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\begin{document}
\title{The Consistency Strength of Choiceless
Failures of SCH
\thanks{2000 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Singular Cardinals Hypothesis (SCH),
core model, parallel Prikry forcing,
symmetric inner model.}
\thanks{Both authors wish to thank the referee, for helpful
comments, suggestions, and corrections which have been
incorporated into the current version of the manuscript.}}
\author{Arthur W.~Apter\thanks{The
first author's research was partially
supported by PSC-CUNY grants,
CUNY Collaborative Incentive
grants, and DFG-NWO Bilateral Grant
KO 1353-5/1/DN 62-630. 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 2009 Scholar Incentive Leave
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\\
Endenicher Allee 60\\
D-53115 Bonn, Germany\\
http://www.math.uni-bonn.de/people/koepke\\
koepke@math.uni-bonn.de}
\date{May 31, 2009\\
(revision as of September 16, 2009)}
\maketitle
%\newpage
\begin{abstract}
We determine exact consistency strengths for various failures of the
Singular Cardinals Hypothesis (${\rm SCH}$) in the setting of the
{{\rm Zermelo-Fraenkel}} axiom system ${\rm ZF}$ without the Axiom of
Choice (${\rm AC}$).
By the new notion of parallel
Prikry forcing that we introduce,
we obtain surjective failures of SCH
using only one measurable cardinal, including
a surjective failure of Shelah's pcf theorem about
the size of the power set of $\ha_\go$.
Using symmetric collapses to $\aleph_{\omega}$,
$\ha_{\go_1}$, or
$\aleph_{\omega_2}$, we show that
injective failures at $\aleph_{\omega}$,
$\ha_{\go_1}$,
or
$\aleph_{\omega_2}$ can have relatively mild
consistency strengths in terms
of Mitchell orders of measurable cardinals.
% we can obtain any surjective failure of SCH using
% only one measurable cardinal, including a surjective
% we can even arrange a surjective
% failure of {{\rm Shelah}}'s pcf theorem at $\aleph_{\omega}$ from
% {{\rm one}} measurable cardinal.
Injective failures of both the aforementioned theorem of
Shelah and Silver's theorem that GCH cannot first fail at
a singular strong limit cardinal of uncountable cofinality
are also obtained.
Lower bounds are shown by core model
techniques and methods due to %Gitik (individually) and
Gitik and Mitchell. % (jointly).
\end{abstract}
\baselineskip=24pt
\section{Introduction}\label{s1}
One of the first applications of {{\rm Paul Cohen}}'s method of
{{\rm forcing}} was given by Easton \cite{E} (see also \cite{J}),
who showed that for {{\rm regular}} cardinals $\kappa$, the
value of the continuum function $2^{\kappa}$ is largely undetermined by the
standard ${\rm ZFC}$ axioms of set theory. To extend those investigations to
{{\rm singular}} cardinals $\kappa$ proved significantly more difficult and
led to the formulation of the {{\rm Singular Cardinals Hypothesis}}
(${\rm SCH}$):
\[ ({\rm cof} (\kappa) < \kappa) \wedge (\forall \nu < \kappa)
[ 2^{\nu} < \kappa]
\implies \kappa^{{\rm cof} (\kappa)} = \kappa^+ . \]
Using the Axiom of Choice (AC), the ${\rm SCH}$ readily implies that
\[({\rm cof} (\kappa) < \kappa) \wedge
(\forall \nu < \kappa) [2^{\nu} < \kappa]
\implies 2^{\kappa} = \kappa^+, \]
i.e., the continuum function at $\kappa$ takes the smallest possible value.
The Singular Cardinals Hypothesis had a decisive impact on the further
development of axiomatic set theory,
leading to sophisticated methods in
combinatorics, forcing, and the theory of inner models.
{{\rm Jack Silver}} (see \cite{Si} and \cite{J})
proved some instances of ${\rm SCH}$ from the ${\rm ZFC}$ axioms and forced
violations of ${\rm SCH}$ in other cases.
{{\rm Ronald Jensen}} showed (see \cite{DeJ})
that violating the ${\rm SCH}$ requires the existence of large cardinals in
inner models of set theory. {{\rm Saharon Shelah}}'s
pcf theory \cite{Sh} extends
{{\rm Silver}}'s analysis also to the case of countable cofinality.
Moti Gitik \cite{G89}, \cite{G91} determined the exact
consistency strength of the negation of ${\rm SCH}$, in the presence of
${\rm AC}$, to be
\[ {\rm Con} ({\rm ZFC} + \neg {\rm SCH}) \iff {\rm Con}
({\rm ZFC} + \exists \kappa [o (\kappa) = \kappa^{+ +}]) . \]
Note that the work of \cite{G89} uses (previously)
unpublished ideas of Woodin.
%On the right hand side, $o (\kappa) = \kappa^{+ +}$ means that $\kappa$ is a
%measurable cardinal with a coherent sequence of measures of maximal length.
The results and arguments mentioned so far essentially involve the
Axiom of Choice. In this paper, we examine the status of $\neg {\rm SCH}$
{{\rm without}} ${\rm AC}$. We will obtain equiconsistencies for
three of our main theorems, and upper and lower bounds in consistency
strength for our fourth main theorem.
There will be sharp differences between the non-AC and AC situations,
as we shall explain shortly.
The logical negation of the ${\rm SCH}$ at $\kappa$ reads
\[ ({\rm cof} (\kappa) < \kappa) \wedge
(\forall \nu < \kappa)[ 2^{\nu} < \kappa]
\wedge (\kappa^{{\rm cof} (\kappa)} \neq \kappa^+) . \]
Without ${\rm AC}$, $\kappa^{{\rm cof} (\kappa)} \neq \kappa^+$ does not
imply that $\kappa^{{\rm cof} (\kappa)}$ is {{\rm larger}} than $\kappa^+$.
So we have to express largeness in terms of cardinality theory without
${\rm AC}$. In the sequel,
we shall distinguish between {{\rm surjective
failures}} of ${\rm SCH}$, e.g.,
\[ ({\rm cof} (\kappa) < \kappa) \wedge
(\forall \nu < \kappa) [2^{\nu} < \kappa]
\wedge \text{ (There is a surjective } f : [\kappa]^{{\rm cof} (\kappa)}
\rightarrow \kappa^{+ +}) \]
and {{\rm injective failures}}, e.g.,
\[ ({\rm cof} (\kappa) < \kappa) \wedge
(\forall \nu < \kappa) [2^{\nu} < \kappa]
\wedge \text{ (There is an injective } f : \kappa^{+ +} \rightarrow
[\kappa]^{{\rm cof} (\kappa)}) . \]
Note that $2^{\nu} < \kappa$ for $\nu < \kappa$ implies that
$\wp(\nu)$ is well-orderable
in some order type less than $\kappa$.
We prove that surjective failures of ${\rm SCH}$
in ${\rm ZF} + \neg{\rm AC}$ are of mild
consistency strength, i.e., only
one measurable cardinal, and that
a surjective failure at
minimal singular cardinals of cofinality $\go$ like
$\aleph_{\omega}$ does not raise the strength.
These surjective failures at $\ha_\go$ may be
beyond what is currently known to be possible
in ZFC, or even contradict what is possible in ZFC.
%In particular, we explicitly prove the following theorems,
%taking $\ha_\go$ as our prototype.
We further construct injective failures of SCH at
$\ha_\go$ that are beyond what is currently
known to be possible in ZFC, or even contradict
what is possible in ZFC, and show that they have
fairly mild consistency strengths as well.
We in addition force injective failures of SCH
at $\ha_{\go_1}$ and $\ha_{\go_2}$ that are
impossible in ZFC, and demonstrate that their
consistency strengths are also quite innocuous.
Specifically, we prove the following theorems,
emphasizing that throughout, whenever we talk about
GCH holding below a cardinal $\gk$, we literally
mean the same thing as when AC is true, i.e., that
for every (well-ordered) cardinal $\nu < \gk$,
$\wp(\nu)$ is well-orderable and has cardinality $\nu^+$.
\begin{theorem}\label{t1} $\ $
For a fixed $\ga \ge 2$,
the following theories are equiconsistent:
\[ {ZFC} + \exists \kappa [\kappa \text{ is measurable}] \]
and
\[ {ZF} + \neg {AC} + \text{GCH holds
below } \ha_\go +
\text{There is a surjective } f : [\aleph_{\omega}]^{\omega}
\rightarrow
\aleph_{\omega + \ga}. \]
\end{theorem}
\begin{theorem}\label{t2} $\ $
For a fixed $n < \go$, $n \ge 1$,
the following theories are equiconsistent:
\[ {ZFC} + \exists \gk[({\rm cof}(\gk) = \go) \wedge
(\forall i < \go) (\forall \gl < \gk)
(\exists \gd < \gk) [(\gd > \gl) \wedge (o(\gd) \ge
\gd^{+i})]] \]
and
\[ {ZF} + \neg {AC} + \text{GCH holds
below } \ha_\go +\text{
There is an injective } f : \aleph_{\omega_n}
\rightarrow [\aleph_{\omega}]^{\omega}. \]
\end{theorem}
When talking about a choiceless injective failure
of SCH for singular cardinals of uncountable
cofinality, the situation is somewhat different.
