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Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
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\title{Forcing the Least Measurable to Violate GCH}
\date{April 14, 1998\\(revised December 22, 1998)}
\author{Arthur W.~Apter
\thanks{Supported by the Volkswagen-Stiftung
(RiP program at Oberwolfach). This research was also
partially supported
by PSC-CUNY Grant 667379. In addition, the author wishes to
thank his RiP partner James Cummings for helpful conversations
on the subject matter of this paper.
Finally, the author wishes to thank the referee, whose
suggestions were incorporated into this
version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA\\}
\begin{document}
\maketitle
\begin{abstract}
Starting with a model for ``GCH + $\gk$ is $\gk^+$ supercompact'',
we force and construct a model for ``$\gk$ is the least measurable
cardinal + $2^\gk = \gk^+$''. This model has the property that
forcing over it with ${\rm Add}(\gk, \gk^{++})$ preserves the fact
$\gk$ is the least measurable cardinal.
\end{abstract}
\baselineskip=24pt
Since the invention in the early 1970s of reverse Easton
forcing (see \cite{J}) by Silver and iterated Prikry
forcing (see \cite{Ma}) by Magidor, it has been fairly
easy to construct a model in which the
least measurable cardinal violates GCH.
The basic technique is to take some ``large'' measurable
cardinal (supercompact or otherwise) satisfying GCH,
force it to violate GCH, and then turn it into the
least measurable cardinal via iterated Prikry forcing
or some other method, such as adding
non-reflecting stationary sets,
to destroy all measurable cardinals below it.
This, however, does not answer the question of whether
it is possible to start with a model in which $\gk$ is
{\it already} the least measurable cardinal and then
force $\gk$ to violate GCH while preserving it is the
least measurable cardinal, since the method just
described requires the use of a forcing construction
in which initially, $\gk$ is much larger than the
least measurable cardinal.
A first answer to the above question was given in
\cite{AG}, in which the author and Gitik showed
Con(ZFC + $\gk$ is supercompact) $\implies$
Con(ZFC + $\gk$ is both strongly compact and
the least measurable cardinal + $2^\gk = \gk^+$ +
$\gk$ is fully Laver indestructible).
If we take as our ground model $V$ a model for
the conclusions of the aforementioned theorem,
then if $\gl > \gk^+$ is a regular cardinal,
we can force over $V$ using, e.g., the usual
Cohen forcing ${\rm Add}(\gk, \gl)$ for
adding $\gl$ subsets to $\gk$ to make
$2^\gk = \gl$. By the indestructibility of
$\gk$ under any $\gk$-directed closed partial
ordering, $\gk$ will remain as the least
measurable cardinal.
The purpose of this paper is to show that full
supercompactness isn't needed to obtain an answer
to the above question, and that in fact, $\gk^+$
supercompactness suffices. Specifically, we
prove the following.
\begin{theorem}\label{thm}
Let $\ov V \models ``$ZFC + GCH + $\gk$ is
$\gk^+$ supercompact''. There is then a partial ordering
$\FP \in \ov V$ so that for $V = \ov V^\FP$,
$V \models ``$ZFC + $2^\gk = \gk^+$ + $\gk$ is the
least measurable cardinal''. Further, for
$\FQ = {({\rm Add}(\gk, \gk^{++}))}^V$,
$V^\FQ \models ``\gk$ is the least measurable
cardinal + $2^\gk = \gk^{++}$''.
\end{theorem}
Note that since forcing a measurable cardinal to
violate GCH requires the use of hypotheses well
beyond measurability (see \cite{G3}, \cite{J},
or \cite{Mi}), starting with a model in which
$\gk$ is the least measurable cardinal and
$2^\gk = \gk^+$ will not in general suffice to
prove Theorem \ref{thm}. It will be necessary
to start with a model in which $\gk$ is a large
cardinal satisfying strong hypotheses and then
somehow transform $\gk$ into the least measurable
cardinal and hope to have preserved enough of
$\gk$'s original properties to be able to do a
further forcing which preserves that $\gk$ is
the least measurable
and causes $\gk$ to violate GCH.
We digress very briefly to give some preliminary
information concerning notation and terminology.
If $\ga < \gb$ are ordinals, then
$[\ga, \gb]$, $[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in standard interval notation.
When forcing, if $V$ is our ground model,
$\FP$ our partial ordering, and
$G \subseteq \FP$ is $V$-generic, then
$V^\FP$ and $V[G]$ will be used interchangeably.
If $p, q \in \FP$, then $q \ge p$ means that
$q$ is stronger than $p$.
For $\varphi$ a formula in the language of
forcing with respect to $\FP$,
$p \decides \varphi$ means that $p$ decides $\varphi$.
If $x \in V[G]$, then $\dot x$ will be a term in
$V$ for $x$. We may, from time to time,
confuse sets with the terms they denote and write
$x$ when we actually mean $\dot x$, especially when
$x$ is in the ground model $V$.
For any further notation or terminology left
unexplained, the reader is urged to consult
\cite{AS1} or \cite{AS2}.
We turn now to the proof of Theorem \ref{thm}.
\begin{proof}
The proof of Theorem \ref{thm} uses ideas from
\cite{AG} in combination with ideas from
\cite{AS1} and \cite{AS2}. The basic method is
to employ a restricted version of the partial ordering
$\FP$ of Theorem \ref{thm} of \cite{AG} and then show that
$\gk^+$ supercompactness suffices to allow the construction
to go through.
Towards this goal, let
$\ov V \models ``$ZFC + GCH + $\gk$ is $\gk^+$
supercompact''. Let
$\la \gd_\ga : \ga < \gk \ra$
enumerate the measurables below $\gk$.
Our partial ordering $\FP$ will be defined as
an Easton support iteration
$\la \la \FP_\ga, \dot \FQ^*_\ga \ra : \ga < \gk \ra$,
where we start with $\FP_0 = \{\emptyset\}$.
