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\author{Arthur W. Apter}
\address{A. W. Apter, The Graduate Center of The City University of New York, Mathematics, 365 Fifth Avenue, New York, NY 10016, USA \& Department of Mathematics, Baruch College, One Bernard
Baruch Way, New York, NY 10010, USA}
\email{awapter@alum.mit.edu}
\urladdr{http://faculty.baruch.cuny.edu/aapter/}
\author{Ioanna M. Dimitriou}
\address{I. M. Dimitriou,
Mathematisches Institut der Universit\"at Bonn, Endenicher Allee 60,
53115 Bonn, Germany}
%Institut f\"ur Medizinische Biometrie, Informatik, und Epidemilogie des Universit\"atsklinikums Bonn, geb\"aude 325, Sigmund-Freud-Str.25, 53127 Bonn, Germany}
\email{ioanna.m.dimitriou@gmail.com}
\urladdr{http://boolesrings.org/ioanna/}
\author{Peter Koepke}
\address{P. Koepke,
Mathematisches Institut der Universit\"at Bonn, Endenicher Allee 60,
53115 Bonn, Germany}
\email{koepke@math.uni-bonn.de}
\urladdr{http://www.math.uni-bonn.de/people/koepke/index.shtml}
\subjclass[2010]{03E25, 03E35, 03E55}
\keywords{Supercompact cardinal, strongly compact cardinal,
Rowbottom cardinal, Rowbottom filter, almost Ramsey cardinal,
almost huge cardinal, Prikry-type forcing, symmetric model.}
\title[All uncountable cardinals in the Gitik model]{All
uncountable cardinals in the Gitik model are almost Ramsey
and carry Rowbottom filters}
\date{\today}
\begin{document}
\thanks{The authors were partially supported by the
DFG-NWO cooperation grant KO 1353/5-1
``Infinitary combinatorics without the Axiom of Choice'',
and the Hausdorff Center for Mathematics of the University of Bonn.
The first author was also partially supported by various PSC - CUNY grants.
The first author wishes to thank the members of the Bonn Logic Group,
and especially his coauthors, for all of the hospitality shown to him
during his visits to Bonn.}
\maketitle
\begin{abstract}
Using the analysis developed in our earlier paper \cite{ApDiKo14},
%``The first measurable cardinal can be the first uncountable regular cardinal at any successor height'',
we show that every uncountable cardinal in Gitik's model of
\cite{Gi80} in which all uncountable cardinals are singular
is almost Ramsey and is also a Rowbottom cardinal
carrying a Rowbottom filter.
We assume that the model of \cite{Gi80} is constructed
from a proper class of strongly compact cardinals,
each of which is a limit of measurable cardinals.
Our work consequently reduces the
best previously known upper bound in consistency
strength for the theory
$\mathsf{ZF}$ +
``All uncountable cardinals are singular'' +
``Every uncountable cardinal is both almost Ramsey and
a Rowbottom cardinal carrying a Rowbottom filter''.
\end{abstract}
\section{Introduction}
In this paper, we analyse the large cardinal properties
that each uncountable cardinal exhibits in Gitik's model
of \cite{Gi80} in which all uncountable cardinals
are singular. Specifically, we will prove the following theorem.
\begin{theorem}\label{main} Let $V\models\mathsf{ZFC}$ +
``There is a proper class of strongly compact cardinals,
each of which is a limit of measurable cardinals'' +
``Every limit of strongly compact cardinals is singular''.
There
is then a proper class partial ordering
${\mathbb P} \subseteq V$ and a symmetric
model $V(G)$ of the theory
$\mathsf{ZF}$ +
``All uncountable cardinals are singular'' +
``Every uncountable cardinal is both almost Ramsey and
a Rowbottom cardinal carrying a Rowbottom filter''.
\end{theorem}
Theorem \ref{main} generalises
earlier work found in our previous paper \cite{ApDiKo14}.
We will take as a convention that the word {\em cardinal}
refers to the well-ordered cardinals, i.e., to the alephs.
