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\begin{document}
\title{Making All Cardinals Almost Ramsey
\thanks{2000 Mathematics Subject Classifications:
03E02, 03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Almost {{Ramsey}} cardinal,
{{Erd\"os}} cardinal, indiscernibles, core model,
supercompact {{Radin}} forcing,
{{Radin}} sequence of measures,
symmetric inner model.}
\thanks{We wish to thank Ralf Schindler for
his insightful comments concerning Lemma \ref{ar2}
and Theorem \ref{t2}.}}
\author{Arthur W.~Apter\thanks{The
first author's research was partially
supported by PSC-CUNY grants and
CUNY Collaborative Incentive
grants. In addition,
the first author wishes to thank
the members of the mathematical
logic group in Bonn
for all of the hospitality shown him
during his spring 2007 sabbatical
visit to the Mathematisches Institut.}\\
%The City University of New York\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu
\\
\\
Peter Koepke\\
Mathematisches Institut\\
Rheinische Friedrich-Wilhelms-Universit\"at\\
Beringstra\ss e 1\\
D-53115 Bonn, Germany\\
http://www.math.uni-bonn.de/people/logic/People/Koepke.html\\
koepke@math.uni-bonn.de}
\date{July 30, 2007\\
(revised July 15, 2008)}
%\title{Almost Ramsey cardinals}\author{Arthur W. Apter and
%Peter Koepke}
\maketitle
\newpage\begin{abstract}
We examine combinatorial aspects
and consistency strength
properties of almost Ramsey cardinals.
Without the Axiom of Choice,
successor cardinals may be almost Ramsey.
From fairly mild supercompactness assumptions,
we construct a model of ZF +
$\neg {\rm AC}_\go$ in which {\it every}
infinite cardinal is almost {{Ramsey}}.
Core model arguments show that strong
assumptions are necessary.
Without successors of singular cardinals,
we can weaken this to an {\it equiconsistency}
of the following theories:
``ZFC + There is a proper class of regular
almost Ramsey cardinals'', and
``ZF + DC + All infinite cardinals except
possibly successors of singular cardinals
are almost Ramsey''.
\end{abstract}
\baselineskip=24pt
\section{Introduction}
{{Erd\"os}} and {{Ramsey}} cardinals are defined by
{\tmem{partition properties}}. A set $X \subseteq \delta$ is
{\tmem{homogeneous for a partition}} $F : [\delta]^{< \omega}
\rightarrow 2$ iff $\forall n (\card{F'' [X]^n} = 1)$; the
{\tmem{partition property}} $\delta \rightarrow (\alpha)^{< \omega}_2$
is
defined as
\[ (\forall F : [\delta]^{< \omega} \rightarrow 2)
(\exists X \subseteq
\delta)
(\tmop{otp} (X) \ge \alpha \wedge X \text{ is homogeneous for
} F) .
\]
An infinite cardinal $\kappa$ is \tmem{$\alpha$-{{Erd\"os}}} iff
$\kappa
\rightarrow (\alpha)^{< \omega}_2$, and it is \tmem{{{Ramsey}}} iff
$\kappa
\rightarrow (\kappa)^{< \omega}_2$. This suggests a natural large
cardinal
notion between {{Erd\"os}} and {{Ramsey}} cardinals.
\begin{definition}
An infinite cardinal $\kappa$ is {\tmem{almost Ramsey}}
iff
$\forall \alpha < \kappa (\kappa \rightarrow (\alpha)^{< \omega}_2)$.
\end{definition}
Almost {{Ramsey}} cardinals were considered before in
unpublished work
of {{H. Friedman}} and by {{J. Vickers}} and {{P.
D.
Welch}} in \cite{VW}.
%[{\tmem{On elementary embeddings from an inner model to the
%universe}}, The Journal of Symbolic Logic, Vol. 66, No. 3 (Sep. 2001),
%pp. 1090-1116].
These cardinals can be viewed as ``diagonal limits'' in the
hierarchy of {{Erd\"os}} cardinals.
For any uncountable almost {{Ramsey}}
cardinal $\kappa$, it can be
shown in ZF that the following
substructure property holds:
if $\lambda, \kappa', \lambda'$ are infinite cardinals
satisfying
$\lambda < \kappa$, $\lambda' \le \kappa' < \kappa$, and
$\lambda' \le \lambda$, then $(\kappa, \lambda) \Rightarrow
(\kappa',
\lambda')$. This means that every first-order structure $(\kappa,
\lambda,
\ldots)$ in a countable language has an elementary substructure $X
\prec
(\kappa, \lambda, \ldots)$ with $\card{X} = \kappa'$ and
$\card{
X \cap \lambda} = \lambda'$.
This can be viewed as a {{Chang}}'s
Conjecture-like two-cardinal
version of the standard downward
{{L\"owenheim-Skolem}} theorem.
It is easy to see that many instances of $(\kappa, \lambda) \Rightarrow
(\kappa', \lambda')$, in particular for successor cardinals $\kappa$,
are
incompatible with the Axiom of Choice. The main result of this paper
yields a
{\tmem{choiceless}} model of $\tmop{ZF}$ in which {\tmem{every}}
infinite cardinal\footnote{For the purposes
of this paper, in a choiceless
model of ZF, infinite cardinals will always
be taken as being well-ordered, i.e., as
being the alephs.}
is almost {{Ramsey}} and in which the generalized
downward
{{L\"owenheim-Skolem}} theorem holds universally.
Specifically, we have the following.
\begin{theorem}\label{t1}
Con(ZFC + There exist cardinals $\gk < \gl$
such that $\gk$ is $2^\gl$ supercompact
where $\gl$ is the least regular almost
Ramsey cardinal greater than $\gk$) $\implies$
Con(ZF + $\neg{AC}_\go$ + Every successor cardinal
is regular + Every %(well-ordered)
infinite %uncountable
cardinal is almost ${Ramsey}$).
\end{theorem}
%\begin{theorem}
% $\tmop{Con} (\tmop{ZFC} + \text{there exists a [ARTHUR please
%amend]})$
% implies $\tmop{Con} (\tmop{ZF} +$every infinite cardinal is almost
% {{Ramsey)}}.
%\end{theorem}
In the construction,
certain large cardinals are collapsed
generically and become the well-ordered cardinals of a symmetric model
of
$\tmop{ZF}$. Due to the strong indestructibility of almost
{{Ramsey}}
cardinals, the large cardinal hypotheses in the ground model can be
taken considerably weaker than
in a similar construction found in \cite{A85}.
(We will discuss this in greater detail
at the end of Section \ref{s4}.)
%in some similar constructions ([ARTHUR please amend]).
Conversely, core model techniques may be used to show that
considerable large cardinal
strength is necessary for the
symmetric model constructions of
this paper.
\begin{theorem}\label{t2}
Assume ZF and that every %(well-ordered)
infinite cardinal is almost
{{Ramsey}}.
Then for every $n < \go$ and every set of ordinals $x$,
$M^\sharp_n(x)$ exists.
%Then there exists an inner model with a
% proper class of strong cardinals.
\end{theorem}
Omitting successors of singular limit cardinals, consistency strengths
go down to what we shall show is
below the existence of {{Ramsey}} cardinals.
\begin{theorem}\label{teq}
The following theories are equiconsistent:
\begin{enumeratealpha}
\item ZFC + There is a proper class of regular almost
{{Ramsey}} cardinals;
\item ZF + DC + All successor cardinals
are regular + All %(well-ordered)
infinite cardinals except possibly
successors of
singular limit cardinals are almost {{Ramsey}}.
\end{enumeratealpha}
\end{theorem}
Before proving our theorems,
we shall show some combinatorial facts
about
almost {{Ramsey}} cardinals and consider some obvious
consistency\break
strength questions.
%More detailed studies of almost {{Ramsey}}
%cardinals and model-theoretic substructure properties will be part of
%the doctoral dissertation of {{I. Dimitriou}}.
\section{Combinatorial aspects}
Almost {{Ramsey}}ness is closely connected to
{{Erd\"os}}-type
partition cardinals.
\begin{definition}
For $\alpha \in Ord$,
%\tmop{Ord}$,
let $\kappa (\alpha)$ be the least
$\kappa$ such
that $\kappa \rightarrow (\alpha)^{< \omega}_2$, if such a $\kappa$
exists.
\end{definition}
The following characterization of almost {{Ramsey}} cardinals as
diagonal limits of the {{Erd\"os}} hierarchy can be verified
easily and
does not involve the Axiom of Choice.
\begin{proposition}\label{p1}
{{(ZF)}} An infinite cardinal $\kappa$ is almost
{{Ramsey}}
iff $\kappa (\alpha)$ is defined for all $\alpha < \kappa$ and
$\kappa =
\bigcup_{\alpha < \kappa} \kappa (\alpha)$.
\end{proposition}
By the classical {{Ramsey}} theorem, $\kappa (\alpha)$ is finite
for
$\alpha < \omega$.
By Proposition \ref{p1}, this immediately
implies that $\go$ is an almost Ramsey cardinal.
Under the Axiom of Choice, for infinite $\alpha$,
$\kappa
(\alpha)$ is strongly inaccessible
(if $\ga$ is also a limit ordinal)
and $\kappa (\alpha + 1) > \kappa
(\alpha)$
(see \cite[Propositions 7.14 and 7.15]{K}).
