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\date{April 20, 2011\\(revised July 13, 2012)}
\title{On Some Questions Concerning Strong Compactness
\thanks{2010 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal,
strongly compact cardinal, GCH, symmetric inner model.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
Brent Cody for helpful conversations
on the subject matter of this paper.
The author also wishes to thank the
second referee for helpful corrections
and suggestions which were incorporated
into the current version of the paper.}\\
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/aapter\\
awapter@alum.mit.edu}
\begin{document}
\maketitle
\begin{abstract}
A question of Woodin asks
if $\gk$ is strongly compact and GCH holds
below $\gk$, then must GCH hold everywhere?
One variant of this question asks
%a question of Woodin asks
if $\gk$ is strongly compact and GCH
fails at every regular cardinal
$\gd < \gk$, then must GCH fail at
some regular cardinal $\gd \ge \gk$?
Another variant asks if it is possible
for GCH to fail at every limit cardinal
less than or equal to
a strongly compact cardinal $\gk$.
We get a negative answer to the first
of these questions and positive answers
to the second of these questions
for a supercompact cardinal $\gk$
in the context of the absence of the full
Axiom of Choice.
%In all of our results, $\gk$ is fully supercompact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite[22.22, page 310]{K}, the following
question is attributed to Woodin:
If $\gk$ is strongly compact and GCH holds
below $\gk$, then must GCH hold everywhere?
Assuming the Axiom of Choice, an easy reflection
argument yields that the answer to this question
must be yes if $\gk$ is supercompact.
However, when full
AC is false, things are very different. Specifically,
we have the following
theorem from \cite{A00}, which provides
a negative answer to Woodin's question in
the context of the absence of AC.
\begin{theorem}\label{t1}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\forall \gd < \gk[DC_{\gd}]$ +
$\gk$ is a strong limit cardinal +
$\forall \gd < \gk[2^\gd = \gd^+]$ + $\gk$ is supercompact +
There is a sequence $\la A_\ga \mid \ga < \gk^{++} \ra$
of distinct subsets of $\gk$''.
\end{theorem}
Woodin's question may be inverted to produce
related questions concerning strongly compact
cardinals and GCH. In particular, one may ask if
$\gk$ is strongly compact and
GCH fails at every regular cardinal $\gd < \gk$,
then must GCH fail at some regular cardinal
$\gd \ge \gk$?
As in Woodin's original question, a simple reflection
argument yields that the answer to this
question must be yes if $\gk$ is supercompact.
On the other hand, it is also possible to ask
about the possibility of $\gk$ being
strongly compact and
GCH failing at every limit cardinal $\gd \le \gk$.
Of course, by Solovay's
celebrated theorem \cite{S}, GCH must always
hold at any singular strong limit cardinal
above a strongly compact cardinal $\gk$.
Consequently, a simple reflection argument now shows
that the answer to this question
must be no if $\gk$ is supercompact.
The purpose of this paper is to
provide answers to these questions
in the context of the absence of full AC
for a supercompact cardinal $\gk$.
%but where $\gk$ is fully supercompact.
We show that as in
\cite{A00}, it is possible to get a negative
answer to the first of the
above questions.
On the other hand, it is also possible
to get a positive answer to the second of
the above questions.
Specifically,
we prove the following two theorems, where
we adopt as our terminology that when
AC is false and $\gd$ is a cardinal\footnote{For
the purposes of this paper, all cardinals will
be well-ordered, i.e., will be alephs.},
``GCH holds at $\gd$'' means that there is an injection
$f : \gd^+ \to \wp(\gd)$, and for every
cardinal $\gl > \gd^+$, there is no injection
$f : \gl \to \wp(\gd)$.
Similarly, in a choiceless context,
``GCH fails at $\gd$'' means that for some
cardinal $\gl > \gd^+$, there is an injection
$f : \gl \to \wp(\gd)$.
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\forall \gd < \gk[DC_{\gd}]$ +
$\gk$ is a strong limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
$\forall \gd < \gk[$If $\gd$ is regular, then
$2^\gd = \gd^{++}$, but if $\gd$ is singular,
then $2^\gd = \gd^+]$ +
GCH holds at every (regular or singular)
cardinal $\gd \ge \gk$''.
\end{theorem}
\begin{theorem}\label{t3}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\neg {AC}_\go$ +
$\gk$ is a limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
GCH fails at every limit
cardinal $\gd \le \gk$ +
GCH holds at every (regular or singular)
cardinal $\gd > \gk$''.
\end{theorem}
We take this opportunity to make a few
brief remarks concerning Theorems \ref{t2} and \ref{t3}.
Note that in the absence of full AC, $\gk$ being
supercompact means that for every cardinal
$\gl \ge \gk$, $P_\gk(\gl)$ carries a
$\gk$-additive, fine, normal ultrafilter, and
$\gk$ being strongly compact means that for every cardinal
$\gl \ge \gk$, $P_\gk(\gl)$ carries a
$\gk$-additive, fine (not necessarily normal) ultrafilter.
Consequently, the conclusions of Theorems \ref{t2} and \ref{t3}
remain valid, with ``$\gk$ is strongly compact'' replacing
``$\gk$ is supercompact''.
%our results are true in a choiceless context
%for strongly compact cardinals as well.
Note also that
as \cite[Example 15.57, pages 259--260]{J} shows, when AC is false,
it is possible for successor cardinals to be singular.
Thus, the fact that every successor cardinal is
regular in the models witnessing the
conclusions of Theorems \ref{t2} and \ref{t3}
is especially significant.
In addition, in Theorem \ref{t2},
in direct analogy to Theorem \ref{t1}, it will
literally be the case that ``$\gk$ is a
strong limit cardinal'',
``For every regular cardinal $\gd < \gk$,
$2^\gd = \gd^{++}$'', and
``For every singular cardinal $\gd < \gk$,
$2^\gd = \gd^+$'' mean the same thing
as when AC is true.
In Theorem \ref{t3}, however,
this won't be the situation. More specifically, GCH
holding and failing will be in the weaker sense
described above, although as is the case when
AC is true, $\gk$ remains a limit cardinal.\footnote{As
\cite[Theorem 21.16, pages 404--406]{J} shows,
without the Axiom of Choice, it is possible for
large cardinals to be successor cardinals.}
Finally, as each of our theorems shows, when
AC is false, a supercompact cardinal need
not possess its full reflection properties.
We mention very briefly some preliminary information.