More explicitly, we have the following
two theorems, which we state in a
very specific, concrete manner for the
sake of exposition.
There are two distinct cases, depending on
whether the singular cardinal has cofinality
at least $\go_2$ or cofinality $\go_1$.
We shall use $\ha_{\go_2}$ and $\ha_{\go_1}$
as our prototypes.
The reason for the split in cases, as well as
the many additional possibilities,
will be discussed in Section \ref{s8}.
%towards the end of the paper.
\begin{theorem}\label{t3}$\ $
The following theories are equiconsistent:
\[ {ZFC} + \exists \kappa [o (\kappa) = \kappa^{++}
+ \go_2] \]
and
\[ {ZF} + \neg {AC} + \text{GCH holds
below } \ha_{\go_2} +\text{
There is an injective } f : \aleph_{\omega_2 + 2}
\rightarrow [\aleph_{\omega_2}]^{\omega_2}. \]
\end{theorem}
\begin{theorem}\label{t4}
{\tmdummy}
\begin{enumeratealpha}
\item\label{t4a} If the theory
\[ {ZFC} + \exists \kappa [o (\kappa) = \kappa^{++}
+ \go_1] \]
is consistent, then so is the theory
\[ {ZF} + \neg {AC} + \text{GCH holds
below } \ha_{\go_1} +\text{
There is an injective } f : \aleph_{\omega_1 + 2}
\rightarrow [\aleph_{\omega_1}]^{\omega_1}. \]
\item\label{t4b}
If the theory
\[ {ZF} + \neg {AC} + \text{GCH holds
below } \ha_{\go_1} +\text{
There is an injective } f : \aleph_{\omega_1 + 2}
\rightarrow [\aleph_{\omega_1}]^{\omega_1} \]
is consistent, then so is the theory
\[ {ZFC} + \exists \gk
[o(\gk) = \gk^{++}]. \]
\end{enumeratealpha}
\end{theorem}
%For $\gl$ the singular cardinal in
In Theorems \ref{t2} -- \ref{t4}, for
$\gl = \ha_\go$, $\gl = \ha_{\go_1}$, or
$\gl = \ha_{\go_2}$,
our proofs will show that the injection into
${[\gl]}^{{\rm cof}(\gl)}$ can be safely replaced with
an injection into $\wp(\gl)$. As we shall discuss
later, the analogous fact for Theorem \ref{t1}
may not hold, i.e.,
%does not necessarily hold, i.e.,
we have not been able to replace
%it does not seem to be possible to replace
the surjection from $[\ha_\go]^\go$
onto $\ha_{\go + \ga}$ with a surjection
from $\wp(\ha_\go)$ onto $\ha_{\go + \ga}$
and obtain the lower bound in consistency
strength of one measurable cardinal.
Also, to avoid trivialities, the surjection
in Theorem \ref{t1} is onto a cardinal greater
than or equal to $\ha_{\go + 2}$, and the
injection in Theorems \ref{t3} and \ref{t4} is
from a cardinal of size at least $\gl^{++}$.
Loosely speaking,
the cardinalities of $[\aleph_{\omega}]^{\omega}$
and $[\aleph_{\omega_1}]^{\omega_1}$
in these situations may be blown up so that they ``contradict'' the
conclusions of the seminal theorems of {{\rm Silver}} \cite{Si} and
{{\rm Shelah}} \cite{Sh} in surjective and injective ways.
In particular, Theorems \ref{t3} and \ref{t4}
provide an injection from $\gl^{+ +}$ into
the power set of a singular cardinal $\gl$ of uncountable
cofinality, together with GCH holding below $\gl$.
This, of course, is in sharp contrast to
Silver's ZFC result \cite{Si} that GCH cannot first
fail at a singular strong limit cardinal
of uncountable cofinality.
In addition, Theorem \ref{t1} yields that it
is possible to have a surjection from
$\wp(\ha_{\go})$ onto any $\ha_\gb$
together with GCH holding below $\ha_\go$,
and Theorem \ref{t2} tells us that it
is possible to have an injection from
$\ha_{\go_n}$ into $\wp(\ha_\go)$ for
any $n \ge 1$, $n < \go$ together with
GCH holding below $\ha_\go$.
We now compare this with the situation in ZFC.
Although it is known (see \cite{G09}) how to
force SCH to fail at $\ha_\go$ with the
size of the power set of $\ha_\go$
arbitrarily large below $\ha_{\go_1}$,
it is currently unknown (see \cite{G09}
for a discussion) whether it is possible
to force SCH to fail at $\ha_\go$ with
$2^{\ha_\go} \ge \ha_{\go_1}$.
It is known, however, that a failure
of SCH at $\ha_\go$ in conjunction with
$2^{\ha_\go} \ge \ha_{\go_1}$ is very
strong, and in fact yields the existence
of an inner model containing a Woodin cardinal
(see \cite{GSS}).
When $n \ge 4$, we get choiceless ``counterexamples''
to Shelah's theorem \cite{Sh} that when
$\ha_\go$ is a strong limit cardinal,
$2^{\ha_\go} < \ha_{\go_4}$.
All of this is once again in sharp contrast to the
situation in ZFC.
A crucial feature of the symmetric
submodels of the parallel {{\rm Prikry}}
forcing or the symmetric collapses will be that
they can be approximated from
within by certain submodels in which
AC holds. The lower bound
on consistency strength of one measurable cardinal
for surjective failures of SCH
will consequently be determined using
the Dodd-Jensen core model \cite{D}, \cite{DJ1}
for sequences of measures.
%and sequences of extenders.
Since core model theory uses the Axiom of Choice, we employ
${\rm HOD}$-like inner models.
Lower bounds on consistency strength for injective
failures of SCH will be obtained by using Gitik
and Mitchell's work \cite{GMi}
(which is also discussed in \cite{G02a}).
This paper is structured as follows.
%We begin in Section \ref{s1} with a discussion of
%our overall goals and background material
%to put our results into context, as well as stating
%our main theorems.
In Section \ref{s2}, we introduce our new
notion of parallel Prikry forcing, which will be
used to construct choiceless, surjective failures
of SCH. We demonstrate some of
its basic properties, and also produce
a choiceless, symmetric submodel
$N$ of a generic extension via
parallel Prikry forcing.
In Section \ref{s3}, we
prove the lemmas necessary to
provide a detailed analysis
of $N$. In Section \ref{s4}, we show how
the appropriate construction of $N$ may be
used to obtain these surjective failures of SCH
at some previously measurable cardinal $\kappa$
which now has cofinality $\go$.
In Section \ref{s5}, we give a general paradigm
for symmetrically collapsing large cardinals at which
SCH fails, either in a model of ZFC or
surjectively in a choiceless model of ZF,
down to small singular limit
cardinals, such as $\ha_\go$, $\ha_{\go_1}$, or $\ha_{\go_2}$.
In Section \ref{s6}, we establish the upper bounds
in consistency strength of our main theorems via forcing,
thereby providing the models in which SCH will fail
either surjectively or injectively at
$\ha_{\go}$, $\ha_{\go_1}$, or $\ha_{\go_2}$.
In Section \ref{s7}, we establish the lower bounds
in consistency strength of our main theorems via
techniques from inner model theory, which completes
the proofs of these theorems.
Section \ref{s8} contains our concluding remarks,
along with a discussion of some generalizations
of the main theorems provable by our methods.
\section{Parallel Prikry forcing}\label{s2}
Parallel {{\rm Prikry}} forcing is a subforcing
of a finite support product
of {{\rm Prikry}} forcings,
where the {{\rm Prikry}} sequences formed are
eventually interlaced in a very systematic fashion.
This prevents the coding
of unwanted information into the generic extension.
Fix a measurable cardinal $\kappa$ and a
normal measure ${\cal U}$ on $\kappa$. Fix in addition a
set $Z \subseteq {\rm Ord}$, which will be the {{\rm support}} of the
subsequent forcing.
We will now define
$Z$-{{\rm fold parallel}} {{\rm Prikry}} {{\rm forcing}} for the
measure ${\cal U}$.\footnote{Parallel Prikry forcing
was first defined using a sequence of pairwise
distinct normal measures on $\gk$. We are indebted to
Gunter Fuchs, who pointed out that a single normal measure suffices.}
We will occasionally write
$(\FP_Z, \le )$, although more often, we
will write $(\FP, \le)$ instead of $(\FP_Z, \le)$ for simplicity.