%(see [{{A. Kanamori}}, {\tmem{The Higher Infinite}},
%Propositions 7.14 and 7.15]).
\begin{proposition}\label{p2}
{{(ZFC)}} Assume $\kappa$ is almost {{Ramsey}}. Then
\begin{enumeratealpha}
\item $\forall \alpha < \kappa (\kappa (\alpha) < \kappa)$;
\item $\kappa$ is a strong limit cardinal.
\end{enumeratealpha}
\end{proposition}
\begin{proof}
a) Let $\alpha < \kappa$, $\ga \ge \go$.
Then $\kappa (\alpha) < \kappa (\alpha + 1)
\le \kappa$. b) is an immediate
consequence of a), keeping in mind that
for every limit ordinal $\ga$,
$\kappa
(\alpha)$ is strongly inaccessible.
\end{proof}
Propositions \ref{p1} and
\ref{p2} yield an {\tmem{indestructibility property}} for
almost
{{Ramsey}} cardinals $\kappa$ under forcing which does not add
bounded
subsets of $\kappa$.
\begin{proposition}\label{p3}
Let $M$ be a transitive model of ``ZFC + $\kappa$ is almost
{{Ramsey}}''. Let $N \supseteq M$ be a transitive model of
{{ZF}} such that $\forall \delta < \kappa(\mathcal{P}(\delta)
\cap M
=\mathcal{P}(\delta) \cap N)$. Then $\kappa$ is almost
{{Ramsey}} in
$N$.
\end{proposition}
\begin{proof}
Let $\alpha < \kappa$. By Proposition \ref{p2}, $(\kappa (\alpha))^M <
\kappa$. \
$\mathcal{P}((\kappa (\alpha))^M) \cap M =\mathcal{P}((\kappa
(\alpha))^M) \cap
N$ implies that $(\kappa (\alpha))^N = (\kappa (\alpha))^M$. Hence
$\kappa =
\bigcup_{\alpha < \kappa} (\kappa (\alpha))^N$,
so by Proposition \ref{p1}, $\kappa$ is almost
{{Ramsey}} in $N$.
\end{proof}
%Propositions \ref{p1} and \ref{p2} also
%tell us that regular almost Ramsey
%cardinals are preserved by small forcing.
We may also infer that almost Ramsey
cardinals are preserved by small forcing.
\begin{proposition}\label{p3a}
Suppose $V \models ``$ZFC +
$\gk$ is almost Ramsey + $\FP$
is a partial ordering such that
$\card{\FP} < \gk$''. Then
$V^\FP \models ``\gk$ is almost Ramsey''.
\end{proposition}
\begin{proof}
By Propositions \ref{p1} and \ref{p2}, write
$\gk = \bigcup_{\ga \in [\ga_0, \gk)} \gk(\ga)$,
where $\ga_0$ is a limit ordinal
with the additional property that
$\card{\FP} < \gk(\ga_0)$. By
\cite[Proposition 7.15 and Exercise 10.16]{K},
for any limit ordinal $\ga \in [\ga_0, \gk)$ and
$\gd(\ga) = {(\gk(\ga))}^V$,
$V^\FP \models ``\gd(\ga) \to {(\ga)}^{<\go}_2$''.
Since $\gk$ therefore remains in $V^\FP$
a limit of cardinals satisfying suitable
partition properties,
$V^\FP \models ``\gk$ is almost Ramsey''.
\end{proof}
In $\tmop{ZFC}$, a regular almost {{Ramsey}} cardinal is
strongly
inaccessible. Singular almost {{Ramsey}} cardinals are much
weaker.
Since $\{\kappa \mid \kappa = \bigcup_{\alpha < \kappa} \kappa (\alpha)\}$
is a
closed class of ordinals, we get the following.
\begin{proposition}
{{(ZFC)}}
\begin{enumeratealpha}
\item Assume $\kappa$ is an uncountable {{\em regular}} almost
{{Ramsey}} cardinal. Then the class of almost
{{Ramsey}}
cardinals is closed unbounded below $\gk$.
\item Assume $\kappa$ is an almost {{Ramsey}} cardinal which
is
{{Mahlo}}. Then the class of {regular} almost
{{Ramsey}}
cardinals is stationary below $\gk$.
\item Assume $\kappa$ is the smallest uncountable regular almost
{{Ramsey}} cardinal. Then $\kappa$ is not {{Mahlo}}.
\item Assume $\gk$ is a {{Ramsey}} cardinal. Then the
class of
almost {{Ramsey}} cardinals is closed unbounded below
$\gk$ and
the class of {{regular}} almost {{Ramsey}} cardinals is
stationary below $\gk$.
\end{enumeratealpha}
\end{proposition}
As a corollary, we get some information on consistency strengths.
\begin{proposition}
{\tmdummy}
\begin{enumeratealpha}
\item ${}{ZFC}$ + There exists an uncountable regular almost
{{Ramsey}} cardinal $\vdash {}{Con} ({}{ZFC}$ + There
exists a
proper class of (singular) almost {{Ramsey}} cardinals).
\item ${}{Con} ({}{ZFC}$ + There exists an uncountable
regular almost
{{Ramsey}} cardinal$) \leftrightarrow {}{Con} ({}{ZFC}
$ + There exists an uncountable regular almost {{Ramsey}}
cardinal
which is not {{Mahlo}}$)$.
\end{enumeratealpha}
\end{proposition}
Let us now work without the Axiom of Choice. The following theorem of
{{J. Silver}}
%(see [{{A. Kanamori}}, {\tmem{The Higher Infinite}}, Theorem 9.3])
(see \cite[Theorem 9.3]{K})
shows that homogeneous sets for partitions are
basically equivalent to sets of {{(order) indiscernibles}} for
first-order
structures.
\begin{proposition}
{{(ZF)}} For infinite ordinals $\alpha$, the partition property
$\kappa \rightarrow (\alpha)_2^{< \omega}$ is equivalent to
the following: for any
first-order structure $\mathcal{M}= (M, \ldots)$ in a countable
language $S$
with $\kappa \subseteq M$, there is a set $X \subseteq \kappa$,
$\tmop{otp}
(X) \ge \alpha$ of {\tmem{indiscernibles}}, i.e., for all
$S$-formulas
$\varphi (v_0, \ldots, v_{n - 1})$,
\break $x_0, \ldots, x_{n - 1} \in X$,
$x_0 < \cdots < x_{n - 1}$, $y_0, \ldots, y_{n - 1} \in X$, $y_0 < \cdots <
y_{n -
1},$
\[ \mathcal{M} \models \varphi (x_0, \ldots, x_{n - 1}) \text{ iff }
\mathcal{M} \models \varphi (y_0, \ldots, y_{n - 1}) . \]
\end{proposition}
For limit $\alpha$, indiscernibility can be strengthened to
{\tmem{good}}
indiscernibility.
\begin{proposition}\label{gia}
{{(ZF)}} Assume $\kappa \rightarrow (\alpha)_2^{< \omega}$,
where
$\alpha$ is a limit ordinal. Then for any first-order structure
$\mathcal{M}= (M, \ldots)$ in a countable language $S$ with $\kappa
\subseteq M$, there is a set $X \subseteq \kappa$, $\tmop{otp} (X)
\ge
\alpha$ of {\tmem{good indiscernibles}}, i.e., for all $S$-formulas
$\varphi
(v_0, \ldots, v_{m - 1}, w_0, \ldots, w_{n - 1})$, $x_0, \ldots, x_{n
- 1}
\in X$, $x_0 < \cdots < x_{n - 1}$, $y_0, \ldots, y_{n - 1} \in X$,
$y_0 <
\cdots < y_{n - 1}$, and $a_0 < \cdots < a_{m - 1} < \min (x_0, y_0)$,
\[ \mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, x_0, \ldots,
x_{n -
1}) \text{ iff } \mathcal{M} \models \varphi (a_0, \ldots, a_{m -
1},
y_0, \ldots, y_{n - 1}) . \]
\end{proposition}
\begin{proof}
We may assume that the structure $\mathcal{M}$ contains a unary
predicate
$\tmop{Ord}$ for the ordinals in $M$
(this includes all ordinals less than $\kappa$)
and a collection of
{{Skolem}} functions for ordinal-valued existential
statements, i.e.,
for every $S$-formula $\varphi (v, \vec{w})$, there is a function $f$
of $\mathcal{M}$
such that
\[ \mathcal{M} \models \forall \vec{w} (\exists v (\tmop{Ord} (v) \wedge
\varphi (v,
\vec{w})) \rightarrow \varphi (f ( \vec{w}), \vec{w})) . \]
Choose a set $X \subseteq \kappa$, $\tmop{otp} (X) = \alpha$
of
indiscernibles for
$\mathcal{M}$ such that its minimum, $\min (X)$, is minimal
for all
such sets of indiscernibles. Assume towards a contradiction that $X$ is
not
good. Then there is an $S$-formula $\varphi (v_0, \ldots, v_{m -
1}, w_0, \ldots, w_{n - 1})$, $x_0,
\ldots, x_{n - 1} \in X$, $x_0 < \cdots < x_{n - 1}$, $y_0, \ldots,
y_{n -
1} \in X$, $y_0 < \cdots < y_{n - 1}$, and \ $a_0 < \cdots < a_{m - 1}
< \min
(x_0, y_0)$ such that
\[ \mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, x_0, \ldots, x_{n - 1})
\text{
and } \mathcal{M} \models \neg \varphi (a_0, \ldots, a_{m - 1}, y_0, \ldots,
y_{n -
1}) . \]
Since $\alpha$ is a limit ordinal, we may take $z_0, \ldots, z_{n - 1}
\in
X$, $z_0 < \cdots < z_{n - 1}$ such that $x_{n - 1} < z_0$ and $y_{n
- 1} <
z_0$.
