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\title{Expanding $\gk$'s Power Set
in its Ultrapowers
\thanks{2000 Mathematics Subject Classifications:
Primary 03E35, 03E55; Secondary 03E05}
\thanks{Keywords: Measurable cardinal, GCH, ultrapower,
elementary embedding, fast function forcing,
Easton support iteration}}
\author{Arthur W.~Apter
\thanks{The author's research
was partially supported
by PSC-CUNY Grant 61449-00-30.}
\thanks{The author wishes to thank the
referee for helpful suggestions which
have been incorporated into this
version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{May 7, 2001\\
(revised June 15, 2001)}
\begin{document}
\maketitle
\begin{abstract}
Starting from a measurable cardinal $\gk$,
we force over $V = L[\mu]$ and construct a model
$\ov V$ in which
$\gk$ is measurable, $2^\gk = \gk^+$,
every elementary embedding witnessing
$\gk$'s measurability is the lift of the
unique $L[\mu]$-elementary embedding
$j : V \to N$ witnessing $\gk$'s measurability,
and all $\ov V$-ultrapowers
witnessing $\gk$'s measurability have the
same cardinals and cofinalities as $N$.
$\ov V$ has the further property that
for any $N$-cardinal $\gd < j(\gk)$, there
are $2^{2^\gk}$ normal measures $\nu$
over $\gk$ such that for
$j_\nu : \ov V \to M_\nu$ the
associated ultrapower embedding,
$M_\nu \models ``2^\gk \ge \gd$''.
\end{abstract}
\baselineskip=24pt
\section{Introduction}
One of the original motivating
factors in the study of measurable
cardinals was the hope that they
could resolve the Generalized
Continuum Hypothesis (GCH).
That this is not the case was shown
by L\'evy and Solovay in \cite{LS},
who demonstrated, among other things,
the relative consistency with a measurable
cardinal $\gk$ of the size of the
%continuum $\mathfrak c$
set of real numbers $\Re$
being virtually arbitrary, subject only to
the restriction that
%${\mathfrak c} < \gk$.
$|\Re| < \gk$.
The work of L\'evy and Solovay, however,
does not imply that there is no relationship
whatsoever between the existence of a
measurable cardinal $\gk$ and GCH.
It is an easy consequence of measurability
that if GCH fails at $\gk$, then GCH must
fail unboundedly often below $\gk$.
Thus, the existence of a measurable cardinal $\gk$
profoundly affects the continuum function
in the sense that it is impossible for $\gk$
to witness the first failure of GCH.
As has turned out to be the case,
the converse of the fact mentioned
above is false.
It is another easy consequence of
measurability
(more specifically, \L os' theorem)
that for $\gk$
a measurable cardinal and
$j : V \to M$ an ultrapower
embedding via any normal measure
$\mu$ witnessing
$\gk$'s measurability,
$M \models ``2^\gk > \gk^+$'' iff
$\{\gd < \gk : 2^\gd > \gd^+\} \in \mu$.
This does not, however, necessarily imply
that $\gk$ itself must witness a violation
of GCH in $V$.
Indeed, Levinski has shown in
\cite{Levinski} the relative consistency,
starting from a measurable cardinal,
of $\gk$ satisfying GCH together with
$2^\gd = \gd^{++}$ for every regular cardinal
$\gd < \gk$.
The purpose of this note is to
expand upon the ideas of \cite{Levinski}
and provide a way of preserving GCH
at a measurable cardinal $\gk$ while
violating GCH at $\gk$ essentially at
will in different ultrapowers.