We then let
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga \ast \dot \FR_\ga =
\FP_\ga \ast \dot \FQ^*_\ga$, where
$\dot \FQ_\ga$ is a term in the language of forcing
with respect to $\FP_\ga$ for
${({\rm Add}(\gd_\ga, \gd^{++}_\ga))}^{V^{\FP_\ga}}$. If
$\forces_{\FP_\ga \ast \dot \FQ_\ga} ``\gd_\ga$
is measurable'', then
$\forces_{\FP_\ga \ast \dot \FQ_\ga} ``\dot \FR_\ga$
is Prikry forcing defined over $\gd_\ga$ with
respect to some normal measure $\dot \mu_\ga$
over $\gd_\ga$''. If
$\forces_{\FP_\ga \ast \dot \FQ_\ga} ``\gd_\ga$
is not measurable'', then
$\forces_{\FP_\ga \ast \dot \FQ_\ga} ``\dot \FR_\ga$
is the trivial partial ordering''.
The ordering $\le$ on $\FP$ is Gitik's modification of
\cite{G1} and \cite{G2} of the usual Easton support
iteration ordering. Although its definition is given
in \cite{G1}, \cite{G2}, and \cite{AG}, for concreteness,
we repeat it here. Specifically, let
$p, q \in \FP$,
$p = \la \dot p_\ga : \ga < \gk \ra$,
$q = \la \dot q_\ga : \ga < \gk \ra$.
Then $q \ge p$
iff $q \ge p$ with respect to the usual Easton support
iteration ordering, but in addition, for some finite
$A \subseteq \support(p)$ and all $\gb \in
\support(p) - A$, for
$\dot q_\gb = \la \dot r_\gb', \dot s_\gb' \ra$,
$\dot p_\gb = \la \dot r_\gb, \dot s_\gb \ra$,
$\la \dot r_\gb', \dot s_\gb' \ra,
\la \dot r_\gb, \dot s_\gb \ra \in \dot \FQ_\gb
\ast \dot \FR_\gb$,
$q \rest \gb \ast \dot r_\gb'
\forces_{\FP_\gb \ast \dot \FQ_\gb} ``$If
$\dot s_\gb'$ and $\dot s_\gb$ are conditions
with respect to Prikry forcing, then $\dot s_\gb'$
is a `direct extension' of $\dot s_\gb'$, i.e.,
$\dot s_\gb'$ is obtained from $\dot s_\gb$ by
shrinking the measure 1 component of $\dot s_\gb$''.
And, as in \cite{G1}, \cite{G2}, and \cite{AG}, if
$A = \emptyset$ in the above, then $q$ is called an
{\it Easton} or {\it direct extension} of $p$.
It is a fundamental result of \cite{G1} and
\cite{G2} that if $p \in \FP$ and
$\varphi$ is a formula in the language of
forcing with respect to $\FP$, then there
exists $q \ge p$, $q$ an Easton extension
of $p$, so that $q \decides \varphi$.
This has as an immediate corollary that
bounded subsets of $\gk$ are added by
initial segments of $\FP$.
\begin{lemma}\label{l1}
$\ov V^\FP \models ``$There are no measurable cardinals
below $\gk$''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} begins as does
the proof of Lemma 1 of \cite{AG}.
Assume towards a contradiction that $\gd < \gk$,
$\gd = \gd_\ga$ is so that
$\ov V^\FP \models ``\gd$ is measurable''. Write
$\FP = \FP_\ga \ast \dot \FQ_\ga \ast \dot \FR_\ga \ast
\dot \FP^\ga$, where $\FP_\ga$ is the forcing up
through stage $\ga$,
$\dot \FQ_\ga \ast \dot \FR_\ga$
is a term for the forcing at stage $\ga$, and
$\dot \FP^\ga$ is a term for the forcing beyond stage $\ga$.
By the definition of $\FP$, since Lemma 1.2 of \cite{G1}
and Lemma 1.4 of \cite{G1} and \cite{G2} imply
$\forces_{\FP_{\ga + 1}} ``\dot \FP^\ga$ adds no new
bounded subsets to the least inaccessible cardinal $> \gd$'',
$\forces_{\FP_{\ga + 1}} ``\gd$ is measurable'' iff
$\forces_{\FP_{\ga + 1} \ast \dot \FP^\ga}
``\gd$ is measurable'', i.e., iff
$\forces_\FP ``\gd$ is measurable''.
However, by the definition of $\FP$,
$\forces_{\FP_{\ga + 1}} ``\gd$ isn't measurable'', a contradiction.
Thus,
$\ov V^\FP \models ``$No $\ov V$-measurable
cardinal $\gd < \gk$ is
measurable''. The proof of Lemma \ref{l1} will therefore
be complete once we have shown there is no cardinal
$\gd < \gk$ so that
$\forces_{\FP} ``\gd$ is measurable''.
To do this, we give an argument similar to the
ones found in the last part of Lemma 8 of \cite{A}
and the last part of Lemma 3 of \cite{AC}.
(See also \cite{H} and \cite{KM}.)
Assume that
$\ov V^\FP \models ``\gd < \gk$ is measurable''.
Since we have just shown that no $\ov V$-measurable
cardinal
$\gd < \gk$ is measurable in $\ov V^\FP$,
again by Lemma 1.2 of \cite{G1} and Lemma 1.4 of \cite{G1} and \cite{G2},
we can write
$\FP = \FP_\gg \ast \dot \FR$, where
$\gd \not\in {\rm field}(\FP_\gg)$ and
$\forces_{\FP_\gg} ``\dot \FR$ adds no new bounded subsets
to the least inaccessible above $\gd$''. Thus, as before,
$\forces_{\FP_\gg} ``\gd$ is measurable'' iff
$\forces_\FP ``\gd$ is measurable'',
so we once again show that
$\forces_{\FP_\gg} ``\gd$ isn't measurable''.