In case
$$\mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, z_0,
\ldots, z_{n
- 1}),$$ one has
\[ \mathcal{M} \models \neg \varphi (a_0, \ldots, a_{m - 1}, y_0, \ldots, y_{n
- 1})
\text{ and } \mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, z_0,
\ldots, z_{n
- 1}), \]
where $y_0 < \cdots < y_{n - 1} < z_0 < \cdots < z_{n - 1}$.
In case
$$\mathcal{M} \models \neg \varphi (a_0, \ldots, a_{m - 1}, z_0, \ldots,
z_{n -
1}),$$ one has
\[ \mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, x_0, \ldots, x_{n - 1})
\text{
and } \mathcal{M} \models \neg \varphi (a_0, \ldots, a_{m - 1}, z_0, \ldots,
z_{n -
1}), \]
where $x_0 < \cdots < x_{n - 1} < z_0 < \cdots < z_{n - 1}$. So in
both
cases, we have an ascending $2 n$-tuble of indiscernibles such that
the
first half behaves differently from the second half with respect to
the
formula $\varphi$ and the parameters $a_0, \ldots, a_{m - 1}$. So
without
loss of generality, we may assume that $x_0 < \cdots < x_{n - 1} < y_0
<
\cdots < y_{n - 1}$ and
\[ \mathcal{M} \models \varphi (a_0, \ldots, a_{m - 1}, x_0, \ldots, x_{n - 1})
\text{
and } \mathcal{M} \models \neg \varphi (a_0, \ldots, a_{m - 1}, y_0, \ldots,
y_{n -
1}) . \]
Write $\vec{x} = x_0, \ldots, x_{n - 1}$ and $\vec{y} = y_0, \ldots,
y_{n -
1}$. Since $\mathcal{M}$ contains {{Skolem}} functions, there are
functions
$f_0, \ldots, f_{m - 1}$ of $\mathcal{M}$ which compute parameters like
$a_0,
\ldots,
a_{m - 1}$:\\
$ \mathcal{M} \models
(\exists v_1 < x_0) (\exists v_2 < x_0) \cdots (\exists
v_{m - 1}
< x_0)
(f_0 ( \vec{x}, \vec{y}) < x_0 \wedge
\break \varphi (f_0 ( \vec{x},
\vec{y}), v_1, \ldots, v_{m - 1}, \vec{x}) \wedge \neg \varphi (f_0
(
\vec{x}, \vec{y}), v_1, \ldots, v_{m - 1}, \vec{y})).$\\
$ \mathcal{M} \models
(\exists v_2 < x_0) \cdots (\exists v_{m - 1} < x_0) (f_0
(
\vec{x}, \vec{y}) < x_0 \wedge
f_1 ( \vec{x}, \vec{y}) < x_0 \wedge
\break \varphi (f_0 ( \vec{x}, \vec{y}), f_1 ( \vec{x}, \vec{y}), \ldots,
v_{m -
1}, \vec{x}) \wedge \neg \varphi (f_0 ( \vec{x}, \vec{y}), f_1 (
\vec{x},
\vec{y}), \ldots, v_{m - 1}, \vec{y})).$\\
$\vdots$ \\
$ \mathcal{M} \models f_0 ( \vec{x}, \vec{y}) < x_0
\wedge \cdots \wedge
f_{m - 1}
( \vec{x}, \vec{y}) < x_0 \wedge
\varphi (f_0 ( \vec{x}, \vec{y}), \break
f_1 (
\vec{x}, \vec{y}), \ldots, f_{m - 1} ( \vec{x}, \vec{y}), \vec{x})
\wedge
\neg \varphi (f_0 ( \vec{x}, \vec{y}), f_1 ( \vec{x}, \vec{y}),
\ldots,
f_{m - 1} ( \vec{x}, \vec{y}), \vec{y}).$\\
Now consider $\vec{z} = z_0, \ldots, z_{n - 1} \in X$, $z_0 < \cdots
< z_{n
- 1}$ such that $y_{n - 1} < z_0$.\\
\noindent (1) There is
$k < m$ such that $f_k ( \vec{x}, \vec{y}) \neq f_k (
\vec{y},
\vec{z})$.\\
% {\tmem{Proof}}.
\begin{proof}
Assume not. Set $\xi_0 = f_0 ( \vec{x}, \vec{y}),
\ldots,
\xi_{m - 1} = f_{m - 1} ( \vec{x}, \vec{y})$. Since
$\vec{x}, \vec{y}$ and $\vec{y}, \vec{z}$
are ascending indiscernible sequences
of the same length, we have
\begin{eqnarray*}
\mathcal{M} & \models & \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{x})
\wedge
\neg \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{y})
\end{eqnarray*}
and
\begin{eqnarray*}
\mathcal{M} & \models & \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{y})
\wedge
\neg \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{z}) .
\end{eqnarray*}
In particular,
\begin{eqnarray*}
\mathcal{M} & \models & \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{y})
\wedge
\neg \varphi (\xi_0, \xi_1, \ldots, \xi_{m - 1}, \vec{y}),
\end{eqnarray*}
which is a contradiction.
\end{proof}(1)
% {\tmem{qed}}(1)
So take $k < m$ such that\\
\noindent (2) $f_k ( \vec{x}, \vec{y}) \neq f_k ( \vec{y}, \vec{z})$.\\
\noindent Let $(\nu_i \mid i < \alpha)$ be a strictly increasing enumeration of the
set
$X$ of indiscernibles, and let $( \vec{x}^{(i)} \mid i < \alpha)$
be a partition of $X$ into ascending sequences of length $n$, with
\[ \vec{x}^{(i)} = \nu_{n \cdot i}, \nu_{n \cdot i + 1}, \ldots,
\nu_{n
\cdot i + n - 1}. \]
We then claim that\\
\noindent (3) $f_k ( \vec{x}^{(0)}, \vec{x}^{(1)}) < f_k ( \vec{x}^{(1)},
\vec{x}^{(2)})$.\\
% {\tmem{Proof}}.
\begin{proof}
By indiscernibility, (2) implies that $f_k (
\vec{x}^{(0)},
\vec{x}^{(1)}) \neq f_k ( \vec{x}^{(1)}, \vec{x}^{(2)})$. Assume towards
a
contradiction that $f_k ( \vec{x}^{(0)}, \vec{x}^{(1)}) > f_k (
\vec{x}^{(1)}, \vec{x}^{(2)})$. Then again by indiscernibility, we
would
obtain a {\tmem{decreasing}} $\in$-sequence
\[ f_k ( \vec{x}^{(0)}, \vec{x}^{(1)}) > f_k ( \vec{x}^{(1)},
\vec{x}^{(2)})
> f_k ( \vec{x}^{(2)}, \vec{x}^{(3)}) > \cdots, \]
which is a contradiction.
\end{proof}(3)
% {\tmem{qed}}(3)
%\noindent
The above now tells us that
\[ f_k ( \vec{x}^{(0)}, \vec{x}^{(1)}) < f_k ( \vec{x}^{(2)},
\vec{x}^{(3)})
< f_k ( \vec{x}^{(4)}, \vec{x}^{(5)}) < \cdots \]
is an {\tmem{ascending}} $\alpha$-sequence of indiscernibles for
$\mathcal{M}$
with
smallest element $f_k ( \vec{x}^{(0)}, \vec{x}^{(1)}) < \nu_0$.
This contradicts the minimal choice of $\min (X)$.
\end{proof}
Recall that $\tmop{HOD}$ is {{G\"odel}}'s model of
{\tmem{hereditarily
ordinal definable sets}}. Let $<_{\tmop{HOD}}$ be the canonical
well-ordering
of $\tmop{HOD}$, defined in the universe $V$
%(see [{{T. Jech}},
%{\tmem{Set Theory - The Third Millennium Edition}}, Lemma 13.25]).
(see \cite[Lemma 13.25]{J}).
\begin{lemma}\label{arhod}
{{(ZF)}} Let $\kappa^+$ be almost {{Ramsey}}. Then
$(\kappa^+)^{\tmop{HOD}} < \kappa^+$.
\end{lemma}
\begin{proof}
Assume towards a contradiction that $(\kappa^+)^{\tmop{HOD}} = \kappa^+$.
For
$\gamma \in [\kappa, \kappa^+)$, choose the $<_{\tmop{HOD}}$-least
bijection
$f_{\gamma} : \gamma \leftrightarrow \kappa$.