We note that our extra assumption (unused by Gitik in \cite{Gi80}) that
there is a proper class of strongly compact cardinals, each of which is
a limit of measurable cardinals, is an entirely reasonable one. It follows,
e.g., if there is a proper class of supercompact cardinals, since any
supercompact cardinal is automatically both strongly compact and
a limit of measurable cardinals.
It is also a theorem of Kimchi and Magidor, unpublished by them but
with a proof appearing in \cite{Ap98}, that it is consistent, relative
to a proper class of supercompact cardinals, for the
proper classes of supercompact and
strongly compact cardinals to coincide precisely, except
at measurable limit points.
In fact, although Magidor's work of \cite{Ma76} shows that it is
consistent, relative to the existence of a strongly compact cardinal,
for the least strongly compact and least measurable cardinal to coincide,
it is not known whether it is even relatively consistent for the first
$\omega$ strongly compact and measurable cardinals to coincide.
(It is a theorem of Magidor, unpublished by him but appearing in
\cite{ApCu00}, that for any fixed $n \in \omega$, it is consistent
relative to the existence of $n$ supercompact cardinals for the first
$n$ strongly compact and measurable cardinals to coincide.)
It is therefore conceivable that it is a theorem of $\mathsf{ZFC}$ that
if there is a proper class of strongly compact cardinals, then there is also
a proper class of strongly compact cardinals, each of which is a limit of
measurable cardinals.
We would also like to take this opportunity to point out that
Theorem \ref{main} dramatically reduces the best previously
known upper bound in consistency strength for %the theory
$\mathsf{ZF}$ +
``All uncountable cardinals are singular'' +
``Every uncountable cardinal is both almost Ramsey and
a Rowbottom cardinal carrying a Rowbottom filter''.
The work of
\cite{Ap85b} and \cite{Ap92b} shows that the consistency of this theory
can be established using
hypotheses strictly between the consistency strength
of %the theories
$\mathsf{ZFC}$ + ``There is a supercompact limit of supercompact cardinals''
and
$\mathsf{ZFC}$ + ``There is an almost huge cardinal''.
These theories are both considerably stronger than
$\mathsf{ZFC}$ + ``There is a proper class of strongly
compact cardinals, each of which is a limit of measurable cardinals''.
(The consistency of %the theory
$\mathsf{ZF}$ + $\neg \mathsf{AC}_\omega$ + ``Every successor cardinal
is regular'' + ``All uncountable cardinals are almost Ramsey''
had previously been shown in \cite{ApKo08} to follow from the consistency of
$\mathsf{ZFC}$ + ``There are cardinals $\kappa < \lambda$ such that
$\kappa$ is $2^\lambda$ supercompact and $\lambda$ is the least
regular almost Ramsey cardinal greater than $\kappa$''.)
We conclude the Introduction by recalling that the cardinal
$\kappa$ is {\em almost Ramsey} if
$\forall \alpha < \kappa [ \kappa \to (\alpha)^{< \omega}_2]$, i.e.,
if given $\alpha < \kappa$ and $f : [\kappa]^{< \omega} \to 2$, there is a
{\em homogeneous set} $X \subseteq \kappa$ having order-type $\alpha$
(so $X$ is such that $|F '' [X]^n| = 1$ for every $n < \omega$).
The cardinal $\kappa$ is {\em Rowbottom} if
$\forall \lambda < \kappa
[\kappa \to [\kappa]^{< \omega}_{\lambda, \omega}]$, i.e., if given
$\lambda < \kappa$ and $f : [\kappa]^{< \omega} \to \lambda$, there is a
{\em homogeneous set} $X \subseteq \kappa$, $|X| = \kappa$ such that
$|f '' [X]^{< \omega}| \le \omega$. If $\kappa$ carries a filter
${\mathcal F}$ such that for any $f : [\kappa]^{< \omega} \to \lambda$,
there is a set $X \in {\mathcal F}$ which is homogeneous for
${\mathcal F}$, then ${\mathcal F}$ is called a {\em Rowbottom filter}.
\section{The Gitik construction}
For our construction,
we follow to a large extent the presentation given
in our earlier paper \cite{ApDiKo14}.
We will frequently quote verbatim from that paper.