Getting specific,
a sequence $p = (s_{\alpha}, A_{\alpha})_{\alpha \in Z}$ is a
{{\rm condition}} in $\FP$ iff
\begin{enumerate}
\item $\forall \alpha \in Z[ (s_{\alpha}
\in [\kappa]^{< \omega}) \wedge
(A_{\alpha} \in {\cal U}) \wedge (\max (s_{\alpha}) <
\min (A_{\alpha}))]$; here, we set
$\max (\emptyset) = - 1$.
\item $\dom(p) := \{\alpha \in Z\mid
A_\ga \neq \gk\}$ is
finite.
\end{enumerate}
We write $(s_{\alpha}, A_{\alpha})$ instead of $(s_{\alpha},
A_{\alpha})_{\alpha \in Z} $.
We will, however, occasionally abuse notation and also
use $(s_\ga, A_\ga)$ to mean an individual component of
$(s_\ga, A_\ga)_{\ga \in Z}$.
Conditions $p' =
(s_{\alpha}', A'_{\alpha})$ and
$p = (s_{\alpha}, A_{\alpha})$ in $\FP$
are partially ordered by $p' \le
p$ (i.e., $p'$ is stronger than $p$)
iff there is an integer $n < \omega$ such that
\begin{enumerate}
\item $\forall \alpha \in \dom (p)
[(\otp (s'_{\alpha} \setminus
s_{\alpha}) = n) \wedge (s'_{\alpha} \setminus
s_{\alpha} \subseteq A_{\alpha}$)].
\item $(\forall \alpha, \beta \in \dom (p))
(\forall \xi \in s'_{\alpha}
\setminus s_{\alpha})
(\forall \zeta \in s_{\beta})[ \xi > \zeta]$.
\item\label{r3} $(\forall \alpha < \beta \in \dom (p))
(\forall i <
n)[ (s'_{\alpha} \setminus s_{\alpha}) [i] <
(s'_{\beta} \setminus s_{\beta})
[i]]$, where $s [i]$ denotes the $i$-th element
of the monotone enumeration
of the set $s$ of ordinals.
\item $(\forall \alpha, \beta \in \dom (p))
(\forall i < n)[(i+ 1 < n) \implies
((s'_{\alpha} \setminus s_{\alpha}) [i] <
(s'_{\beta} \setminus s_{\beta}) [i+ 1])]$.
\item $\forall \alpha \in \dom (p)
[ A'_{\alpha} \subseteq A_{\alpha}] $.
\end{enumerate}
Intuitively, $p' \le p$ means that on the
domain of $p$, the following hold:
\begin{enumerate}
\item The stems $s_{\alpha}$
% (the black squares in the picture)
are extended into the
corresponding reservoir sets $A_{\alpha}$
%(blobs with white interior) in a
in a systematic fashion.
\item The extension points
%(grey squares)
are chosen greater
than all of the previous stem points.
\item There are the same number of new points at
all indices in $\dom (p)$,
and these are chosen in layers which are
strictly ascending.
\end{enumerate}
Moreover, reservoirs may be thinned out,
%(grey blobs) and
and new stems outside the old domain may be grown.
%\includegraphics{0.7-1.eps}
Let $G$ be $\FP$-generic over $V$.
$G$ adjoins a system $(C_{\alpha} \mid \alpha
\in Z)$, where for a fixed $\ga$,
\[ C_{\alpha} = \bigcup \{s_{\alpha} \mid (s_{\beta},
A_{\beta})_{\beta \in Z}
\in G\}. \]
Density arguments show that the $C_{\alpha}$ are distinct.
Lemma \ref{l1} shows that the $C_\ga$
are Prikry sequences for the measure ${\cal U}$.
\begin{lemma}\label{l1}
{\tmdummy}
\begin{enumeratealpha}
\item\label{l1a} Let $\gamma \in Z$. Then $C_{\gamma}$ is a Prikry
sequence for ${\cal U}$, i.e.,
\[ \forall X \in \wp(\kappa) \cap V [(X \in {\cal U}) \iff
(C_{\gamma} \setminus X \text{ is finite})] . \]
This implies that $C_{\gamma}$ is cofinal in $\kappa$ of order
type $\omega$.
\item Let $\gamma, \delta \in Z$, $\gamma < \delta$. Then $C_{\gamma} \cap
C_{\delta}$ is finite, and therefore $C_{\gamma} \Delta C_{\delta}$ is
infinite.
\end{enumeratealpha}
\end{lemma}
\begin{proof}
a) Let $X \in \wp(\kappa) \cap V$. Assume that $X \in {\cal U}$.
Take $p =
(s_{\alpha}, A_{\alpha}) \in G$ such that $A_{\gamma} \subseteq X$.
By the
definition of $C_{\gamma} $, $C_{\gamma} \setminus s_{\gamma} \subseteq
A_{\gamma} \subseteq X$.
Hence, $C_{\gamma} \setminus X \subseteq s_{\gamma}$
is finite.
For the converse,
assume that $X \not\in {\cal U}$.
We show that $C_{\gamma} \setminus
X$ is cofinal in $\kappa$. Let $\nu < \kappa$. Take $p = (s_{\alpha},
A_{\alpha}) \in G$ such that
$A_{\gamma} \subseteq (\kappa \setminus X) \cap
(\kappa \setminus \nu)$. By the definition of $C_{\gamma} $,
\[ C_{\gamma} \setminus s_{\gamma} \subseteq A_{\gamma} \subseteq (\kappa
\setminus X) \cap (\kappa \setminus \nu), \]
and by density, $C_{\gamma} \setminus s_{\gamma} \neq \emptyset$.
Say $\xi
\in C_{\gamma} \setminus s_{\gamma} $.
Then $\xi \in C_{\gamma} \setminus X$
and $\xi \ge \nu$, as required.
\hfill$\square$\\
b) Take $p = (s_{\alpha}, A_{\alpha}) \in G$ such that
$\gamma, \delta \in
\dom (p)$, $\gg < \gd$.
It suffices to show that $C_{\gamma} \cap C_{\delta}
\subseteq s_{\gamma} \cap s_{\delta} $. Consider $\xi \in C_{\gamma} \cap
C_{\delta} $. Take $p' = (s'_{\alpha}, A'_{\alpha}) \in G$ such that $p'
\le p$ and $\xi \in s_{\gamma}' \cap s_{\delta}' $. By
requirement (\ref{r3}) of the definition of
$\le$, $(s'_{\gamma} \setminus s_{\gamma}) \cap s'_{\delta} =
\emptyset$ and $(s'_{\delta} \setminus s_{\delta}) \cap s'_{\gamma} =
\emptyset$. This implies that $\xi \in s_{\gamma} \cap s_{\delta} $.
\end{proof}
\begin{lemma}\label{l2}
$(\FP, \le)$ satisfies the $\kappa^+$-chain condition.
\end{lemma}
\begin{proof}
Let $\{(s^i_{\alpha},
A^i_{\alpha}) \mid i < \kappa^+ \} \subseteq \FP$. We want
to show that at least two of the $(s^i_{\alpha}, A^i_{\alpha})$ are
compatible in $\FP$. By a $\Delta$-system argument,
we may assume that the
domains $\dom ((s^i_{\alpha}, A^i_{\alpha}))$ form a $\Delta$-system
with kernel $Z_0 \in [Z]^{< \omega}$. By a pigeonhole argument,
we may assume
that there are $i < j < \kappa^+$
such that $(s^i_{\alpha})_{\alpha \in Z_0}
= (s^j_{\alpha})_{\alpha \in Z_0} $. Then
$(s^i_{\alpha}, A^i_{\alpha})$ and
$(s^j_{\alpha}, A^j_{\alpha})$
have a common refinement $(t_{\alpha},
B_{\alpha})$ defined by
\[ t_{\alpha} = \left\{ \begin{array}{l}
s_{\alpha}^i \text{, if } \alpha \in \dom ((s^i_{\alpha},
A^i_{\alpha})),\\
s_{\alpha}^j \text{, if } \alpha \in \dom ((s^j_{\alpha},
A^j_{\alpha})),\\
\emptyset \text{, otherwise} ,
\end{array} \right. \]
and
\[ B_{\alpha} = A^i_{\alpha} \cap A^j_{\alpha} . \]
\end{proof}
So forcing with $\FP$ preserves
all cardinals greater than or equal to
$\kappa^+$. On the other
hand, in case $Z$ is infinite,
the measurable cardinal $\kappa$ is made
countable. To see this,
let $(C_{\alpha})_{\alpha \in Z}$ be the system of {{\rm Prikry}}
sequences added by the forcing.