Define $F : [\gk^+ \setminus \kappa]^3
\rightarrow 2$ by
\[ F (\{\alpha, \beta, \gamma\}) = \left\{ \begin{array}{l}
0 \text{ iff } f_{\gamma} (\alpha) < f_{\gamma} (\beta)\\
1 \text{ iff } f_{\gamma} (\alpha) > f_{\gamma} (\beta)
\end{array} \right. \text{, for } \alpha < \beta < \gamma . \]
Take $X \subseteq \kappa^+$ homogeneous for $F$, with $\tmop{otp} (X)
=
\kappa + 2$. Let $\gamma = \max (X)$. Then define $h : \kappa + 1
\rightarrow \kappa$ by $h (\xi) = f_{\gamma} (\alpha_{\xi})$, where
$\alpha_{\xi}$ is the $\xi$-th element of $X$.
\bigskip\noindent
{\rm{Case}} 1: $\forall x \in [X]^3(F (x) = 0)$. Then for $\xi <
\zeta <
\kappa + 1$, we have that
$\alpha_{\xi} < \alpha_{\zeta} < \gamma$ and
$\{\alpha_{\xi}, \alpha_{\zeta}, \gamma\} \in [X]^3$.
Since $F
(\{\alpha_{\xi},
\alpha_{\zeta}, \gamma\}) = 0$, it follows that
\[ h (\xi) = f_{\gamma} (\alpha_{\xi}) < f_{\gamma} (\alpha_{\zeta})
= h
(\zeta) . \]
Thus $h : \kappa + 1 \rightarrow \kappa$ is order preserving, which
is
impossible.
\bigskip\noindent
{\rm{Case}} 2: $\forall x \in [X]^3(F (x) = 1)$. Then for $\xi <
\zeta <
\kappa + 1$, we have that
$\alpha_{\xi} < \alpha_{\zeta} < \gamma$ and
$\{\alpha_{\xi}, \alpha_{\zeta}, \gamma\} \in [X]^3$.
Since $F
(\{\alpha_{\xi},
\alpha_{\zeta}, \gamma\}) = 1$, it follows that
\[ h (\xi) = f_{\gamma} (\alpha_{\xi}) > f_{\gamma} (\alpha_{\zeta})
= h
(\zeta) . \]
Thus $h : \kappa + 1 \rightarrow \kappa$ is a strictly descending
$\kappa +
1$ chain in the ordinals, which is a contradiction.
\end{proof}
\section{Almost {{Ramsey}} cardinals and the
{{Dodd-Jensen}}
core model}\label{s3}
We shall show that almost {{Ramsey}} cardinals are almost
{{Ramsey}} in appropriate {\tmem{core models}}, in particular in
the
{{Dodd-Jensen}} core model $K^{\tmop{DJ}}$
which is presented in \cite{D}
and \cite{DJ1}. Since core model
theory
usually assumes the Axiom of Choice, we shall also use the inner model
$\tmop{HOD}$
%of {\tmem{hereditarily ordinal definable sets}}
or extensions
$\tmop{HOD} [a]$ of $\tmop{HOD}$ by sets $a \subseteq \tmop{HOD}$. The
following proposition is found in \cite{AK06}.
%[{{A. Apter}} and {{P.
%Koepke}}, {\tmem{The Consistency Strength of $\aleph_{\omega}$ and
%$\aleph_{\omega_1}$ being Rowbottom Cardinals without the Axiom of
%Choice}}, Archive for Mathematical Logic]:
\begin{proposition}\label{pak06}
{{(ZF)}} Let $a \subseteq \tmop{HOD}$ be a set. Then
\begin{enumeratealpha}
\item $\tmop{HOD} [a]$ is a set-generic extension of $\tmop{HOD}$,
so
$\tmop{HOD} [a] \models \tmop{ZFC}$.
\item $(K^{\tmop{DJ}})^{\tmop{HOD}} = (K^{\tmop{DJ}})^{\tmop{HOD}
[a]}$;
moreover this equality holds for every level of the hierarchy,
i.e.,
$(K_{\alpha}^{\tmop{DJ}})^{\tmop{HOD}} =
(K_{\alpha}^{\tmop{DJ}})^{\tmop{HOD} [a]}$ for every $\alpha \in
\tmop{Ord}$.
\end{enumeratealpha}
\end{proposition}
By Proposition \ref{pak06}, we may define $K^{\tmop{DJ}} =
(K^{\tmop{DJ}})^{\tmop{HOD}}$ in models without choice. The following
{\tmem{indiscernibles lemma}} by {{T. Dodd}} and {{R.
Jensen}} (see \cite{VW}, as well
as \cite{D} and \cite{DJ1})
%}}(see [{{J. Vickers}} and {{P. D. Welch}}, {\tmem{On
%elementary
%embeddings from an inner model to the universe}}, The Journal of
%Symbolic
%Logic, Vol. 66, No. 3 (Sep. 2001), pp. 1090-1116])
is used to find homogeneous sets inside $K^{\tmop{DJ}}$.
\begin{proposition}\label{gi}
Let $\kappa$ be an infinite cardinal. Suppose $A \in K^{\tmop{DJ}}
\cap
\mathcal{P}(K_{\kappa}^{\tmop{DJ}})$ and that $I$, an
infinite
set of good indiscernibles for $\mathcal{A}= (K_{\kappa}^{\tmop{DJ}},
A)$, is
such that $\tmop{cof} (\tmop{otp} (I)) > \omega$. Then there is $I'
\in
K^{\tmop{DJ}}$, $I' \supseteq I$ a set of good indiscernibles for
$\mathcal{A}$.
\end{proposition}
\begin{lemma}\label{ar1}
{{(ZF)}} Let $\kappa > \aleph_1$ be almost {{Ramsey}}.
Then
$\kappa$ is almost {{Ramsey}} in $K^{\tmop{DJ}}$.
\end{lemma}
\begin{proof}
Let $F : [\kappa]^{< \omega} \rightarrow 2$, $F \in K^{\tmop{DJ}}$ be
a
partition. Let $\alpha < \kappa$. Then $\alpha + \aleph_1 < \kappa$.
By
Proposition \ref{gia},
take a set $X \subseteq \kappa$ of {\tmem{good}}
indiscernibles for the structure
$\mathcal{M} = (K^{\tmop{DJ}}_{\kappa}, F)$,
with
$\tmop{otp} (X) \ge \alpha + \aleph_1$. Let $X'$ be the initial
segment of $X$ of order type $(\alpha + \aleph_1)^{\tmop{HOD} [X]}$.
In
the model $\tmop{HOD} [X]$, $X'$ is a set of good indiscernibles for
$\mathcal{M}$
such that $\tmop{cof} (\tmop{otp} (X')) > \omega$. By
Proposition \ref{gi}
%indiscernibles lemma
applied inside $\tmop{HOD} [X]$, there is a set $Y \supseteq
X'$, $Y
\in K^{\tmop{DJ}}$ which is a set of good
indiscernibles for $\mathcal{M}$.
Then $Y$ is
also
homogeneous for the partition $F$ of order type
greater than or equal to $\ga$.
\end{proof}
We are now able to prove the inner model direction of Theorem
\ref{teq}.
\begin{lemma}
$\tmop{Con} ({ZF}$ + All infinite cardinals except possibly
successors
of singular limit cardinals are almost {{Ramsey}}$)$ $\implies$
$\tmop{Con} ({ZFC}$ + There is a proper class of regular almost
{{Ramsey}} cardinals$)$.
\end{lemma}
\begin{proof}
Assume Con($\tmop{ZF}$ + All infinite cardinals except possibly successors
of
singular limit cardinals are almost {{Ramsey}}). If there is a
proper
class of {\tmem{regular}} almost {{Ramsey}} cardinals,
then by Lemma \ref{ar1}, we are
done.
So assume that this is not the case, and let the cardinal $\theta$ be
an
upper bound for the set of regular almost {{Ramsey}}
cardinals. Then
$\theta^{+ +}$ and $\theta^{+ + +}$ are not successors of limit
cardinals.
By assumption, $\theta^{+ +}$ and $\theta^{+ + +}$ are almost
{{Ramsey}}. By the definition of $\theta$, $\theta^{+ +}$ and
$\theta^{+ + +}$ must be singular. By \cite{Sc99},
% [{{R. Schindler}},
% {\tmem{Successive weakly compact or singular cardinals}}, Journal of
%Symb. Logic 64 (1999), pp. 139-146]
this implies consistency strength far above
measurable cardinals, and hence the
consistency of a proper class of regular
almost Ramsey cardinals.
\end{proof}
We briefly outline now how
to prove the forcing direction of
Theorem \ref{teq}. Specifically,
we wish to show that
Con(ZFC + There is a proper class of
regular almost Ramsey cardinals) $\implies$
Con(ZF + DC + All successor cardinals are regular +
All infinite cardinals except possibly successors
of singular limit cardinals are almost Ramsey).
To do this,
we construct the model $N$ of Theorem 1 of \cite{A83},
using the class of regular almost Ramsey cardinals
in place of the class of supercompact cardinals.
(We refer readers of this paper to
\cite{A83} for the exact definition of $N$.)
The proofs of Lemmas 1.1--1.7 of \cite{A83}
then show that $N \models
``$ZF + DC + All successor
cardinals are regular + All successor
cardinals except possibly successors
of singular limit cardinals are
almost Ramsey'', assuming that
each regular almost Ramsey cardinal is
indestructible under forcing with arbitrary
L\'evy collapses, and each regular almost
Ramsey cardinal is preserved by small forcing.
However, these facts just follow from
Propositions \ref{p3} and \ref{p3a}.
