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\title{Sandwiching the Consistency Strength of
Two Global Choiceless Cardinal Patterns
\thanks{2000 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal, supercompact
Radin forcing, Radin sequence of measures, symmetric
inner model.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}\\
% and CUNY Collaborative Incentive grants.}\\
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}
\date{October 21, 2009\\
(revised November 21, 2009)}
\begin{document}
\maketitle
\begin{abstract}
We provide upper and lower bounds in consistency
strength for the theories
``ZF + $\neg {\rm AC}_\go$ +
All successor cardinals except successors of
uncountable limit cardinals are regular + Every
uncountable limit cardinal is singular +
The successor of every uncountable
limit cardinal is singular of cofinality $\go$'' and
``ZF + $\neg {\rm AC}_\go$ +
All successor cardinals except successors of
uncountable limit cardinals are regular + Every
uncountable limit cardinal is singular +
The successor of every uncountable
limit cardinal is singular of cofinality $\go_1$''.
In particular, our models for both of these
theories satisfy ``ZF + $\neg {\rm AC}_\go$ +
$\gk$ is singular iff $\gk$ is either an
uncountable limit cardinal or the successor of
an uncountable limit cardinal''.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s1}
There are many instances in the literature where
choiceless large cardinal patterns are initially forced
from strong hypotheses which
one later sees can be weakened somewhat.
For example, it is shown in \cite{A92a} that
the models constructed in \cite{G85}
from an almost huge cardinal
can actually be built from a cardinal which is
intermediate in consistency strength between a
supercompact limit of supercompact cardinals and
an almost huge cardinal.
The purpose of this paper is to continue
in this vein by proving two
``sandwich theorems'', where a sandwich theorem
traps the consistency strength of a particular
statement between two distinct large cardinal axioms.
In particular, we begin by
providing a smaller new upper bound in consistency strength and
a new lower bound in consistency strength for a
choiceless cardinal pattern that is a corollary of
the work of \cite{G85} and \cite{A92a}.
We then show how the methods developed can be used to
prove a sandwich theorem for a choiceless cardinal
pattern that does not follow explicitly from the
work of \cite{G85} and \cite{A92a}.
%but can be established via the techniques we develop.
Specifically, we have the following theorems.
\begin{theorem}\label{t1}
$\ $
\begin{enumerate}
\item\label{i1a} Suppose $V \models ``$ZFC + $\gk$ is
$2^{[\beth_\go(\gk)]^{< \gk}}$ supercompact''.
There is then a partial ordering $\FP \in V$ and a
symmetric submodel $N \subseteq V^\FP$
of height $\gk$ such that
$N \models ``$ZF + $\neg AC_\go$ + All successor
cardinals except successors of uncountable
limit cardinals are regular +
Every uncountable limit cardinal is singular +
The successor of every uncountable
limit cardinal is singular
of cofinality $\go$''.
\item\label{i1b} Assume ZF and that
all uncountable limit cardinals are
singular and the successor of every
uncountable limit cardinal is
singular of cofinality $\go$.
Then for every $n < \go$ and every set of ordinals
$x$, $M^\sharp_n(x)$ exists.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{t2}
$\ $
\begin{enumerate}
\item\label{i2a} Suppose $V \models ``$ZFC + $\gk$ is
$2^{[\beth_{\go_1}(\gk)]^{< \gk}}$ supercompact''.
There is then a partial ordering $\FP \in V$ and a
symmetric submodel $N \subseteq V^\FP$
of height $\gk$ such that
$N \models ``$ZF + $\neg AC_\go$ + All successor
cardinals except successors of uncountable
limit cardinals are regular +
Every uncountable limit cardinal is singular +
The successor of every uncountable
limit cardinal is singular
of cofinality $\go_1$''.
\item\label{i2b} Assume ZF and that
all uncountable limit cardinals are
singular and the successor of every
uncountable limit cardinal is
singular of cofinality $\go_1$.
Then for every $n < \go$ and every set of ordinals
$x$, $M^\sharp_n(x)$ exists.
\end{enumerate}
\end{theorem}
We note that Theorems \ref{t1}(\ref{i1b}) and
\ref{t2}(\ref{i2b}) have an additional strong consequence.