Then a simple density argument shows that the
function
\[ \alpha \mapsto \min (C_{\alpha}) \]
maps any countable subset of $Z$ onto $\kappa$.
Therefore, we shall work
instead with the symmetric submodel
\[ N = {\rm HOD}^{V[G]}
( \bigcup_{\alpha \in Z} \tilde{C}_{\alpha} \cup \{(
\tilde{C}_{\alpha} \mid \alpha \in Z)\}), \]
where $\tilde{C}_{\alpha} =\{C \in \wp(\kappa) \mid C \Delta C_{\alpha}
\text{ is finite} \}$.
This is the class of sets which are hereditarily definable in the generic
extension from finitely many parameters
from the class ${\rm Ord} \cup
\{C_{\alpha} \mid \alpha \in Z\}
\cup \{( \tilde{C}_{\alpha} \mid \alpha \in Z)\}$.
If, e.g., $Z = \kappa^{+ +}$,
as we shall show in Section \ref{s4},
the following will give a surjection
contradicting ${\rm SCH}$.
\begin{lemma}\label{l3}
In $N$, there is a surjection $f : [\gk]^\go \rightarrow
Z$.
\end{lemma}
\begin{proof}
Define $f$ using the parameter
$( \tilde{C}_{\alpha} \mid \alpha \in Z)$ by
\[ X \mapsto \left\{ \begin{array}{l}
\text{The unique } \alpha \in Z \text{ such that } X \in
\tilde{C}_{\alpha} \text{, if that exists,}\\
0, \text{ otherwise} .
\end{array} \right. \]
It follows that $f$ is surjective, since
$f(C_\alpha) = \alpha$ by Lemma \ref{l1}\ref{l1a}).
\end{proof}
\section{Finite support approximations}\label{s3}
The model $N$ will be analyzed using {{\rm finite support approximations}}.
We show that parallel {{\rm Prikry}} forcing with a finite support $Z
\subseteq {\rm Ord}$ is equivalent to standard {{\rm Prikry}} forcing.
Note that standard {{\rm Prikry}} forcing
corresponds to the forcing $\FP_1 =
\FP_{\{0\}}$. For simplicity,
we consider sets $Z = \ell$, where $\ell < \go$.
\begin{lemma}\label{l4}
Let $G$ be $\FP_Z$-generic for $V$,
where $Z = \ell < \go$.
Then $V [G]$ is an extension of $V$ by Prikry forcing $\FP_1 $.
Therefore, by the properties of
standard Prikry forcing, $V [G]$ has
the same bounded subsets as $V$.
\end{lemma}
\begin{proof}
Choose a condition $p = (s_{i}, A) \in G$
such that $\dom (p) =
Z$. Note that we may densely assume
that the finitely many $A_{i}$ are
all equal to $A$, and that $A \neq \kappa$. Then $V [G]$ is a generic
extension of $V$ by the restricted partial
ordering $\FP_Z' =\{q \in \FP_Z \mid q
\le p\}$. Observing that $(\emptyset, A) \in \FP_1$,
define $\FP'_1 =\{r
\in \FP_1 \mid r \le (\emptyset, A)\}$. It suffices to define a dense
embedding $\pi$ from $\FP'_Z$ into $\FP_1'$.
Consider $(s'_{i}, B_i) \in \FP'_Z
$. For $i \in Z$, let
\[ s'_{i} \setminus s_{i} = \{\xi^0_{i}, \xi^1_{i}, \ldots,
\xi^{n - 1}_{i} \}, \]
where
\[ \xi^0_0 < \xi^0_1 < \cdots <
\xi^0_{\ell - 1} < \xi^1_0 < \xi^1_1 <
\cdots < \xi^1_{\ell - 1} < \cdots < \xi^{n - 1}_0 < \xi^{n - 1}_1 <
\cdots < \xi^{n - 1}_{\ell - 1}. \]
% < \xi^n_0 < \xi^n_1 < \cdots < \xi^n_{\ell - 1} . \]
Then let $\pi ((s'_{i}, B_i)) = (t, \bigcap_{i < \ell} B_i)$, where
\[ t =\{\xi^0_0, \xi^0_1, \ldots, \xi^0_{\ell - 1}, \xi^1_0, \xi^1_1,
\ldots, \xi^1_{\ell - 1}, \ldots, \xi^{n - 1}_0, \xi^{n - 1}_1, \ldots,
\xi^{n - 1}_{\ell - 1} \}. \]
% \xi^n_0, \xi^n_1, \ldots, \xi^n_{l - 1} \}. \]
This obviously defines a dense embedding.
\end{proof}
Let us again consider an arbitrary support $Z \subseteq {\rm Ord}$ and a
finite subset $Z_0 \subseteq Z$. We define restrictions to $Z_0$ by
\[\FP \rest Z_0 = \{p \rest Z_0 \mid p \in \FP\} \]
and
\[G \rest Z_0 = \{p \rest Z_0 \mid p \in G\}. \]
%$ \begin{eqnarray*}
% \FP \upharpoonright Z_0 & = & \{p \upharpoonright Z_0 \mid p \in \FP\},\\
% G \upharpoonright Z_0 & = & \{p \upharpoonright Z_0 \mid p \in G\}.
% \end{eqnarray*}$
\begin{lemma}
Let $G$ be $\FP$-generic.
% Restrict to the support $Z_0$ by
Then $G \upharpoonright Z_0$ is $\FP \upharpoonright Z_0$-generic.
\end{lemma}
\begin{proof}
Easy.
\end{proof}
The approximation of the model ${\rm HOD}^{V[G]}
( \bigcup_{\alpha \in Z}
\tilde{C}_{\alpha} \cup \{( \tilde{C}_{\alpha} \mid \alpha \in Z)\})$
by finite
support parallel {{\rm Prikry}} extensions will be based on certain
symmetries of the partial ordering $(\FP, \le)$.
\begin{lemma}\label{l6}
Let $p = (s_{\alpha}, A_{\alpha}) \in \FP$.
Set $p^- = (\emptyset, A_{\alpha})
\in \FP$, $\FP_p =\{q \in \FP\mid q \le p\}$,
and $\FP_{p^-} =\{q \in \FP\mid q
\le p^- \}$. Then the following hold:
\begin{enumeratealpha}
\item The map
\[ \pi : (t_{\alpha}, B_{\alpha}) \mapsto (t_{\alpha} \setminus
s_{\alpha}, B_{\alpha}) \]
is an order isomorphism between $(\FP_p, \le)$ and $(\FP_{p^-},
\le)$.
\item $D$ is dense in $\FP$ below $p^-$ iff $\pi^{- 1} [D]$
is dense in $\FP$
below $p$.
\item $H$ is $\FP$-generic below $p^-$ iff $\pi^{- 1} [H]$
is $\FP$-generic
below p.
\item If $H$ is $\FP$-generic below $p^-$,
then for every sequence
$(s'_{\alpha})_{\alpha \in Z}$ with $\max (s'_{\alpha}) < \min
(A_{\alpha})$, the set
\[ \{(s'_{\alpha} \cup u_{\alpha}, B_{\alpha}) \mid
(u_{\alpha}, B_{\alpha})
\in H\} \]
is $\FP$-generic with $(s_{\alpha}', A_{\alpha}) \in \{(s'_{\alpha} \cup
u_{\alpha}, B_{\alpha}) \mid (u_{\alpha}, B_{\alpha}) \in H\}$.
\end{enumeratealpha}
\end{lemma}
\begin{proof}
Obviously, the map having domain
$\FP_{p^{-}}$ defined by
\[ (u_{\alpha}, B_{\alpha}) \mapsto (s_{\alpha}
\cup u_{\alpha}, B_{\alpha})
\]
is the inverse of $\pi$, and hence $\pi$ is a bijection.
The definition of the
order relation $\le$ implies immediately that $\pi$ is
order preserving. Then $\pi$ and $\pi^{- 1}$ preserve density and
genericity. Property d) follows directly from c).
\end{proof}
Let $\dot{C}_{\alpha}$ be a canonical name for the $\alpha$-th
{{\rm Prikry}} sequence added by forcing with $\FP$,
and let $\dot{D}$
be a canonical name for the sequence
$( \tilde{C}_{\alpha} \mid \alpha \in Z)$
used in the definition of $N$.
\begin{lemma}\label{l7}
Let $\varphi$ be an $\in$-formula and $\varphi (
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0},
\ldots, \dot{C}_{\alpha_{n
- 1}}, \dot{D})$ be a forcing sentence.