We will show, roughly speaking, that
starting from a measurable cardinal,
it is possible to force over $L[\mu]$
and obtain a model $\ov V$
in which the measurable cardinal
$\gk$ carries the largest number of normal
measures possible, $2^\gk = \gk^+$ in $\ov V$,
every elementary embedding
$\ov j : \ov V \to M$
witnessing $\gk$'s measurability lifts $j$
where $j : L[\mu] \to L[\mu]^\gk/\mu$ is
the unique elementary embedding witnessing
$\gk$'s measurability,
yet there are always the maximal number of
ultrapowers, all of which have the same
cardinals and cofinalities as
$L[\mu]^\gk/\mu$, in which $2^\gk$ is at least
as large as any $L[\mu]^\gk/\mu$-cardinal
$\gd < j(\gk)$.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Assume
$V = L[\mu] \models ``\gk$ is measurable +
$\gl > \gk^+$ is a regular cardinal +
$j : L[\mu] \to L[\mu]^\gk/\mu$ is
the unique elementary embedding witnessing
$\gk$'s measurability''. There is then a
cardinal and cofinality preserving partial ordering
$\FP \in V$ such that for $\ov V = V^\FP$,
$\ov V \models ``$ZFC + $2^\gk = \gk^+$ +
$2^{2^\gk} = 2^{\gk^+} = \gl$ +
If $M$ is an ultrapower via a normal measure
over $\gk$ and
$\ov j : \ov V \to M$ is the associated
elementary embedding,
$\ov j$ lifts $j$
and $M$ has the same cardinals and cofinalities as
$L[\mu]^\gk/\mu$
+ For every
$L[\mu]^\gk/\mu$-cardinal
$\gd < j(\gk)$, there are
$2^{2^\gk}$ normal measures $\nu$ over
$\gk$ and associated ultrapowers $M_\nu$ and
elementary embeddings
$j_\nu : \ov V \to M_\nu$ such that
$M_\nu \models ``2^\gk \ge \gd$''.
\end{theorem}
Our notation and terminology are
largely standard.
We do specifically mention that
as is customary, we always
identify an ultrapower with its
transitive collapse.
For $\ga < \gb$ ordinals,
$[\ga, \gb]$, $[\ga, \gb)$, $(\ga, \gb]$, and
$(\ga, \gb)$ are as in standard interval notation.
When forcing $q \ge p$ means that
$q$ is stronger than $p$.
A forcing partial ordering $\FP$
is $\gk$-closed if every increasing
chain of length $\gk$ has an upper bound.
$\FP$ is $\gk$-directed closed if
every directed subset of $\FP$ of
length less than $\gk$ has an upper bound.
If $G \subseteq \FP$ is $V$-generic,
$V[G]$ and $V^\FP$ are used interchangeably
to denote the generic extension obtained by
forcing with $\FP$.
For $\gk$ a regular cardinal and
$\gl$ an ordinal, $\add(\gk, \gl)$
denotes the standard Cohen partial
ordering for adding $\gl$ subsets of $\gk$.
The elementary embedding
$j : V \to N$ lifts to
$\ov j : V[G] \to N[\ov j(G)]$
(or equivalently, $\ov j$ is a lift of $j$) if
$j$ agrees with $\ov j$ on the common domain $V$.
\section{The Proof of Theorem \ref{t1}}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let
$V = L[\mu] \models ``\gk$ is measurable +
$\gl > \gk^+$ is a regular cardinal +
$j : L[\mu] \to L[\mu]^\gk/\mu$ is
the unique elementary embedding witnessing
$\gk$'s measurability''.
We define the desired partial ordering
$\FP = \FP^0 \ast \dot \FP^1
\ast \dot \FP^2 \ast \dot \FP^3 \ast \dot \FP^4$, where
$\FP^0 \ast \dot \FP^1 =
\add(\go, 1) \ast \dot \add(\gk^+, \gl)$.
Working in
$V^{\FP^0 \ast \dot \FP^1} = V_1$,
we define $\FP^2$ as Woodin's notion
of forcing which adds a fast function
to $\gk$.
Although the definition of
$\FP^2$ is given on page 106 of
\cite{H4}, for the convenience
of readers, we repeat it here as well.
Specifically, $\FP^2$, which is
ordered by inclusion, is defined
as the set of all partial functions $g$ from
$\gk$ to $\gk$ satisfying the
following properties:
\begin{enumerate}
\item $\card{\dom(g)} < \gk$.
\item $\dom(g)$ is composed of inaccessible cardinals.
\item For every $\gg \in \dom(g)$, $g''\gg \subseteq \gg$.
\item For every $\gg \in \dom(g)$, $\card{\dom(g)
\cap \gg} < \gg$.
\end{enumerate}
Working in $V^{\FP^2}_1 = V_2$,
let $f$ be the generic fast function
added by $\FP^2$.