This is that for every $n < \go$, there is an inner
model with $n$ Woodin cardinals, i.e., that PD holds
in all set generic extensions.
Theorems \ref{t1}(\ref{i1b}) and \ref{t2}(\ref{i2b})
%(and the aforementioned strong consequence)
are due to Busche and Schindler \cite{BuSc}.
They follow from \cite[Section 3.1]{BuSc},
and are stated explicitly as \cite[Theorem 5]{AK}.
We will explicitly prove Theorem \ref{t1}. However, since
the proof of Theorem \ref{t2} is virtually identical
to the proof of Theorem \ref{t1}, we will only indicate
the minor modifications to the proof of Theorem \ref{t1}
that need to be made in order to establish
Theorem \ref{t2}.
Also, we observe that our models witnessing the
conclusions of Theorems \ref{t1}(\ref{i1a}) and
\ref{t2}(\ref{i2a}) satisfy the theory
``ZF + $\neg {\rm AC}_\go$ +
$\gk$ is singular iff $\gk$ is either an
uncountable limit cardinal or the successor of
an uncountable limit cardinal''.
\begin{pf}
Our proof uses Gitik's techniques of \cite{G85}.
Our presentation of Gitik's
techniques is based on the one given in
\cite{A92}, but also follows the ones given in
\cite{A85}, \cite{A96}, \cite{AK}, and \cite{A10}.
All of these rely heavily on \cite{G85}.
As the necessary facts about Radin
forcing are distributed throughout
the literature, our
bibliographical citations will reflect
this.
Our witnessing model $N$ for Theorem \ref{t1}(\ref{i1a})
%and \ref{t2}
is the specific version of the model
$N_A$ of \cite{A92} described
at the end of that paper, except that $\gl$ is now
the least singular strong limit cardinal of cofinality
$\go$ above $\gk$ instead of the least measurable
cardinal above $\gk$.
(See also \cite{A10}.)
%, only constructed using a Radin sequence of
%measures of length $\ha_1$ instead of $\gk^+$
%and not truncating the universe at $\gk$.
We explicitly give the construction below.
%Let $\gl = [\beth_\go(\gk)]^{< \gk}$.
Let $\gl = \beth_\go(\gk)$.
Let $j : V \to M$ be an elementary
embedding witnessing the
$2^{[\gl]^{{<} \gk}}$ supercompactness of $\gk$.
Our first step is to define a
{{Radin}} sequence of measures $\mu_{< {{\gk^+}}} = \la \mu_\ga \mid
\ga < {{\gk^+}} \ra$
appropriate for supercompact Radin forcing over $P_{\gk}(\gl)$.
Specifically, if
$\ga = 0$, $\mu_\ga$ is given by $X \in \mu_\ga$ iff
$\la j(\gb) \mid \gb < \gl \ra \in j(X)$,
and if $\ga > 0$, $\ga < \gk^+$, $\mu_\ga$ is
given by $X \in \mu_\ga$ iff $\la \mu_\gb \mid \gb < \ga \ra
=_{\hbox{\rm df}} \mu_{< \ga} \in j(X)$.
Since $M^{2^{[\gl]^{{<} \gk}}} \subseteq M$,
$\mu_{< {{\gk^+}}}$ is well-defined.
%Next, using $\mu_{< {{\gk^+}}}$, we let $\FR_{< {{\gk^+}}}$ be
%supercompact {{Radin}} forcing defined over $V_{\gk} \times
%P_{\gk}(\gl)$.
Next, we let $\FR_{< {{\gk^+}}}$ be supercompact Radin
forcing over $P_\gk(\gl)$ defined using $\mu_{< {{\gk^+}}}$.
The particulars of the definition
are virtually identical
to the ones found in \cite{A85}, \cite{A92},
\cite{A96}, \cite{AK}, and \cite{A10},
but for clarity, we repeat them here.
%(which may be taken as our standard references for what follows),
$\FR_{< {{\gk^+}}}$ is composed of all
finite sequences of the form $\la \la p_0, u_0, C_0, \ra, \ldots,
\la p_n, u_n, C_n \ra , \la \mu_{< {{\gk^+}}}, C \ra \ra$ 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
${\rm otp}(p) < q \cap \gk$.
%($\overline{p}$ is the order type of $p$.)