Finally, since by Proposition \ref{p1},
ZF $\vdash$ ``Any limit of almost Ramsey
cardinals is an almost Ramsey cardinal'',
$N \models ``$All infinite cardinals
except possibly successors of singular
limit cardinals are almost Ramsey''.
This completes our discussion of
Theorem \ref{teq}.
\hfill$\square$\medskip
In the following, we apply the core model below a
proper class of strong cardinals,
denoted by
the class term $K$ (see \cite{Sc}).
%[Schindler, {\tmem{The core model up to one strong
%cardinal}}, doctoral dissertation, Bonn, check Ralf's publications]).
As with
the {{Dodd-Jensen}} core model, we get the following.
\begin{proposition}
{{(ZF)}} Let $a \subseteq \tmop{HOD}$ be a set. Then
$K^{\tmop{HOD}}
= K^{\tmop{HOD} [a]}$.
\end{proposition}
If there is no inner model with a
proper class of strong cardinals and the Axiom of Choice
holds, then the core model $K$ satisfies
certain covering properties. In particular,
Theorem 8.18 of \cite{Sc} tells us that if
%$0^{\tmop{hand grenade}}$ does not exist and
$\gk \ge \go_2$, then
$\tmop{cof}^V({(\gk^+)}^K) \ge \tmop{Card}^V(\gk)$.
%i.e., for
%sufficiently large singular cardinals $\kappa$, we have $\kappa^+ =
%(\kappa^+)^K$.
\begin{lemma}\label{ar2}
{{(ZF)}} Let $\kappa^+$ be almost {{Ramsey}}, where
$\kappa$ is
a singular cardinal. Then there is an inner
model with
a proper class of strong cardinals.
\end{lemma}
\begin{proof}
Assume that there is no inner model
with a proper class of strong cardinals. By Lemma
\ref{arhod},
$(\kappa^+)^{\tmop{HOD}} < \kappa^+$. Since $K \subseteq \tmop{HOD}$,
$(\kappa^+)^K < \kappa^+$. Choose a bijection $f : \kappa
\leftrightarrow
(\kappa^+)^K$ and a cofinal subset $Z \subseteq \kappa$ such that
$\tmop{otp} (Z) < \kappa$. The class $\tmop{HOD} [f, Z]$ is a model
of
$\tmop{ZFC}$ which satisfies that $\kappa$ is a singular cardinal
such that
$(\kappa^+)^K < \kappa^+$.
Then inside the model $\tmop{HOD} [f, Z]$,
$\tmop{cof}({(\gk^+)}^K) < \gk$.
This contradicts the above mentioned
covering property of the core model.
\end{proof}
We are now able to give a straightforward
proof of the following simplified
version of Theorem \ref{t2}.
\begin{theorem}\label{t2a}
Assume ZF and that every %(well-ordered)
infinite cardinal is almost
{{Ramsey}}.
%Then for every $n < \go$ and every set of ordinals $x$,
%$M^\sharp_n(x)$ exists.
Then there exists an inner model with a
proper class of strong cardinals.
\end{theorem}
%We are now able to prove Theorem \ref{t2}.
%\begin{lemma}
% Assume ZF and that every infinite cardinal is almost
% {{Ramsey}}. Then there exists an inner model with a
%proper class of strong cardinals.
%\end{lemma}
\begin{proof}
By assumption, $\aleph_{\omega + 1}$ is almost {{Ramsey}} and
the
successor of the singular cardinal $\aleph_{\omega}$.
Lemma \ref{ar2} now implies the desired conclusions.
\end{proof}
We conclude Section \ref{s3} with a brief
discussion of generalized versions of Lemma \ref{ar2},
Theorem \ref{t2}, and Theorem \ref{t2a}, which
were pointed out to us by Schindler.
%by remarking that Schindler has pointed out to us that
%the conclusions of
%both Lemma \ref{ar2} and
%Theorem \ref{t2} can be strengthened.
For Lemma \ref{ar2}, by recent work of Jensen
and Steel \cite{JeSt}, the conclusions may
be improved to inferring that
there is an inner model containing a Woodin cardinal.
Theorems \ref{t2} and \ref{t2a}
may in fact be strengthened to the following.
%may be %restated and generalized to the following.
\begin{theorem}\label{tsc}
Suppose $\Phi$ is a large cardinal concept
such that $\Phi(\gd) \implies$ There is
no inner model of ZFC in which $\gd$ is
a successor cardinal. Suppose further that
there is a model $N$ of ZF with a
proper class of cardinals $\gk$ such that in $N$,
$\gk$ is singular and $\Phi(\gk^+)$ is true.
Then for every $n < \go$ and every set of ordinals $x$,
$M^\sharp_n(x)$ exists.
\end{theorem}
%\noindent
Thus, Theorems \ref{t2} and \ref{t2a}
follow from Theorem \ref{tsc},
with $\Phi(\gd)$ interpreted as ``$\gd$ is almost Ramsey''.
Further, Theorem \ref{tsc} implies
that for every $n < \go$, there is actually an
inner model with $n$ Woodin cardinals, i.e.,
that PD holds in all set generic extensions.
Theorem \ref{tsc} may be proven using exactly
the same arguments as found in Section 3.1 of
\cite{BuSc}.
\section{A model in which every infinite cardinal
is almost Ramsey}\label{s4}
From certain fairly mild supercompactness
assumptions, we construct a model in which
all infinite cardinals
are almost {{Ramsey}}.
We have already stated the relevant theorem
%we are about to prove
as Theorem \ref{t1}.
We restate it here in
the form in which it will be proven.
%a slightly more precise form.
%Specifically, we prove the following.
\begin{theorem}\label{t1a}
Let $V \models ``$ZFC + $\gk < \gl$
are such that $\gk$ is $2^\gl$
supercompact where $\gl$ is the
least regular almost {{Ramsey}} cardinal
greater than $\gk$''. There is then
a model $N$ of height $\gk$ such that
$N \models ``$ZF + $\neg {AC}_\go$ +
Every successor cardinal is regular +
Every infinite cardinal is almost {{Ramsey}}''.
\end{theorem}
\begin{proof}
Our proof uses {{Gitik}}'s techniques of \cite{G85}.
As in \cite{A96}, we differ from
\cite{G85}, \cite{A85}, and \cite{A92}
only in the length of the {{Radin}} sequence of
measures used.
Our presentation follows
the ones given in \cite{A85},
\cite{A92}, and \cite{A96}
(all of which are based on \cite{G85}),
suitably modified to our current context
of almost Ramseyness.
As the necessary facts about Radin
forcing are distributed throughout
the literature, our
bibliographical citations will reflect
this.
%and will in general be presented either in terms of relevance
%or in chronological order.
Let $j : V \to M$ be an elementary
embedding witnessing the
$2^\gl$ supercompactness of $\gk$.
Our first step is to define a
{{Radin}} sequence of measures $\mu_{< \gr} = \la \mu_\ga \mid
\ga < \gr \ra$ over $P_{\gk}(\gl)$.
%where $\gr$ is
%such that $\mu_{\gr}$ is the first repeat point for
%$\mu_{< \gr}$.
Specifically, if
$\ga = 0$, $\mu_\ga$ is defined by $X \in \mu_\ga$ iff
$\la j(\gb) \mid \gb < \gl \ra \in j(X)$,
and if $\ga > 0$, $\mu_\ga$ is
defined by $X \in \mu_\ga$ iff $\la \mu_\gb \mid \gb < \ga \ra
=_{\hbox{\rm df}} \mu_{< \ga} \in j(X)$.
$\gr$ is then defined as the first ordinal such that $A \in
\mu_{\gr}$ implies that
for some $\ga < \gr$, $A \in \mu_\ga$, i.e.,
$\gr$ is the first {\em repeat point}.
By cardinality considerations
and the fact that $M^{2^\gl} \subseteq M$
(see \cite{CW}, \cite{R},
\cite{G85}, \cite{A85}, \cite{A96}, \cite{A92}, or
\cite{G07}),
this definition makes sense, and $\gr$ exists.
Next, using $\mu_{< \gr}$, we let $\FR_{< \gr}$ be
supercompact {{Radin}} forcing defined over $V_{\gk} \times
P_{\gk}(\gl)$. The particulars of the definition
are virtually identical
to the ones found in \cite{A85}, \cite{A92},
and \cite{A96}, but for clarity, we repeat them here.
%(which may be taken as our standard references for what follows),
$\FR_{< \gr}$ is composed of all
finite sequences of the form $\la \la p_0, u_0, C_0, \ra, \ldots,
\la p_n, u_n, C_n \ra , \la \mu_{< \gr}, C \ra \ra$ such
that the following conditions hold:
\begin{enumerate}
\item For $0 \le i < j \le n$, $p_i \smag p_j$, where for
$p, q \in P_{\gk}(\gl)$, $\ p \smag q$ means $p \subseteq q$ and
$\tmop{otp}(p) < q \cap \gk$.
%($\overline{p}$ is the order type of $p$.)
\item For $0 \le i \le n$, $p_i \cap \gk$ is a
measurable cardinal.
\item $\tmop{otp}(p_i)$ is the least cardinal greater than
$p_i \cap \gk$ which is a regular almost {{Ramsey}}
cardinal. In analogy to the notation of \cite{G85},
\cite{A85}, \cite{A92}, and \cite{A96},
we write $\tmop{otp}(p_i) = {(p_i \cap \gk)}^*$.