We define
$\FP^3$ as the Easton support iteration
of length $\gk$ which, for every
$\gg \in \dom(f)$, forces with
$\add(\gg, f(\gg))$.
We can now complete the definition of $\FP$
by forcing over $V^{\FP^3}_2 = V_3$ with
$\FP^4 = \add(\gk, \gk^+)$.
Before continuing with the proof of
Theorem \ref{t1}, we take this opportunity
to give a brief explanation of the roles
the various components of our iteration
play in the definition of $\FP$.
$\FP^0$ is used so that Hamkins' gap
forcing results of \cite{H1}, \cite{H2},
and \cite{H3} can be applied to deduce that
every elementary embedding in the final
generic extension lifts a ground model
elementary embedding.
$\FP^1$ then increases the power set of
$\gk^+$ to the value, $\gl$, which will
end up being the total number of normal
measures $\gk$ carries in our ultimate model.
After we have forced with $\FP^1$,
we are now ready to begin the forcing
which allows us to expand $\gk$'s
power set in its different ultrapowers yet
preserve GCH at $\gk$ in the underlying universe.
$\FP^2$ introduces
the fast function $f$, which indicates
the number of subsets added to each
cardinal in the preparatory iteration $\FP^3$.
Since a key feature of a fast function is that,
in Hamkins' terminology of \cite{H4}, its
value is highly mutable in an ultrapower,
the power set of $\gk$ in an appropriately
chosen ultrapower can be made as large as desired
using the generic subsets of $\gk$ introduced by
$\FP^4$ in tandem with a coding given by $f$.
By the nature of their definitions, forcing with
$\FP^2 \ast \dot \FP^3 \ast \dot \FP^4$ over
$V_1$ won't destroy GCH at $\gk$.
%the final forcing
%component $\FP^4$, which doesn't
%destroy GCH, is employed so that
%by the correct coding,
\begin{lemma}\label{l2}
$V^\FP \models ``2^\gk = \gk^+$ $+$
$2^{2^\gk} = 2^{\gk^+} = \gl$''.
\end{lemma}
\begin{proof}
Since $\card{\FP^0} = \go < \gk$,
the L\'evy-Solovay results
\cite{LS} in tandem with basic
cardinal arithmetic imply that
$V^{\FP^0} \models ``$GCH + $\gk$ is measurable''.
Because
$\forces_{\FP^0} ``\dot \FP^1$ is
$\gk^+$-closed'',
the definition of $\FP^1$ then implies that
$V_1 \models ``$GCH holds for all cardinals
up to and including $\gk$ + $\gk$ is measurable
+ $2^{2^\gk} =
2^{\gk^+} = \gl$''.
As
$V_1 \models ``\card{\FP^2} = \gk$'',
$V_2 \models ``2^\gk = \gk^+$
+ $2^{2^\gk} =
2^{\gk^+} = \gl$''.
Further, by Theorem 1.5 of \cite{H4},
$V_2 \models ``\gk$ is measurable''.
Thus, by the definition of
$\FP^3$ in $V_2$ as an Easton support
iteration of length $\gk$,
$V_2 \models ``\card{\FP^3} = \gk$'', and
$V_3 \models ``\gk$ is regular +
$2^\gk = \gk^+$
+ $2^{2^\gk} =
2^{\gk^+} = \gl$''.
Since $\FP^4$ is ${(\add(\gk, \gk^+))}^{V_3}$,
$V^{\FP^4}_3 = \ov V \models
``2^\gk = \gk^+$
+ $2^{2^\gk} =
2^{\gk^+} = \gl$''.
This proves Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l2a}
$V$ and $V^\FP$ have the same
cardinals and cofinalities.
\end{lemma}
\begin{proof}
By the definitions of $\FP^0$ and $\FP^1$,
$V$ and $V_1$ have the same cardinals and
cofinalities.
