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\title{On the Consistency Strength of
Two Choiceless Cardinal Patterns
\thanks{2000 Mathematics Subject Classifications:
primary 03E45, 03E55; secondary 03E40}}
\date{May 25, 1999\\
(revised May 14, 2000 and July 12, 2000)}
\author{Arthur W.~Apter
\thanks{The author owes a huge debt of gratitude
to the referee, whom he wishes to thank
for patience shown towards the author's questions,
for thoroughly reading the first and second
versions of the manuscript
and pointing out numerous
errors and suggesting numerous improvements,
and for providing the proof for the
improved form of Theorem \ref{t2} given at
the end of the paper. The referee's influence
is omnipresent in this version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\
awabb@cunyvm.cuny.edu\\
http://math.baruch.cuny.edu/$\sim$apter}
\begin{document}
\maketitle
\begin{abstract}
Using work of Devlin and Schindler
in conjunction with work on Prikry
forcing in a choiceless context
done by the author, we show
that two choiceless cardinal patterns have
consistency strength of at least one
Woodin cardinal.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
One of the most intriguing problems in large
cardinals without the Axiom of Choice is
to obtain a model for the theory $T_1$ =
``ZF + ${\rm DC}_\gk$ +
$\gk$ is a strong limit cardinal + Both
$\gk$ and $\gk^+$ are measurable cardinals''.
It was partially with this motivation
that Bull, in his 1976 MIT thesis,
proved the following theorem (published
in \cite{B}):
\begin{theorem}\label{t1}
Con(ZFC + $\gk < \gl$ are cardinals so
that $\gk$ is $\gl$ supercompact and
$\gl$ is measurable) $\implies$
Con(ZF + $DC_\gk$ +
$\gk$ is both a strong limit and Ramsey cardinal +
$\gk^+$ is measurable).
\end{theorem}
\setlength{\parindent}{0in}
For the duration of this paper,
we will refer to the weakening
of $T_1$ for which the relative
consistency was obtained in Theorem \ref{t1}
as $T_2$.
\setlength{\parindent}{1.5em}
It is unfortunately the case, as Bull showed in
\cite{B}, that $\gk$ isn't measurable in the
model for Theorem \ref{t1}.
Nonetheless, Bull's result was the starting
point for many investigations into the
relative consistency of sequences of
consecutive large cardinals, such as those
carried out and exposited in
\cite{A83}, \cite{A85}, and \cite{AHe1}.
It is interesting to note that Bull's
technique and the generalizations thereof
given in \cite{A83} and \cite{A85}
require the use of supercompactness.
This issue is addressed in greater
detail in \cite{AHa}.
This still does not address, however, the
exact consistency strength of either of
the theories $T_1$ or $T_2$.
Indeed, since neither of the cardinal
patterns given by $T_1$ and $T_2$
follows from AD, and both of these
cardinal patterns appear to have consistency
strength much stronger than AD,
it is quite conceivable that supercompactness
will turn out to be necessary in establishing
the relative consistency of both
$T_1$ and $T_2$.
The purpose of this note is to show that
earlier work on Prikry forcing in a
choiceless context given in \cite{A96}
combined with techniques Devlin
uses to prove Theorems 1 and 2 of
\cite{D} and Schindler uses to prove
Theorem 1 of \cite{S} yield,
modulo an assumption necessary to
carry out certain core model arguments,
that the consistency strength of a
weakening of $T_1$
different from either of the
hypotheses used in \cite{S}
is at least one Woodin cardinal.
Specifically, we prove the following.
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZF + DC +
$\gk$ is a limit cardinal +
Unboundedly many in $\gk$
successor cardinals $\gd < \gk$
are regular + $\gk$ carries a shrinking
filter + $\gk^+$ is measurable +
$\Omega > \gk^+$ is inaccessible
and $V^{\rm HOD}_\Omega$ is
closed under sharps''.
There is then an inner model with one
Woodin cardinal.
\end{theorem}
As an immediate corollary to Theorem
\ref{t2}, we obtain:
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZF + $\gk$ is a limit
cardinal + Unboundedly many in $\gk$
successor cardinals
$\gd < \gk$ are regular + $\gk$
carries a shrinking filter +
$\gk^+$ is singular +
$\Omega > \gk^+$ is inaccessible
and $V^{\rm HOD}_\Omega$ is
closed under sharps''.
There is then an inner model with one
Woodin cardinal.