By the L\'evy-Solovay results \cite{LS},
it cannot be the case that $|\FP_\gg| < \gd$,
so we assume without loss of generality that
$|\FP_\gg| = \gd$.
This has as an immediate consequence that
$\gg = \gd$.
Note now that since
$\ov V^{\FP_\gd} \models ``\gd$ is Mahlo'',
$\ov V \models ``\gd$ is Mahlo''.
Next, let $p \in \FP_\gd$ be so that
$p \forces ``\dot \mu$ is a measure over $\gd$''.
We show there is some $q \ge p$, $q \in \FP_\gd$ so that
for every $X \in {(\wp(\gd))}^{\ov V}$,
$q \decides ``X \in \dot \mu$''.
To do this, we build in $\ov V$ a binary tree
${\cal T}$ of height $\gd$, assuming no such $q$ exists.
The root of our tree is $\la p, \gd \ra$.
At successor stages $\gb + 1$, assuming
$\la r, X \ra$ is on the $\gb^{{\rm th}}$ level of ${\cal T}$,
$r \ge p$, and $X \subseteq \gd$, $X \in \ov V$ is so that
$r \forces ``X \in \dot \mu$'', we let
$X = X_0 \cup X_1$ be such that
$X_0, X_1 \in \ov V$,
$X_0 \cap X_1 = \emptyset$, and for
$r_0 \ge r$, $r_1 \ge r$ incompatible,
$r_0 \forces ``X_0 \in \dot \mu$'' and
$r_1 \forces ``X_1 \in \dot \mu$''.
We can do this by our hypothesis of the
non-existence of a $q \in \FP_\gd$ as mentioned earlier.
We place both $\la r_0, X_0 \ra$ and $\la r_1, X_1 \ra$ in
${\cal T}$ at height $\gb + 1$ as the successors of $\la r, X \ra$.
At limit stages $\gl < \gd$, for each branch ${\cal B}$ in
${\cal T}$ of height $\le \gl$, we take the intersection of
all second coordinates along ${\cal B}$.
The result is a partition of $\gd$ into $\le 2^\gl$ many sets,
so since $\gd$ is Mahlo, $2^\gl < \gd$, i.e.,
the partition is into $< \gd$ many sets. Since
$\ov V^{\FP_\gd} \models ``\gd$ is measurable'',
there is at least one element $Y$ of this partition
resulting from a branch of height $\gl$ and a condition
$s \ge p$ so that
$s \forces ``Y \in \dot \mu$''.
For all such $Y$, we place a pair of the form
$\la s, Y \ra$ into ${\cal T}$ at level $\gl$
as the successor of each element of the branch
generating $Y$.
Work now in $\ov V^{\FP_\gd}$.
Since $\gd$ is measurable in $\ov V^{\FP_\gd}$,
$\ov V^{\FP_\gd} \models ``\gd$ is weakly compact''.
By construction, ${\cal T}$ is a tree having $\gd$
levels so that each level has size $< \gd$.
Thus, by the weak compactness of $\gd$ in
$\ov V^{\FP_\gd}$, we can let
${\cal B} = \la \la r_\gb, X_\gb \ra : \gb < \gd \ra$
be a branch of height $\gd$ through ${\cal T}$.
If we define, for $\gb < \gd$, $Y_\gb = X_\gb - X_{\gb + 1}$,
then since
$\la X_\gb : \gb < \gd \ra$ is so that
$0 \le \gb < \rho < \gd$ implies
$X_\gb \supseteq X_\rho$, for
$0 \le \gb < \rho < \gd$,
$Y_\gb \cap Y_\rho = \emptyset$.
Since by the construction of ${\cal T}$,
at level $\gb + 1$, the two second coordinate
portions of the successor of $\la r_\gb, X_\gb \ra$ are
$X_{\gb + 1}$ and $Y_\gb$, for the $s_\gb$ so that
$\la s_\gb, Y_\gb \ra$ is at level $\gb + 1$ of ${\cal T}$,
$\la s_\gb : \gb < \gd \ra$ must form in
$\ov V^{\FP_\gd}$ an antichain of size $\gd$ in $\FP_\gd$.
In $\ov V^{\FP_\gd}$,
$\FP_\gd$ is embeddable as a subordering of the Easton support product
$\prod_{\ga < \gd} \FQ^*_\ga$ as calculated in
$\ov V^{\FP_\gd}$. As
$\ov V^{\FP_\gd} \models ``\gd$ is Mahlo'',
this immediately implies that
$\ov V^{\FP_\gd} \models ``\FP_\gd$ is $\gd$-c.c.'',
contradicting that
$\la s_\gb : \gb < \gd \ra$ is in
$\ov V^{\FP_\gd}$ an antichain of size $\gd$.
Thus, there is some $q \ge p$ so that for every
$X \in {(\wp(\gd))}^{\ov V}$,
$q \decides ``X \in \dot \mu$'', i.e.,
$\gd$ is measurable in $\ov V$.
This contradiction proves Lemma \ref{l1}.
\end{proof}
We remark that the proof of Lemma \ref{l1}
essentially contains a proof of the useful
general fact that if
$\FP^* \times \FP^*$ is $\gd$-c.c$.$ and
$\forces_{\FP^*} ``\gd$ is measurable'', then
$\gd$ is measurable in the ground model.
Let $V = \ov V^\FP$. Since $\ov V \models {\rm GCH}$
and $\gk$ has in $\ov V$ a normal measure
concentrating on non-measurables,
Lemma 1.5 of \cite{G1} immediately yields
$V \models ``\gk$ is the least measurable cardinal'',
and since
$\ov V \models ``|\FP| = \gk$'',
$V \models ``2^\gl = \gl^+$ for any cardinal
$\gl \ge \gk$''. Take
$\FQ = {({\rm Add}(\gk, \gk^{++}))}^V$.