\item For $0 \le i \le n$, $p_i \cap \gk$ is a
measurable cardinal.
\item ${\rm otp}(p_i)$ is the least strong limit
cardinal greater than $p_i \cap \gk$
(which of course is $\beth_\go(p_i \cap \gk)$).
%which is a measurable cardinal carrying exactly
%$({\rm otp}(p_i))^+$ many normal measures.
In analogy to the notation of \cite{G85},
\cite{A85}, \cite{A92}, \cite{A96}, \cite{AK}, and \cite{A10},
we write ${\rm otp}(p_i) = {(p_i \cap \gk)}^*$.
%By extension of this notation, $\gl = \gk^*$.
\item For $0 \le i \le n$, $u_i$ is a
{{Radin}} sequence of measures
%over $V_{p_i \cap \gk} \times P_{p_i \cap \gk}({\rm otp}(p_i))$
appropriate for supercompact Radin forcing over
$P_{p_i \cap \gk}({\rm otp}(p_i))$
with ${(u_i)}_0$, the
$0$th coordinate of $u_i$, a supercompact measure over
$P_{p_i \cap \gk}({\rm otp}(p_i))$.
\item $C_i$ is a sequence of measure 1 sets for $u_i$.
\item $C$ is a sequence of measure 1
sets for $\mu_{< {{\gk^+}}}$.
\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 = p_n \smag p$.
\item For each $p \in {(C)}_0$, ${\rm otp}(p) =
{(p \cap \gk)}^*$ 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
the successor of every limit cardinal
is singular of cofinality $\go$.
That they may be included and have
the Radin forcing attain its desired goals
follows by the fact that
$V \models ``\gk$ is supercompact and $\gl$
is the least strong limit cardinal
greater than $\gk$''.
Thus, by closure,
$M \models ``\gk$ is measurable and
$\gl$ is the least strong limit cardinal
greater than $\gk$''.
This means that by reflection,
$\{ p \in P_{\gk}(\gl) \mid p \cap \gk$ is a measurable
cardinal and ${\rm otp}(p)$ is the
least strong limit 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
$\FR_{< {{\gk^+}}}$. If $\pi_0 =
%\break
\la \la p_0, u_0, C_0 \ra ,
\ldots , \la p_n, u_n, C_n \ra ,
\la \mu_{< {{\gk^+}}},
C \ra \ra$ and $\pi_1 = \la \la q_0, v_0, D_0 \ra , \ldots ,%\break
\la q_m, v_m, D_m \ra , \la \mu_{< {{\gk^+}}} , D \ra \ra$, 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 $\la \mu_{< {{\gk^+}}}, C \ra$) 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 $\FR_{< {{\gk^+}}}$ (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{AK}, \cite{A10}, \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{t1}(\ref{i1a}). It is given by
the finite support product
ordered componentwise
$$\prod_{\{\la \ga, \gb \ra \mid
\go \le \ga < \gb < \gk \ {\rm are} \
{\rm regular} \ {\rm cardinals}\}}
{\rm Coll}(\ga, {<} \gb) \times
\FR_{< {{\gk^+}}},$$
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 $\FR_{< {{\gk^+}}}$.
For any condition $\pi \in \FR_{< {{\gk^+}}}$, 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
$p \in {\rm p-part}(\pi)$, $p \cap \gk = \ga\}$,
$E_1 = \{\ga < \gk \mid \ga$ is a limit point of $E_0\}$, and
$E_2 = E_1 \cup \{\go\} \cup \{\gb \mid
\exists \ga [\ga$ is a limit point of
$E_1$ and $\gb = \ga^*]\}$.
Let $\la \ga_\nu \mid \nu < \gk \ra$ be the
continuous, increasing enumeration of $E_2$,
and let $\nu = \nu' + n$ for some $n \in \go$.
For $\gb$ where
$\gb \in [\ga_\nu, \ga_{\nu + 1})$ in the
first case and $\gb = \ga_{\nu + 1}$
in the second and third cases,
define sets $C_i(\ga_\nu, \gb)$
for $i = 1, 2, 3$
according to specific conditions on $\nu$, $\nu'$, and $n$
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
$h_{p(\ga_\nu)} : p(\ga_\nu) \to {\rm otp}(p(\ga_\nu))$ be
the order isomorphism between $p(\ga_\nu)$ and
${\rm otp}(p(\ga_\nu))$. 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 $(\nu = \nu' + n$, $\nu' > 0$,
and $2 \le n < \go)$ or
$(\nu' = 0$ and $n \in \go)$, i.e., $\gn$ is neither a
limit ordinal nor the successor of a limit ordinal.
