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\title{Inaccessible Cardinals,
Failures of GCH, and
Level by Level Equivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, inaccessible cardinal, GCH,
level by level equivalence between strong
compactness and supercompactness.}}
\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 discussions on
the subject matter of this paper.}
\thanks{The author wishes to thank the
referee, for helpful comments and suggestions
which have been 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}
\date{April 16, 2012\\
(revised February 16, 2013)}
\begin{document}
\maketitle
\begin{abstract}
We construct models
for the level by level equivalence between
strong compactness and supercompactness
containing failures of GCH at inaccessible cardinals.
In one of these models, no cardinal is supercompact
up to an inaccessible cardinal, and for every
inaccessible cardinal $\gd$, $2^\gd > \gd^{++}$.
In another of these models, no cardinal is
supercompact up to an inaccessible cardinal,
and the only inaccessible cardinals at
which GCH holds are also measurable.
These results extend and generalize \cite[Theorem 3]{A02a}.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{A02a}, the following theorem was proven.
\begin{theorem}{\bf(\cite[Theorem 3]{A02a})}\label{t1a}
Suppose $V \models ``$ZFC + $\K \neq \emptyset$
is the class of supercompact cardinals''. There is then
a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\K$ is the class of supercompact
cardinals''. In $V^\FP$,
$2^\gd = \gd^{++}$ if $\gd$ is inaccessible,
and $2^\gd = \gd^+$ if $\gd$ is not inaccessible.
Further, in $V^\FP$,
for every pair of regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$
supercompact, except possibly if $\gk$ is a
measurable limit of cardinals $\gd$ which are
$\gl$ supercompact, or $\gl$ is inaccessible.
\end{theorem}
%In Theorem \ref{t1a} above, suppose our
%ground model $V$ is such that
%$V \models ``$ZFC + $\gk$ is supercompact +
%No cardinal is supercompact up to an
%inaccessible cardinal''.
%This then yields the next result as an immediate corollary.
If our ground model $V$ satsifies
level by level equivalence between strong
compactness and supercompactness and
in addition is such that
$V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal is supercompact up to an inaccessible cardinal'',
then we have the next result as an immediate corollary.
\begin{theorem}\label{t1b}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence between strong compactness and
supercompactness holds.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact''. In $V^\FP$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence between strong compactness
and supercompactness holds.
Further, in $V^\FP$, for every inaccessible
cardinal $\gd$, $2^\gd = \gd^{++}$, and for every cardinal
$\gd$ which is not inaccessible, $2^\gd = \gd^+$.
\end{theorem}
The techniques of \cite{A02a}, however,
will not produce models analogous to the one
for Theorem \ref{t1b} in which
$2^\gd > \gd^{++}$ for every inaccessible cardinal $\gd$.
%, e.g., for every inaccessible cardinal $\gd$,
%$2^\gd = \gd^{+++}$, $2^\gd = \gd^{+ 4}$, etc.
In addition, \cite[Theorem 3]{A02a} says nothing
about whether it is possible to obtain similar models in which
%build a model along the same lines in which
%have a theorem analogous to \cite[Theorem 3]{A02a} in which
only certain inaccessible cardinals violate GCH.
This raises the following two questions.
\bigskip
\noindent {\bf Question 1:} Is it possible to construct models
such as those of Theorem \ref{t1b} in which, for
every inaccessible cardinal $\gd$, $2^\gd > \gd^{++}$?
\bigskip
\noindent {\bf Question 2:} Is it possible to construct models
analogous to those of \cite[Theorem 3]{A02a} in which only
certain inaccessible cardinals violate GCH?
\bigskip
The purpose of this paper is to answer the above
questions in the affirmative. Specifically,
we will prove the following theorems.
\begin{theorem} \label{t2}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence between strong compactness and
supercompactness holds.
Let $h : \gk + 1 \to {\hbox{\rm Ord}}$
satisfy the following four conditions.
\begin{enumerate}
\item\label{i1} $h(\gd) = 0$ if $\gd$ is not an inaccessible cardinal.
