N \models ``$ZF + DC + If
$\la S_n : 1 \le n < \go \ra$ is a sequence
of sets such that
%for $n \in \go$,
$S_n \subseteq \ha_n$ and $S_n$ is
stationary, then $\la S_n : 1 \le n < \go \ra$ is
mutually stationary''.
\end{theorem}
Before beginning the proof of
Theorem \ref{t1}, we note that
our terminology and notation
will be fairly standard.
For anything left unexplained,
readers are urged to consult \cite{A98}.
We do wish to explicitly state and prove,
however, the following theorem
from \cite{CFM}, which will be
key to the proof of Theorem \ref{t1}.
\begin{theorem}\label{t2}
{(\bf Foreman and Magidor --- Theorem 5.2 of \cite{CFM})}
Let $\la \gk_i : i < \gd \ra$ be
an increasing sequence of measurable
cardinals, where $\gd < \gk_0$ is
a regular cardinal.
Let $S_i \subseteq \gk_i$ be
stationary for each $i < \gd$.
It is then the case that
$\la S_i : i < \gd \ra$ is
mutually stationary.
\end{theorem}
\begin{proof}
The proof we give is taken from
\cite{CFM}. We will
quote from \cite{CFM} almost verbatim,
feeling free to make minor modifications
as necessary.
We thank James Cummings for his kind
permission to include the argument.
Note that the hypotheses imply that for all
$i<\gd$, $\kappa_i> \sup(\seq{\kappa_j:j* \max(a_i)$.
It must be that $t(a_0, \ldots, a_j) < \gb$,
for if not, an application of indiscernibility shows that
every element of $J_i$ which is greater
than $\max(a_i)$ is bounded by $t(a_0, \ldots, a_j)$.
This is impossible, since $J_i$ is unbounded in $\gk_i$.
This shows that $t(a_0, \ldots, a_j) < \sup(z_i)$, so
$\sup({\mathfrak B} \cap \gk_i) = \sup(z_i)$.
%and we are done.
This completes the proof of Theorem \ref{t2}.
\end{proof}
%\noindent For a proof of Theorem \ref{t2},
%we refer readers to \cite{CFM}.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V \models ``$ZFC +
$\la \gk_n : n < \go \ra$ is a
sequence of supercompact cardinals
whose supremum is $\gl$''.
%Let $V$ be as in the hypotheses of Theorem \ref{t1}.
Let $\FQ$ be the iteration of either \cite{A83} or
\cite{A98} which makes the supercompactness of each $\gk_n$
Laver indestructible \cite{L} under
$\gk_n$-directed closed partial
orderings. Without loss of generality, as in either \cite{A83} or
\cite{A98}, we can assume that $V$ is a model constructed via
forcing over an earlier model with $\FQ$, i.e., that $V \models
``$The supercompactness of each $\gk_n$ is indestructible under
$\gk_n$-directed closed forcing''.
Take as before $\gk_{-1} = \go$.
%Let $\FP_0 = {\rm Coll}(\go, {<} \gk_0)$.
For each $n \ge 0$, let
$\FP_n = {\rm Coll}(\gk_{n - 1}, {<} \gk_n)$, i.e., the
usual L\'evy collapse of all cardinals in the interval
$(\gk_{n - 1}, \gk_n)$ to
$\gk_{n - 1}$. We then define
$\FP = \prod_{n < \omega} \FP_n$.
Let $G$ be $V$-generic over $\FP$. $V[G]$, being a model of AC, is
%not our desired model $N$.
not the model witnessing the
conclusions of Theorem \ref{t1}.
In order to define $N$, we first note that
by the Product Lemma, $G_n$,
the projection of $G$ onto $\FP_n$, is
$V$-generic over $\FP_n$, and $\prod_{j < n} G_j$ is
$V$-generic over $\prod_{j < n} \FP_j$. Next,
%take $\gk_{-1} = \go$, and
let ${\cal F} = \{f : f :
\omega \to \gl$
%\sup_{n < \omega}(\k_n)$
is a function for which $f(n) \in
(\gk_{n - 1}, \gk_{n})\}$.
For any $f \in {\cal F}$,
for $n < \go$, define
$G_n \rest f(n) = \{p \in G_n :
\dom(p) \subseteq \gk_{n - 1}
\times f(n)\}$, and take
$G \rest f = \prod_{n < \go} (G_n \rest f(n))$. Note
that by the Product Lemma and the properties of the L\'evy collapse,
$G \rest f$ is $V$-generic over $\prod_{n < \omega} (\FP_n \rest f(n))$,
where $\FP_ n \rest f(n) =
\{p \in \FP_n : \dom(p) \subseteq \gk_{n -1}
\times f(n)\}$. $N$ can now intuitively be
described as the least model
of ZF extending $V$ which contains, for every $f \in {\cal F}$, the set
$G \rest f$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified sublanguage of the forcing language $\cal L$ with respect to
$\FP$ which contains symbols $\check v$ for each $v \in V$, a unary
predicate symbol $\check V$ (to be interpreted
$\check V(\check v) \iff v \in
V$, i.e., $\check V$ allows us to
determine members of the ground model),
%those members of $N$ which are present in the ground model),
and symbols $\dot G \rest f$
for each $f \in {\cal F}$.