By the fact $V \models ``\FQ$ is $\gk$-directed closed'',
$V^\FQ \models ``$There are no measurable cardinals below $\gk$''.
We will thus have completed the proof of Theorem \ref{thm}
once we have shown the following.
\begin{lemma}\label{l2}
$V^\FQ \models ``\gk$ is measurable''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} uses the ideas found
in the proof of Lemma 2 of \cite{AG} in tandem with
ideas found in the last part of the proofs of
Lemma 9 of \cite{AS1} and \cite{AS2}.
First, let $j : \ov V \to M$ be an
elementary embedding witnessing the
$\gk^+$ supercompactness of $\gk$.
In $M$, $j(\FP \ast \dot \FQ) =
\FP \ast \dot \FQ_\gk \ast \dot \FR_\gk
\ast \dot \FR \ast j(\dot \FQ) =
\FP \ast \dot \FQ \ast \dot \FR_\gk
\ast \dot \FR \ast j(\dot \FQ) =
j(\FP) \ast j(\dot \FQ)$.
Let $\ov V^{\FP \ast \dot \FQ} =
\ov V[G \ast H]$, where
$G$ is $\ov V$-generic over $\FP$,
and $H$ is $\ov V[G]$-generic over
$\FQ$. In Lemma 2 of \cite{AG},
it was shown that there exists a term
$\tau \in M$ for a ``master condition''
for the appropriate $\FQ$, i.e., a term
$\tau \in M$ in the language of forcing
with respect to $j(\FP)$ so that in $M$,
$\forces_{j(\FP)} ``\tau$ extends every
$j(\dot q)$ for $\dot q \in \dot H$''. Since
$\ov V[G \ast H] \models ``|H| = \gk^{++}$'' and
$M^{\gk^{++}} \not\subseteq M$, we cannot under
the current circumstances infer the existence of
an analogous term for a ``master condition'' for
our $\FQ$. We can, however, in analogy to
Lemma 2 of \cite{AG},
infer the existence of a sequence
$\la \dot q_\ga : \ga \in (\gk^+, \gk^{++}) \ra \in \ov V$
for ``master conditions'' for certain partial orderings
which are restrictions of $\FQ$. Specifically, for
$\ga \in (\gk^+, \gk^{++})$ and $p \in \FQ$, let
$p \rest \ga = \{\la \la \rho, \sigma \ra, \eta \ra \in
p : \sigma < \ga\}$,
$\FQ_\ga = \{p \rest \ga : p \in \FQ\}$, and
$H_\ga = \{p \rest \ga : p \in H\}$. Clearly,
$\ov V[G \ast H] \models ``|H_\ga| = \gk^+$'' for
all $\ga \in (\gk^+, \gk^{++})$'', and
$H_\ga$ is composed of compatible conditions.
Also, since $\FP$ is $\gk$-c.c$.$ in both
$\ov V$ and $M$ and
$\forces_\FP ``\dot \FQ$ is $\gk^+$-c.c$.$''
in both $\ov V$ and $M$, the usual arguments show
$M[G \ast H]$ remains $\gk^+$ closed with respect to
$\ov V[G \ast H]$. In addition, as always, $j(\gk) > \gk^{++}$.
This means, for each
$\ga \in (\gk^+, \gk^{++})$,
$T_\ga = \{j(\dot q) : \exists p \in G
[\la p, q \ra \in G \ast H_\ga]\} \in
M[G \ast H]$ has a name $\dot T_\ga \in M$
so that in $M$,
$\forces_{j(\FP)} ``|\dot T_\ga| = \gk^+$,
any two elements of $\dot T_\ga$ are
compatible, and $\dot T_\ga$ is a subset of
a partial ordering which is $j(\gk)$-directed
closed''. Thus, since $M^{\gk^+} \subseteq M$,
$\forces_{j(\FP)} ``\cup \dot T_\ga$
is an upper bound for each element of
$\dot T_\ga$''. A term $\dot q_\ga$ for
$\cup \dot T_\ga$
is a term for a ``master condition'' for
$\FQ_\ga$, and our definition ensures
$\la \dot q_\ga : \ga \in (\gk^+, \gk^{++}) \ra \in \ov V$.
We proceed now in analogy to \cite{AS1}, pages 119-120
and \cite{AS2}, pages 2024-2025.
Note that if $\ga \in (\gk^+, \gk^{++})$,
$j(\dot \FQ_\ga) = j(\dot \FQ)_{j(\ga)}$
is a term for a well-defined partial ordering in
$M^{j(\FP)}$ so that
$\forces_{j(\FP)} ``\dot q_\ga \in j(\dot \FQ_\ga)$''.
Also, since $M^{\gk^+} \subseteq M$ and
$M[G \ast H]$ is closed under $\gk^+$ sequences
with respect to $\ov V[G \ast H]$,
$\forces_{j(\FP)} ``\cup_{\gb < \ga} \dot q_\gb
\in j(\dot \FQ)$'' whenever
$\ga \in (\gk^+, \gk^{++})$. We are now able
to infer that for $\ga \in (\gk^+, \gk^{++})$,
$\forces_{j(\FP)} ``$If $\la \rho, \sigma \ra
\in \dom(\dot q_\ga) - \dom(\cup_{\gb < \ga} \dot q_\gb)$, then
$\sigma \in [\cup_{\gb < \ga} j(\gb), j(\ga))$''.
If not, then we can find $\la \rho, \sigma \ra$ and a minimal
$\gb$ so that for some $p_0 \in j(\FP)$,
$p_0 \forces ``\sigma < j(\gb)$ and
$\la \rho, \sigma \ra \in \dom(\dot q_\ga) -
\dom(\cup_{\gg < \ga} \dot q_\gg)$''.