Let
$H(\ga_\nu, \ga_{\nu + 1})$ be the projection of $G$ onto
${\rm Coll}(\ga_\nu, {<} \ga_{\nu + 1})$. Then
$C_2(\ga_\nu, \ga_{\nu + 1}) = H(\ga_\nu, \ga_{\nu + 1})$.
\item $\nu = \nu' + 1$ for $\nu' > 0$,
i.e., $\nu$ is the successor of a limit ordinal.
Let
$H(\ga^+_\nu, \ga_{\nu + 1})$ be the projection of $G$ onto
${\rm Coll}(\ga^+_\nu, {<} \ga_{\nu + 1})$. Then
$C_3(\ga^+_\nu, \ga_{\nu + 1}) = H(\ga^+_\nu, \ga_{\nu + 1})$.
\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
$\la \ga_\gg \mid \gg < \nu \ra$.
$C_2(\ga_\nu, \ga_{\nu + 1})$ is used
to collapse $\ga_{\nu + 1}$ to be the
successor of $\ga_\nu$ when $\nu$ is
neither a limit ordinal nor the
successor of a limit ordinal, and
$C_3(\ga^+_\nu, \ga_{\nu + 1})$
is used to collapse $\ga_{\nu + 1}$ to be
the successor of $\ga^+_\nu$ when
$\nu$ is the successor of a limit ordinal.
Intuitively, the symmetric inner model $N \subseteq V[G]$
witnessing the conclusions of Theorem \ref{t1}(\ref{i1a})
is $V_\gk$ of the least model of ZF
extending $V$ which contains
$C_1(\ga_\nu, \gb)$ if $\nu$
is a limit ordinal and
$\gb \in [\ga_\nu, \ga_{\nu + 1})$,
$C_2(\ga_\nu, \ga_{\nu + 1})$ if
$\nu$ is neither a limit ordinal
nor the successor of a limit ordinal, and
$C_3(\ga^+_\nu, \ga_{\nu + 1})$ if $\nu$
is the successor of a limit ordinal.
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 $i = 1$ and
$\underline{C_i(\nu, \nu + 1)}$ for $i = 2, 3$.
%for the three sets just described.
Recall that it is
possible to decide $p(\ga_\nu)$ (and hence
${\rm otp}(p(\ga_\nu))$) 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
$\pi = \la{\la p_{ij_i}, u_{ij_i}, C_{ij_i} \ra}_{i \le m,
1 \le j_i\le n_i}, \la \mu_{< {{\gk^+}}}, C \ra \ra$ 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, $D_\nu = \{ r \in \FP \mid r \rest
\FR_{< {{\gk^+}}}$ 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},
\cite{A96}, \cite{AK}, and \cite{A10},
$\underline{\ga_\nu} = \{ \la r, \check \ga_\nu(r)
\ra \mid r \in D_\nu \}$, where $\ga_\nu(r)$ is the $\ga_\nu$ determined
by the condition $r$.
The canonical names $\underline{C_i(\nu, \gb)}$
for $i = 1$ and
$\underline{C_i(\nu, \nu + 1)}$ for $i = 2, 3$
are defined in a manner so as to
be invariant under the appropriate group of
automorphisms. Specifically, there are three 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},
\cite{A96}, \cite{AK}, and \cite{A10},
$\ga_{\nu + 1}$ is determined by $D_\nu$.
Further, we adopt
throughout each of the three cases
the notation of \cite{G85}, \cite{A85},
\cite{A92}, \cite{A96}, \cite{AK}, and \cite{A10}.
\begin{enumerate}
\item $\nu' = \nu \ne 0$
and $n=0$, i.e.,
$\nu$ is a limit ordinal.