\item\label{i2} For $\gd$ an inaccessible cardinal,
$h(\gd)$ has the properties that
$h(\gd) > \gd^+$, $h(\gd)$ is the
successor of a cardinal of cofinality
greater than $\gd$, and $h(\gd)$
is below the least inaccessible cardinal above $\gd$.
\item\label{i3} Let $\gr_\gd$ be the
cardinal predecessor of $h(\gd)$.
If $\gd < \gk$ is
$\gr_\gd$ supercompact, there is
$j_\gd : V \to M$ witnessing the
$\gr_\gd$ supercompactness
of $\gd$ which is generated by a
supercompact ultrafilter over $P_\gd(\gr_\gd)$ with
$j_\gd(h)(\gd) = h(\gd)$.
\item\label{i4} If $\gd \le \gk$ is $\gg$
supercompact and $\gg \ge h(\gd)$,
there is
$j_{\gd, \gg} : V \to M$ witnessing the $\gg$
supercompactness of $\gd$ which is generated
by a supercompact ultrafilter over
$P_\gd(\gg)$ with $j_{\gd, \gg}(h)(\gd) = h(\gd)$.
\end{enumerate}
\noindent There is then
a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\gk$ is supercompact + No cardinal
is supercompact up to an inaccessible cardinal''.
In $V^\FP$, level by level equivalence between
strong compactness and supercompactness holds,
and for every $\gd \le \gk$ which is
an inaccessible cardinal,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$. Further,
in $V^\FP$, $\gd$ is
$\gr_\gd$ supercompact
if $\gd$ is a measurable cardinal.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + $\K \neq \emptyset$
is the class of supercompact cardinals''. There is then
a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\K$ is the class of supercompact
cardinals''. In $V^\FP$,
$2^\gd = \gd^{++}$ if $\gd$ is a nonmeasurable inaccessible
cardinal, and
$2^\gd = \gd^+$ if $\gd$ is a measurable cardinal.
Further, in $V^\FP$,
for every pair of regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$
supercompact, except possibly if $\gk$ is a
measurable limit of cardinals $\gd$ which are
$\gl$ supercompact, or $\gl$ is a nonmeasurable
inaccessible cardinal.
\end{theorem}
As a corollary to the proof of Theorem \ref{t3},
we will also have the following theorem.
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$,
no cardinal is supercompact up to an inaccessible cardinal, and
level by level equivalence between strong compactness and
supercompactness holds.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact''. In $V^\FP$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence between strong compactness
and supercompactness holds. Further, in $V^\FP$,
for every nonmeasurable inaccessible
cardinal $\gd$,
$2^\gd = \gd^{+19}$, and for every measurable cardinal
$\gd$, $2^\gd = \gd^+$.
\end{theorem}
We take this opportunity to make
some remarks concerning
the above theorems.
First, we note that although
%We note that although
the conditions on $h$ in Theorem \ref{t2} appear
to be rather technical in
nature, they are actually
satisfied by many naturally
occurring functions.
For instance, if $\gd$ is an
inaccessible cardinal
and $h(\gd)$ is
defined as $\gd^{+ 19}$, the
successor of the least $V$-strong
limit cardinal greater than $\gd$ of
cofinality $\gd^{++}$,
$\gd^{+ \go + 5}$, etc.,
then $h$ satisfies the conditions
of Theorem \ref{t2}.
In addition, we observe that
in Theorems \ref{t1b}, \ref{t2}, and \ref{t4},
it immediately follows
that $\gk$ is the
least supercompact cardinal.
This is because in each case, in $V$ and
$V^\FP$, no cardinal is supercompact up to an
inaccessible cardinal.
%Further, whereas for \cite[Theorem 3]{A02a},
%it is not possible to infer that
%$V^\FP \models ``$If $\gk < \gl$ are such that
%$\gk$ is $\gl$ strongly compact, and $\gl$ is
%measurable, then $\gk$ is $\gl$ supercompact'',
Finally, in Theorem \ref{t3}, we explicitly note
that our techniques require it must
be the case that $2^\gd = \gd^{++}$ if
$\gd$ is an inaccessible cardinal which
is not also measurable. However, as our proof will show,
there are many different possible values for
$2^\gd$ in Theorem \ref{t4}
(e.g., $\gd^{++}$, $\gd^{+ 5}$, the successor of
the first $\ha$ fixed point above $\gd$, etc.)
if $\gd$ is an inaccessible
cardinal which is not also measurable.