Let $p = (s_{\alpha}, A_{\alpha}) \in \FP$, $q
= (t_{\alpha}, B_{\alpha}) \in \FP$ be such that
$p \upharpoonright \{\alpha_0,
\ldots, \alpha_{n - 1} \}= q \upharpoonright \{\alpha_0, \ldots,
\alpha_{n -
1} \}$. Then we cannot have that $p \Vdash \varphi (
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0},
\ldots, \dot{C}_{\alpha_{n
- 1}}, \dot{D})$ and $q \Vdash \neg
\varphi ( \overrightarrow{\check{\xi}},
\dot{C}_{\alpha_0}, \ldots, \dot{C}_{\alpha_{n - 1}}, \dot{D})$
simultaneously.
\end{lemma}
\begin{proof}
Assume towards a contradiction that $p \Vdash \varphi (
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0},
\ldots, \dot{C}_{\alpha_{n
- 1}}, \dot{D})$ and \break
$q \Vdash \neg \varphi ( \overrightarrow{\check{\xi}},
\dot{C}_{\alpha_0}, \ldots, \dot{C}_{\alpha_{n - 1}}, \dot{D})$.
Set $r^- =
(\emptyset, A_{\alpha} \cap B_{\alpha})$. Take a set $H$ which is
$\FP$-generic below $r^-$.
Set
\[ H_p =\{(s_{\alpha} \cup u_{\alpha}, E_{\alpha}) \mid (u_{\alpha},
E_{\alpha}) \in H\} \text{ and } H_q =\{(t_{\alpha} \cup u_{\alpha},
E_{\alpha}) \mid (u_{\alpha}, E_{\alpha}) \in H\}. \]
By Lemma \ref{l6},
$H_p$ and $H_q$ are $\FP$-generic, with $p \in H_p$ and
$q \in H_q $ respectively. Hence,
\begin{equation}
V [H_p] \models \varphi ( \vec{\xi}, ( \dot{C}_{\alpha_0})^{H_p}, \ldots,
( \dot{C}_{\alpha_{n - 1}})^{H_p}, \dot{D}^{H_p}) \text{ and } V [H_q]
\models \neg \varphi ( \vec{\xi}, ( \dot{C}_{\alpha_0})^{H_q}, \ldots, (
\dot{C}_{\alpha_{n - 1}})^{H_q}, \dot{D}^{H_q}).
\end{equation}
For $i < n$, $s_{\alpha_i} = t_{\alpha_i}$ and $( \dot{C}_{\alpha_i})^{H_p}
= ( \dot{C}_{\alpha_i})^{H_q}$. For $\alpha \not\in \{\alpha_0, \ldots,
\alpha_{n - 1} \}$, $( \dot{C}_{\alpha})^{H_p} \Delta (
\dot{C}_{\alpha})^{H_q} \subseteq s_{\alpha} \cup t_{\alpha}$ is finite.
Thus, $\dot{D}^{H_p} = \dot{D}^{H_q}$. Since the generic sets $H$, $H_p $,
and $H_q$ differ only by finite sets in $V$, $V [H] = V [H_p] = V [H_q]$.
Then (1) leads to the contradiction
\[ V [H] \models \varphi ( \vec{\xi},
( \dot{C}_{\alpha_0})^{H_p}, \ldots, (
\dot{C}_{\alpha_{n - 1}})^{H_p}, \dot{D}^{H_p}) \text{ and } V [H]
\models \neg \varphi ( \vec{\xi}, ( \dot{C}_{\alpha_0})^{H_p}, \ldots, (
\dot{C}_{\alpha_{n - 1}})^{H_p}, \dot{D}^{H_p}) . \]
\end{proof}
\begin{lemma}\label{l8}
Let $G$ be $\FP$-generic,
with $C_{\alpha} = ( \dot{C}_{\alpha})^G$ for $\alpha
\in Z$ and $D = \dot{D}^G$. Let $X \in V [G]$ be defined by
\[ X =\{\zeta \in {\rm Ord} \mid V [G] \models \varphi (\zeta, \vec{\xi},
C_{\alpha_0}, \ldots, C_{\alpha_{n - 1}}, D)\} \]
where $\alpha_0, \ldots, \alpha_{n - 1} \in Z$.
Then $X \in V [G \upharpoonright
\{\alpha_0, \ldots, \alpha_{n - 1} \}]$.
\end{lemma}
\begin{proof}
Let $Z_0 =\{\alpha_0, \ldots, \alpha_{n - 1} \}$. We only present the case
$Z_0 \neq \emptyset$. Define
% \begin{equation}
$X' =\{\zeta \in {\rm Ord} \mid \text{For all }
k < \omega$, $\exists p
= (s_{\alpha}, A_{\alpha}) \in \FP [(Z_0 \subseteq \dom (p)) \wedge
(\otp (s_{\alpha_0}) \ge k) \wedge
(p \upharpoonright Z_0 \in G
\upharpoonright Z_0) \wedge (p \Vdash \varphi ( \check{\zeta},
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0}, \ldots,
\dot{C}_{\alpha_{n - 1}}, \dot{D}))]\} \in V [G \upharpoonright Z_0] $.
% \end{equation}
We claim that $X = X'$. To see this,
if $\zeta \in X$,
then there is $p = (s_{\alpha}, A_{\alpha}) \in G$ such
that $p \Vdash \varphi ( \check{\zeta}, \overrightarrow{\check{\xi}},
\dot{C}_{\alpha_0}, \ldots, \dot{C}_{\alpha_{n - 1}}, \dot{D})$.
By density,
it is possible to
assume that $Z_0 \subseteq \dom (p)$. Since $G$ contains
conditions where the $\alpha_0$-th stem
is of arbitrary finite order type,
it is possible also to
arrange that $\otp (s_{\alpha_0}) \ge k$ for any
given $k < \omega$. Hence, $\gz \in X'$.
% $\zeta$ is an element of the abstraction term in
% (3).
Conversely, assume that $\zeta \not\in X$.
Take $q = (t_{\alpha}, B_{\alpha})
\in G$ to be
such that \break $q \Vdash \neg \varphi ( \check{\zeta},
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0},
\ldots, \dot{C}_{\alpha_{n
- 1}}, \dot{D})$. We may assume that $Z_0 \subseteq \dom (q)$. Suppose
that there were $p = (s_{\alpha}, A_{\alpha}) \in \FP$ such that
\begin{equation}
( Z_0 \subseteq \dom (p)) \wedge
(\otp (s_{\alpha_0}) \ge
\otp (t_{\alpha_0}))
\wedge (p \upharpoonright Z_0 \in G
\upharpoonright Z_0) \wedge (p \Vdash \varphi ( \check{\zeta},
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0}, \ldots,
\dot{C}_{\alpha_{n - 1}}, \dot{D})) .
\end{equation}
Take some $p' = (s'_{\alpha}, A'_{\alpha}) \in G$ such that $p'
\upharpoonright Z_0 = p \upharpoonright Z_0 $.
Since $G$ is a generic filter,
there is \ $p'' = (s''_{\alpha}, A''_{\alpha}) \in G$ such that $(s''_a,
A''_{\alpha}) \le (t_{\alpha}, B_{\alpha})$ and $(s''_a, A''_{\alpha})
\le (s'_{\alpha}, A'_{\alpha})$. Then $t_{\alpha_0}$ and
$s'_{\alpha_0}$ are both initial segments of $s''_{\alpha_0} $, and
$t_{\alpha_0}$ is an initial segment of $s'_{\alpha_0} = s_{\alpha_0} $.
Let
$\ell = \otp (s_{\alpha_0} \setminus t_{\alpha_0})$. Define a condition
$q^{\ast} = (s^{\ast}_{\alpha}, A''_{\alpha}) \le (t_{\alpha},
B_{\alpha}) = q$ by
\[ s^{\ast}_{\alpha} = \left\{ \begin{array}{l}
%s''_{\alpha}
t_{\alpha} \cup
\{(s''_{\alpha} \setminus t_{\alpha}) [i] \mid i < \ell\}
\text{ if } \alpha \in \dom (q),\\
\emptyset \text{ otherwise,}
\end{array} \right. \]
where $(s''_{\alpha} \setminus t_{\alpha}) [i]$ is the $i$-th
element of the
monotone enumeration of $s''_{\alpha} \setminus t_{\alpha} $.
Note that for $\alpha \in {\rm dom}(q)$,
$s^*_\alpha$ may be a proper initial segment of $s''_{\alpha}$.
%Then the
Since $p'' = (s''_{\alpha}, A''_{\alpha}) \in G$,
$p' = (s'_{\alpha}, A'_{\alpha}) \in G$,
$p'' \le p'$,
$p' \upharpoonright Z_0 = p \upharpoonright Z_0$,
and $G$ is a generic filter,
the conditions $q^{\ast}$ and $p$
have the same stems on the support $Z_0 $. By
thinning out reservoir sets,
we can further assume that $q^{\ast}
\upharpoonright Z_0 = p \upharpoonright Z_0 $.