$\underline{C_1(\nu, \gb)} = \{
\la r, (\check r \rest \FR_{< {{\gk^+}}}) \rest (\ga_\nu(r), \gb) \ra
\mid r \in
D_\nu \}$, where for $r \in \FP$,
$\pi = r \rest \FR_{< {{\gk^+}}}$, $\pi \rest
(\ga_\nu(r), \gb) = \{ {{h_{p(\ga_\nu)(r)}}}''p\cap \gb \mid p \in$ 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 $(\nu = \nu' + n$, $\nu' > 0$,
and $2 \le n < \go)$ or
$(\nu' = 0$ and $n \in \go)$, i.e., $\gn$ is neither a
limit ordinal nor the successor of a limit ordinal.
$\underline{C_2(\nu, \nu + 1)} =
\{ \la r, (\check r \rest
{\hbox{\rm Coll}}(\ga_\nu(r), {<} \ga_{\nu + 1}(r)))
%\rest \gb
\ra \mid r \in D_\nu \}$.
\item $\nu = \nu' + 1$ for $\nu' > 0$,
i.e., $\nu$ is the successor of a limit ordinal.
$\underline{C_3(\nu, \nu + 1)} =
\{ \la r, (\check r \rest
{\hbox{\rm Coll}}(\ga^+_\nu(r), {<} \ga_{\nu + 1}(r)))
%\rest \gb
\ra \mid r \in D_\nu \}$.
\end{enumerate}
\noindent As in \cite{G85}, \cite{A85}, \cite{A92},
\cite{A96}, \cite{AK}, and \cite{A10}, since for any
$r , r' \in D_\nu \cap G$, $p(\ga_\nu)(r) =
p(\ga_\nu)(r')$, each of the definitions
just given is unambiguous.
Let ${\cal G}$ be the group of automorphisms of \cite{G85},
and let $\underline{C(G)} =
\{\psi(\underline{C_1(\nu,\gb)}) \mid \psi
\in {\cal G}$, $0 \le \nu < \gk$, and $\gb \in [\nu,\gk)$ is a
cardinal$\} \cup
\bigcup_{i = 2, 3} \{
\psi(\underline{C_i(\nu,\nu + 1)}) \mid \psi
\in {\cal G}$ and $0 \le \nu < \gk\}$.
$C(G) =
\{i_G(\psi(\underline{C_1(\nu,\gb)})) \mid \psi
\in {\cal G}$, $0 \le \nu < \gk$, and $\gb \in [\nu,\gk)$ is a
cardinal$\} \cup
\bigcup_{i = 2, 3} \{
i_G(\psi(\underline{C_i(\nu,\nu + 1)})) \mid \psi
\in {\cal G}$ and $0 \le \nu < \gk\}
= i_G(\underline{C(G)})$.
$N$ 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.,
$N = V^{ {\rm HVD} (C(G)) }_{\gk}$.
Let $\la \gd_\nu \mid \nu < \gk \ra$ be the
continuous, increasing enumeration of
$\{\ga_\nu \mid \nu < \gk\} \cup
\{(\ga^+_\nu)^V \mid \nu < \gk$ is the successor
of a limit ordinal$\}$.
The arguments of \cite{G85}
and \cite[Lemma 1]{A92} allow us to conclude that
$N \models ``$ZF + $\neg {\rm AC}_\go$ +
For every limit ordinal $\nu$, $\gd_\nu = \ga_\nu = \ha_\nu$
is singular + If $\nu = \nu' + 1$ and $\nu'$ is
not a limit ordinal, then $\gd_\nu$ is a regular cardinal +
$\forall \nu[\gd_\nu \le \ha_\nu]$''.
%Every successor cardinal is regular
%Every successor cardinal is regular +
%Every limit cardinal is singular + The
%successor of every singular cardinal is measurable''.
In addition, we
know that for any ordinal
$\gamma$ and any set $x \subseteq \gamma$, $x \in N$,
$x = \{\ga<\gamma \mid
V[G] \models
\phi(\ga, i_G(\psi_1(\underline{C_{i_1}(\nu_1,\gb_1)})),\ldots,
i_G(\psi_n(\underline{C_{i_n}(\nu_n,\gb_n)})), C(G))\}$,
where $i_j$ is
an integer, $1 \le j \le n$, $1 \le i_j \le 3$, each $\psi_i
\in {\cal G}$, each $\gb_i$ is an appropriate ordinal for $\nu_i$,
and $\phi(x_0, \ldots, x_{n+1})$ is a formula which may also
contain some parameters from $V$ which we shall suppress.