We assume a basic knowledge of set theoretic
terminology and large cardinals and
forcing, as provided, e.g., by \cite{J}.
In particular, when $\FP$ is our forcing
partial ordering and $G$ is $V$-generic over $\FP$,
we will abuse notation somewhat and use both
$V^\FP$ and $V[G]$ to denote the generic extension by $\FP$.
We will also frequently abuse notation by writing
$x$ instead of $\check x$ for ground model sets.
We note in addition that for $\gk$ a regular cardinal and
$\ga$ an ordinal, $\add(\gk, \ga)$ is the standard
partial ordering for adding $\ga$ many Cohen
subsets of $\gk$, i.e.,
$\add(\gk, \ga) = \{f : \gk \times \ga \to \{0, 1\}
\mid \card{\dom(f)} < \gk\}$, ordered by inclusion.
For $\gk$ a regular cardinal and $\gl > \gk$ an
inaccessible cardinal, ${\rm Coll}(\gk, {<} \gl)$
is the standard L\'evy collapse partial ordering
for collapsing $\gl$ to $\gk^+$, i.e.,
${\rm Coll}(\gk, {<} \gl) = \{f : \gk \times \gl
\to \gl \mid \card{\dom(f)} < \gk$, and for every
$\la \ga, \gb \ra \in \dom(f)$,
$f(\la \ga, \gb \ra) < \gb\}$, ordered by inclusion.
\section{The Proofs of Theorems \ref{t2} and \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
We will be constructing a symmetric model of
``ZF + $\forall \d < \k[{\rm DC}_\gd]$'' in which
$\k$ is supercompact, $\k$ is a strong limit cardinal,
every successor cardinal is regular,
GCH fails at every regular cardinal $\gd < \gk$, and
GCH holds at all other cardinals.
\begin{proof}
The proof of Theorem \ref{t2} will be similar to
the proof of Theorem \ref{t1} found in \cite{A00}.
We will therefore freely quote (sometimes verbatim
when appropriate) from \cite{A00}.
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Let $\la \gd_\ga \mid \ga < \gk \ra$ enumerate in
increasing order the regular cardinals less than $\gk$.
For each ordinal $\a < \gk$, let
$\FP_\a = \add(\d_\a, \d_\a^{++})$.
The partial ordering $\FP$ with which we force is
then the Easton support product $\prod_{\a < \gk} \FP_\a$.
Let $G$ be $V$-generic over $\FP$. The full generic extension
$V[G]$ is not our desired model $N$.
In order to define $N$, we first let $G_\a$
for any $\a < \gk$ be the
projection of $G$ onto $\prod_{\gb < \a} \FP_\gb = \FQ_\a$.
By the Product Lemma, $G_\a$ is $V$-generic over $\FQ_\a$.
We can now intuitively describe $N$ as the least
model of ZF extending $V$ which contains,
for each $\a < \gk$, the set $G_\a$.
In order to define $N$ more formally, 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 $\dot G_\a$ for each
ordinal $\a < \gk$. $N$ can then
be defined inside $V[G]$ 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
\noindent Standard arguments show $N \models {\rm ZF}$.
\begin{lemma}\label{l1}
Let $\l$ be an ordinal. If $x \subseteq \l$,
$x \in N$, then $x \in V[G_\a]$ for some $\a < \k$.
\end{lemma}
\begin{proof}
%We follow the proof of \cite[Lemma 1]{A00}.
We slightly modify the proof of \cite[Lemma 1]{A00}.
Let $\tau$ be a term for $x$ such that
$p \forces ``\tau \subseteq \l$''. Without loss of generality,
by coding if necessary, we can assume $\tau$ mentions only
one term of the form $\dot G_\a$ for some $\a < \k$.
For $q \in \FP$, $q = \la q_\b \mid \b < \k \ra$, define
$q \rest \a = \la q^*_\b \mid \b < \k \ra$ by
$q^*_\b = q_\b$ if $\b < \a$ and $q^*_\b = 0$
(the trivial condition) otherwise.
We can now define a term $\sigma$ by
$q \forces ``\g \in \sigma$'' iff $q$ extends $p$ and
%$q \forces ``\g \in \tau$'', and
$q \rest \a \forces ``\g \in \tau$''. It is clear that
$p \forces ``\sigma \subseteq \tau$''. We show in addition that
$p \forces ``\tau \subseteq \sigma$''.
To see that this is true,
let $q$ extending $p$,
$q = \la q_\b \mid \b < \k \ra$ be such that
$q \forces ``\g \in \tau$'', and assume towards a contradiction that
$q \rest \a \not\forces ``\g \in \tau$''.
Let $r$ extending $q \rest \a$,
$r = \la r_\b \mid \b < \k \ra$ be such that
$r \forces ``\g \not\in \tau$''. If we define
$s = \la s_\b \mid \b < \k \ra$ by $s_\b = r_\b$ for $\b < \a$
and $s_\b = q_\b$ otherwise, then by definition, $s$ extends
$q$ and $s \forces ``\g \in \tau$''.
Let $r_\b$ and $s_\b$ be such that $r_\b$ and $s_\b$ are
incompatible. Since $r_\b, s_\b \in \FP_\b$ and
$\FP_\b = \add(\d_\b, \d_\b^{++})$,
there is an automorphism
$\psi_\b : \FP_\b \to \FP_\b$ generated by a permutation of
$\d_\b$ such that $\psi_\b(r_\b)$ is compatible with $s_\b$.
(Any $t \in \add(\d_\b, \d_\b^{++})$ is a collection of
ordered triples of the form
$\la \xi_0, \xi_1, \xi_2 \ra$,
where $\xi_0 < \d_\b$, $\xi_1 < \d_\b^{++}$, and
$\xi_2 \in \{0, 1\}$. This means that we can let
$t$'s first domain
$\dom_1(t) = \{\xi < \d_\b \mid
\exists \xi_1 < \d_\b^{++} \exists \xi_2 \in \{0, 1\}
[\la \xi, \xi_1, \xi_2 \ra \in t]\}$.
Let $\eta < \d_\b$ be an ordinal greater than
$\max(\sup(\dom_1(s_\b)), \sup(\dom_1(r_\b)))$.
$\eta$ exists since for any condition $t \in \FP_\b$,
$\card{\dom_1(t)} < \d_\b$ and $\d_\b$ is a regular cardinal.