\end{theorem}
The referee has pointed out that
it is possible to weaken the
hypotheses of Theorem \ref{t2} to
``ZF +
$\gk$ is a limit cardinal +
Unboundedly many in $\gk$
successor cardinals $\gd < \gk$
are regular + $\gk$ carries a shrinking
filter + $\gk^+$ is weakly compact +
$\Omega > \gk^+$ is inaccessible
and $V^{\rm HOD}_\Omega$ is
closed under sharps''.
We will provide details of the
referee's proof in Section \ref{s3}.
The terminology we use in Theorems
\ref{t2} and \ref{t3} is that given
in \cite{D} and \cite{S}.
Thus, as in \cite{S}, in a choiceless
context, a cardinal $\gd$ is inaccessible
iff for no cardinal $\ga < \gd$ does a
cofinal map
$f : V_\ga \to \gd$ exist.
As in \cite{D}, the cardinal $\gd$ carries
a shrinking filter $\cal D$ iff whenever
$A \in {\cal D}$ and
$f : {[A]}^{< \omega} \to \ga^+ < \gd$,
there is $B \subseteq A$, $B \in {\cal D}$
so that $|f''{[B]}^{< \omega}| \le \ga$.
Rowbottom's theorem implies that if
$\gd > \ha_1$ is a measurable cardinal carrying
a normal measure, then $\gd$ carries
a shrinking filter, i.e., the
normal measure itself.
However, as shown by Prikry in \cite{P},
it is possible for $\gd > \ha_1$ to carry a
shrinking filter and be singular.
To prove Theorems \ref{t2} and \ref{t3},
we will need the following two lemmas.
Lemma \ref{l1} appears as Lemma 3 of
\cite{S} and is a generalization of
a theorem of Vop\v enka.
Vop\v enka's original theorem can be found as
Theorem 65, pages 293--294 of \cite{J}.
Lemma \ref{l2} is a slight strengthening
of Lemma 4 of \cite{S}.
The results given in the first and
second sentences
of the conclusions of Lemma \ref{l2} are
due to Steel and can be found in
\cite{Sc} and \cite{St}.
%The result given in the second sentence
%of the conclusions of Lemma \ref{l2} is
%due to Steel and can be found in \cite{St}.
The results given in the third and fourth
sentences of the conclusions of Lemma \ref{l2}
are due to Mitchell, Schimmerling, and Steel
and can be found in \cite{MSS} and \cite{MS}.
%from \cite{S}.
\begin{lemma}\label{l1}
%(Lemma 3 of \cite{S})
Let $x \subseteq {\rm HOD}[a] \subseteq V$
for some $a$ and $x$.
There is then a partial ordering
$\FP \in {\rm HOD}[a]$ so that
$x$ is ${\rm HOD}[a]$-generic over $\FP$.
Further, if
$x \subseteq V^{{\rm HOD}[a]}_\ga$ for some
$\ga < \gd$, where
$V \models ``\gd$ is inaccessible'', then
$\FP$ can be chosen so that
$\FP \in V^{{\rm HOD[a]}}_\gd$.
\end{lemma}
\begin{lemma}\label{l2}
%(Lemma 4 of \cite{S})
Let $N \subseteq V$,
$N \models {\rm ZFC}$
be an inner model of $V$ so that
$N \models ``\gd$ is inaccessible
and $V_\gd$ is closed under sharps''.
Suppose there is no inner model of $V$ with a
Woodin cardinal.
Then the core model
$K = K^N$ inside $N$ up to $\gd$ exists and is
$\gd$ iterable.
Also, if $M = N^\FP$ for some
$\FP \in V^N_\gd$, then $K^M = K$.
Further, if
$N \models ``\ha_2 \le \ga < \gd$ is a cardinal'', then
$N \models ``{\rm cof}({(\ga^+)}^K) \ge \ga$''.
In particular, if
$N \models ``\ha_2 \le \ga < \gd$ is a singular
cardinal'', then
$N \models ``{(\ga^+)}^K = \ga^+$''.
\end{lemma}
\section{The Proofs of Theorems \ref{t2}
and \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
We assume $V$ is as in the hypotheses of
Theorem \ref{t2}.
Since $V \models ``$ZF + DC +
$\gk^+$ is measurable'', we know that
$\gk^+$ carries a normal measure.
Thus, by the work of Section 1 of
\cite{A96}, it is possible to
define a Prikry partial ordering
$\FP_1 \in V$ so that
$V_1 = V^{\FP_1} \models ``$The bounded
subsets of $\gk^+$ are the same
as in $V$ + $\gk^+$ is a singular
cardinal of cofinality $\omega$''.