$N$ is
then defined as follows.
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gd = \bigcup_{\ga < \gd} N_\ga$ if $\gd$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}
The standard arguments show $N \models {\rm ZF}$.
In addition, the current $N$ is
the model $N$ of Theorem 1 of \cite{A83}
constructed using an $\go$ sequence of
indestructible supercompact cardinals.
Therefore, by the proofs of Lemma 1.7 and
Lemmas 1.1 - 1.4 of \cite{A83},
$N \models ``$DC + For each $n$ such
that $0 \le n < \go$,
$\gk_{n - 1} = \ha_{n}$ +
$\ha_n$ is a regular cardinal for
$0 \le n < \go$''.
In fact, by the proof of Lemma 1.2
of \cite{A83} (see also the discussion given
in the proof of Theorem 4 of \cite{AH}),
$N \models ``\ha_n$ is a Ramsey cardinal for
$1 \le n < \go$''.
The proof of Theorem \ref{t1}
is now completed by the following lemma.
\begin{lemma}\label{l1}
In $N$, any sequence of sets
stationary in the $\ha_n$'s for
$1 \le n < \go$ is mutually stationary.
\end{lemma}
\begin{proof}
Suppose that
$N \models ``\mathfrak A$ is an
algebra on $\gl$ and $\la S_n : n < \go \ra$
is such that $S_n \subseteq \gk_n$
and $S_n$ is stationary in $\gk_n$''.
Note that each of $\mathfrak A$
and $\la S_n : n < \go \ra$
may be coded by sets of ordinals.
Therefore, by Lemma 1.1 of \cite{A83},
there is some $f \in {\cal F}$
for which both
$\la S_n : n < \go \ra \in
V[G \rest f]$ and
${\mathfrak A} \in V[G \rest f]$.
We observe that for each $n < \go$,
$V[G \rest f] \models ``\gk_n$ is measurable''.
We show this by writing
$\FP = (\prod_{n + 1 \le m < \go}
(\FP_m \rest f(m)) \times
(\prod_{0 \le m \le n}
(\FP_m \rest f(m)) = \FQ_1 \times \FQ_0$.
By its definition, $\FQ_1$ is $\gk_n$-directed
closed, so by indestructibility,
$V[\prod_{n + 1 \le m < \go} (G_m \rest f(m))] \models
``\gk_n$ is supercompact''.
In particular,
$V[\prod_{n + 1 \le m < \go} (G_m \rest f(m))] \models
``\gk_n$ is measurable''.
Further, by its definition,
$\card{\FQ_0} < \gk_n$ in both
$V$ and $V[\prod_{n + 1 \le m < \go} (G_m \rest f(m))]$,
so by the L\'evy-Solovay results \cite{LS},
$V[(\prod_{n + 1 \le m < \go} (G_m \rest f(m)) \times
(\prod_{0 \le m \le n} (G_m \rest f(m))] = V[G \rest f] \models
``\gk_n$ is measurable''.
In addition, for each $n < \go$,
$V[G \rest f] \models ``S_n$ is a
stationary subset of $\gk_n$''.
To see this, suppose that
$C \in V[G \rest f] \subseteq N$ is a closed,
unbounded subset of $\gk_n$ in
$V[G \rest f]$.
Because the notion of closed, unbounded
subset of $\gk_n$ is upwards absolute,
$C$ is also a closed,
unbounded subset of $\gk_n$ in $N$, i.e.,
since $S_n$ is stationary in
$\gk_n$ in $N$, $S_n \cap C \neq \emptyset$.
Thus, as $\mathfrak A$ is by
definition still an algebra on $\gl$ in
$V[G \rest f]$, by Theorem \ref{t2},
we can let
${\mathfrak B} \prec {\mathfrak A}$,
${\mathfrak B} \in V[G \rest f]$
be an elementary substructure of $\mathfrak A$
such that for all
$n < \go$,
%with $\gk_n \in {\mathfrak B}$,
$\sup({\mathfrak B} \cap \gk_n) \in S_n$.
This means $\la S_n : n < \go \ra$
is mutually stationary in $V[G \rest f]$.
Since the facts mentioned in the preceding
two sentences all remain true in $N$ as well
as in $V[G \rest f]$,
$\la S_n : n < \go \ra$ is mutually
stationary in $N$.
This completes the proof of
Lemma \ref{l1} and Theorem \ref{t1}.
\end{proof}
\end{proof}
We remark that the techniques used
in the proof of Theorem \ref{t1}
will yield a generalized version of
Theorem \ref{t1}, if we begin with
stronger hypotheses.