If $\la \rho_i \mid i < \zeta \ra$ enumerates
$\dom_1(r_\b)$ and $\la \rho_i' \mid i < \zeta \ra$ enumerates
the first $\zeta$ ordinals greater than $\eta$, then
$\psi^*_\b : \d_\b \to \d_\b$ given by
$\psi^*_\b(\rho_i) = \rho_i'$,
$\psi^*_\b(\rho_i') = \rho_i$,
and $\psi^*_\b$ is the identity otherwise is the
desired permutation. The automorphism $\psi_\b$ is defined by
applying $\psi^*_\b$ to each element
of a condition's first domain, i.e.,
for $t \in \FP_\gb$, $\psi_\b(t) =
\{\la \psi^*_\b(\xi_0), \xi_1, \xi_2 \ra \mid
\la \xi_0, \xi_1, \xi_2 \ra \in t \}$.)
Thus, if $\pi = \la \pi_\b \mid \b < \k \ra$ is defined by
$\pi_\b = \psi_\b$ if $r_\b$ and $s_\b$ are incompatible and
$\psi_\b$ is as just described
and $\pi_\b$ is the identity otherwise, $\pi$ generates an
automorphism of $\FP$ such that
$\pi(r)$ is compatible with $s$.
Note now that $\pi_\b$ is the identity for $\b < \a$.
Since terms for ground model sets and terms mentioning only
$\dot G_\a$ can be assumed to be invariant under automorphisms
of $\FP$ not changing the value of $G_\a$,
$\pi(r) \forces ``\g \not\in \tau$'',
$\pi(r)$ is compatible with $s$, and
$s \forces ``\g \in \tau$''. This contradiction means that
$q \rest \a \forces ``\g \in \tau$'', i.e.,
$p \forces ``\tau \subseteq \sigma$'', i.e.,
$p \forces ``\tau = \sigma$''.
Since $\sigma$ can clearly be realized in $V[G_\a]$,
$x \in V[G_\a]$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$N \models ``\forall \d < \k
[{\rm DC}_\d]$''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} is
identical to the proof of \cite[Lemma 2]{A00}.
For completeness, we present it here.
Fix $\d < \k$ a cardinal in $N$.
Recall that ${\rm DC}_\d$ is the statement that
whenever $X$ is a set and $R \subseteq
[X]^{< \gd} \times X$ is a relation
such that for all
$\vec y \in [X]^{< \gd}$, there is $z \in X$
such that $\vec y \ R \ z$, then there is a
$\gd$ sequence $\vec Y$ such that for all
$\ga < \gd$, $\vec Y \rest \ga \ R \ Y(\ga)$.
Consequently, working inductively, assume
$p \forces ``\dot X \in \dot N$ is a set,
$\dot R \in \dot N$, $\dot R \subseteq
{[\dot X]}^{< \d} \times \dot X$ is a relation,
$\la \tau_\a \mid \a < \b < \d \ra \in \dot N$ is a sequence of
elements of $\dot X$, and for
$\la \tau_\a \mid \a < \g < \b \ra$,
$\la \tau_\a \mid \a < \g \ra \ \dot R \ \tau_\g$''.
We show how to define $\tau_\b$. Work in $V$. Let
$\eta = \sup(\{\a \mid \exists \g < \b [\dot G_\a$
occurs in $\tau_\g]\})$.
Since each $\tau_\g$ for $\g < \b$ can be assumed to be
an element of ${\cal L}_1$, and since $\k$ is a regular
limit cardinal, $\eta < \k$, so
$\la \tau_\a \mid \a < \b \ra$ can be defined using only
$\dot G_\eta$ and hence is an element of ${\cal L}_1$.
By AC in $V$,
since $\FP$ is an Easton support product of the appropriate
Cohen partial orderings, $\FP$ is $\k$-c.c. Thus,
again by AC in $V$, there is
%let ${\cal B} \subseteq \FP$,
${\cal B}$ with $|{\cal B}| < \k$,
${\cal B} = \{\la p_\rho, \sigma_\rho \ra \mid
\rho < \g^* < \k \}$ such that
%and ${\cal B}$ has the property that
${\cal A} = \{p_\rho \mid \rho < \g^*\}$
forms a maximal antichain of conditions extending $p$ and
$p_\rho \forces ``\la \tau_\a \mid \a < \b \ra \ \dot R
\ \sigma_\rho$''. As before,
$\eta^* = \sup(\{\a \mid \exists \g < \g^*
[\dot G_\a$ occurs in $\sigma_\g]\})$ is such that
$\eta^* < \k$, meaning ${\cal B}$ can be used to define
a term $\tau_\b \in {\cal L}_1$ such that
$p \forces ``\la \tau_\a \mid \a < \b \ra \ \dot R \ \tau_\b$''.
Since $\d < \k$, as before,
$\la \tau_\a \mid \a < \d \ra \in {\cal L}_1$. By the fact
$\la \tau_\a \mid \a < \d \ra$ can be realized in $N$,
$\la \tau_\a \mid \a < \d \ra$ will denote in $N$ a
${\hbox{\rm DC}}_\d$ sequence for $\dot R$ and $\dot X$.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$N \models ``\gk$ is a limit cardinal +
Every successor cardinal is regular''.
\end{lemma}
\begin{proof}
Standard arguments (see \cite{J}) in
conjunction with the fact that $\FP$
is the Easton support product of
$\add(\d_\a, \d_\a^{++})$ where $\a < \k$
show that $V$ and $V[G]$ have the same cardinals
and cofinalities.
%By the definition of $\FP$, $V$ and $V[G]$ have
%the same cardinals and cofinalities.
Therefore, since $V \subseteq N \subseteq V[G]$,
$N$ also has the same cardinals and cofinalities
as do $V$ and $V[G]$. In particular,
$N \models ``\gk$ is a limit cardinal +
Every successor cardinal is regular''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$N \models ``\forall \d < \k[$If $\d$ is regular, then
$2^\d = \d^{++}$, but if $\d$ is singular, then
$2^\d = \d^+]$''.
\end{lemma}
\begin{proof}
%We use ideas from the proof of \cite[Lemma 4]{A00}.
Let $\d < \k$ be a (regular or singular) cardinal.
Let $\gl$ be the least inaccessible cardinal greater than $\d$.
%As in the proof of Lemma \ref{l3},
Write
$\FP = \FQ_\gl \times \FQ^\gl$ and $G = G_\gl \times G^\gl$,
where $\FQ^\gl = \prod_{\gl \le \ga < \gk} \FP_\ga$ and
$G^\gl$ is the projection of $G$ onto $\FQ^\gl$.