Hence,
${(\gk^+)}^{V_1} =
{(\gk^+)}^V$, and since by hypothesis
$V \models ``\gk$ carries
a shrinking filter ${\cal D}$'',
$V_1 \models ``{\cal D}$ is a shrinking
filter'' as well.
Let $\FP_0 \in V_1$ be Prikry forcing over
$\gk$ defined using ${\cal D}$.
By Lemma 1.3 of \cite{A96},
which does not require that ${\cal D}$
be a normal measure,
$V^{\FP_0}_1 = V_2 \models
``\gk^+ = {(\gk^+)}^{V_1}$ and hence
is a singular cardinal (of cofinality
$\omega$)''.
We show now that
$V_2 \models ``\gk$ is a singular cardinal of
cofinality $\omega$''.
Clearly, the standard density arguments yield
$V_2 \models ``{\rm cof}(\gk) = \omega$'',
so it suffices to prove
$V_2 \models ``\gk$ is a cardinal''.
We do this by using the argument Devlin
employs in the proof of Theorem 1 of
\cite{D}. Specifically, we show
forcing with $\FP_0$ preserves
that every regular successor
cardinal below $\gk$ is a cardinal. Since
$V_1 \models ``\gk$ is a limit cardinal
and unboundedly many in $\gk$ successor
cardinals $\gd < \gk$ are regular'',
this will yield that forcing with
$\FP_0$ over $V_1$ preserves that
$\gk$ is a cardinal.
Therefore, if $\gd = \gl^+ < \gk$ is
the least $V_1$-regular successor
cardinal so that for some
$p = \la s_0, S_0 \ra \in \FP_0$,
$p \forces ``\gd$ isn't a cardinal'',
then we can assume without loss
of generality that
$p \forces ``\dot f : \gl \to \gd$ is
a surjection''.
For each $s \in {[S_0]}^{< \omega}$, let
$T_s = \{\gb < \gd$ : For
some condition
$q = \la s_0 {}^\frown s, A \ra$
extending $p$ and some $\ga < \gl$,
$q \forces ``\dot f(\ga) = \gb$''$\}$.
Note that if
$\la t, A \ra, \la t, B \ra \in \FP_0$,
$\la t, A \cap B \ra$ extends both of
these conditions. Thus, $T_s$ is
well-defined and has cardinality at most $\gl$.
Therefore, by our hypothesis that
$\gd$ is regular in $V_1$,
$F : {[S_0]}^{< \omega} \to \gd$ given by
$F(s) = \sup(T_s)$ is such that
$F(s)$ defines an ordinal below $\gd$.
Since ${\cal D}$ is a shrinking filter in both
$V$ and $V_1$,
this means that for some $S_1 \subseteq S_0$,
$S_1 \in {\cal D}$,
$|F''{[S_1]}^{< \omega}| \le \gl$ and
$\gg = \sup(F''{[S_1]}^{< \omega}) < \gd$. However,
$p' = \la s, S_1 \ra$ extends $p$ and
$p' \forces ``\dot f '' \gl \subseteq \gg + 1
< \gd$'',
a contradiction.
Thus, all regular successor
cardinals $\gd < \gk$ in
$V$ and $V_1$ remain cardinals in $V_2$.
We continue by using a modification
of the argument employed by Schindler in the
proof of Theorem 1 of \cite{S}.
We assume that there is no inner model of
$V_2$ with a Woodin cardinal, which immediately
implies that there is no inner model of either
$V_1$ or $V$ with a Woodin cardinal.
Let $t_1$ and $t_2$ be the cofinal $\omega$
sequences present in $V_2$ through $\gk$ and
$\gk^+$ respectively,
and let $M = {\rm HOD}^{V_2}$.
Note that $\Omega$ remains inaccessible in
$V_2$. This is since
$\FP_1 \ast \FP_0 \in V_\Omega$
and every $x \in V^{V_2}_\Omega$ has a name
$\dot x \in V_\Omega$, which then
implies that there are no
$p \in \FP_1 \ast \FP_0$, $\tau$, and
$\ga < \Omega$ with
$p \forces ``\tau : V^{V_2}_\ga \to \Omega$
is a cofinal map''.
Further, by the homogeneity of
$\FP_1 \ast \FP_0$, $M$ is a definable inner
model of $V$, since a set $x$ of ordinals is
ordinal definable in $V_2$ iff
there is a
formula $\varphi$ and an ordinal sequence
$\vec{\ga}$ such that
$\xi \in x$ $\Leftrightarrow$
$\forces_{\FP_1 \ast \FP_0} \varphi(\xi, \vec{\ga})$.