Specifically,
we assume knowledge of forcing as presented in
\cite[Ch. VII]{Ku80}, \cite[Ch. 14]{Je03}, and of symmetric
submodels as presented in \cite[Ch.15]{Je03}.
By a result of Felgner \cite{Fe71}, we also assume that our
ground model $V$ satisfies the global Axiom of Choice (global AC).
We will use global AC when choosing the relevant ultrafilters
in the discussion to be given below about type 1 and type 2
regular cardinals.
\\
We start with an increasing sequence of
cardinals,
\[ \langle\kappa_\xi\ ; \ \xi \in \mathsf{Ord} \rangle, \]
such that for every $\xi\in\mathsf{Ord}$,
$\kappa_\xi$ is both strongly compact and a limit of measurable cardinals,
and such that the sequence has no regular limits.
We will construct a model in which all uncountable cardinals
are singular.
We will do
this by slightly
modifying and adapting Gitik's construction of \cite{Gi80},
whose notation and terminology we freely use.
Our construction is a finite support product of Prikry-like forcings
which are interweaved in order to
prove a Prikry-like lemma for that part of the forcing. \\
Call $\mathsf{Reg}^{}$ the class of regular cardinals
in $V$. For convenience call
$\omega=:\kappa_{-1}$. For an $\alpha\in\mathsf{Reg}^{}$
we define a $\mathsf{cf}'\alpha$ to distinguish between the
following categories.
\begin{itemize}
\item[(type 1)] This occurs if there is a
largest
$\kappa_\xi\leq\alpha$
(i.e., $\alpha\in[\kappa_\xi,\kappa_{\xi+1})$). We then define
$\mathsf{cf}'\alpha:=\alpha$.
If $\alpha=\kappa_\xi$ and $\xi\not=-1$, then
let $\Phi_{\kappa_\xi}$ be a measure for
$\kappa_\xi$ (which need not be normal).
If $\alpha=\omega$ then let $\Phi_{\omega}$ be
any uniform ultrafilter over $\omega$.
If $\alpha>\kappa_\xi$ is inaccessible,
then let $H_\alpha$ be a $\kappa_\xi$-complete fine ultrafilter
over $\mathcal{P}_{\kappa_\xi}(\alpha)$ and let $h_\alpha:
\mathcal{P}_{\kappa_\xi}(\alpha)\to \alpha$ be a bijection. Define
\[ \Phi_\alpha: = \{X\subseteq\alpha \ ; \ h_\alpha^{-1}``X
\in H_\alpha\}. \]
This is a uniform $\kappa_\xi$-complete ultrafilter over $\alpha$.
If $\alpha>\kappa_\xi$ is not inaccessible, then let $\Phi_\alpha$ be any $\kappa_\xi$-complete
uniform ultrafilter over $\alpha$.
\item[(type 2)] This occurs if there is no largest strongly
compact $\leq\alpha$. We then let $\beta$ be the largest (singular)
limit of strongly compacts
$\leq\alpha$. Define $\mathsf{cf}'\alpha := \mathsf{cf}\beta$. Let
\[ \langle\kappa^\alpha_\nu \ ; \ \nu <\mathsf{cf}'\alpha
\rangle \label{type2}\]
be a fixed ascending sequence of strongly compacts
$\geq\mathsf{cf}'\alpha$ such that
$\beta =\bigcup\{\kappa^\alpha_\nu \ ; \ \nu <\mathsf{cf}'\alpha\}$.
If $\alpha$ is inaccessible, then for each
$\nu<\mathsf{cf}'\alpha$, let $H_{\alpha,\nu}$ be
a fine ultrafilter over $\mathcal{P}_{\kappa_\nu^\alpha}(\alpha)$ and
$h_{\alpha,\nu}:\mathcal{P}_{\kappa_\nu}(\alpha)\to \alpha$ a bijection. Define
\[ \Phi_{\alpha,\nu}:= \{X\subseteq\alpha \ ; \ h_\alpha^{-1}``X
\in H_{\alpha,\nu}\}. \]
Again, $ \Phi_{\alpha,\nu}$ is a $\kappa^\alpha_\nu$-complete
uniform ultrafilter over $\alpha$.