Since $V[G^\gl]$ and $V$ contain the same bounded
subsets of $\gl$ and $V[G_\gl] \subseteq N$, it suffices
to show that $V[G_\gl] \models ``2^\d = \d^{++}$ if
$\d$ is regular, but $2^\d = \d^+$ if $\d$ is singular''.
However, once again, standard arguments (see \cite{J})
in conjunction with the fact that $\FQ_\gl$ is the
Easton support product of $\add(\d_\a, \d_\a^{++})$ where
$\a < \l$ yield that
$V[G_\gl] \models ``2^\d = \d^{++}$ if $\d$ is regular,
but $2^\d = \d^+$ if $\d$ is singular''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
We remark that Lemmas \ref{l3}
and \ref{l4} show $N \models ``\k$ is
a strong limit cardinal''.
Also, note that
$N \models \neg {\hbox{\rm AC}}_\k$.
To see this, we follow the remark found
after the proof of \cite[Lemma 4]{A00}.
Define in $N$ for each $\a < \k$ the set
$X_\a = \{x \subseteq \d_\a^{++} \mid x$ codes a $\d_\a^{++}$
sequence of subsets of $\d_\a\}$.\footnote{By the proof
of Lemma \ref{l3}, $\d_\a$ is regular in
$V$, $N$, and $V[G]$.}
Although
$\la X_\a \mid \a < \k \ra \in N$, $
({\prod_{\a < \k} X_\a})^N
= \emptyset$. This follows since an element $y$ of $
({\prod_{\a < \k} X_\a})^N$
may be thought of as a set of ordinals, so by Lemma \ref{l1},
$y \in V[G_\b]$ for some $\b < \k$.
This, however, is impossible, as $\card{\FQ_\b} < \k$,
so a final segment of the sequence
of regular cardinals below $\k$ satisfies GCH in $V[G_\b]$.
\begin{lemma}\label{l5}
$N \models ``$GCH holds at every
(regular or singular) cardinal $\gd \ge \gk$''.
\end{lemma}
\begin{proof}
Since $\card{\FP} = \k$ and $V \models {\rm GCH}$,
$V[G] \models ``$GCH holds at every
(regular or singular) cardinal $\gd \ge \gk$''.
The fact that $V \subseteq N \subseteq V[G]$
then immediately implies that
$N \models ``$For every (regular or singular) cardinal
$\gd \ge \gk$, there is an injection $f : \d^+ \to \wp(\d)$,
but for every (regular or singular) cardinal $\gd \ge \gk$
and every cardinal $\gl > \gd^+$, there is no injection
$f : \gl \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$N \models ``\k$ is supercompact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6} is virtually identical to the proof of
\cite[Lemma 5]{A00}.
As before, for completeness, we include it here.
Fix $\l \ge \k$ and ${\cal U}$
a $\k$-additive, fine, normal ultrafilter
over $P_\k(\l)$ in $V$. Working
in $N$, let ${\cal U}' = \{x \subseteq
{(P_\k(\l))}^N
\mid \exists y \in \U[y \subseteq x]\}$. We show that
$N \models ``{\cal U}'$ is a $\k$-additive,
fine, normal ultrafilter over
${(P_\k(\l))}^N$''.
To see this, fix $x \subseteq
{(P_\k(\l))}^N
$, $x \in N$, and let $\tau$ be a term for $x$ mentioning
only $\dot G_\a$. Contained in the proof of Lemma \ref{l1}
is the fact that $y = \{p \in
{(P_\k(\l))}^V
\mid p \in x\}$
is actually a set in $V[G_\a]$. This follows since
the proof of Lemma \ref{l1} really shows that for a term
$\tau^*$ as just described and an element
$z \in V$, the statement $``z \in \tau^*$'' is decidable in
$V[G_\a]$. Thus, since $|\FQ_\a| < \k$, the
L\'evy-Solovay arguments \cite{LS}
show that in $V[G_\a] \subseteq N$,
either $y$ or $(P_\k(\l))^V - y$ contains a set in $\U$.
%its complement contains a ${\cal U}$ measure 1 set.
Further, if
$N \models ``\la x_\b \mid \b < \g < \k \ra$ is a sequence
such that each $x_\b \in {\cal U}'$'', then
let $\tau_1$ be such that $\tau_1$ denotes
$\la x_\b \mid \b < \g < \k \ra$ and mentions only $\dot G_\a$.
The methods of \cite{LS} yield that
for every $\b < \g$, there is
%a condition $p_\b \in G_\a$ and
a set $y_\b \in {\cal U}$ definable in $V$ such that
$\forces_{\FQ_\a} ``y_\b \subseteq \{p \in
{(P_\k(\l))}^V \mid p \in \dot x_\b\}$''.
%and $y_\b \in {\cal U}$.
%it is possible to define in $V[G_\a]$ a sequence
%$\la y_\b \mid \b < \g < \k \ra$ such that for each $\b < \g$,
%The methods of \cite{LS} then imply that
%$\bigcap_{\b < \g} y_\b$ contains a
%${\cal U}$ measure 1 set in $V[G_\a]$, so
%$\bigcap_{\b < \g} x_\b$ contains a ${\cal U}$
%measure 1 set in $N$.
Since $y^* = \bigcap_{\b < \g} y_\b \in {\cal U}$,
$N \models ``\exists y \in \U[y \subseteq
\bigcap_{\b < \g} x_\b]$''.
%so because
%$V[G_\a] \subseteq N$, this same statement is true in $N$.
Finally, if $N \models ``f :
{(P_\k(\l))}^N
\to \l$ is a choice function'', then if $\dot f$
denotes $f$ and mentions only $\dot G_\a$,
it is possible to define in $V[G_\a] \subseteq N$ the function
$g = f \rest {(P_\k(\l))}^V$. Once more, the results of
\cite{LS} show that for some $x \in {\cal U}$,
$V[G_\a] \models ``g$ is constant on $x$''. Thus,
$N \models ``{\cal U}'$ is a $\k$-additive,
fine, normal ultrafilter over
${(P_\k(\l))}^N$''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l1} -- \ref{l6} and the intervening remarks
complete the proof of Theorem \ref{t2}.
\end{proof}
Having completed the proof of Theorem \ref{t2},
we turn now to the proof of Theorem \ref{t3}.