It must thus be the case that there is $p_1 \ge p_0$,
$p_1 \forces ``$For some $\dot p \in \dot H_\ga$,
$\la \rho, \sigma \ra \in \dom(j(\dot p))$''.
Since by elementarity and the definitions of
$H_\gb$ and $H_\ga$, for $\dot p \rest \gb = \dot q$
a term for an element of $H_\gb$,
$p_1 \forces ``j(\dot q) = j(\dot p \rest \gb) =
j(\dot p) \rest j(\gb)$'',
it must be the case that
$p_1 \forces ``\la \rho, \sigma \ra \in \dom(j(\dot q))$''.
This means
$p_1 \forces ``\la \rho, \sigma \ra \in \dom(\dot q_\gb)$'',
a contradiction.
Note that for $\ga \in (\gk^+, \gk^{++})$,
$j(\dot \FQ_\ga)$ is a $j(\FP)$ term in $M$
for the Cohen partial ordering for adding $j(\ga)$
subsets to $j(\gk)$, i.e., a term for the partial ordering
$\{f : j(\gk) \times \ga \to \{0, 1\} : f$ is a function
so that $|\dom(f)| < j(\gk)\}$, ordered by inclusion.
Also, since
$\forces_{j(\FP)} ``2^\gl = \gl^+$ for all cardinals
$\gl \ge j(\gk)$'',
$\forces_{j(\FP)} ``j(\dot \FQ)$ is $j(\gk^+)$-c.c$.$
and $j(\dot \FQ)$ has $j(\gk^{++})$ many
antichains''. This means that if
$\forces_{j(\FP)} ``\dot {\cal A}$ is an
antichain of $j(\dot \FQ)$'', then for some
term $\tau$ so that
$\forces_{j(\FP)} ``\tau \in (j(\gk^+), j(\gk^{++}))$'',
$\forces_{j(\FP)} ``\dot {\cal A} \subseteq
j(\dot \FQ)_\tau$''.
As $j(\FP)$ is $j(\gk)$-c.c$.$ in $M$,
$\tau$ can be taken as an actual ordinal in the interval
$(j(\gk^+), j(\gk^{++}))$.
Also, since we can assume $M$ is generated by an
ultrapower of $\ov V$ via a supercompact ultrafilter over
${(P_\gk(\gk^+))}^{\ov V}$
and $\ov V \models {\rm GCH}$,
$\ov V \models ``|j(\gk^{++})| = \gk^{++}$''.
Thus, we can let
$\la \dot {\cal A}_\ga : \ga \in (\gk^+, \gk^{++}) \ra
\in \ov V$
be an enumeration of the canonical $M$ terms in the
language of forcing with respect to $j(\FP)$ for all
maximal antichains of $j(\dot \FQ)$.
We define in $\ov V$ a sequence
$\la \dot r_\ga : \ga \in (\gk^+, \gk^{++}) \ra$
of canonical $M$ terms in the language of forcing
with respect to $j(\FP)$ for elements of
$j(\dot \FQ)$ so that
$\forall \ga \in (\gk^+, \gk^{++})
[\forces_{j(\FP)} ``$If $\gb \le \gg$,
$\dot r_\gb \le \dot r_\gg$,
$\dot r_\ga \ge \dot q_\ga$, and
$\dot r_\ga \in j(\dot \FQ)_{j(\ga)}$''$]$
and so that
$\forall \dot {\cal A} \in
\la \dot {\cal A}_\ga : \ga \in (\gk^+, \gk^{++}) \ra
\exists \gb \in (\gk^+, \gk^{++})
[\forces_{j(\FP)} ``\dot r_\gb$ extends some element of
$\dot {\cal A}$''$]$.
Assuming we have such a sequence, we will use it in
place of the term for the ``master condition'' of
\cite{AG} to define an ultrafilter
${\cal U} \in V^\FQ$ witnessing the measurability of
$\gk$ in $V^\FQ$. To define
$\la \dot r_\ga : \ga \in (\gk^+, \gk^{++}) \ra$,
if $\ga$ is a limit,
we can let $\dot r_\ga$ be a term for
$\cup_{\gb < \ga} \dot r_\gb$.
As before, since
$M^{\gk^+} \subseteq M$, this definition is valid.
Assuming now $\dot r_\ga$ has been defined and
we wish to define
$\dot r_{\ga + 1}$, let
$\la \dot {\cal B}_\gb : \gb < \eta \le \gk^+ \ra$
be the subsequence of
$\la \dot {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each term so that
$\forces_{j(\FP)} ``\dot {\cal A} \subseteq
j(\dot \FQ)_{j(\ga) + 1}$''. Since
$\forces_{j(\FP)} ``\dot q_\ga, \dot r_\ga \in
j(\dot \FQ)_{j(\ga)}$ and
$\dot q_{\ga + 1} \in j(\dot \FQ)_{j(\ga) + 1}$'' and
$j(\ga) < j(\ga + 1)$,
$\dot r_{\ga + 1}' = \dot r_\ga \cup \dot q_{\ga + 1}$
is well-defined,
as by our earlier observations,
$\forces_{j(\FP)} ``$If $\la \rho, \sigma \ra \in
\dom(\dot q_\gg) - \dom(\cup_{\gb < \gg} \dot q_\gb)$, then
$\sigma \in [\cup_{\gb < \gg} j(\gb), j(\gg))$''.