If $\alpha$ is not inaccessible, then for each $\nu<\mathsf{cf}'\alpha$, let $\Phi_{\alpha,\nu}$ be
any $\kappa_\nu^\alpha$-complete uniform ultrafilter over $\alpha$.
\end{itemize}
This $\mathsf{cf}'\alpha$ will be used when we want to organise the choice of ultrafilters
for the type 2 cardinals. \\
We use the fine ultrafilters $H_\alpha$ and $H_{\alpha,\nu}$ to make
sure that in the end only the strongly compacts and their singular
limits remain cardinals in our final
symmetric model $V(G)$. For type 1 ordinals
we will do some tree-Prikry like forcings to singularise in
cofinality $\omega$. Type 1 cardinals in the open intervals
$(\kappa_\xi,\kappa_{\xi+1})$ will be collapsed to $\kappa_\xi$
because enough of these forcings will be isomorphic to strongly
compact Prikry forcings (or ``fake'' strongly compact Prikry
forcings in the case of $\xi=-1$). This is why we use fine
ultrafilters for these cardinals.
To singularise type 2 ordinals Gitik used a technique he credits in
\cite{Gi80} to Magidor, a Prikry-type forcing that relies on the
countable cofinal sequence $\vec c$ that we build for
$\mathsf{cf}'\alpha$ to pick a countable sequence of ultrafilters
$\langle \Phi_{\vec c (n)} \ ; \ n\in\omega \rangle$. To show that
the type 2 cardinals are collapsed, we use again the fine
ultrafilters.
As usual with Prikry-type forcings, one has to prove a Prikry-like
lemma (see \cite[Lemma 5.1]{Gi80}). For the arguments one
requires the forcing conditions to grow nicely.
These conditions can be viewed as trees. These trees will grow from
``left to right'' in order to ensure that a type 2 cardinal $\alpha$
will have the necessary information from the Prikry
sequence\footnote{For the rest of this paper, we will abuse
terminology by using phrases like ``Prikry sequence'' and ``Prikry
forcing'' when referring to our Prikry-like forcing notions.} at
stage $\mathsf{cf}'\alpha$. Let us take a look at the definition of
the stems of the Prikry sequences to be added.
\begin{definition} For a set $t\subseteq\mathsf{Reg}^{}\times
\omega\times\mathsf{Ord}$ we define the sets
\begin{align*}
\mathsf{dom}(t):= & \{ \alpha\in \mathsf{Reg}^{} \ ; \
\exists m\in\omega \exists\gamma\in\mathsf{Ord}((\alpha,m,
\gamma)\in t)\},
\text{ and} \\
\mathsf{dom}^2(t):= & \{ (\alpha,m)\in\mathsf{Reg}^{}\times
\omega \ ; \ \exists \gamma\in\mathsf{Ord}((\alpha,m,\gamma)\in t)\}.
\end{align*}
Let $P_1$ be the class of all finite subsets $t$ of
$\mathsf{Reg}^{}\times\omega\times\mathsf{Ord}$, such
that for every $\alpha\in\mathsf{dom}(t)$,
$t(\alpha):=\{(m,\gamma)\ ; \ (\alpha,m,\gamma)\in t\}$ is an
injective function from some
finite subset of $\omega$ into $\alpha$.
\end{definition}
To add a Prikry sequence to a type 2 cardinal $\alpha$, we want to
have some information on the Prikry sequence of the cardinal
$\mathsf{cf}'\alpha$. We also want to make sure that these stems are
appropriately ordered for the induction in the proof of the
aforementioned Prikry-like lemma. So we define the following.
\begin{definition}\label{P2def}Let $P_2$ be the class of all $t\in P_1$
such that the following hold.
\begin{enumerate}
\item For every $\alpha\in\mathsf{dom}(t)$, $\mathsf{cf}'\alpha\in
\mathsf{dom}(t)$
and $\mathsf{dom}(t(\mathsf{cf}'\alpha))\supseteq\mathsf{dom}(t(\alpha))$.