Let $$\overline \FP =
\prod_{i_j = 2, j \le n}
{\rm Coll}(\ga_{\nu_j}, {<} \ga_{\nu_j + 1})
\times
\prod_{i_j = 3, j \le n}
{\rm Coll}(\ga^+_{\nu_j}, {<} \ga_{\nu_j + 1})
\times \FR_{< {{\gk^+}}}.$$
For $\pi \in \FR_{< {{\gk^+}}}$,
%and $\gamma$ an arbitrary ordinal,
let $\pi \rest \gamma = \{\la q,
u, C \ra \in \pi \mid q \cap \gk \le \gamma \}$,
and let $\FR_\gamma =
\{\pi \rest \gamma \mid \pi \in \FR_{< {{\gk^+}}}\}$.
For
$p \in \overline{\FP}$, $p = \la p_1, \ldots, p_m, \pi \ra$, $m \le n$,
$\pi \in \FR_{< {{\gk^+}}}$, let
$p \rest \gamma = \la q_1, \ldots, q_m, \pi \rest
\gamma \ra$, where $q_j = p_j$ if $\ga_{\nu_j} \le \gamma$
and $q_j = \emptyset$ otherwise. In
other words, $p \rest \gamma$ is the part of p below or at
$\gamma$. Without loss of generality,
we ignore the empty coordinates
and let $\overline{\FP} \rest \gamma
= \{p \rest \gamma \mid p \in \overline{\FP} \}$. Let
$G \rest \gamma$ be the projection of
$G$ onto $\overline{\FP} \rest \gamma$. An
analogous fact to \cite[Theorem 3.2.11]{G85}
holds, using the same
proof as in \cite{G85},
%namely for any $x \subseteq \gamma$,
namely $x \in
V[G \rest \gamma]$. In addition, the elements
of $\overline{\FP} \rest \gamma$
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
$\varphi$ is any formula mentioning
only (terms for ground model sets and)
$\underline{C_{i_1}(\nu_1,\gb_1)},
\ldots, \underline{C_{i_n}(\nu_n,\gb_n)}$, and
$\underline{C(G)}$, $p \decides \varphi$
(i.e., $p$ {\em decides} $\varphi$), and
$q \sim p$, then $q \decides \varphi$ in
the same way that $p$ does.
It thus follows as an immediate
corollary of the work of \cite{G85}
that if we define
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra} =
\{{[p]}_\sim \mid p \in G \rest \gg\}$, then
$x \in V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}]$
and
$V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}]
\subseteq N$.
It further follows by the work of \cite{G85}
that if $\nu < \gk$ is a limit ordinal
and $\gg = \ga_{\nu + 1}$, then since
$\gd_{\nu + 1} = \ga_{\nu + 1} = \ga^*_\nu$ and $\ga^*_\nu$
is a strong limit cardinal,
there are fewer than $\ga_{\nu + 1}$
many such equivalence classes.
In other words,
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}$
is $V$-generic over a partial ordering forcing
equivalent to a partial ordering
$\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}$
such that
$\card{\FQ^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}}
< \ga_{\nu + 1}$.
It is this last fact, in tandem with the
way in which $N$ is defined, which allows
us to show that $N$ is our desired model.
In particular, the following two lemmas
complete the proof of Theorem \ref{t1}(\ref{i1a}).
\begin{lemma}\label{l1}
If $\nu < \gk$ is a limit ordinal, then
$N \models ``\ga_{\nu + 1}$ is a singular
cardinal having cofinality $\go$''.
\end{lemma}
\begin{proof}
Since $\nu < \gk$ is a limit ordinal,
%by the definition of the sequence $\la \ga_\gb \mid \gb < \gk \ra$,
as was mentioned above,
$\gd_{\nu + 1} = \ga_{\nu + 1} = \ga^*_\nu$. By
the definition of $\ga^*_\nu$,
$V \models ``\ga^*_\nu$ is a strong limit cardinal
having cofinality $\go$'',
%{\rm cof}(\ga^*_\nu) = \go$'',
so since $V \subseteq N$,
$N \models ``{\rm cof}(\ga_{\nu + 1}) = \go$''.