Once more using the fact
$M^{\gk^+} \subseteq M$, we can define by induction in $M$
a sequence of terms
$\la \dot s_\gb : \gb < \eta \ra$ for elements of
$j(\dot \FQ)_{j(\ga) + 1}$
so that
$\forces_{j(\FP)} ``$If $\gb < \gg$,
$\dot s_\gb \le \dot s_\gg$,
$\dot s_0 \ge \dot r_{\ga + 1}'$,
$\dot s_\rho = \cup_{\gb < \rho} \dot s_\gb$ if
$\rho$ is a limit, and
$\dot s_{\gb + 1} \ge \dot s_\gb$ is so that
$\dot s_{\gb + 1}$ extends some element of
$\dot {\cal B}_\gb$''.
The just mentioned closure fact implies
$\dot r_{\ga + 1} = \cup_{\gb < \eta} \dot s_\gb$
is well-defined.
To see that
$\forall \dot {\cal A} \in
\la \dot {\cal A}_\ga : \ga \in (\gk^+, \gk^{++}) \ra
\exists \gb \in (\gk^+, \gk^{++})
[\forces_{j(\FP)} ``\dot r_\gb$ extends some element of
$\dot {\cal A}$''$]$,
note that for the same reasons as in \cite{AS1} and \cite{AS2},
$\la j(\ga) : \ga < \gk^{++} \ra$
is unbounded in $j(\gk^{++})$.
Specifically, if
$\gb < j(\gk^{++})$ is an ordinal, then for some
$f : \gk^+ \to M$ representing $\gb$, we can assume that
for all $\gg < \gk^+$, $f(\gg) < \gk^{++}$.
Thus, by the regularity of $\gk^{++}$ in $\ov V$,
$\gb_0 = \cup_{\gg < \gk^+} f(\gg) < \gk^{++}$, and
$j(\gb_0) > \gb$.
This means by our earlier remarks that if
$\dot {\cal A} \in \la \dot {\cal A}_\ga :
\ga \in (\gk^+, \gk^{++}) \ra$,
$\dot {\cal A} = \dot {\cal A}_\rho$, then we can let
$\gb \in (\gk^+, \gk^{++})$ be so that
$\forces_{j(\FP)} ``\dot {\cal A} \subseteq
j(\dot \FQ)_{j(\gb)}$''.
By construction, for $\eta > \max(\gb, \rho)$,
$\forces_{j(\FP)} ``\dot r_\eta$ extends some
element of $\dot {\cal A}$''.
Work for the moment in $M$.
Let $\la \dot p, \dot r \ra$ be a
$\FP \ast \dot \FQ \ast \dot \FR_\gk$
name for an element of
$\FR \ast j(\dot \FQ)$, where $\dot r$
is a term for one of the $\dot r_\ga$ s
as defined above. If $\varphi$ is a formula
in the language of forcing with respect to
$\FR \ast j(\dot \FQ)$,
then since
$\FR \ast j(\dot \FQ)$
is an Easton support iteration of the type described earlier,
by Lemma 1.4 of \cite{G1} and \cite{G2}, there is a term
$\la \dot p', \dot r' \ra$ for a direct extension of
$\la \dot p, \dot r \ra$ deciding $\varphi$.
By the definition of direct extension,
$\la \dot p', \dot r \ra$
will be a name for a condition so that
$\la \dot p', \dot r' \ra$
is a name for a direct extension of
$\la \dot p', \dot r \ra$ deciding $\varphi$. Thus,
$\dot {\cal A}' =
\{\dot q : \dot q$ is a
$\FP \ast \dot \FQ \ast \dot \FR_\gk \ast \dot \FR
= j(\FP)$ name for a condition extending $\dot r$ so that
$\la \dot p', \dot q \ra$ decides $\varphi\}$
generates a $j(\FP)$ term for a maximal antichain
$\dot {\cal A}$ of $j(\dot \FQ)$.
By the preceding paragraph and the earlier construction,
there must be some
$\eta \in (\gk^+, \gk^{++})$ so that
$\forces_{j(\FP)} ``\dot r_\eta \ge \dot r$ and
$\dot r_\eta$ extends an element of $\dot {\cal A}$''.
This just means that for any appropriate $\varphi$
and any
$\FP \ast \dot \FQ \ast \dot \FR_\gk$ name
$\la \dot p, \dot r \ra$ for an element of
$\FR \ast j(\dot \FQ)$,
where $\dot r$ is a term for one of the above
$\dot r_\ga$~s, there is a name
$\la \dot p', \dot r' \ra$
for a condition deciding $\varphi$ and extending
$\la \dot p, \dot r \ra$ so that
$\dot p'$ is a name for a direct extension of
$\dot p$ and $\dot r'$ is a name for some
$\dot r_\eta$.
We continue arguing now as in \cite{AG}, which is
in analogy to \cite{G1}, pages 291-292.
We must take into account, however, that unlike in
\cite{AG} but as in \cite{G1},
$M$ has only a restricted amount of closure. Let
$\la \dot A_\ga : \ga < \gk^{++} \ra$
enumerate in $\ov V$ the canonical
$\FP \ast \dot \FQ$ names of subsets of $\gk$
found in
$\ov V[G \ast H]$. Since
$M^{\gk^+} \subseteq M$,
any initial segment of the sequence of statements
$\la \gk \in j(\dot A_\ga) : \ga < \gk^{++} \ra$
is an element of $M$. Thus, we can define in
$\ov V$ a sequence of $\FP \ast \dot \FQ \ast \dot \FR_\gk$
names of elements of
$\FR \ast j(\dot \FQ)$,
$\la \dot p_\ga : \ga < \gk^{++} \ra$,
so that the following three conditions hold.
\begin{enumerate}
\item $\dot p_0$ is a term for
$\la 0, \dot r_0 \ra$, where $0$
represents the trivial condition with respect to
$\FR$.
\item Let
$\dot p_\ga = \la \dot q_\ga, \dot r_{\eta_\ga} \ra$, where
$\eta_\ga \in (\gk^+, \gk^{++})$.