\item For $0 \le i \le n$, $u_i$ is a
{{Radin}} sequence of measures over $V_{p_i \cap \gk} \times
P_{p_i \cap \gk}(\tmop{otp}(p_i))$ with ${(u_i)}_0$, the
$0$th coordinate of $u_i$, a supercompact measure over
$P_{p_i \cap \gk}(\tmop{otp}(p_i))$.
\item $C_i$ is a sequence of measure 1 sets for $u_i$.
\item $C$ is a sequence of measure 1
sets for $\mu_{< \gr}$.
\item For each $p \in {(C)}_0$, where ${(C)}_0$ is the
coordinate of $C$ such that ${(C)}_0 \in \mu_0$, $
\bigcup_{i \in \{0, \ldots, n\}} p_i \smag p$.
\item For each $p \in {(C)}_0$, $\tmop{otp}(p) =
{(p \cap \gk)}^*$ and $p \cap \gk$ is a
measurable cardinal.
\end{enumerate}
Conditions (5) and (6) are both standard
to any definition of {{Radin}} forcing.
Conditions (1), (2), (4), and (7)
are all standard to any definition of
{\em supercompact} Radin forcing.
Conditions (3) and (8) are
used because of our ultimate
aim of constructing a model in which
all infinite cardinals are almost Ramsey.
That they may be included and have
the Radin forcing attain its desired goals
follows by the fact that
$V \models ``\gk$ is $2^\gl$ supercompact
where $\gl$ is the least regular almost
{{Ramsey}} cardinal above $\gk$''.
Thus, by closure,
$M \models ``\gk$ is measurable and
$\gl$ is the least regular
almost {{Ramsey}} cardinal above $\gk$''.
This means that by reflection,
$\{ p \in P_{\gk}(\gl) \mid p \cap \gk$ is a measurable
cardinal and $\tmop{otp}(p)$ is the
least regular
almost {{Ramsey}} cardinal greater than $p \cap \gk
\} \in \mu_0$.
This will ensure that the Radin
sequence of cardinals eventually produced
can be used in our final symmetric inner model $N$.
%We therefore will be able to
For completeness of exposition,
we recall now the definition of the ordering on
$\FR_{< \gr}$. If $\pi_0 =
%\break
\la \la p_0, u_0, C_0 \ra ,
\ldots , \la p_n, u_n, C_n \ra ,
\la \mu_{< \gr},
C \ra \ra$ and $\pi_1 = \la \la q_0, v_0, D_0 \ra , \ldots ,%\break
\la q_m, v_m, D_m \ra , \la \mu_{< \gr} , D \ra \ra$, then
$\pi_1$ extends $\pi_0$ iff the following conditions hold.
\begin{enumerate}
\item For each $\la p_j, u_j, C_j \ra$
which appears in $\pi_0$, there is a $\la q_i, v_i, D_i \ra$ which
appears in $\pi_1$ such that $\la q_i, v_i \ra = \la p_j, u_j \ra$
and $D_i \subseteq C_j$, i.e., for each coordinate ${(D_i)}_\ga$
and ${(C_j)}_\ga$, ${(D_i)}_\ga \subseteq {(C_j)}_\ga$.
\item $D \subseteq C$.
\item $n \le m$.
\item If $\la q_i, v_i, D_i \ra$ does not appear in $\pi_0$, let
$\la p_j, u_j, C_j \ra$ (or $\la \mu_{< \gr}, C \ra$) be
the first element of $\pi_0$ such that $p_j \cap \gk >
q_i \cap \gk$. Then
\begin{enumerate}
\item $q_i$ is
order isomorphic to some $q \in {(C_j)}_0$.
\item There exists
an $\ga < \ga_0$, where $\ga_0$ is the length
of $u_j$, such that $v_i$ is isomorphic ``in a natural way'' to an
ultrafilter sequence $v \in {(C_j)}_\ga$.
\item For $\gb_0$ the length of $v_i$, there is a function
$f: \gb_0 \to \ga_0$ such that for $\gb < \gb_0$, ${(D_i)}_\gb$ is a
set of ultrafilter sequences such that for some subset
${(D_i)}'_\gb $ of ${(C_j)}_{f(\gb)}$, each ultrafilter sequence
in ${(D_i)}_\gb$ is isomorphic ``in a natural way'' to an ultrafilter
sequence in ${(D_i)}'_\gb $.
\end{enumerate}
\end{enumerate}
\noindent For further information on the definition of the ordering
on $\FR_{< \gr}$ (including the meaning of ``in a natural
way'') and more facts about {{Radin}} forcing in general,
readers are referred to \cite{A85}, \cite{A92},
\cite{A96}, \cite{CW}, \cite{FW}, \cite{G85},
\cite{G07}, and \cite{R}.
We are now ready to define the partial
ordering $\FP$ used in the proof
of Theorem \ref{t1a}. It is given by
the finite support product
ordered componentwise
$$\prod_{\{\la \ga, \gb \ra \mid
\ga < \gb < \gk \ {\rm are} \
{\rm regular} \ {\rm cardinals}\}}
{\rm Coll}(\ga, {<} \gb) \times
\FR_{< \gr},$$
where ${\rm Coll}(\ga, {<} \gb)$ is the
L\'evy collapse of all cardinals of size
less than $\gb$ to $\ga$.
Let $G$ be $V$-generic over $\FP$, and let
$G_0$ be the projection of $G$ onto $\FR_{< \gr}$.
For any condition $\pi \in \FR_{< \gr}$, call
$\la p_0, \ldots p_n \ra$ {\em the
$p$-part of $\pi$}. Let $R = \{p \mid
\exists \pi \in G_0[p \in
{\rm p-part}(\pi)]\}$, and let
$R_\ell = \{p \mid p \in R$ and $p$
is a limit point of $R\}$. Define three sets
$E_0$, $E_1$, and $E_2$ by
$E_0 = \{\ga \mid$ For some
$\pi \in G_0$ and some
$p \in {\rm p-part}(\pi)$, $p \cap \gk = \ga\}$,
$E_1 = \{\ga \mid \ga$ is a limit point of $E_0\}$, and
$E_2 = E_1 \cup \{\go\} \cup \{\gb \mid
\exists \ga \in E_1[\gb = \ga^*]\}$.
$E_2$ will be the set of cardinals in
our symmetric inner model $N$.
Let $\la \ga_\nu \mid \nu < \gk \ra$ be the
continuous, increasing enumeration of $E_2$, and let
$\nu = \nu' + n$ for some $n \in \go$,
where $\nu'$ is either a limit ordinal
or $0$.
For $\gb \in [\ga_\nu, \ga_{\nu + 1})$, define sets
$C_i(\ga_\nu, \gb)$ for $i = 1, 2$
according to specific conditions on $\nu$ and $\nu'$
in the following manner:
\begin{enumerate}
\item $\nu = \nu' \neq 0$ and $n = 0$, i.e.,
$\nu$ is a limit ordinal. Let
$p(\ga_\nu)$ be the element $p$ of $R$ such that
$p \cap \gk = \ga_\nu$, and let
$h_{p(\ga_\nu)} : p(\ga_\nu) \to \tmop{otp}(p(\ga_\nu))$ be
the order isomorphism between $p(\ga_\nu)$ and
$\tmop{otp}(p(\ga_\nu))$. Then
$C_1(\ga_\nu, \gb) = \{{h_{p(\ga_\nu)}} '' p \cap \gb \mid
p \in R_\ell$, $p \subseteq p(\ga_\nu)$, and
$h^{-1}_{p(\ga_\nu)}(\gb) \in p\}$.
\item $(\nu = \nu' + n$ and $n \ne 0)$ or
$(\nu = \nu' = 0)$, i.e.,
($\nu$ is a successor ordinal)
or $(\nu = 0)$. Let
$H(\ga_\nu, \ga_{\nu + 1})$ be the projection of $G$ onto
${\rm Coll}(\ga_\nu, {<} \ga_{\nu + 1})$. Then
$C_2(\ga_\nu, \gb) = H(\ga_\nu, \ga_{\nu + 1})
\rest \gb$, i.e.,
$C_2(\ga_\nu, \gb) =
\{p \in H(\ga_\nu, \ga_{\nu + 1}) \mid
\dom(p) \subseteq \ga_\nu \times \gb\}$.
\end{enumerate}
$C_1(\ga_\nu, \gb)$ is used to collapse
$\gb$ to $\ga_\nu$ when $\nu$ is a
limit ordinal, and is also used to
generate the closed, cofinal sequence
$\la \ga_\gb \mid \gb < \nu \ra$.
%when $\nu$ is a limit ordinal.
$C_2(\ga_\nu, \gb)$ is used to collapse
$\gb$ to $\ga_\nu$ when $\nu$ is a successor ordinal
or $\nu = 0$.
%$C_1(\ga_\nu, \gb)$ is also used to
Intuitively, the symmetric inner model $N \subseteq V[G]$
witnessing the conclusions of Theorem \ref{t1a}
is $V_\gk$ of the least model of ZF
extending $V$ which contains,
for $\gb \in [\ga_\nu, \ga_{\nu + 1})$,
$C_1(\ga_\nu, \gb)$ if $\nu$
is a limit ordinal, and
$C_2(\ga_\nu, \gb)$ if $\nu$
is a successor ordinal or $\nu = 0$.