We will be constructing a symmetric model of
``ZF + $\neg {\rm AC}_\go$'' in which
$\k$ is supercompact, $\k$ is a limit cardinal,
every successor cardinal is regular,
GCH fails at every limit cardinal $\gd \le \gk$, and
GCH holds at all cardinals above $\gk$.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
We define a partial ordering $\ov \FQ$ such that
$V^{\ov \FQ} = \ov V \models ``$ZFC + $\k$ is supercompact +
$2^\k = \k^{++}$ + $2^\d = \d^+$ for every cardinal
$\d \ge \k^+$ + There is a club $C \subseteq \k$
composed of inaccessible cardinals and their limits with
$2^\d = 2^{\d^+} = \d^{++}$ for every $\gd \in C$''.
To obtain $\ov \FQ$, let $\FQ_1$ be Laver's partial ordering
of \cite{L} which makes $\k$'s supercompactness indestructible
under $\k$-directed closed forcing.
Since $\FQ_1$ may be defined so that $\card{\FQ_1} = \k$,
it is then the case that
$V^{\FQ_1 \ast \dot \add(\k, \k^{++})} = V_2 \models ``$ZFC +
$\k$ is supercompact + $2^\k = \k^{++}$ + $2^\gd = \gd^+$ for
every cardinal $\gd \ge \k^+$''.
Let $\FQ_3 \in V_2$ be Radin forcing defined over
$\gk$ using one repeat point (see either \cite{G10} or
\cite{R} for the precise definition of $\FQ_3$).
Standard facts about Radin forcing (see \cite{A91},
\cite{G10}, and \cite{R}) then show that
$V^{\FQ_3}_2 = V^{\FQ_1 \ast \dot \add(\k, \k^{++}) \ast \dot \FQ_3} =
\ov V \models
``$ZFC + $\k$ is supercompact +
$2^\k = \k^{++}$ + $2^\d = \d^+$ for every cardinal
$\d \ge \k^+$ + There is a club $C \subseteq \k$
composed of inaccessible cardinals and their limits with
$2^\d = 2^{\d^+} = \d^{++}$ for every $\gd \in C$''.
With an abuse of notation, we now let $\ov V = V$.
Let $\la \gk_i \mid i < \gk \ra \in V$ be the
continuous, increasing enumeration of $C \cup \{\go\}$.
For $i < \gk$, let $\FP_i = {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$.
The partial ordering $\FP$ with which we force is then the
Easton support product $\FP = \prod_{i < \k} \FP_i$.
%We now define $\FP = \prod_{i < \k} \FP_i$ with Easton support.
%\footnote{As our proof will show, the suport used
%can actually be arbitrary.}.
Let $G$ be $V$-generic over $\FP$.
$V[G]$, being a model of AC, is once more not our
desired model $N$. In order to define $N$, we
first note that as before, by the Product Lemma,
for $i < \gk$, $G_i$, the projection of $G$ onto $\FP_i$,
is $V$-generic over $\FP_i$.
Again by the Product Lemma, $G_I = \prod_{i \in I} G_i$
is $V$-generic over $\FP_I = \prod_{i \in I} \FP_i$.
We can now intuitively describe $N$ as the least model of
ZF extending $V$ which contains, for each finite set of
ordinals $I \subseteq \gk$, the set $G_I$.
In order to define $N$ more formally, 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 $\dot G_I$ for each
finite set of ordinals $I \subseteq \gk$. $N$ can then
be defined inside $V[G]$ 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
\noindent As in the proof of Theorem \ref{t2},
standard arguments show $N \models {\rm ZF}$.\footnote{Although
defining $N$ using $G_i$ for every $i < \k$
is equivalent to our presentation, it is not
as useful for the arguments we are about to give.}
%this definition of $N$
%using G_I$ for every finite $I \subseteq \k$
\begin{lemma}\label{l7}
Let $\l$ be an ordinal. If $x \subseteq \l$,
$x \in N$, then $x \in V[G_I]$ %V[\prod_{i \in I} G_i]$
for some finite set of ordinals $I \subseteq \gk$.
\end{lemma}
\begin{proof}
Suppose $i < \gk$.
It is a standard fact (see, e.g., \cite[Lemma 5.2]{B78})
that since $\FP_i$ is a version of the L\'evy collapse,
for any $p, q \in \FP_i$, there is an automorphism
$\pi_i : \FP_i \to \FP_i$ such that $\pi_i(p)$ is
compatible with $q$. The proof of Lemma \ref{l7} is now
essentially the same as the proof of Lemma \ref{l1}, with
each occurrence of ``$\ga$'' in Lemma \ref{l1} replaced
by an occurrence of ``$I$''.
This completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
%If $N \models ``\gd < \gk$ is a successor cardinal'', then
%either $\gd = \gk_i$, $\gd = \gk^+_i$, or $\gd = \gk^{++}_i$
%for some $i < \gk$.
$N \models ``$Every successor cardinal is regular''.
\end{lemma}
\begin{proof}
Suppose first that $N \models ``\gd > \gk$ is a successor cardinal''.
Since $\FP = \prod_{i < \k} \FP_i$ is an Easton support product,
$\FP$ is $\gk$-c.c. This means that $V$ and $V[G]$ have the same
cardinals and cofinalities at and above $\gk$. Therefore, since
$V \subseteq N \subseteq V[G]$, $V$, $N$, and $V[G]$ all have
the same cardinals and cofinalities at and above $\gk$. In particular,
$N \models ``\gd$ is regular''.
Suppose next that $N \models ``\gd < \gk$ is a
(successor or limit) cardinal''.
We claim that
either $\gd = \gk_i$,
$\gd = (\gk^+_i)^V$, or
$\gd = (\gk^{++}_i)^V$
for some $i < \gk$.
To see this, since $C$ is club in $\gk$, we can let
$k < \gk$ be such that $\gk_{k + 1}$ is the least
member of $C$ greater than $\gd$.
If the claim is false, then because
$\gd \neq \gk_k$, $\gd \neq (\gk^+_k)^V$, and
$\gd \neq (\gk^{++}_k)^V$, $\gd \in ((\gk^{++}_k)^V, \gk_{k + 1})$.
However, since $G_k$ is $V$-generic over
${\rm Coll}(\gk^{++}_k, {<} \gk_{k + 1})$,
$V[G_k] \models ``\gd$ is not a cardinal''.
Consequently, because $V[G_k] \subseteq N$,
$N \models ``\gd$ is not a cardinal'', a contradiction
to the assumption that
$\gd \neq \gk_k$, $\gd \neq (\gk^+_k)^V$, and
$\gd \neq (\gk^{++}_k)^V$.