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``\dot p_{\ga + 1}$ is a term for a condition of the form
$\la \dot q_{\ga + 1}, \dot r_{\eta_{\ga + 1}} \ra$ deciding
$`\gk \in j(\dot A_\ga)$', where
$\dot q_{\ga + 1}$ is a direct extension of
$\dot q_\ga$ and
$\eta_{\ga + 1} \ge \eta_\ga$,
$\eta_{\ga + 1} \in (\gk^+, \gk^{++})$''.
By the work just done, $\dot p_{\ga + 1}$ exists.
\item For a limit $\gl < \gk^{++}$,
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``\dot p_\gl = \la \dot q_\gl,
\cup_{\ga < \gl} \dot r_{\eta_\ga} \ra$
is a term so that
$\dot q_\gl$ is a direct extension of each
member of the sequence
$\la \dot q_\ga : \ga < \gl \ra$''.
\end{enumerate}
Since $M^{\gk^+} \subseteq M$,
Lemma 1.2 of \cite{G1} and the definitions of
$\dot \FR$ and $j(\dot \FQ)$ ensure
$\dot p_\gl$ exists. Further,
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``\dot p_\gl$ decides each statement
$`\gk \in j(\dot A_\ga)$' for $\ga < \gl$''.
If in $M$,
$\forces_{\FP \ast \dot \FQ} ``\dot \FR_\gk$ is trivial'',
then in the following argument,
$\dot \FR_\gk$ can essentially be ignored.
Thus, assume without loss of generality that
$\forces_{\FP \ast \dot \FQ} ``\dot \FR_\gk$
is Prikry forcing over $\gk$''. Work in
$\ov V[G \ast H]$. Define a set
${\cal U} \subseteq \wp(\gk)$ by
$C \in {\cal U}$ iff $C \subseteq \gk$ and for some
$\la r, q \ra \in G \ast H$ and some
$\dot q' = \la \emptyset, \dot B \ra$
a term for an element of $\dot \FR_\gk$, in $M$,
$\la r, \dot q, \dot q', \dot q_\ga, \dot r_{\eta_\ga} \ra =
\la r, \dot q, \dot q', \dot p_\ga \ra \forces
``\gk \in j(\dot C)$''
for some $\ga < \gk^{++}$ and some name $\dot C$ of $C$.
Note that the definition of ${\cal U}$ is independent of
a particular name $\dot C$ for $C$, since
$j''G \subseteq G$ and in $M$,
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``$If $\ga \le \gb$, then $\dot p_\ga \le \dot p_\gb$''.
Also, the definition of ${\cal U}$ is independent of
choice of $\dot q'$, since if
$\la \emptyset, \dot B \ra$ and
$\la \emptyset, \dot B' \ra$
are terms for two distinct conditions in $\FR_\gk$, then
$\la \emptyset, \dot B \cap \dot B' \ra$
is a term for a condition extending them both.
It is clear from the definition of ${\cal U}$ that
${\cal U}$ extends the $\gk$-additive normal ultrafilter
${\cal U}' \in \ov V$ over $\gk$ given by
$X \in {\cal U}'$ iff $\gk \in j(X)$. Thus,
${\cal U}$ is a filter in $\ov V[G \ast H]$.
To show that ${\cal U}$ is a $\gk$-additive
ultrafilter over $\gk$ in $\ov V[G \ast H]$,
as in \cite{AG}, we mimic the argument of
Lemma 2.2 of \cite{G1}. Let, for some limit $\ga < \gk$,
$\forces_{\FP \ast \dot \FQ} ``\la \dot C_\gb :
\gb < \ga \ra$ is a sequence of distinct subsets of
$\gk$ so that $\cup_{\gb < \ga} \dot C_\gb = \gk$''.
Then in $M$,
$\forces_{j(\FP) \ast j(\dot \FQ)}
``\la j(\dot C_\gb) : \gb < \ga \ra$ is a sequence of
subsets of $j(\gk)$ so that
$\cup_{\gb < \ga} j(\dot C_\gb) = j(\gk)$''.
For every $\gb < \ga$, there are
$\la r, q \ra \in G \ast H$ and $\ga_\gb < \gk^{++}$ so that
$\la r, \dot q \ra \forces ``\dot C_\gb =
\dot A_{\ga_\gb}$''.
Thus, by the regularity of $\gk^{++}$ in both
$\ov V$ and $\ov V[G \ast H]$, for $\gl$ defined in
$\ov V[G \ast H]$ by
$\gl = \cup_{\gb < \ga} \ga_\gb$, $\gl < \gk^{++}$. Hence, since
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``\dot p_\gl$ decides each statement
$` \gk \in j(\dot A_\gg)$' for $\gg < \gl$'',
$\forces_{\FP \ast \dot \FQ \ast \dot \FR_\gk}
``\dot p_\gl$ decides
$` \gk \in j(\dot C_\gb)$' for every $\gb < \ga$''.
Consider now in
$M[G \ast H]$ the sequence of statements
$\la \varphi_\gb : \gb < \ga \ra$, where each
$\varphi_\gb$ is the statement
$``\dot p_\gl \forces_{\dot \FR \ast j(\dot \FQ)}
`\gk \in j(\dot C_\gb)$' ''.
Since $\FR_\gk$ is in $M[G \ast H]$ Prikry forcing over $\gk$,
by $\ga$ applications in $M$ of the Prikry lemma, there is a term
$\dot q' = \la \emptyset, \dot B \ra$ for an element of
$\FR_\gk$ so that in $M[G \ast H]$,
$q'$ decides each $\varphi_\gb$ for $\gb < \ga$.
Thus, there must be some $\gg < \ga$ so that
$q' \forces_{\FR_\gk} ``\dot p_\gl
\forces_{\dot \FR \ast j(\dot \FQ)}
`\gk \in j(\dot C_\gg)$' ''.