But then, $q^{\ast} \Vdash \neg \varphi
( \check{\zeta}, \overrightarrow{\check{\xi}},
\dot{C}_{\alpha_0}, \ldots,
\dot{C}_{\alpha_{n - 1}}, \dot{D})$ and
$p \Vdash \varphi ( \check{\zeta},
\overrightarrow{\check{\xi}}, \dot{C}_{\alpha_0}, \ldots,
\dot{C}_{\alpha_{n
- 1}}, \dot{D})$. However, this contradicts Lemma \ref{l7}.
Thus, there are no
$p$ satisfying (2), and hence $\zeta\not\in X'$.
\end{proof}
\section{Applications to the Singular Cardinals Hypothesis}\label{s4}
\begin{theorem}\label{t5}
Assume that
$V \models ``$ZFC + $\gk$ is a measurable cardinal''.
Then there is a partial ordering $\FP \in V$
and a symmetric submodel $N \subseteq V^\FP$ in which
there is a surjective failure of the Singular Cardinals
Hypothesis at $\gk$.
In particular, for $\gb \ge 2$, there is
such a model $N$ with a surjection from
$[\gk]^{\go}$ onto $\gk^{+ \gb}$ in $N$, and
$N \models ``$GCH holds below $\gk$''.
% is an extension $N \supseteq V$ which is a model of ${\rm ZF}$ and in $N$
% the Singular Cardinals Hypothesis is violated at $\kappa$.
\end{theorem}
\begin{proof}
Without loss of generality,
by forcing or using an appropriate constructible inner model,
we may also assume that $V \models {\rm GCH}$.
Define the forcing $(\FP, \le) = (\FP_Z, \le)$
with $Z = \kappa^{+ \gb}$ as above.
Let $V [G]$ be a generic extension of $V$
by $\FP$, and let $(C_{\alpha})_{\alpha < \kappa^{+ \gb}}$
be the sequence of
{{\rm Prikry}} sequences adjoined by $G$. Then form the model
\[ N = {\rm HOD}^{V[G]}
(\{C_{\alpha} \mid \alpha < \kappa^{+ \gb} \} \cup \{(
\tilde{C}_{\alpha} \mid \alpha < \kappa^{+ \gb})\}) . \]
Every set of ordinals in $N$ is of the form
\[ X =\{\zeta \in {\rm Ord} \mid V [G] \models \varphi (\zeta, \vec{\xi},
C_{\alpha_0}, \ldots, C_{\alpha_{n - 1}},
( \tilde{C}_{\alpha} \mid \alpha <
\kappa^{+ \gb}))\} \]
for some $\in$-formula $\varphi$, $\vec{\xi} \in {\rm Ord}$, and $\alpha_0,
\ldots, \alpha_{n - 1} < \kappa^{+ \gb} $. By Lemma \ref{l8},
\[ X \in V [G \upharpoonright \{\alpha_0, \ldots, \alpha_{n - 1} \}] . \]
By Lemma \ref{l4},
finite support parallel {{\rm Prikry}} forcings do not add bounded
subsets of $\kappa$. Hence, $\kappa$ is a singular cardinal in $N$,
and $N \models ``$GCH holds below $\gk$''.
% \forall \nu \in [\omega, \kappa) 2^{\nu} = \nu^+$.
By Lemma \ref{l3}, there is a
surjection $f : [\gk]^{\go} \rightarrow (\kappa^{+ \gb})^V$ in $N$.
By Lemma \ref{l8} and Lemma \ref{l2},
$(\kappa^{+ \gb})^V = (\kappa^{+ \gb})^N$.
Therefore, $f$ is a choiceless, surjective failure of ${\rm SCH}$.
\end{proof}
Since by Lemma \ref{l8} and Lemma \ref{l4},
$N$ and $V$ contain the same bounded subsets of $\gk$,
$(V_\gk)^V = (V_\gk)^N$.
From this,
it is possible to infer that
any $x \in (V_\gk)^N$ is well-orderable.
Further, Lemma \ref{l8} and Lemma \ref{l4},
together with the fact that in $N$, ${\rm cof}(\gk) = \go$,
tell us that in $N$, there is a sequence of inaccessible cardinals
$\la \gk_i \mid i < \go \ra$ whose limit is $\gk$.
These observations will be used in the construction
of the witnessing model for Theorem \ref{t1}
to be given in Section \ref{s6}.
\section{Collapsing cardinals}\label{s5}
We briefly describe the general method
we shall use for
symmetrically collapsing a singular
cardinal $\kappa$ which is a limit
of inaccessible cardinals down to small cardinals like
$\aleph_{\omega}$, $\ha_{\go_1}$, or $\aleph_{\omega_2}$.
Most of what we are about to discuss is found in
\cite[Section 4, pages 730 -- 732]{AK06},
whose presentation we closely follow.
In particular, our construction will result
in a choiceless, symmetric inner model of
a generic extension $V[G]$.
Assume that $\gl$ represents in our ground
model $V$ one of the cardinals $\go$,
$\go_1$, or $\go_2$.
Let $V \models ``$ZF + $\la \gk_i \mid i < \gl \ra$
is a sequence of inaccessible cardinals whose limit is $\gk$''.
Note that it may or may not be the case that AC is true in $V$.
Even if AC is false in $V$,
by our remarks above, it will be possible
to assume that $V_\gk$ is well-orderable.
Thus, ``$\gk_i$ is an inaccessible cardinal''
will have the same meaning as when AC is true.
%, i.e., that $\gk_i$ is a regular limit cardinal
%such that for every cardinal $\gd < \gk_i$, $\wp(\gd)$
%is well-orderable and has cardinality less than $\gk_i$.
%As will be seen later, it will be possible to
%make this assumption when forcing over the
%relevant choiceless model of ZF.
Take $\la \gl_i \mid i < \gl \ra$
as the sequence
$\la \gk_i \mid i < \gl \ra$, together
with its limit points less than $\gk$.
(If $\gl = \go$, then
$\la \gl_i \mid i < \gl \ra = \la \gk_i \mid i < \gl \ra$.)
Let $I = \{i < \gl \mid i$ is
either a successor ordinal or $0\}$.
For $i \in I$, let
$\FP_i = {\rm Coll} ( \gl^+_{i - 1}, {<} \gl_i)$.
Note that we take $\gl^+_{-1} = \gl^{+ m}$
for some fixed $1 \le m < \go$, and that
${\rm Coll} ( \gl^+_{i - 1}, {<} \gl_i)$ is the
L\'evy collapse of all cardinals in the open interval
$(\gl^+_{i - 1}, \gl_i)$ down to $\gl^+_{i - 1}$.
We then define $\mathbb{P}= \prod_{i < \gl} \mathbb{P}_i$
with full support.
Let $G$ be $\FP$-generic over $V$,
and for $i \in I$, let $G_i$ be
the projection of $G$ onto $\FP_i$.
For $j \in I$, let
$\FQ_j = \prod_{i \le j, i \in I} \FP_i$ and
$H_j = \prod_{i \le j, i \in I} G_i$.
It is the case, by the
properties of the L\'evy collapse
and the Product Lemma, that
$H_j$ is $\FQ_j$-generic over $V$.
Our symmetric inner model $N \subseteq V[G]$
can now be intuitively
described as the least model of ZF
extending $V$ which contains, for
every $j \in I$, the set $H_j$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified sublanguage of the forcing language $\cal L$ with respect to
$\FP$ which contains symbols $\check v$ for each $v \in V$, a unary
predicate symbol $\check V$
(to be interpreted $\check V(\check v) \iff v \in
V$), and symbols $\dot H_j$ for every $j \in I$.
$N$ is then defined as follows.\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\gl$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}\bigskip
The relevant arguments found on
\cite[page 732]{AK06} remain valid,
regardless if GCH is true in $V$,
since we can always assume that $V_\gk$ is well-orderable
and has cardinality $\gk$.
Therefore, it will be the case that $\gk$ becomes in $N$
either $\ha_\go$, $\ha_{\go_1}$, or $\ha_{\go_2}$
(depending on whether initially, $\gl = \go$,
$\gl = \go_1$, or $\gl = \go_2$). Since
we will always be able to assume that
$V \models ``$GCH holds below $\ha_\go$'',
$N \models ``$GCH holds below $\gk$''.
Further, as in \cite{AK06}, if $x \in N$ is a set of ordinals, then
$x \in V[H_j]$ for some $j < \gl$.
Hence, since $\FQ_j \in V_\gk$, which means that
$\FQ_j$ is well-orderable, the cardinal and cofinality
structure in $N$ above $\gk$ is the same as in $V$.