%The proof of our claim will thus be complete
%once we have shown
We next claim that for any $i < \gk$,
$N \models ``\gk_i$, $(\gk^+_i)^V$, and $(\gk^{++}_i)^V$
are all cardinals''. However, since any collapse map $f$
would have to be coded by a set of ordinals, if this were false,
then by Lemma \ref{l7}, there would have to be some
finite set of ordinals $I \subseteq \gk$ such that
$f \in V[G_I] = V[\prod_{j \in I} G_j]$.
Because $\prod_{j \in I} G_j$ is $V$-generic over
$\prod_{j \in I} \FP_j =
\prod_{j \in I} {\rm Coll}(\gk^{++}_j, {<} \gk_{j + 1})$
and $I$ is finite, this is impossible.
We now know that if
$N \models ``\gd < \gk$ is a successor cardinal'',
then there must be some $i < \gk$ such that
either $\gd = \gk_i$, $\gd = (\gk^+_i)^V$,
or $\gd = (\gk^{++}_i)^V$.
Assume that $\gd = \gk_i$.
It must be the case that $i$ is a successor ordinal.
This is since
if $i$ were a limit ordinal, then
$N \models ``\gk_i = \sup_{k < i} \gk_k$ and $\gk_k$
for $k < i$ is a cardinal'', i.e.,
$N \models ``\gk_i$ is a limit cardinal''.
Consequently, because $i$ is a successor ordinal,
$\gk_i$ is a successor member of the Radin generic
club $C$. This means that
$V \models ``\gk_i$ is inaccessible'', so in particular,
$V \models ``\gk_i$ is a regular cardinal''. If
$N \models ``\gk_i$ is singular'', then let
$S \subseteq \gk_i$, $S \in N$ be a witness to this fact.
Again by Lemma \ref{l7}, there must be some
finite set of ordinals $I \subseteq \gk$ such that
$f \in V[G_I] = V[\prod_{j \in I} G_j]$. Once more,
because $\prod_{j \in I} G_j$ is $V$-generic over
$\prod_{j \in I} \FP_j =
\prod_{j \in I} {\rm Coll}(\gk^{++}_j, {<} \gk_{j + 1})$
and $I$ is finite, this is impossible.
Hence, $N \models ``\gd$ is a regular cardinal''.
Assume finally that
either $\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
Clearly, since $V \models {\rm ZFC}$, it is also true that
$V \models ``\gk^+_i$ and $\gk^{++}_i$ are regular cardinals''.
The same contradiction as obtained in the preceding paragraph
again yields that
$N \models ``\gd$ is a regular cardinal''.
This completes the proof of Lemma \ref{l8}.
\end{proof}
As has just been noted in the the proof of Lemma \ref{l8},
$N \models ``\gd < \gk$ is a cardinal'' iff either
$\gd = \gk_i$, $\gd = (\gk^+_i)^V$, or $\gd = (\gk^{++}_i)^V$
for some $i < \gk$.
%if $i < \gk$, then $N \models ``\gk_i$ is a cardinal''.
Therefore, since $\gk = \sup_{i < \gk} \gk_i$,
$N \models ``\gk$ is a limit cardinal''.
\begin{lemma}\label{l9}
$N \models ``$GCH fails at every limit cardinal $\gd \le \gk$''.
\end{lemma}
\begin{proof}
Suppose first that $\gd = \gk$.
As noted in the proof of Lemma \ref{l8},
$V$, $N$, and $V[G]$ all have the same cardinals
and cofinalities at and above $\gk$.
In addition, $V \models ``2^\gk = \gk^{++}$''.
%Further, since $\card{\FP} = \gk$ and
%$V \models ``2^\gk = \gk^{++}$'', $V[G] \models ``2^\gk = \gk^{++}$''.
Therefore, because $V \subseteq N$, %\subseteq V[G]$,
$N \models ``$There is an injection
$f : \gk^{++} \to \wp(\gk)$''.
Suppose now that $\gd < \gk$.
By the second paragraph of the proof of Lemma \ref{l8},
there must be some $i < \gk$ such that either
$\gd = \gk_i$, $\gd = (\gk^+_i)^V$, or $\gd = (\gk^{++}_i)^V$.
Since $V \subseteq N$, it cannot be the case that
either $\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
This means we can let $i < \gk$ be such that $\gd = \gk_i$.
As $\gd \in C$, $V \models ``2^\gd = \gd^{++}$''.
Consequently, since
$V \subseteq N$ and the third paragraph of the proof
of Lemma \ref{l8} implies that
both $\gd = (\gk^+_i)^V$ and $\gd = (\gk^{++}_i)^V$
remain cardinals in $N$,
$N \models ``$There is an injection $f : \gd^{++} \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l9}.
\end{proof}
\begin{lemma}\label{l10}
$N \models ``$GCH holds at every (regular or singular)
cardinal $\gd > \gk$''.
\end{lemma}
\begin{proof}
%As has been observed,
Since $V$, $N$, and $V[G]$ all have the same cardinals
and cofinalities at and above $\gk$,
$\card{\FP} = \gk$, and
$V \models ``2^\gd = \gd^{+}$
for every cardinal $\gd \ge \gk^+$'',
$V[G] \models ``2^\gd = \gd^{+}$
for every cardinal $\gd \ge \gk^+$''.
Therefore, again because $V \subseteq N \subseteq V[G]$,
$N \models ``$For every (regular or singular)
cardinal $\gd > \gk$, there is an injection
$f : \gd^+ \to \wp(\gd)$, and for every cardinal
$\gl > \gd^{+}$, there is no injection
$f : \gl \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l10}.
\end{proof}
\begin{lemma}\label{l11}
$N \models \neg AC_\go$.
\end{lemma}
\begin{proof}
We follow the remarks given after the proofs of
\cite[Lemma 4]{A00} and Lemma \ref{l4}, making
the appropriate modifications in proof.
Define in $N$ for each $n < \go$ the set
$X_n = \{x \subseteq (\gk^{++}_n)^V \mid x$
codes a well-ordering of $(\gk^{+ 3}_n)^V$ of
order type $(\gk^{++}_n)^V\}$.\footnote{Since
$(\gk^{++}_n)^V = (\gk^{++}_n)^N$, it is
also possible to define $X_n$ in $N$ as
$X_n = \{x \subseteq \gk^{++}_n \mid x$
codes a well-ordering of $(\gk^{+ 3}_n)^V$ of
order type $\gk^{++}_n\}$.}
Although each $X_n \neq \emptyset$ and
$\la X_n \mid n < \go \ra \in N$,
$(\prod_{n < \go} X_n)^N = \emptyset$.