\item If $\{\alpha_0,\dots,\alpha_{n-1}\}$ is an increasing enumeration
of $\mathsf{dom}(t)
\setminus\kappa_0$, then there are $m,j\in\omega$, such that $m\geq
1$, $j\leq n-1$ with the properties that
\begin{itemize}
\item for every $k \omega$ is a cardinal''.
By \cite[Corollary 2.9]{ApDiKo14}, %if $\kappa$ is a successor
%cardinal in $V(G)$, then
$\kappa = \delta_\alpha$
for some ordinal $\alpha$.
%and if $\kappa$ is a
%limit cardinal in $V(G)$, then for some limit
%ordinal $\lambda$, $\kappa$ is the limit of the sequence
%$\langle \kappa_\alpha \ ; \ \alpha < \lambda \rangle$.
To show that $V(G) \models ``\kappa$ is almost Ramsey'', let
$f : [\kappa]^{< \omega} \to 2$, $f \in V(G)$.
Since $f$ can be coded by a subset of $\kappa$, by
Lemma \ref{app}, $f \in V[G\upharpoonright^* E_e]$ for some
$e \subseteq \delta_{\alpha + 1}$.
Again by the proof of \cite[Theorem 2.5]{ApDiKo14} (including the
work of the appendix), we can write
$V[G\upharpoonright^* E_e] = V[H_0][H_1]$,
where $H_0$ is $V$-generic over a partial ordering $\mathbb E$
such that $|\mathbb E| < \delta_\alpha$, and
$H_1$ is $V[H_0]$-generic over a partial ordering $\mathbb Q$
which adds no bounded subsets of $\delta_\alpha$.
Note now that in $V$, $\kappa$ is a limit of both
(non-measurable) Ramsey and measurable cardinals. This is
since either $\kappa$ is strongly compact or
a limit of strongly compacts. By the L\'evy-Solovay results
\cite{LeSo67}, in $V[H_0]$, $\kappa$ is a limit of
(non-measurable) Ramsey and
measurable cardinals as well. Further, because forcing over
$V[H_0]$ with $\mathbb Q$ adds no bounded subsets of
$\kappa = \delta_\alpha$, in $V[H_0][H_1] = V[G\upharpoonright^* E_e]$,
$\kappa$ is also a limit of Ramsey cardinals.
Consequently, by \cite[Proposition 1]{ApKo08}, $\kappa$ is almost Ramsey in
$V[G\upharpoonright^* E_e]$. This means that for
every $\beta < \kappa$, there is a set $X_\beta \in
V[G\upharpoonright^* E_e]$ which is homogeneous for $f$ and
has order type at least $\beta$.
As $V[G\upharpoonright^* E_e] \subseteq V(G)$, $X_\beta \in V(G)$.
Since $f$ was arbitrary, it therefore follows that
$V(G) \models ``\kappa$ is almost Ramsey''.
To show that $V(G) \models ``\kappa$ is a Rowbottom
cardinal carrying a Rowbottom filter'', we first note that
there is some $E_{e_0} \in I$ such that
$V[G\upharpoonright^* E_{e_0}] \models
``\mathsf{cf}(\kappa) = \omega$ and $\kappa =
\sup(\langle \zeta_i \ ; \ i < \omega \rangle)$, where each
$\zeta_i$ is measurable''.
To see this, observe that because (by hypothesis)
$V \models ``\kappa$ is a limit of measurable cardinals'' and
$V(G) \models ``$All uncountable cardinals are singular'',
by Lemma \ref{app}, there is
some $e_0 \subseteq \delta_{\alpha + 1}$ and $E_{e_0} \in I$ such that
$V[G\upharpoonright^* E_{e_0}] \models
``\mathsf{cf}(\kappa) = \omega$ and $\kappa =
\sup(\langle \zeta_i \ ; \ i < \omega \rangle)$, where each
$\zeta_i$ is measurable in $V$''.
%$V \models ``\kappa$ is a limit of measurable cardinals'',
However, as in the preceding paragraph, we can write
$V[G\upharpoonright^* E_{e_0}] = V[H_0][H_1]$,
where $H_0$ is $V$-generic over a partial ordering $\mathbb E$
such that $|\mathbb E| < \delta_\alpha = \kappa$, and
$H_1$ is $V[H_0]$-generic over a partial ordering $\mathbb Q$
which adds no bounded subsets of $\delta_\alpha$.