Further, standard arguments show that the
Singular Cardinals Hypothesis (SCH) is preserved
in $V_1$.\footnote{The Singular Cardinals
Hypothesis (SCH) is the set theoretic
statement that if $\gd$ is a singular
strong limit cardinal,
%and $2^\eta = \eta^+$ for every $\eta < \gd$,
then $2^\gd = \gd^+$.}
Thus, since $V_2$ is a generic extension of $V_1$,
a model for SCH,
by adding a fast function,
Theorem 1.3 of \cite{H4} implies that
$V_1$ and $V_2$ have the same cardinals,
cofinalities, and continuum function.
Then, since $\gk$ remains measurable in $V_2$,
$\FP^3$ is the Easton support iteration
of length $\gk$ which forces with
$\add(\gg, f(\gg))$ at each stage,
and by the definition of $\FP^2$, for
$\gg_0$, $\gg_1$ successive elements of
$\dom(f)$, $f(\gg_0) < \gg_1$,
$V_2$ and $V_3$ have the same cardinals and cofinalities.
Since $V_3 \models ``2^\gk = \gk^+$''
and $\FP^4$ is ${(\add(\gk, \gk^+))}^{V_3}$,
$\ov V$ and $V_3$ have the same
cardinals and cofinalities.
This proves Lemma \ref{l2a}.
\end{proof}
\begin{lemma}\label{l1}
$\ov V = V^\FP \models
``$If $M$ is an ultrapower via a normal measure
over $\gk$ and
$\ov j : \ov V \to M$ is the associated
elementary embedding,
$\ov j$ lifts
$j : L[\mu] \to L[\mu]^\gk/\mu$
and $M$ has the same cardinals and cofinalities as
$L[\mu]^\gk/\mu$''.
\end{lemma}
\begin{proof}
Let
$\ov j : \ov V \to M$ be an
elementary embedding witnessing
$\gk$'s measurability. By the
definition of $\FP$, we can write
$\FP = \FP^0 \ast \dot \FQ$, where
$\card{\FP^0} = \go$ and
$\forces_{\FP^0} ``\dot \FQ$ is
$\ha_1$-closed''.
In the terminology of Hamkins
\cite{H1}, \cite{H2}, and \cite{H3},
$\FP$ ``admits a gap at $\ha_1$'', so
by the results of \cite{H1}, \cite{H2},
and \cite{H3},
$V^\FP \models ``$Any elementary embedding
witnessing $\gk$'s measurability
must be the lift of an
elementary embedding in $V$ witnessing
$\gk$'s measurability''.
Since $V = L[\mu]$, this means that
$\ov j$ must lift $j$, i.e.,
$\ov j \supseteq j$ and
$\ov j : L[\mu][G] \to L[\mu]^\gk/\mu[\ov j(G)]$
for $G$ $L[\mu]$-generic over $\FP$.
To see that $M$ and $L[\mu]^\gk/\mu$
have the same cardinals and cofinalities,
we note that by Lemma \ref{l2a},
$V = L[\mu]$ and $\ov V$ have the same cardinals
and cofinalities.
By elementarity, it then immediately
follows that $M$ and $L[\mu]^\gk/\mu$
have the same cardinals and cofinalities.
This proves Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$For every
$L[\mu]^\gk/\mu$-cardinal
$\gd < j(\gk)$, there are
$2^{2^\gk}$ normal measures $\nu$ over
$\gk$ and associated ultrapowers $M_\nu$ and
elementary embeddings
$j_\nu : \ov V \to M_\nu$ such that
$M_\nu \models ``2^\gk \ge \gd$''.
\end{lemma}
\begin{proof}
We begin by working in $V_1$.
Since $\FP^0 \ast \dot \FP^1$
is such that
$\card{\FP^0} = \go$ and
$\forces_{\FP^0} ``\dot \FP^1$ is
$\ha_1$-closed'', we can infer
as in the proof of Lemma \ref{l1}
that any elementary embedding
$i : V_1 \to M^*$ witnessing
$\gk$'s measurability in $V_1$ must be
a lift of $j$.
Take $i : V_1 \to M^*$ to be an
elementary embedding witnessing
$\gk$'s measurability in $V_1$
which is generated by a normal
measure over $\gk$ in $V_1$.
%Let $f \in V_2$ be the fast function
%added by forcing over $V_1$ with $\FP^2$.