In addition, standard arguments (see \cite{AK06})
show that either $N \models \neg {\rm AC}_\go$,
$N \models {\rm DC}$, or $N \models {\rm DC}_{\go_1}$,
depending if $\gl = \go$, $\gl = \go_1$, or
$\gl = \go_2$ respectively.
\section{Upper bounds}\label{s6}
In this section, we prove the forcing direction
for Theorems \ref{t1} -- \ref{t4}.
This will establish the upper bounds in
consistency strength for each of these theorems.
We will present a uniform proof throughout.
%for all of our theorems.
Our general paradigm will be to force over the
appropriate model $V'$, which is obtained
from our ground model $V$,
using the collapses described
in Section \ref{s5}. We then build the choiceless inner model
$N$ described in Section \ref{s5}, which will
end up being our witnessing model.
%In keeping with this paradigm, we will present a
%uniform proof for all of our theorems.
%More specifically, we first describe how
%to construct the model $V'$.
%which will function as the
%model over which the collapsing forcing will be done
For Theorem \ref{t1}, we let
$V \models ``$ZFC + GCH + $\gk$ is a measurable cardinal''.
Fix an arbitrary $\ga \ge 2$.
By Theorem \ref{t5} and the discussion found
in Section \ref{s4}, we may assume that there is
a partial ordering $\FP \in V$ and a choiceless, symmetric
submodel $V' \subseteq V^\FP$ such that
$V' \models ``$GCH holds below $\gk$ + ${\rm cof}(\gk) = \go$ +
There is a surjection $f : [\gk]^{\go} \to \gk^{+ \ga}$ +
$V_\gk$ is well-orderable + There is a sequence of
inaccessible cardinals $\la \gk_i \mid i < \go \ra$
whose limit is $\gk$''.
For Theorem \ref{t2},
let $n < \go$, $n \ge 2$ be fixed, and
suppose that
$V \models ``$ZFC +
$\exists \gk[({\rm cof}(\gk) = \go) \wedge
(\forall i < \go) (\forall \gl < \gk)
(\exists \gd < \gk) [(\gd > \gl) \wedge (o(\gd) \ge
\gd^{+i})]]$''.
Once again, by passing to the appropriate
inner model if necessary, we may assume that
$V \models {\rm GCH}$ as well.
By the work of \cite{G07}
(see also \cite{G02}), we may assume that $V$
has been generically extended to a model $V'$ such that
$V' \models ``$ZFC + GCH holds
below $\gk$ + $2^\gk = \gk^{+ \go_n}$, and
there is an injective $f : \gk^{+ \go_n} \to [\gk]^{\go}$ +
There is a sequence of
inaccessible cardinals $\la \gk_i \mid i < \go \ra$
whose limit is $\gk$''.
For Theorems \ref{t3} and \ref{t4}, suppose that
$V \models ``$ZFC + $\exists \gk [o(\gk) =
\gk^{++} + \gz]$''.
Here, $\gz = \go_2$ for Theorem \ref{t3}, and
$\gz = \go_1$ for Theorem \ref{t4}.
As above, by passing to the appropriate
inner model if necessary, we may assume that
$V \models {\rm GCH}$ as well.
By the work of \cite{Mc} and \cite{Ma}
(see also the remark immediately
following the statement of Theorem 1
on \cite[page 274]{GMi}), by first forcing
to make $2^\gk = \gk^{++}$ while preserving that
$o(\gk) = \gz$, and then forcing to change
$\gk$'s cofinality to $\gz$, we may assume that
$V$ has been generically extended to a model $V'$ such that
$V' \models ``$ZFC + GCH holds below $\gk$ +
${\rm cof}(\gk) = \gz$ + $2^{\gk} = \gk^{++}$, and
there is an injective $f : \gk^{+ +} \to [\gk]^{\gz}$ +
There is a sequence of
inaccessible cardinals $\la \gk_i \mid i < \gz \ra$
whose limit is $\gk$''.
We may now collapse over every $V'$
as in Section \ref{s5}
and build a choiceless, symmetric submodel
$N$ of the generic extension via the collapses.
For Theorem \ref{t2}, we take $m = n$, where
$m$ is as mentioned in Section \ref{s5}.
For Theorems \ref{t1}, \ref{t3}, and \ref{t4},
the value of $m$ is irrelevant.
In each instance, $N$ witnesses
the forcing direction of either Theorem \ref{t1},
Theorem \ref{t2}, Theorem \ref{t3}, or Theorem \ref{t4}.
\hfill$\square$
%\section{Getting a measurable cardinal
%from a choiceless, surjective failure of SCH}\label{s7}
\section{Lower bounds and the proofs of
Theorems \ref{t1} -- \ref{t4}}\label{s7}
In this section, we prove the inner model
portions of Theorems \ref{t1} -- \ref{t4},
i.e., we obtain lower bounds in consistency strength
for each of these theorems.
Once this has been done, the proofs of
Theorems \ref{t1} -- \ref{t4} will be complete.
We begin by establishing the lower bound in
consistency strength for Theorem \ref{t1}.
\begin{theorem}\label{t6}
Assume that SCH fails in a
surjective way in a model $V$ of ZF.
Then there is an inner model of ZFC with a measurable
cardinal.
\end{theorem}
\begin{proof}
Let $\gk$ be a singular cardinal such that
$(\forall \nu < \gk)[2^\nu < \gk]$, and let
$f : [\gk]^{{\rm cof}(\gk)} \to \gk^{+ +}$ be a surjection.
Let $\lambda = {\rm cof} (\kappa) + \aleph_2$.
Then because $\gk$ is a limit cardinal, $\gl < \gk$.
Assume towards a
contradiction that there is no inner model
of ZFC with a measurable cardinal. Let
$K$ be (the canonical term for) the {{\rm Dodd-Jensen}} core model $K$
(see \cite{D} and \cite{DJ1} for further details)
below a measurable cardinal.
For $Y \subseteq {\rm Ord}$, take $g_Y : \otp(Y) \leftrightarrow Y$
to be the uniquely defined order preserving map
between $\otp(Y)$ and $Y$.
Consider $X \in [\gk]^{{\rm cof}(\gk)}$.
Let ${\rm HOD}[X]$ be the smallest inner model such that
${\rm HOD} \cup \{X\} \subseteq {\rm HOD}[X]$.
By \cite[Proposition 1.1(1)]{AK06}, ${\rm HOD}[X]$ is a
set generic extension of HOD, so ${\rm HOD}[X] \subseteq V$
is a model of ZFC. Further, in ${\rm HOD}[X]$, there is
no inner model of ZFC with a measurable cardinal.
By the
{{\rm Dodd-Jensen}} covering theorem in ${\rm HOD}[X]$,
there is a covering set
$Y \in K^{{\rm HOD}[X]}$,
$X \subseteq Y \subseteq \kappa$, $\otp (Y) < \lambda$.
Let $Z = g_Y^{- 1}
[X] \in \wp(\lambda)$. Then
\begin{equation}
X = g_Y [Z] \text{ for some } Y \in \wp(\kappa)
\cap K^{{\rm HOD}[X]} \text{ and
} Z \in \wp(\lambda) .
\end{equation}
By the absoluteness properties of the {{\rm Dodd-Jensen}}
core model, $K^{{\rm HOD}[X]} = K^{\rm HOD}$. Consequently,
\begin{equation}
X = g_Y [Z] \text{ for some } Y \in \wp(\kappa)
\cap K^{{\rm HOD}} \text{ and
} Z \in \wp(\lambda) .
\end{equation}
Since ${\rm GCH}$ holds in $K^{\rm HOD}$,
take a surjective $k : \kappa^+ \rightarrow
\wp(\kappa) \cap K^{\rm HOD}$. Since $2^\gl < \gk$,
take a surjective $h : \gk
\rightarrow \wp(\lambda)$. By (4), the map
\[ (\gamma, \eta) \mapsto f (g_{k(\gamma)} [h (\eta)]) \]
is a surjection from $\kappa^+ \times \gk$ onto $\kappa^{+ +}$.
This contradiction completes the proof of
Theorem \ref{t6}.
\end{proof}
We establish the lower bounds in consistency strength for
Theorems \ref{t2} -- \ref{t4} by using
Gitik's work of \cite{G02}
(see also \cite{G07}) and
Gitik and Mitchell's work of \cite{GMi}.
For Theorem \ref{t2},
we work from a choiceless model $N$
of ZF such that
$N \models ``$GCH holds below $\ha_\go$ +
There is an injective $f : \ha_{\go_n}
\to [\ha_\go]^{\go}$, where
$1 \le n < \go$''.
In particular, we can let
$x = \la x_\ga \mid \ga < (\ha_{\go_n})^N \ra \in N$
be a sequence of distinct subsets of $(\ha_{\go})^N$.