Then, to see that $N \models ``\ga_{\nu + 1}$ is a
cardinal'', suppose that $\gr < \ga_{\nu + 1}$ and
$N \models ``f : \gr \to
\ga_{\nu + 1}$ is a function''. Since $f$ may be
coded by a subset of $\ga_{\nu + 1}$,
by the preceding paragraph,
$f \in V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}]
\subseteq N$ for the appropriate generic object
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}$.
Because
$G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}$
is $V$-generic
over a partial ordering forcing equivalent
to a partial ordering having cardinality less than
$\ga_{\nu + 1}$,
$f$ cannot witness that $\ga_{\nu + 1}$
is no longer a cardinal.
This completes the proof of Lemma \ref{l1}.
%Since
%$V[G^{\la \la \nu_1, \gb_1 \ra, \ldots, \la \nu_n, \gb_n \ra \ra}]
%\models ``\ga_{\nu + 1}$ is a cardinal'',
%this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$N \models ``$All successor cardinals except
successors of uncountable limit cardinals are regular +
Every uncountable limit cardinal is singular +
The successor of every uncountable limit cardinal
is singular of cofinality $\go$''.
\end{lemma}
\begin{proof}
By our earlier remarks, $N \models ``\gd_\nu$
is a cardinal, except possibly if $\nu = \nu' + 1$
and $\nu'$ is a limit ordinal''. By Lemma \ref{l1},
for such a $\nu$, $N \models ``\gd_\nu = \ga_\nu$ is
a singular cardinal of cofinality $\go$''. Since as
we have already observed, $N \models ``\forall \nu
[\gd_\nu \le \ha_\nu]$'', an easy induction in
conjunction with the fact that
$N \models ``$For every limit
ordinal $\nu$, $\gd_\nu =
\ga_\nu = \ha_\nu$ is singular +
If $\nu = \nu' + 1$ and $\nu'$ is not a limit ordinal,
then $\gd_\nu$ is a regular cardinal'' now shows that
$N \models ``\forall \nu [\gd_\nu = \ha_\nu]$''.
In particular, we may also infer that
$N \models ``$All successor cardinals except
successors of uncountable limit cardinals are regular +
Every uncountable limit cardinal is singular +
The successor of every uncountable limit cardinal
is singular of cofinality $\go$''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
Lemmas \ref{l1} and \ref{l2} complete the proof of
Theorem \ref{t1}(\ref{i1a}).
\end{pf}
\begin{pf}
To prove Theorem \ref{t1}(\ref{i1b})
(and thereby complete the proof of
Theorem \ref{t1}),
we assume ZF and that all uncountable limit
cardinals are singular and the successor of every
uncountable limit cardinal is singular. Let
$\Phi(\gr)$ be the statement ``$\gr$ is a singular cardinal''.
Since ZFC $\vdash$ ``All successor cardinals are regular'',
if $\Phi(\gr)$ holds, there is no inner model
of ZFC in which $\gr$ is a successor cardinal.
Further, by assumption, there is a proper class of
cardinals $\gd$ in which $\gd$ is singular and
$\Phi(\gd^+)$ is true. Therefore, by
\cite[Theorem 5]{AK} (see also the work of
\cite[Section 3.1]{BuSc}, from which
\cite[Theorem 5]{AK} is derived), for every $n < \go$
and every set of ordinals $x$, $M^\sharp_n(x)$ exists.
This completes the proof of both Theorem \ref{t1}(\ref{i1b})
and Theorem \ref{t1}.
\end{pf}
\begin{pf}
To sketch the proof of Theorem \ref{t2},
suppose $V \models ``$ZFC + $\gk$ is
$2^{[\beth_{\go_1}(\gk)]^{< \gk}}$ supercompact''.
%Let $\gl = {[\beth_{\go_1}(\gk)]^{< \gk}}$, and
Let $\gl = \beth_{\go_1}(\gk)$, and
let $\mu_{{<} \gk^+}$ and $\FR_{{<} \gk^+}$ be
defined exactly as in the proof
of Theorem \ref{t1}(\ref{i1a}) using this
value of $\gl$.