By Theorem 1.5 of \cite{H4},
for any $\gd < i(\gk)$, there is a lift
$\ov i : V_2 \to M$ of $i$ such that
$\ov i$ is generated by a normal
measure over $\gk$ in $V_2$ and
$\ov i(f)(\gk) = \gd$. Since what we have
just done shows
$\ov i(\gk) = i(\gk) = j(\gk)$,
this means that for any $\gd < j(\gk)$,
there is an elementary embedding
$\ov i$ lifting $i$ witnessing $\gk$'s measurability
which is generated by a normal measure
over $\gk$ in $V_2$
such that $\ov i(f)(\gk) = \gd$.
Fix $\gd \in [\gk^+, j(\gk))$ and
$k : V_2 \to M$ an elementary embedding
witnessing $\gk$'s measurability
generated by a normal measure over $\gk$ in $V_2$ such that
$k(f)(\gk) = \gd$.
Let $G_0$ be $V_2$-generic over
$\FP^3$ and $G_1$ be $V_2[G_0]$-generic over
$\FP^4$. We show that $k$ lifts in $V_2[G_0][G_1] = \ov V$ to
$\ov k : V_2[G_0][G_1] \to M[G_0][G_2][G_3][G_4]$, where
$\ov k '' G_0 \ast G_1 \subseteq G_0
\ast G_2 \ast G_3 \ast G_4$,
$G_2$, $G_3$, and $G_4$ are appropriately generic objects
constructed in $V_2[G_0][G_1]$, and there are
$2^{2^\gk} = 2^{\gk^+} = \gl$ different choices for $G_3$.
Assuming this is the case,
the normal measures $\nu$ over $\gk$ defined as
$A \in \nu$ iff $\gk \in \ov k(A)$
for each different choice of $G_3$
in aggregate have size $2^{2^\gk}$, since as in
Lemma 1.1 of \cite{A01}, $\ov k = j_\nu$ for
$j_\nu$ the elementary embedding generated by $\nu$,
and $\ov k(G_0 \ast G_1) =
G_0 \ast G_2 \ast G_3 \ast G_4$.
We construct now the required generic objects.
We begin by noting that by the choice of
$k$ and the definition of $\FP^3 \ast \dot \FP^4$,
$k(\FP^3 \ast \dot \FP^4) =
\FP^3 \ast \dot \add(\gk, \gd) \ast \dot
\FQ^1 \ast \dot \FQ^2$, where
$\dot \FQ^1$ is a term for the portion of
$k(\FP^3 \ast \dot \FP^4)$ defined between
stages $\gk$ and $k(\gk)$, and
$\dot \FQ^2$ is a term for (the appropriate version of)
$\add(k(\gk), k(\gk^+))$.
Thus, to build $G_2$, it is necessary to
find an
$M[G_0]$-generic object for
${(\add(\gk, \gd))}^{M[G_0]}$.
However, since
$k$ is
generated via a normal measure over $\gk$
and $\gd \in [\gk^+, j(\gk))$,
%$V_2 \models ``\card{\gd} =
%|\{f : f : \gk \to \gd$ is a function$\}| =
%\card{{[\gk^+]}^\gk} = \gk^+$'',
we know that
$V_2 \models ``\card{\gd} = \gk^+$''.
Thus, we can let $g \in V_2[G_0]$,
$g : \gk^+ \to \gd$ be a bijection.
$g$ generates in $V_2[G_0]$ an isomorphism between
${(\add(\gk, \gk^+))}^{V_2[G_0]}$ and ${(\add(\gk, \gd))}^{M[G_0]}$
%(as computed in $M[G_0]$)
via the map which,
for any $p \in {(\add(\gk, \gk^+))}^{V_2[G_0]}$, sends
$\la \la \ga, \gb \ra, \gg \ra \in p$ to
$\la \la \ga, g(\gb) \ra, \gg \ra$.
The isomorphism $\pi$ so generated can
easily be verified to be such that
$\pi '' G_1 = G_2$ is
$M[G_0]$-generic over
${(\add(\gk, \gd))}^{M[G_0]}$.