Hence, for some $\la r, q \ra \in G \ast H$, in $\ov V$,
$\la r, \dot q \ra \forces_{\FP \ast \dot \FQ}
``$In $M^{\FP \ast \dot \FQ}$, $\dot q' \forces_{\dot \FR_\gk}
``\dot p_\gl \forces_{\dot \FR \ast j(\dot \FQ)}
`\gk \in j(\dot C_\gg)$' '' '', i.e.,
$\la r, \dot q, \dot q', \dot p_\gl \ra \forces
``\gk \in j(\dot C_\gg)$'' in $M$.
This means that ${\cal U}$ is a $\gk$-additive
ultrafilter over $\gk$ in $\ov V[G \ast H]$.
This proves Lemma \ref{l2}.
\end{proof}
Lemmas \ref{l1} and \ref{l2} complete the
proof of Theorem \ref{thm}.
\end{proof}
In conclusion, we make several remarks.
We note that since
$V^\FQ \models ``2^{\gk^+} = \gk^{++}$'',
if we redefine ${\cal U}$ in the proof of
Lemma \ref{l2} by $C \in {\cal U}$ iff
$C \subseteq {(P_\gk(\gk^+))}^{\ov V[G \ast H]}$
and for some $\la r, q \ra \in G \ast H$ and some
$\dot q' = \la \emptyset, \dot B \ra \in \FR_\gk$, in $M$,
$\la r, \dot q, \dot q', \dot p_\ga \ra \forces
``\la j(\gb) : \gb < \gk^+ \ra \in j(\dot C)$'',
then by following the above proof, as in \cite{AG},
we can show that
$V^\FQ \models ``\gk$ is $\gk^+$ strongly compact''.
Interestingly, though,
something like either the proof of Lemma 1.5 of \cite{G1}
or the proof of Lemma \ref{l2} above does not yield that
$\ov V^\FP \models ``\gk$ is $\gk^+$ strongly compact''.
This is since the proof of Lemma 1.5 of \cite{G1} requires,
for showing that $\gk$ remains measurable in $\ov V^\FP$,
both a measurable embedding $j : \ov V \to M$
witnessing the non-measurability of $\gk$ in $M$
and an enumeration in $\ov V$ of length
$\gk^+$ of all canonical terms for subsets of $\gk$
in $\ov V^\FP$. The $\gk$ closure of $M$ then
allows the appropriate definition for a measure over
$\gk$ in $\ov V^\FP$ to be given. Whereas it is
possible to construct a $\gk^+$ strongly compact embedding
$j : \ov V \to M$ witnessing either the non-$\gk^+$
strong compactness or non-measurability of $\gk$ in $M$
(simply compose an embedding $j_0 : \ov V \to N$
witnessing the $\gk^+$ supercompactness of $\gk$ with
a measurable embedding $j_1 : N \to M$ witnessing the
non-measurability of $\gk$ in $M$ to get $j$),
the enumeration in $\ov V$ of all canonical terms for subsets of
${(P_\gk(\gk^+))}^{\ov V^\FP}$
will have length $\gk^{++}$. Since $M$ is only $\gk$ closed,
there won't be enough closure to allow anything like the
proof of Lemma 1.5 of \cite{G1} to go through for $\gk^{++}$
many terms.
Also, since the forcing $\FP$ does nothing at stage $\gk$,
the proof of Lemma \ref{l2} is irrelevant
under these circumstances.
We encounter these same problems (and worse) if
$j : \ov V \to M$ witnesses only the measurability of $\gk$.
Finally, if $j : \ov V \to M$ witnesses the
$\gk^+$ supercompactness of $\gk$, then since
$M$ is $\gk^+$ closed,
$\gk$ will be measurable in $M$, and
we will be required to add
$\gk^{++}$ many new subsets to $\gk$ over $\ov V^\FP$
as well as over $M^\FP$, something we obviously
do not wish to do. Thus, whether
$\ov V^\FP \models ``\gk$ is $\gk^+$ strongly compact''
remains an intriguing open question.
We finish by observing that trying to weaken the
hypothesis of Theorem \ref{thm} seems to be very difficult.
If we attempt to push through an amalgamation of Woodin's
unpublished ideas (see also \cite{AC2} and
\cite{G3}) for forcing
the violation of GCH at a measurable $\gk$ using a
model $V$ for
``GCH + $\gk$ is $\wp^2(\gk)$-hypermeasurable'' together with the
ideas of this paper, then we run into trouble.
This is due to the lack of closure of the inner model
$M$ associated with a $\wp^2(\gk)$-hypermeasurable embedding
$j : V \to M$ and the fact that Woodin's methods for
overcoming this lack of closure fall far short of
being applicable to non-closed partial orderings
such as Prikry forcing.
On the other hand, if we just wish to force and
construct a model $V$ in which the least
measurable cardinal $\gk$ is such that
$2^\gk = \gk^+$ and there is a
$\gk$-directed closed partial ordering
$\FQ^* \in V$ so that
$V^{\FQ^*} \models ``2^\gk = \gk^{++}$ and
$\gk$ is measurable'', then the work of
\cite{AC2} shows that this can be done
starting from a ground model
$\ov V \models {\rm GCH}$
with a measurable cardinal $\gl$ satisfying
$o(\gl) = \gl^{++}$.
Note, however, that it is not necessarily
the case that $\gk = \gl$, nor is it true that
$\FQ^*$ is ${\rm Add}(\gk, \gk^{++})$.
Thus, we can ask if the
hypothesis of ``GCH + $\gk$ is $\gk^+$ supercompact''
is really necessary to prove the theorem of this paper,
or if it is possible to get by with hypotheses on
the order of hypermeasurability or measurable
cardinals of high Mitchell order.
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\end{document}