Define $\FP$ as the finite support product
ordered componentwise
$$\prod_{\{\la \ga, \gb \ra \mid
\go_1 \le \ga < \gb < \gk \ {\rm are} \
{\rm regular} \ {\rm cardinals}\}}
{\rm Coll}(\ga, {<} \gb) \times
\FR_{< {{\gk^+}}},$$
i.e., $\FP$ is defined as in the proof
of Theorem \ref{t1}(\ref{i1a}), except that the
smallest regular cardinal to which another
regular cardinal may be collapsed is
${(\go_1)}^V$.\footnote{This idea was also
used in the proof of the main theorem of \cite{A10}.}
Since this definition of
$\FP$ ensures that ${(\go_1)}^V$ is not
collapsed (and so remains a regular cardinal),
if we construct $N$ in the analogous manner
to the construction given in the proof
of Theorem \ref{t1}(\ref{i1a}),
we may use the same arguments as given in the
proof of Theorem \ref{t1}(\ref{i1a}) to prove
Theorem \ref{t2}(\ref{i2a}). The proof of
Theorem \ref{t2}(\ref{i2b}) is then exactly
the same as the proof of Theorem \ref{t1}(\ref{i1b}).
This completes the proof sketch of Theorem \ref{t2}.
\end{pf}
We note that in the proof of Theorem \ref{t2},
there is nothing special about cofinality $\go_1$.
Other uncountable cofinalities, e.g.,
$\go_2$, $\go_3$, etc$.$ are also possible,
with the appropriate further modifications
analogous to those given in the proof of Theorem \ref{t2}.
We leave it to interested readers to
work out the details for themselves.
There is of course a vast disparity between the
upper and lower bounds in consistency strength
given in Theorems \ref{t1} and \ref{t2}.
We conclude by asking if it is possible to prove
equiconsistencies for both of these theorems,
and not just ``sandwich theorems''.
Unfortunately, such results remain outside the
reach of current set theoretic technology.
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\end{document}
There are many instances in the literature where
%an upper bound in consistency strength is provided for
a choiceless large cardinal pattern is forced
from fairly strong hypotheses without discussing
lower bounds in consistency strength.
The purpose of this paper is to prove a
``sandwich theorem'', i.e., to provide
a smaller new upper bound in consistency strength and
a new lower bound in consistency strength for a
choiceless large cardinal pattern that follows from
the work of \cite{G85} and \cite{A92}.
Specifically, we will prove the following theorems.
We follow the proof of \cite[Lemma 1]{A92},
quoting freely (and sometimes verbatim) from
this proof when applicable.
In order to prove Lemma \ref{l1}, we must first
ascertain the nature of the cardinal structure
of $N$. Specifically, we show that all
(well-ordered) cardinals of $N$ are either an
$\ga_\nu$ or an ${(\ga^+_\nu)}^V$ if
$\nu = \gl + 1$ and $\gl$ is a limit ordinal.
Thus, we begin by showing that any $\gg$ for
$\gg = \ga_\nu$ or $\gg = {(\ga^+_\nu)}^V$ if
$\nu = \gl + 1$ and $\gl$ is a limit ordinal
remains a cardinal in $N$.
Let $\gg$ be as just stated. If
$x \subseteq \gg$, $x \in N$, then
as mentioned above, $x \in V[G \rest \gg]$, where
$G \rest \gg$ is $V$-generic over $\ov \FP \rest \gg$
and $\ov \FP \rest \gg$ and $G \rest \gg$ are as
previously described. Thus, it suffices to show that
$\gg$ remains a cardinal in $V[G \rest \gg]$.
To see that this is true, observe that it is
possible to write $\ov \FP \rest \gg =
\FQ_0 \times \FQ_1$, where $\FQ_0$ is a partial
ordering (possibly trivial) defined over $\gg$ and
some ordinal $\gb > \gg$, and $\FQ_1$ is the rest
of $\ov \FP \rest \gg$.
Since by the definition of $N$, $\FQ_0$ will
be either trivial (if $\gg = \ga_{\nu + 1}$ and
$\nu$ is a limit ordinal), a partial ordering
of the form ${\rm Coll}(\gg, {<} \gb)$, or
\begin{lemma}\label{l2}
$N \models ``\gg$ is a singular cardinal'' iff
for $\nu < \gk$ a limit ordinal,
either $\gg = \ga_\nu$ or $\gg = \ga_{\nu + 1}$.
\end{lemma}