By the results of \cite{LeSo67}, it is still the case that
$V[H_0] \models ``\kappa$ is a limit of measurable cardinals''.
Because forcing with $\mathbb Q$ adds no bounded subsets of $\kappa$,
$V[H_0][H_1] = V[G\upharpoonright^* E_{e_0}]$ is as desired.
Consequently, for the remainder of the proof of Lemma \ref{l1}, we fix
$E_{e_0}$, $\langle \zeta_i \ ; \ i < \omega \rangle
\in V[G\upharpoonright^* E_{e_0}]$, and
$\langle \mu_i \ ; \ i < \omega \rangle \in
V[G\upharpoonright^* E_{e_0}]$ such that
$V[G\upharpoonright^* E_{e_0}] \models ``\kappa =
\sup(\langle \zeta_i \ ; \ i < \omega \rangle)$, where
each $\zeta_i$ is a measurable cardinal, and each
$\mu_i$ is a normal measure over $\zeta_i$''.
We also define ${\mathcal F} \in V[G\upharpoonright^* E_{e_0}]$ by
${\mathcal F} = \{X \subseteq \kappa \ ; \ \exists n < \omega
\forall i \ge n [X \cap \zeta_i \in \mu_i]\}$.
Clearly, ${\mathcal F}$ generates a filter (in any model of
$\mathsf{ZF}$ in which it is a member).
Now, let
$f : [\kappa]^{< \omega} \to \lambda$, $f \in V(G)$,
with $\omega \le \lambda < \kappa$ a cardinal in $V(G)$.
As before,
since $f$ can be coded by a subset of $\kappa$, by
Lemma \ref{app}, $f \in V[G\upharpoonright^* E_e]$ for some
$e \subseteq \delta_{\alpha + 1}$.
Without loss of generality, by coding if necessary,
we may assume in addition that
$V[G\upharpoonright^* E_e] \supseteq V[G\upharpoonright^* E_{e_0}]$.
%Because the argument in the preceding paragraph works for any $E_a \in I$
%such that $V[G\upharpoonright^* E_{a}] \models
%``\mathsf{cf}(\kappa) = \omega$ and $\kappa =
%\sup(\langle \rho_i \ ; \ i < \omega \rangle)$, where each
%$\rho_i$ is measurable in $V$'',
%we know that
%$V[G\upharpoonright^* E_{e}] \models ``\kappa =
%\sup(\langle \zeta'_i \ ; \ i < \omega \rangle)$, where
%each $\zeta'_i$ is a measurable cardinal, and each
%$\mu'_i$ is a normal measure over $\zeta'_i$''.
As in the preceding paragraph,
$V[G\upharpoonright^* E_{e}] = V[H^*_0][H^*_1]$, where
$H^*_0$ is $V$-generic over a partial ordering having
cardinality less than $\kappa$. Therefore, by the results of \cite{LeSo67},
we may further assume that in $V[G\upharpoonright^* E_{e}]$, a
%$\langle \zeta'_i \ ; \ i < \omega \rangle$ is a
final segment ${\mathbb F}$ of $\langle \zeta_i \ ; \ i < \omega \rangle$ is
composed of measurable cardinals, and that
for any $i$ such that $\zeta_i \in {\mathbb F}$,
%is a member of this final segment,
$\mu'_i$ defined in $V[G\upharpoonright^* E_{e}]$ by
$\mu'_i = \{X \subseteq \zeta_i \ ; \ \exists Y \in \mu_i
[Y \subseteq X]\}$ is a normal measure over
$\zeta_i$. %in $V[G\upharpoonright^* E_{e}]$.
Let $n_0$ be least such that
$\zeta_{n_0} \in {\mathbb F}$. %is a member of this final segment.
%measurable in $V[G\upharpoonright^* E_{e}]$.