To build the
$2^{2^\gk} = 2^{\gk^+} = \gl$ different
possibilities for $G_3$, we note that
by standard arguments,
$M[G_0][G_2]$ is $\gk$-closed with respect to
$V_2[G_0][G_1]$. Also,
$M[G_0][G_2] \models ``\card{\FQ^1} = k(\gk)$'',
and by elementarity and the fact that
$M \models ``\card{\FP^3 \ast \dot \add(\gk, \gd)}
< k(\gk)$'',
$M[G_0][G_2] \models ``\card{\wp(\FQ^1)} =
2^{k(\gk)} = k(\gk^+)$''. Since
$V_2 \models ``\card{k(\gk^+)} =
|\{f : f : \gk \to \gk^+$ is a function$\}| =
\card{{[\gk^+]}^\gk} = \gk^+$'',
we can let
$\la D_\ga : \ga < \gk^+ \ra$ enumerate in
$V_2[G_0][G_1]$ all dense open subsets of
$\FQ^1$ present in $M[G_0][G_2]$.
The $\gk$-closure of $M[G_0][G_2]$ with
respect to $V_2[G_0][G_1]$ and the fact
$M[G_0][G_2] \models ``\FQ^1$ is $\gk$-closed''
now allow us, as in Lemma 1.1 of \cite{A01},
to build in $V_2[G_0][G_1]$ a tree
${\mathfrak T}$ of height $\gk^+$
satisfying the following properties:
\begin{enumerate}
\item The root of $\T$ is the empty condition.
\item If $p$ is an element at level $\ga < \gk^+$ of
$\T$, then the successors of $p$ at level
$\ga + 1$ are two incompatible extensions of $p$
which are elements of $D_\ga$.
\item If $\eta < \gk^+$ is a limit ordinal, then
the elements of $\T$ at height $\eta$ are upper
bounds to any path through $\T$ of height $\eta$.
\end{enumerate}
It is easily seen that any path through $\T$ generates an
$M[G_0][G_2]$-generic object over $\FQ^1$.
Since there are
$2^{\gk^+} = 2^{2^\gk} = \gl$ such paths through
$\T$, there are $2^{\gk^+} = 2^{2^\gk} = \gl$
many different possibilities for an
$M[G_0][G_2]$-generic object $G_3$ over $\FQ^1$.
And, for any choice of $G_3$, standard arguments
once again show that
$M[G_0][G_2][G_3]$ remains $\gk$-closed
with respect to
$V_2[G_0][G_1][G_3] = V_2[G_0][G_1]$.
To obtain $G_4$, we use an argument that
has appeared elsewhere in different guises,
including
%Lemma 3.1 of \cite{A01a} and
pages 119--120 of \cite{AS97a}.
We first fix a generic object $G_3$ as
in the above paragraph, and then note that
$k$ lifts to
$k' : V_2[G_0] \to M[G_0][G_2][G_3]$
in $V_2[G_0][G_1]$. Also, for simplicity
in notation, for the remainder of the
proof of Lemma \ref{l3}, we fix
$\add(\gk, \gk^+)$ as meaning
${(\add(\gk, \gk^+))}^{V_2[G_0]}$,
$\add(k(\gk), k(\gk^+))$ as meaning
${(\add(k(\gk), k(\gk^+)))}^{M[G_0][G_2][G_3]}$, and
$\add(k(\gk), \ga)$ for $\ga \in (k(\gk), k(\gk^+))$
as meaning
${(\add(k(\gk), \ga))}^{M[G_0][G_2][G_3]}$.