In particular, in Theorem 1 of \cite{A85},
starting from a model of ZFC containing
an almost huge cardinal $\gk$, a model
$N$ of height $\gk$ is constructed
(in the notation of \cite{A85},
the model $N_A$ in which $B = \emptyset$)
such that
$N \models ``$ZF + $\neg \rm{AC}_\go$ +
Every successor cardinal is a Ramsey cardinal
(and hence is regular) + Every limit cardinal
is a singular Jonsson cardinal''.
(The initial hypothesis of almost hugeness
was weakened somewhat in \cite{A92}).
Let ${\cal S} = \la \gk_\ga : \ga < \gk \ra$ be the
sequence of cardinals which becomes
the successor cardinals of $N$.
Suppose further that
${\cal S}' = \la \gl_\ga : \ga < \gb \ra \in N$
is a subsequence of $\cal S$ having
supremum $\gl$ such that
$\gb < \gl_0$ is a regular
cardinal in $N$, ${\mathfrak A} \in N$
is an algebra on $\gl$, and
$\la S_\ga : \ga < \gb \ra \in N$ is
such that in $N$,
$S_\ga \subseteq \gl_\ga$
and $S_\ga$ is stationary in $\gl_\ga$.
By Lemma 1.3 of \cite{A85} and
the symmetric properties of $N$,
there
is a submodel $V[H] \subseteq N$
satisfying the Axiom of Choice such that
${\mathfrak A}, {\cal S}', \la S_\ga :
\ga < \gb \ra \in V[H]$ and
$V[H] \models ``$Each
member of ${\cal S}'$ is measurable''.
Therefore, the proof of Lemma \ref{l1}
%Theorem \ref{t1} of this paper
yields the following theorem.
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC +
$\gk$ is almost huge''.
There is then a partial ordering
$\FP \in V$ and a symmetric inner model
$N \subseteq V^\FP$ having height $\gk$
such that
$N \models ``$ZF + $\neg${AC}${}_\go$ +
Every successor cardinal is a Ramsey cardinal +
Every limit cardinal
is a singular Jonsson cardinal''.
Further, in $N$, if
$\la \gl_\ga : \ga < \gb \ra$ is
a sequence of regular cardinals such that
$\gb < \gl_0$ and
$\la S_\ga : \ga < \gb \ra$ is a sequence
of sets such that each $S_\ga$ is
a stationary subset of $\gl_\ga$, then
$\la S_\ga : \ga < \gb \ra$ is mutually stationary.
\end{theorem}
\noindent Of course, neither Theorem \ref{t2} nor
Theorem \ref{t3} answers the Question posed
in Section \ref{s1}, which unfortunately
remains open.
\begin{thebibliography}{99}
\bibitem{A98} A.~Apter,
``Laver Indestructibility and the Class of
Compact Cardinals'', {\it Journal of
Symbolic Logic 63}, 1998, 149--157.
\bibitem{A83} A.~Apter, ``Some
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\end{thebibliography}
\end{document}
\begin{graveyard}
Many interesting properties of mutual
stationarity were proven by Foreman
and Magidor in \cite{FM}.
In particular, Theorem 7 of
\cite{FM} tells us that if
$\la \gk_\ga : \ga < \gb \ra$ is
an increasing sequence of regular cardinals
and $S_\ga \subseteq \gk_\ga$ is stationary
and consists of points of cofinality
$\go$, then $\la S_\ga : \ga < \gb \ra$
is mutually stationary.
Theorem 24 of \cite{FM} indicates,
however, that in $L$,
the analogous property fails if $S_n$
consists of points of uncountable cofinality.
In particular, there is a sequence of
stationary sets $\la S_n : 1 \le n < \go \ra$
such that $S_n \subseteq \ha_n$, $S_n$ is
stationary and consists of points having
cofinality $\ha_1$, yet
$\la S_n : 1 \le n < \go \ra$ isn't
mutually stationary.
This result was generalized by Schindler
\cite{Sc} to show that the same fact
is true in higher core models.
It is easy to see that if
$\la S_\gk : \gk \in {\cal K} \ra$
is mutually stationary, then
each $S_\gk$ is a stationary
subset of $\gk$.
The converse of this statement, however,
has turned out to be of great interest.
To see this, suppose that
$C \in V[G \rest f] \subseteq N$ is a closed,
unbounded subset of $\gk_n$ in
$V[G \rest f]$
and
$p \in \FP/(G \rest f)$ is such that
in $V[G \rest f]$,
$p \forces ``\la \gb_\ga : \ga < \gg \ra$
is an increasing sequence in $N$ of elements
of $C$ whose supremum is $\gb$''.
Since $\gk_n$ is regular in both
$V[G \rest f]$ and $N$, in $V[G \rest f]$,
$\{\gd < \gb : \gd \in C\}$ will contain
an increasing sequence of elements of $C$
whose supremum, which is of course a
member of $C$, must be $\gb$.
Hence, $C$ is also a closed,
unbounded subset of $\gk_n$ in $N$, i.e.,
since $S_n$ is stationary in
$\gk_n$ in $N$, $S_n \cap C \neq \emptyset$.
\end{graveyard}
*