Assume that
$x \subseteq \Omega$ is bounded and ordinal
definable in $V_2$.
Work for the time being in $V$.
By the preceding paragraph,
$x \in V_\Omega$, so by Lemma \ref{l1},
$x \in {\rm HOD}^\FQ$
for some $\FQ \in V^{\rm HOD}_\Omega$.
Thus, since the
$V_\Omega$ of every small generic
extension of ${\rm HOD}$ is closed under sharps,
we are now able to infer that
$V_2 \models ``\Omega$ is inaccessible and
$V^{\rm HOD}_\Omega$ (as computed in $V_2$)
is closed under sharps''.
Therefore,
$M \models ``\Omega$ is inaccessible and
$V_\Omega$ is closed under sharps'', so
by Lemma \ref{l2}, $K = K^M$ inside
$M$ up to $\Omega$ exists and is $\Omega$
iterable in $M$.
We claim now that ${(\gk^+)}^K = {(\gk^+)}^V =
{(\gk^+)}^{V_1} = {(\gk^+)}^{V_2}$.
If not, then let $f : \gk \to {(\gk^+)}^K$,
$f \in V_2$ be
a bijection. By Lemma \ref{l1},
$M[\la t_1, f \ra]$ is a generic extension
of $M$ via a partial ordering which is an
element of $V^M_\Omega$.
Hence, Lemma \ref{l2} implies that inside
$M[\la t_1, f \ra]$
and up to $\Omega$,
$K = K^M = K^{M[\la t_1, f \ra]}$.
Since $\gk$ is a singular cardinal in
$M[\la t_1, f \ra]$,
another application of Lemma \ref{l2} then
yields that ${(\gk^+)}^K =
{(\gk^+)}^{M[\la t_1, f \ra]}$.
On the other hand, it is
of course the case that
$M[\la t_1, f \ra] \models
``{(\gk^+)}^K < \gk^+$''.
From this contradiction, we therefore know that
${(\gk^+)}^K = \gk^+$, and so
$M[\la t_2, t_1 \ra] \models
``$ZFC + ${(\gk^+)}^K = \gk^+$ +
$\gk^+$ is a singular cardinal''.
%This, however, contradicts that
This final contradiction proves Theorem \ref{t2}.
\end{proof}
As indicated in Section \ref{s1},
Theorem \ref{t3} is an immediate
corollary of Theorem \ref{t2}.
Instead of choosing $t_2$ as being
Prikry generic for the measurable
cardinal $\gk^+$, one simply chooses
$t_2$ as being a cofinal sequence
through $\gk^+$ witnessing that
$\gk^+$ is singular.
The remaining details of the proof of
Theorem \ref{t3} are then virtually
identical to the proof of Theorem
\ref{t2}, as readers can easily verify
for themselves.
\section{Concluding Remarks}\label{s3}
In conclusion to this note,
we remark that in the first
version of this paper, we asked if
the hypotheses of Theorem \ref{t2} could
be weakened so that the assumption of
DC were dropped and $\gk^+$ were just
assumed to be weakly compact and not measurable.
As we mentioned in Section \ref{s1}, the
referee has provided a proof that this
is indeed possible. Here are the details:
Let $\FP_0$ be as in the proof of
Theorem \ref{t2}.
Force with $\FP_0$ over $V$.
By the arguments given in the proof of
Theorem \ref{t2}, $K$ can be constructed
in ${\rm HOD}^{V^{\FP_0}}$, and we will as before have that
${(\gk^+)}^K = {(\gk^+)}^V = {(\gk^+)}^{V^{\FP_0}}$.
Thus, ${(\gk^+)}^V$ is a successor cardinal in $K$.
On the other hand,
by the local definability of $K$ (see \cite{St}, \S 6)
and the homogeneity of $\FP_0$, it will be the case
that $x$ is a set of ordinals in $K$ iff
there is a certain
formula $\varphi$ and an ordinal sequence
$\vec{\ga}$ such that
$\xi \in x$ $\Leftrightarrow$
$\forces_{\FP_0} \varphi(\xi, \vec{\ga})$.
This implies that $K$ is a definable inner
model of $V$.
This means we can then use Lemma 1(d) of
\cite{S} to infer that ${(\gk^+)}^V$,
which by hypothesis is weakly compact, is
inaccessible in $K$.
This contradiction proves the improved
version of Theorem \ref{t2}.
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\end{document}