For $\ga \in (\gk, \gk^+)$ and
$p \in \add(\gk, \gk^+)$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_1 \rest \ga = \{p \rest \ga : p \in G_1\}$. Clearly,
$V_2[G_0][G_1] \models ``|G_1 \rest \ga| \le \gk$
for all $\ga \in (\gk, \gk^+)$''. Thus, since
$\add(k(\gk), k(\gk^+))$ is
$k(\gk)$-directed closed,
$k(\gk) > \gk^+$, and
$M[G_0][G_2][G_3]$ is $\gk$-closed with respect to
$V_2[G_0][G_1]$,
$q_\ga = \bigcup\{k'(p) : p \in G_1 \rest \ga\}$ is
well-defined and is an element of
$\add(k(\gk), k(\gk^+))$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} k(\gb), k(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} k(\gb)$. Let
$\gb$ be minimal such that $\gs < k(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \ga$,
$\la \rho, \gs \ra \in \dom(k'(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \ga$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$k'(q) = k'(p) \rest k'(\gb) = k'(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(k'(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since the fact
$M[G_0][G_2] \models ``2^{k(\gk)} = k(\gk^+)$ +
$\card{\FQ^1} = k(\gk)$'' implies that
$M[G_0][G_2][G_3] \models
``2^{k(\gk)} = k(\gk^+)$'',
$M[G_0][G_2][G_3] \models ``\add(k(\gk),
k(\gk^+))$ is
$k(\gk^+)$-c.c$.$ and has
$k(\gk^+)$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_2][G_3]$ is a
maximal antichain of $\add(k(\gk), k(\gk^+))$,
${\cal A} \subseteq \add(k(\gk), \gb)$ for some
$\gb \in (k(\gk), k(\gk^+))$. Thus, since
$V_2 \models ``|k(\gk^+)| = \gk^+$'', we can let
$\la {\cal A}_\ga : \ga \in (\gk, \gk^+) \ra \in
V_2[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(k(\gk), k(\gk^+))$
present in
$M[G_0][G_2][G_3]$.
Working in $V_2[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gk, \gk^+) \ra$ of
elements of $\add(k(\gk), k(\gk^+))$ such that
$\forall \ga \in (\gk, \gk^+) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(k(\gk), k(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gk, \gk^+) \ra
\exists \gb \in (\gk, \gk^+)
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_4 = \{p \in \add(k(\gk), k(\gk^+)) :
\exists r \in \la r_\ga : \ga \in (\gk, \gk^+)
[r \ge p]$ is an
$M[G_0][G_2][G_3]$-generic object over
$\add(k(\gk), k(\gk^+))$. To define
$\la r_\ga : \ga \in (\gk, \gk^+) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gk, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gk, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_2][G_3]$ is
$\gk$-closed with respect to
$V_2[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gk \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(k(\gk), k(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(k(\gk), k(\ga))$,
$q_{\ga + 1} \in \add(k(\gk), k(\ga + 1))$, and
$k(\ga) < k(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_2][G_3]$ is closed under
$\gk$ sequences with respect to
$V_2[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb : \gb < \eta \ra$ such that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_4$ is
$M[G_0][G_2][G_3]$-generic over
$\FQ^2 = \add(k(\gk), k(\gk^+))$,
%$\FQ^2 = {(\add(k(\gk), k(\gk^+)))}^{M[G_0][G_2][G_3]}$,
we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gk, \gk^+) \ra
\exists \gb \in (\gk, \gk^+)
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la k(\ga) : \ga < \gk^+ \ra$ is
unbounded in $k(\gk^+)$. To see this, if
$\gb < k(\gk^+)$ is an ordinal, then for some
$f : \gk \to M$ representing $\gb$,
we can assume that for $\gg < \gk$,
$f(\gg) < \gk^+$. Thus, by the regularity of
$\gk^+$ in $V_2$,
$\gb_0 = \bigcup_{\gg < \gk} f(\gg) <
\gk^+$, and $k(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gk^+ \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gk, \gk^+)$ be such that
${\cal A} \subseteq \add(k(\gk), k(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gk, \gk^+)$ is such that for some
$\ga \in (\gk, \gk^+)$, $p = p \rest \ga$,
$G_4$ is such that if
$p \in G_1$, $k'(p) \in G_4$.
Thus, we can fully lift $k$ in
$V_2[G_0][G_1]$ to
$\ov k : V_2[G_0][G_1] \to M[G_0][G_2][G_3][G_4]$
such that
$\ov k '' G_0 \ast G_1 \subseteq G_0 \ast G_2
\ast G_3 \ast G_4$.
Since $M[G_0][G_2] \models ``2^\gk \ge \gd$'',
and by the definitions of fast function forcing and
$\FQ^1$ and $\FQ^2$, $\FQ^1 \ast \dot \FQ^2$ is
$\gd$-closed in $M[G_0][G_2]$,
this proves Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l2} - \ref{l3} complete the
proof of Theorem \ref{t1}.
\end{proof}
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\end{document}