To define $N$ more precisely, it is necessary to define
canonical names $\underline{\ga_\nu}$ for the $\ga_\nu$'s and canonical
names $\underline{C_i(\nu, \gb)}$
for the two sets just described. Recall that it is
possible to decide $p(\ga_\nu)$ (and hence
$\tmop{otp}(p(\ga_\nu))$) by
writing $\omega \cdot \nu = \omega^{\sigma_0} \cdot n_0 +
\omega^{\sigma_1} \cdot n_1 + \cdots + \omega^{\sigma_m} \cdot
n_m$ (where $\sigma_0 > \sigma_1 > \cdots > \sigma_m$ are
ordinals, $n_0, \ldots, n_m > 0$ are integers, and $+$,
$\cdot$, and exponentiation are the
ordinal arithmetical operations), letting
$\pi = \la{\la p_{ij_i}, u_{ij_i}, C_{ij_i} \ra}_{i \le m,
1 \le j_i\le n_i}, \la \mu_{< \gr}, C \ra \ra$ be such that
min($p_{i1} \cap \gk, \omega^{{\hbox{\rm length}}(u_{i1})}) =
\sigma_i$ and length($u_{ij_i}) = {\hbox{\rm min}}(p_{i1} \cap \gk,
{\hbox{\rm length}}(u_{i1}))$ for $1 \le j_i \le n_i$, and letting
$p(\ga_\nu)$ be $p_{mn_m}$.
Further, $D_\nu = \{ r \in \FP \mid r \rest
\FR_{< \gr}$ extends a condition $\pi$ of the above form$\}$
is a dense open subset of $\FP$. $\underline{\ga_\nu}$ is the name of
the $\ga_\nu$ determined by any element of $D_\nu \cap G$; in the
notation of \cite{G85}, \cite{A85}, \cite{A92}, and \cite{A96},
$\underline{\ga_\nu} = \{ \la r, \check \ga_\nu(r)
\ra \mid r \in D_\nu \}$, where $\ga_\nu(r)$ is the $\ga_\nu$ determined
by the condition $r$.
The canonical names $\underline{C_i(\nu, \gb)}$
are defined in a manner so as to
be invariant under the appropriate group of
automorphisms. Specifically, there are two cases to
consider. We again write $\nu = \nu' + n$,
where $n \in \go$ and $\nu'$ is either
a limit ordinal or $0$,
and let $\gb$ be as
before. We also assume without loss of generality that as in
\cite{G85}, \cite{A85}, \cite{A92}, and \cite{A96},
$\ga_{\nu + 1}$ is determined by $D_\nu$. Further, we adopt
throughout each of the two cases
the notation of \cite{G85}, \cite{A85}, \cite{A92}, and \cite{A96}.
\begin{enumerate}
\item $\nu' = \nu \ne 0$ and $n=0$.
$\underline{C_1(\nu, \gb)} = \{
\la r, (\check r \rest \FR_{< \gr}) \rest (\ga_\nu(r), \gb) \ra \mid r \in
D_\nu \}$, where for $r \in \FP$, $\pi = r \rest \FR_{< \gr}$, $\pi \rest
(\ga_\nu(r), \gb) = \{ {{h_{p(\ga_\nu)(r)}}}''p\cap \gb \mid p \in$ p-part$
(\pi)$, $ p \subseteq p(\ga_\nu)(r)$, $ p \in R_\ell \rest \pi$, and
$h^{-1}_{p(\ga_\nu)(r)}(\gb) \in p\} $.
\item ($\nu = \nu' + n$ and $n \neq 0$)
or $(\nu = \nu' = 0)$.
$\underline{C_2(\nu, \gb)} =
\{ \la r, (\check r \rest
{\hbox{\rm Coll}}(\ga_\nu(r), {<} \ga_{\nu + 1}(r)))
\rest \gb \ra \mid r \in D_\nu \}$.
\end{enumerate}
\noindent As in \cite{G85}, \cite{A85}, \cite{A92},
and \cite{A96}, since for any
$r , r' \in D_\nu \cap G$, $p(\ga_\nu)(r) =
p(\ga_\nu)(r')$, both of the definitions
just given are unambiguous.
Let ${\cal G}$ be the group of automorphisms of \cite{G85},
and let $\underline{C(G)} = \break
\bigcup_{i = 1, 2} \{
\psi(\underline{C_i(\nu,\gb)}) \mid \psi
\in {\cal G}$, $0 \le \nu< \gk$, and $\gb \in [\nu,\gk)$ is a
cardinal$\}$.
$C(G) = \bigcup_{i = 1 , 2}
\{i_G(\psi(\underline{C_i(\nu,\gb)}))
\mid \psi \in {\cal G}$, $ 0 \le \nu< \gk$,
and $\gb \in [\nu,\gk)$ is a cardinal$\} = i_G(\underline{C(G)})$.
$N$ is then the set of all
sets of rank less than
$\gk$ of the model consisting of all sets
which are hereditarily $V$ definable from $C(G)$, i.e.,
$N = V^{ \tmop{HVD} (C(G)) }_{\gk}$.
The arguments of \cite{G85} allow us to conclude that
$N \models ``$ZF + $\neg {\rm AC}_\go$ +
$\la \ha_\nu \mid \nu \in {\rm Ord} \ra =
\la \ga_\nu \mid \nu \in {\rm Ord} \ra$ +
Every successor cardinal is regular''.
In addition, we
know that for any ordinal
$\gl$ and any set $x \subseteq \gl$, $x \in N$, $x = \{\ga<\gl \mid
V[G] \models
\varphi(\ga, i_G(\psi_1(\underline{C_{i_1}(\nu_1,\gb_1)})),\ldots,
i_G(\psi_n(\underline{C_{i_n}(\nu_n,\gb_n)})), C(G))\}$,
where $i_j$ is
an integer, $1 \le j \le n$, $1 \le i_j \le 2$, each $\psi_i
\in {\cal G}$, each $\gb_i$ is an appropriate ordinal for $\nu_i$,
and $\varphi(x_0, \ldots, x_{n+1})$ is a formula which may also
contain some parameters from $V$ which we shall suppress.
Let $$\overline \FP =
\prod_{i_j = 2}
{\rm Coll}(\ga_{\nu_j}, {<} \gb_j)
\times \FR_{< \gr}.$$
For $\pi \in \FR_{< \gr}$,
%and $\gl$ an arbitrary ordinal,
let $\pi \rest \gl = \{\la q,
u, C \ra \in \pi \mid q \cap \gk \le \gl \}$.
%and let $\FR_\gl = \{\pi \rest \gl \mid \pi \in \FR_{< \gr}\}$.
For
$p \in \overline{\FP}$, $p = \la p_1, \ldots, p_m, \pi \ra$, $m \le n$,
$\pi \in \FR_{< \gr}$, let $p \rest \gl = \la q_1, \ldots, q_m, \pi \rest
\gl \ra$, where $q_j = p_j$ if $\ga_{\nu_j} \le \gl$
and $q_j = \emptyset$ otherwise. In
other words, $p \rest \gl$ is the part of p below or at
$\gl$. Without loss of generality, we ignore the empty coordinates
and let $\overline{\FP} \rest \gl = \{p \rest \gl \mid p \in \overline{\FP} \}$. Let
$G \rest \gl$ be the projection of $G$ onto $\overline{\FP} \rest \gl$. An
analogous fact to Theorem 3.2.11 of \cite{G85} holds, using the same
proof as in \cite{G85},
%namely for any $x \subseteq \gl$,
namely $x \in
V[G \rest \gl]$. In addition, the elements of $\overline{\FP} \rest \gl$
can be partitioned into equivalence classes (the ``almost
similar'' equivalence classes of \cite{G85}) with respect to
$\underline{C_{i_1}(\nu_1,\gb_1)}, \ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$
via an equivalence relation to be called $\sim$
such that if $\sigma < \gl$, $\tau$ is a
suitable term for $x$, and $p
\forces ``\sigma \in \tau$'', then for any $q \sim p$,
$q \forces ``\sigma \in \tau$''. It thus follows as an immediate
corollary of the work of \cite{G85} that if we define $
\widetilde{G \rest \gl}
= \{ {[p]}_\sim \mid p \in G \rest \gl \}$,
then $x \in
V[\widetilde{G \rest \gl}]
$ and $V[
\widetilde{G \rest \gl}
] \subseteq N$.
Further, if $\gl = \ga_\nu$ and either
$\nu$ is a successor ordinal
or $\nu = 0$,
then the work of \cite{G85}
also tells us that
%$\ov \FP \rest \gl$ is forcing equivalent to a partial ordering of the form
$\widetilde{G \rest \gl} =
G_0 \times G_1$ is
$V$-generic over a partial ordering
of the form $\FP_0 \times \FP_1$,
where $\FP_1 = {\rm Coll}(\gl, {<} \gb)$
for some $\gb$, and $\FP_0$ is forcing
equivalent to a partial ordering $\FP^*$
such that $\card{\FP^*} < \gl$.
In what follows, we will slightly abuse
notation and denote $V[G_0][G_1]$ and
$V[G_1]$ by $V^{\FP_0 \times \FP_1}$ and
$V^{\FP_1}$ respectively.