Let $y \in N$ be a countable sequence of ordinals
having order type $\go$ cofinal in $\gl = (\ha_\go)^N$.
In $L[x, y] \models {\rm ZFC}$, $\gl$ is a singular strong limit
cardinal of cofinality $\go$.
Further, it is the case that
$L[x, y] \models ``2^\gl \ge \gl^{+ \go_n}$''
(since $(\go_n)^N \ge (\go_n)^{L[x, y]}$).
By \cite[Theorem 1, clause 3]{GMi} %and its proof
and the remark at the bottom of \cite[page 1]{G02},
in the core model $(K)^{L[x, y]}$,
$\{\gd < \gl \mid o(\gd) \ge \gd^{+ i}\}$ is cofinal
in $\gl$ for each $i < \go$.
If ${\rm cof}(\gl) = \go$ in $(K)^{L[x, y]}$,
then we are done. If not, then ${\rm cof}(\gl) \ge \go_1$
in $(K)^{L[x, y]}$. Work in the core model, and
for each $i < \go$, let
$S_i = \{\gd < \gl \mid o(\gd) \ge \gd^{+ i}\}$.
Define $\gd_0$ as the minimal member of $S_0$,
and for $1 \le i < \go$, take $\gd_i$ as the minimal
member of $S_i$ greater than $\gd_{i - 1}$.
Then $\gz = \sup(\{\gd_i \mid i < \go\}) < \gl$ is such that
${\rm cof}(\gz) = \go$ and
$\{\gd < \gz \mid o(\gd) \ge \gd^{+ i}\}$ is cofinal
in $\gz$ for each $i < \go$.
We have consequently
established the existence of a model, i.e.,
$(K)^{L[x, y]}$, containing
the cardinal $\gk$ mentioned in the hypotheses for
Theorem \ref{t2}.
For Theorems \ref{t3} and \ref{t4}b),
we work from a choiceless model $N$
of ZF such that
$N \models ``$GCH holds below $\ha_\gz$ +
There is an injective $f : \ha_{\gz + 2}
\to [\ha_\gz]^{\gz}$''.
Here, $\gz = ({\go_2})^N$ for Theorem \ref{t3},
and $\gz = ({\go_1})^N$ for Theorem \ref{t4}b).
In particular, we can let
$x = \la x_\ga \mid \ga < (\ha_{\gz + 2})^N \ra \in N$
be a sequence of distinct subsets of $(\ha_{\gz})^N$.
Let $y \in N$ be an uncountable sequence of ordinals
having order type $\gz$ cofinal in $\gl = (\ha_\gz)^N$.
In $L[x, y] \models {\rm ZFC}$, $\gl$ is a singular strong limit
cardinal of uncountable cofinality
such that $2^\gl \ge \gl^{+ +}$
(since $(\gl^{++})^N \ge (\gl^{++})^{L[x, y]}$).
As above, let $(K)^{L[x, y]}$ be the core model.
By \cite[Theorem 1, clause 1]{GMi}, when $\gz = (\go_2)^N$, since
$L[x, y] \models ``\gl$ is singular and
${\rm cof}(\gl) = \gg \ge (\go_2)^N \ge \go_2$'',
$(K)^{L[x, y]} \models ``o(\gl) \ge \gl^{+ +} + \gg$''.
By \cite[Theorem 1, clause 2]{GMi}, when $\gz = (\go_1)^N$, since
$L[x, y] \models ``\gl$ is singular and
${\rm cof}(\gl) = \gg \ge (\go_1)^N \ge \go_1$'',
$(K)^{L[x, y]} \models ``o(\gl) \ge \gl^{+ +}$''.
For either value of $\gz$, we have once again
established the existence of a model containing
%i.e., $(K)^{L[x, y]}$, containing
the cardinal $\gk$ mentioned in the
hypothesis or conclusion of
Theorems \ref{t3} and \ref{t4}b).
Theorem \ref{t6} and the discussion
in the preceding two paragraphs establish the
lower bounds in consistency strength for
Theorems \ref{t1} -- \ref{t4}, thereby
completing the proofs of these theorems.
\hfill$\square$
\section{Concluding remarks}\label{s8}
In conclusion, we would like to make several
remarks concerning Theorems \ref{t1} -- \ref{t4}.
To begin, we observe that there is nothing
special about $\ha_{\go}$, $\ha_{\go_1}$, and $\ha_{\go_2}$
in Theorems \ref{t1} -- \ref{t4}. It is certainly
possible to collapse the cardinal $\gk$ in question
to other singular limit cardinals, such as
$\ha_{\go + \go}$, $\ha_{\ha_\go}$, $\ha_{\go_5}$,
$\ha_{\ha_{\go_6}}$, etc., and obtain similar
equiconsistencies. We leave it to readers to
work out the exact details for themselves.
We observe that the dichotomy between
Theorems \ref{t3} and \ref{t4} comes from the fact that
\cite[Theorem 1]{GMi} splits into the cases
of a strong limit cardinal $\gk$ of uncountable cofinality having
cofinality at least $\go_2$ and cofinality $\go_1$.
In the latter situation, we only know that in the core model,
$o(\gk)$ is greater than or equal to the size of the
power set of $\gk$.
This does not appear to be a strong enough
hypothesis in order to do the forcing
necessary to construct the model containing
the injection found in Theorem \ref{t4}.
Our methods also make it possible to blow
up the power sets of the singular limit
cardinals in the injective failures of
SCH given by Theorems \ref{t2} -- \ref{t4}
arbitrarily high. For instance, starting with,
e.g., a model for GCH containing a
cardinal $\gk$ which is $\gl$ strong,
where $\gl > \gk$ is inaccessible, it is
possible (see \cite{G09}) to force
$2^\gk = \gl$ while simultaneously changing $\gk$'s
cofinality to $\go$.
Using a similar assumption, e.g., a model for
GCH with a cardinal $\gk$ such that $\gl > \gk$
is inaccessible and $o(\gk) = \gl + \go_1$
or $o(\gk) = \gl + \go_2$, it
is possible to force $2^\gk = \gl$ and then
change $\gk$'s cofinality to $\go_1$ or $\go_2$.
One may then symmetrically collapse $\gk$
as in Theorems \ref{t2} -- \ref{t4} to be
$\ha_\go$, $\ha_{\go_1}$, $\ha_{\go_2}$, etc.,
thereby producing a choiceless injective failure
of SCH in which there is an injection from
a regular limit cardinal into
$\wp(\ha_\go)$, $\wp(\ha_{\go_1})$, $\wp(\ha_{\go_2})$, etc.
However, under the circumstances just
described, it may not be
possible to obtain an equiconsistency.
For instance, assume that an injection from
a regular limit cardinal into $\wp(\ha_\go)$
is obtained. It does not yet seem to be possible to
force the existence of the cardinal used
in our construction, i.e., a singular strong limit
cardinal $\gd$ of cofinality $\go$ such that
$2^\gd$ is a regular limit cardinal, starting with
an inaccessible cardinal above
the cardinal $\gk$ mentioned in the hypotheses
of Theorem \ref{t2}.
(See \cite[Section 4]{G07} for the largest
possible value to which the size of the
power set of a cardinal
$\gk$ as in the hypotheses of Theorem \ref{t2}
can currently be blown up.)
By \cite[Theorem 1, clause 3]{GMi}
and the remark at the bottom of \cite[page 1]{G02},
$\gd$ will satisfy the properties of the
cardinal of the hypotheses of Theorem \ref{t2}
in the core model.
Of course, because of the dichotomy described above,
an equiconsistency also does not appear to be possible
when forcing an injection
from a regular limit cardinal
into $\wp(\ha_{\go_1})$,
although there is no problem in obtaining an
equiconsistency when forcing an injection
from a regular limit cardinal into
$\wp(\ha_{\go_2})$, $\wp(\ha_{\go_3})$, etc.
As we noted earlier, we have not been able
to replace in Theorem \ref{t1} the
surjection from $[\ha_\go]^\go$ onto $\ha_{\go + 2}$
with a surjection from $\wp(\ha_\go)$ onto $\ha_{\go + 2}$.
The reason is that the proof of
Theorem \ref{t6} looks as though it requires
the former sort of surjection.
We therefore ask if the equiconsistency of
Theorem \ref{t1} can be obtained using
only the latter kind of surjection.
Finally, we ask if it is possible to prove
analogues of Theorem \ref{t1} for singular
strong limit cardinals of uncountable cofinality.
Magidor's forcing of \cite{Ma} for changing
to an uncountable cofinality does not appear
to be amenable to the analyses found in
Sections \ref{s2} -- \ref{s4}, which
seems to pose a formidable barrier.
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\end{document}