This follows since an element $y$ of $(\prod_{n < \go} X_n)^N$
is a set of ordinals, so by Lemma \ref{l7},
$y \in V[G_I] = V[\prod_{i \in I} G_i]$ for some finite set of
ordinals $I \subseteq \gk$.
Let $m$ be the maximum integer which is an element of $I$.
Write $I = I_0 \cup I_1$, where
$I_0 = \{i \in I \mid i \le m\}$ and $I_1 = I - I_0 =
\{i \in I \mid i > m\} = \{i \in I \mid i \ge \go\}$.
By the closure properties of the L\'evy collapse,
each member of the sequence $\la (\gk^{+ 3}_n)^V \mid n < \go \ra$
remains a cardinal in $V[\prod_{i \in I_1} G_i]$. Since
$\prod_{i \in I_0} \FP_i =
\prod_{i \in I_0} {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$
and $I_0$ is finite, there is some $j < \go$ such that
for all $\ell \ge j$, $\card{\prod_{i \in I_0} \FP_i} < \gk_\ell$.
Thus, a final segment of the sequence
$\la (\gk^{+ 3}_n)^V \mid n < \go \ra$ remains a sequence
of cardinals in $V[\prod_{i \in I_1} G_i][\prod_{i \in I_0} G_i] =
V[G_I]$, which is impossible.
This completes the proof of Lemma \ref{l11}.
\end{proof}
The proof that $N \models ``\gk$ is supercompact''
is the same as in Lemma \ref{l6}, with each occurrence
of ``$\ga$'' replaced by an occurrence of ``$I$''.
Lemmas \ref{l7} -- \ref{l11} and the intervening remarks
consequently complete the proof of Theorem \ref{t3}.
\end{proof}
The proof of Lemma \ref{l9} indicates that for any $i < \gk$,
GCH fails at $\gk_i$.
Since it is possible to show that
any $\gk_i$ for $i$ a successor ordinal is a
successor cardinal in $N$ (its predecessor in $N$
must be $\gk^{++}_{i - 1}$), there are many successor
cardinals below $\gk$ violating GCH.
We remark that by slightly changing the definition of $N$,
it is possible to obtain a model of ZF + $\neg{\rm AC}_\go$
satisfying the conclusions of Theorem \ref{t3} in which GCH
holds at every successor cardinal $\gd < \gk$. Specifically,
we have the following theorem.
\begin{theorem}\label{t4}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\neg {AC}_\go$ +
$\gk$ is a limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
GCH fails at every limit
cardinal $\gd \le \gk$ +
GCH holds at every (regular or singular)
cardinal $\gd > \gk$, as well as at
every successor cardinal $\gd < \gk$''.
\end{theorem}
\begin{sketch}
We suppose as in the proof of Theorem \ref{t3} that
$V \models ``$ZFC + $\gk$ is supercompact +
$2^\gk = \gk^{++}$ + $2^\gd = \gd^+$ for every
cardinal $\gd \ge \gk^+$ + There is a club
$C \subseteq \gk$ composed of inaccessible cardinals
and their limits with $2^\gd = 2^{\gd^+} = \gd^{++}$ for
every $\gd \in C$''. Once again, let
$\la \gk_i \mid i < \gk \ra \in V$ be the continuous,
increasing enumeration of $C \cup \{\go\}$.
Change the definition of
$\FP_i$ so that $\FP_i = {\rm Coll}(\gk_i, {<} \gk_{i + 1})$
if $i < \gk$ is either $0$ or a successor ordinal, but
$\FP_i = {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$ if
$i < \gk$ is a limit ordinal.
The remainder of the definition of $\FP$ is as
before, i.e., $\FP = \prod_{i < \gk} \FP_i$ with Easton support.
Let $G$ be $V$-generic over $\FP$, and for each
$i < \gk$, let $G_i$ be the projection of
$G$ onto $\FP_i$.
$N$ is then constructed
as in the proof of Theorem \ref{t3}.
The same argument as given in the first paragraph of the
proof of Lemma \ref{l8} shows that
$N \models ``$Every successor cardinal $\gd > \gk$ is regular''.
The proofs of the natural analogues of Lemmas \ref{l6},
\ref{l7}, and \ref{l10} are as before.
The proof of the natural analogue of Lemma \ref{l11}
is as before, with the definition of $X_n$ changed to
$X_n = \{x \subseteq \gk_n \mid x$ codes a well-ordering of
$(\gk^+_n)^V$ of order type $\gk_n\}$.
This shows that
$N \models ``$ZF + $\neg {\rm AC}_\go$ + GCH holds at every
(regular or singular) cardinal $\gd > \gk$ + $\gk$
is supercompact''.
%The natural analogue of the argument found in the second
%paragraph of the proof of Lemma \ref{l8} shows that
%for any $i < \gk$, $N \models ``\gk_i$ is a cardinal''.
The natural analogue of the argument found in the second
paragraph of the proof of Lemma \ref{l8} shows that if
$N \models ``\gd < \gk$ is a (successor or limit) cardinal'', then
either $\gd = \gk_i$ for some $i < \gk$, or
for some limit ordinal $i < \gk$, either
$\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
(As in the proof of Lemma \ref{l8}, let $k < \gk$
be such that $\gk_{k + 1}$ is the least member of
$C$ greater than $\gd$. If this is false, then
either $\gd \in ((\gk^{++}_k)^V, \gk_{k + 1})$ if $k$ is a
limit ordinal, or $\gd \in (\gk_k, \gk_{k + 1})$ if $k$ is
either a successor ordinal or $0$. In each case, in
$V[G_k] \subseteq N$
%(a submodel of $N$)
and $N$, $\gd$ is not a cardinal.)
The same argument as given in the proof of Lemma \ref{l8}
now shows that $\gk_i$ for any $i < \gk$ and
both $\gk^+_i$ and $\gk^{++}_i$ for $i < \gk$
a limit ordinal remain cardinals in $N$.
From this, we may infer as in the proofs of
Lemmas \ref{l8} and \ref{l9} and the intervening remark that
$N \models ``\gk$ is a limit cardinal + Every successor cardinal
$\gd < \gk$ is regular + If
$i < \gk$ is a limit ordinal, then
$\gk_i$ is a limit cardinal + GCH fails at every
limit cardinal $\gd \le \gk$''.