The discussion of the proof of
Theorem \ref{t1a} will now
be completed by the following lemma.
\begin{lemma}\label{l1}
$N \models ``$Every $\ga_\nu$
is an almost {{Ramsey}} cardinal''.
\end{lemma}
\begin{proof}
Fix $\nu < \gk$ and $\gl = \ga_\nu$.
Since by Proposition \ref{p1},
ZF $\vdash$ ``Any limit of
%suitably Erd\"os or
almost
{{Ramsey}} cardinals is an almost
{{Ramsey}} cardinal'', without loss
of generality, we may assume that
$\nu$ is a successor ordinal.
Further, by the properties of
the {{Radin}} forcing $\FR_{< \gr}$,
we may
also assume without loss of generality
that in $V$, $\gl$ is a regular
almost {{Ramsey}} cardinal.
Suppose
$N \models ``f : {[\gl]}^{< \go} \to 2$
is a partition''.
%{{Gitik}}'s methods tell us roughly speaking that
Note that $f$ may be coded by
a subset of $\gl$. Therefore,
as in our discussion above,
$f \in V^{\FP_0 \times \FP_1}$, where
$\FP_0$ is forcing equivalent to
a partial ordering $\FP^*$ such that
$\card{\FP^*} < \gl$, and $\FP_1$ is
${\rm Coll}(\gl, {<} \gb)$ for some $\gb$.
Since by Proposition \ref{p3}, any
regular almost {{Ramsey}} cardinal $\gd$
is automatically
indestructible under
%$\gd$-directed closed forcing
${\rm Coll}(\gd, {<} \gb)$ (and much more),
$V^{\FP_1} \models ``\gl$ is almost {{Ramsey}}''.
Since $\card{\FP^*} < \gl$ in both
$V$ and $V^{\FP_1}$,
by Proposition \ref{p3a},
%and the fact the
%L\'evy-Solovay results \cite{LS}
%show that both the properties
%of being suitably Erd\"os
%and almost Ramseyness are preserved
%by small forcing,
$\gl$ is almost
{{Ramsey}} in $V^{\FP_1 \times \FP_0} =
V^{\FP_0 \times \FP_1}$. Thus, for every
$\ga < \gl$, there is
$X \in {[\gl]}^\ga$, $X \in V^{\FP_0 \times \FP_1}$
which is homogeneous for $f$. Since
$V^{\FP_0 \times \FP_1} \subseteq N$, $X \in N$
as well.
This completes the proof of
Lemma \ref{l1}.
\end{proof}
Lemma \ref{l1} completes
the proof of Theorem \ref{t1a}.
\end{proof}
We note that the
properties of {{Radin}} forcing,
together with {{Gitik}}'s methods,
allow us to infer that since
the {{Radin}} forcing $\FR_{< \gr}$ is defined
using a long enough sequence of measures $\mu_{< \gr}$,
$N$ will contain regular limit cardinals.
Also, the arguments of \cite{A85} suitably
modified tell us that
$N \models ``$Every singular limit
cardinal is a Jonsson cardinal''.
We conclude by remarking that in the
models of \cite{G85}
and \cite{A85}, it is the case that
all infinite cardinals are almost
{{Ramsey}}.
The methods of proof are similar to those
found in this paper.
The constructions use an almost huge cardinal,
but the consistency strength of the assumptions
employed was reduced in \cite{A92a} to
something in consistency strength strictly
in between a supercompact limit of
supercompact cardinals and an almost huge cardinal.
Our hypotheses employed for Theorem
\ref{t1a}, of course, are considerably weaker than
those of \cite{G85}, \cite{A85}, or \cite{A92a}.
There are two main reasons for this.
One is that we do not have any singular
successor cardinals in our desired model $N$.
The other is that, roughly speaking,
as Proposition \ref{p3} shows, almost {{Ramsey}} cardinals
$\gl$ are automatically indestructible under
%$\gl$-directed closed forcing,
forcing not adding any bounded
subsets of $\gl$,
meaning that
no additional preparation is required prior
to the construction of $N$.
We conjecture that in {{Gitik}}'s model of
\cite{G80} in which all uncountable cardinals
are singular, built using a proper class of
strongly compact cardinals, all
infinite cardinals are also almost {{Ramsey}}.
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\begin{eqnarray*}
M & \models & \exists v_1 < x_0 \exists v_2 < x_1 \ldots \exists
v_{m - 1}
< x_0
(f_0 ( \vec{x}, \vec{y}) < x_0 \wedge \varphi (f_0 ( \vec{x},
\vec{y}), v_1, \ldots, v_{m - 1}, \vec{x}) \wedge \neg \varphi (f_0
(
\vec{x}, \vec{y}), v_1, \ldots, v_{m - 1}, \vec{y}))\\
M & \models & \exists v_2 < x_0 \ldots \exists v_{m - 1} < x_0 (f_0
(
\vec{x}, \vec{y}) < x_0 \wedge f_1 ( \vec{x}, \vec{y}) < x_0 \wedge
\varphi (f_0 ( \vec{x}, \vec{y}), f_1 ( \vec{x}, \vec{y}), \ldots,
v_{m -
1}, \vec{x}) \wedge \neg \varphi (f_0 ( \vec{x}, \vec{y}), f_1 (
\vec{x},
\vec{y}), \ldots, v_{m - 1}, \vec{y}))\\
& \vdots & \\
M & \models & f_0 ( \vec{x}, \vec{y}) < x_0 \wedge \ldots \wedge
f_{m - 1}
( \vec{x}, \vec{y}) < x_0 \wedge \varphi (f_0 ( \vec{x}, \vec{y}),
f_1 (
\vec{x}, \vec{y}), \ldots, f_{m - 1} ( \vec{x}, \vec{y}), \vec{x})
\wedge
\neg \varphi (f_0 ( \vec{x}, \vec{y}), f_1 ( \vec{x}, \vec{y}),
\ldots,
f_{m - 1} ( \vec{x}, \vec{y}), \vec{y})
\end{eqnarray*}
$\tmop{Con} ({ZFC}$ + There is a proper class of regular almost
{{Ramsey}} cardinals$)$ $\implies$
$\tmop{Con} ({ZF}$ + All infinite cardinals except possibly
successors
of singular limit cardinals are almost {{Ramsey}}$)$
Our hypotheses employed for Theorem
\ref{t1a}, of course, are considerably weaker than
those of \cite{G85}, \cite{A85}, or \cite{A92a}.
The main reason for this is that, roughly speaking,
as Proposition \ref{p3} shows, almost {{Ramsey}} cardinals
$\gl$ are automatically indestructible under
%$\gl$-directed closed forcing,
forcing not adding any bounded
subsets of $\gl$,
meaning that
no additional preparation is required prior
to the construction of our desired model $N$.
We conjecture that in {{Gitik}}'s model of
\cite{G80} in which all uncountable cardinals
are singular, built using a proper class of
strongly compact cardinals, all
infinite cardinals are also almost {{Ramsey}}.
We consider the notion of almost {{Ramsey}}
cardinal. In particular, we examine both their
combinatorial properties and consistency strengths.
We also construct a model of ZF +
$\neg {\rm AC}_\go$ in which every %uncountable
%(well-ordered)
infinite cardinal is almost {{Ramsey}}.
If there is no inner model with a
proper class of strong cardinals and the Axiom of Choice
holds, then the core model $K$ satisfies
a form of covering. In particular,
Theorem 8.18 of \cite{Sc} tells us that if
$0^{\tmop{hand grenade}}$ does not exist and
$\gk \ge \go_2$, then
$\tmop{cof}^V({(\gk^+)}^K) \ge \tmop{Card}^V(\gk)$.
%i.e., for
%sufficiently large singular cardinals $\kappa$, we have $\kappa^+ =
%(\kappa^+)^K$.
\begin{proof}
Assume that there is no inner model
with a proper class of strong cardinals. By Lemma
\ref{arhod},
$(\kappa^+)^{\tmop{HOD}} < \kappa^+$. Since $K \subseteq \tmop{HOD}$,
$(\kappa^+)^K < \kappa^+$. Choose a bijection $f : \kappa
\leftrightarrow
(\kappa^+)^K$ and a cofinal subset $Z \subseteq \kappa$ such that
$\tmop{otp} (Z) < \kappa$. The class $\tmop{HOD} [f, Z]$ is a model
of
$\tmop{ZFC}$ which satisfies that $\kappa$ is a singular cardinal
such that
$(\kappa^+)^K < \kappa^+$. But this contradicts
% the covering theorem below $0^{\tmop{hand grenade}}$
Theorem 8.18 of \cite{Sc}
inside the model $\tmop{HOD} [f, Z]$,
which by \cite{Sc} immediately implies
the existence of an inner model with a
proper class of strong cardinals.
\end{proof}
We conclude Section \ref{s3} by remarking
that Schindler has pointed out to us that
the conclusions of
Theorem \ref{t2} can be strengthened.
In particular,
the proof of Corollary 5 of \cite{ScZe}
(which is found in \cite{ScSt}), combined
with the proof of Lemma \ref{arhod} of this paper,
actually show that if the successor
of a singular cardinal $\gk$ is almost Ramsey, then
for every $n < \go$ and every $x \subseteq \gk$,
$M^\sharp_n(x)$ exists.
Thus, for every $n < \go$, there is an
inner model with $n$ Woodin cardinals.