By a theorem of Prikry (see \cite[Theorem 8.7]{Ka03}),
${\mathcal F^*} = \{X \subseteq \kappa \ ; \ \exists n \ge n_0
\forall i \ge n [X \cap \zeta_i \in \mu'_i]\} \in
V[G\upharpoonright^* E_e]$ is such that for some $Z^* \in {\mathcal F}^*$,
$Z^*$ is homogeneous for $f$.
By the definitions of ${\mathcal F}$ and ${\mathcal F^*}$ and the fact that
%${\mathcal F} \subseteq {\mathcal F^*}$,
every $\mu'_i$ measure $1$ set contains a $\mu_i$
measure $1$ set for $\zeta_i \in {\mathbb F}$, %$i \ge n_0$,
it then immediately follows that for some $Z \in {\mathcal F}$,
$Z \subseteq Z^*$, $Z$ is homogeneous for $f$.
Thus, ${\mathcal F} \in V[G\upharpoonright^* E_{e_0}] \subseteq V(G)$ generates
a Rowbottom filter for $\kappa$ in $V(G)$.
This completes the proof of both Lemma \ref{l1} and Theorem \ref{main}.
\end{proof}
We conclude by asking whether it is possible to remove the
additional assumption that every strongly compact cardinal
is a limit of measurable cardinals.
We conjecture that this is indeed possible, although with
a fair bit of work.
\bibliographystyle{alpha}
\bibliography{ioanna}
\end{document}
Now, let
$f : [\kappa]^{< \omega} \to \lambda$, $f \in V(G)$,
with $\omega \le \lambda < \kappa$ a cardinal in $V(G)$.
As before,
since $f$ can be coded by a subset of $\kappa$, by
Lemma \ref{app}, $f \in V[G\upharpoonright^* E_e]$ for some
$e \subseteq \delta_{\alpha + 1}$.
Without loss of generality, by coding if necessary,
we may assume in addition that
$V[G\upharpoonright^* E_e] \supseteq V[G\upharpoonright^* E_{e_0}]$.
Because the argument in the preceding paragraph works for any $E_a \in I$
such that
$V[G\upharpoonright^* E_{a}] \models
``\mathsf{cf}(\kappa) = \omega$ and $\kappa =
\sup(\langle \rho_i \ ; \ i < \omega \rangle)$, where each
$\rho_i$ is measurable in $V$'',
we know that
$V[G\upharpoonright^* E_{e}] \models ``\kappa =
\sup(\langle \zeta'_i \ ; \ i < \omega \rangle)$, where
each $\zeta'_i$ is a measurable cardinal, and each
$\mu'_i$ is a normal measure over $\zeta'_i$''.
Since as in the preceding paragraph,
$V[G\upharpoonright^* E_{e}] = V[H^*_0][H^*_1]$, where
$H^*_0$ is $V$-generic over a partial ordering having
cardinality less than $\kappa$, by the results of \cite{LeSo67},
we may assume that $\langle \zeta'_i \ ; \ i < \omega \rangle$ is a
final segment of $\langle \zeta_i \ ; \ i < \omega \rangle$, and that
for the $j$ such that $\zeta'_i = \zeta_j$,
$\mu'_i = \{X \subseteq \zeta_j \ ; \ \exists Y \in \mu_j
[Y \subseteq X]\}$. By a theorem of Prikry (see \cite[Theorem 8.7]{Ka03}),
${\mathcal F^*} = \{X \subseteq \kappa \ ; \ \exists n < \omega
\forall i \ge n [X \cap \zeta'_i \in \mu'_i]\} \in
V[G\upharpoonright^* E_e]$ is such that for some $Z^* \in {\mathcal F}^*$,
$Z^*$ is homogeneous for $f$.
By the definitions of ${\mathcal F}$ and ${\mathcal F^*}$ and the fact that
%${\mathcal F} \subseteq {\mathcal F^*}$,
every $\mu'_i$ measure $1$ set contains a $\mu_j$
measure $1$ set for the appropriate $j$,
it then immediately follows that for some $Z \in {\mathcal F}$,
$Z \subseteq Z^*$, $Z$ is homogeneous for $f$.
Thus, ${\mathcal F} \in V[G\upharpoonright^* E_{e_0}] \subseteq V(G)$ generates
a Rowbottom filter for $\kappa$ in $V(G)$.