It remains to show that
$N \models ``$GCH holds at every successor cardinal $\gd < \gk$''.
To see this, suppose first that $\gd = \gk_i$
for some $i < \gk$. As we have already observed, $i$ must be a
successor ordinal. As a consequence, $G_i$ must be $V$-generic over
${\rm Coll}(\gk_i, {<} \gk_{i + 1})$, so since
$V[G_i] \subseteq N$ and $N \models ``\gk_{i + 1}$
is a cardinal'',
$N \models ``\gk_{i + 1} = \gk^+_i = \gd^+$, and
there is an injection $f : \gd^+ \to \wp(\gd)$''.
Because $G_{i + 1}$ is $V$-generic over
${\rm Coll}(\gk_{i + 1}, {<} \gk_{i + 2})$,
$V[G_{i + 1}] \subseteq N$, and
$N \models ``\gk_{i + 2}$ is a cardinal'',
$N \models ``\gk_{i + 2} =
\gk^+_{i + 1} = \gd^{++}$''.
Assume now that $N \models ``$There is an injection
$f : \gd^{++} \to \wp(\gd)$'', i.e., that
$N \models ``$There is an injection
$f : \gk_{i + 2} \to \wp(\gk_i)$''.
Since $N \subseteq V[G]$, it must therefore be
the case that
$V[G] \models ``$There is an injection
$f : \gk_{i + 2} \to \wp(\gk_i)$''.
By the properties of the L\'evy collapse, however,
this is impossible.
%, then we can let
%$A = \la A_j \mid j < \gk_{i + 2} \ra \in N$ be such that
%each $A_j \subseteq \gk_i$.
%Since $A$ may be coded as a set of ordinals,
%there must be some finite
Suppose finally that either $\gd = (\gk^+_i)^V$ or
$\gd = (\gk^{++}_i)^V$ where $i < \gk$ is a limit ordinal.
If $\gd = (\gk^+_i)^V$, then since
$V \models ``2^{\gk^+_i} = \gk^{++}_i$'' and
$(\gk^{++}_i)^V$ remains a cardinal in $N$,
$(\gk^{++}_i)^V = (\gd^+)^N$, and
$N \models ``$There is an injection $f : \gd^+ \to \wp(\gd)$''.
Because $G_i$ is $V$-generic over
${\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$,
$V[G_i] \subseteq N$, and
$N \models ``\gk_{i + 1}$ is a cardinal'',
$N \models ``\gk_{i + 1} = ((\gk^{++}_i)^V)^+ = \gd^{++}$''.
If $\gd = (\gk^{++}_i)^V$, then since $G_i$ is $V$-generic over
${\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$,
$V[G_i] \models ``((\gk^{++}_i)^V)^+ = \gk_{i + 1}$ and
$2^{(\gk^{++}_i)^V} = \gk_{i + 1}$'', i.e.,
$V[G_i] \models ``2^\gd = \gd^+$''.
Because $V[G_i] \subseteq N$ and
$N \models ``\gk_{i + 1}$ is a cardinal'',
$N \models ``$There is an injection $f : \gd^+ \to \wp(\gd)$''.
As $G_{i + 1}$ is $V$-generic over
${\rm Coll}(\gk_{i + 1}, {<} \gk_{i + 2})$,
$V[G_{i + 1}] \subseteq N$, and
$N \models ``\gk_{i + 2}$ is a cardinal'',
$N \models ``\gk_{i + 2} = \gk^+_{i + 1} = \gd^{++}$''.
In either case, if $N \models ``$There is an injection
$f : \gd^{++} \to \wp(\gd)$'', then since
$N \subseteq V[G]$, we obtain a contradiction as
before to the properties of the L\'evy collapse.
This completes the sketch of the proof of Theorem \ref{t4}.
\end{sketch}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we note that it is possible to modify
%we make several remarks.
%We begin by noting that it is possible to modify
Theorem \ref{t2} and its proof so that the behavior of
the continuum function at regular cardinals below
$\gk$ is given by a fixed ground model Easton function.
We leave it to readers to fill in the details.
However, the methods of this paper do not seem to
allow us to use a ground model Easton function
%defined on all cardinals below $\gk$
to control the behavior of the continuum function
on all cardinals at and below $\gk$ (in either the strong
sense of Theorem \ref{t2} or the weaker sense of
Theorems \ref{t3} and \ref{t4}) while having
GCH hold above $\gk$.
We ask if this is possible.
In particular, is is possible to construct
a model analogous to the ones for either
Theorem \ref{t2} or Theorem \ref{t3}
in which GCH fails everywhere below $\gk$?
Since AC fails completely in the models
witnessing the conclusions of Theorems \ref{t3} and \ref{t4},
we ask if it is possible to construct analogues
of these models in which some weak version of AC holds.
More generally, we finish by asking if it is possible
to prove analogues of Theorems \ref{t2} -- \ref{t4},
or the generalizations to which we have just alluded,
in the context of the full Axiom of Choice.
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\end{document}
\begin{lemma}\label{l3}
$N \models ``\k$ is a limit cardinal''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3} is
identical to the proof of \cite[Lemma 3]{A00}.
Once again, for completeness, we present it here.
Since $\k$ is measurable in $V$, there is a normal measure
$\mu \in V$ over $\k$ such that
$\{\d < \k \mid \FQ_\d$ is an Easton support product and $\d$ is
Mahlo$\} \in \mu$. For any such $\d$, write
$\FP = \FQ_\d \times \FQ^\d$ and $G = G_\d \times G^\d$, where
$\FQ^\d = \prod_{\a \le \d} \FP_\a$ and $G^\d$
is the projection of $G$ onto $\FQ^\d$. By the definition of
each $\FP_\a$,
$V[G^\d] \models ``\d$ is Mahlo and $\FQ_\d$ is an
Easton support product''. Thus,
$V[G^\d] \models ``\FQ_\d$ is $\d$-c.c.'', so
$V[G^\d][G_\d] = V[G] \models ``\d$ is a cardinal''.
As $V \subseteq N \subseteq V[G]$,
$N \models ``\d$ is a cardinal'', so because there are
unboundedly many in $\k$ such cardinals,
$N \models ``\k$ is a limit cardinal''.
This completes the proof of Lemma \ref{l3}.
\end{proof}