We now give some
preliminary information
concerning notation and terminology.
For anything left unexplained,
readers are urged to consult
\cite{AS97a} or \cite{AS97b}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
%and $p \decides \varphi$ means that $p$ decides $\varphi$.
For $\gk$ a regular cardinal and $\gl$
an ordinal,
$\add(\gk, \gl)$ is the
standard partial ordering for adding
$\gl$ many Cohen subsets of $\gk$.
For $\ga < \gb$ ordinals,
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
%If $\FP$ is a reverse Easton iteration
%such that at stage $\ga$, a nontrivial
%forcing is done adding a subset
%of $\gd$, then we will say that
%$\gd$ is in the field of $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
The partial ordering
$\FP$ is {\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
$ \FP$ is {\em ${<}\gk$-strategically closed}
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
It is in addition the case that if $\FP$
is $\gk$-directed closed, then $\FP$ is
${<}\gk$-strategically closed.
Suppose $V$ is a model of ZFC
in which for all regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
Such a universe will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
%Any model witnessing level by
%level equivalence between strong
%compactness and supercompactness
%also witnesses the Kimchi-Magidor
%property \cite{KM} that the classes
%of strongly compact and supercompact
%cardinals coincide precisely,
%except at measurable limit points.
The exception
is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
%Models in which level by level
%equivalence between strong compactness
%and supercompactness holds nontrivially
%were first constructed in \cite{AS97a}.
%We take this opportunity to
%elaborate a bit more on the notion of
%level by level equivalence between strong
%compactness and supercompactness.
Any model of ZFC with this property
also witnesses the Kimchi-Magidor
property \cite{KM} that the classes
of strongly compact and supercompact
cardinals coincide precisely,
except at measurable limit points.
Models in which GCH and level by level
equivalence between strong compactness
and supercompactness hold nontrivially
were first constructed in \cite{AS97a}.
We assume familiarity with the
large cardinal notions of
measurability, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{K} for further details.
%We do mention, however, that
%the cardinal $\gk$ is ${<} \gl$
%strongly compact or ${<} \gl$ supercompact if
%$\gk$ is $\gg$ strongly compact or $\gg$
%supercompact for every $\gg < \gl$.
We do note, however, that
we will say {\em $\gk$ is supercompact
up to the inaccessible cardinal $\gl$} if
$\gk$ is $\gd$ supercompact for every
$\gd < \gl$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2} and \cite{H3}
will be employed in the
proofs of Theorems \ref{t2} -- \ref{t4}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2} and \cite{H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{H3},
$\FP$ is {\em mild}
with respect to a cardinal $\gk$
iff every set of ordinals $x$ in
$V^\FP$ of size less than $\gk$ has
a ``nice'' name $\tau$
in $V$ of size less than $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2}, \cite{H3}, and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2} and \cite{H3}
we will be using
is then the following.
\begin{theorem}\label{gf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
Finally, at several junctures throughout
the course of this paper, we will mention the
``standard lifting techniques'' for lifting a
$\gl$ supercompactness embedding $j : V \to M$
generated by a supercompactness measure over $P_\gk(\gl)$
to a generic extension given by a
suitably defined Easton support iteration.
Although there are numerous references to this
in the literature, we will use the argument found in
\cite[Theorem 4]{A01} as the basis for the sketch
we are about to present.
Very briefly, this argument assumes the following.
\begin{enumerate}
\item $V \models {\rm GCH}$.
\item $\gl$ is a regular cardinal.
\item $\FP \ast \dot \FQ =
\la \la \FP_\ga, \dot \FQ_\ga \ra : \ga \le \gk \ra$
is an Easton support iteration
having length $\gk + 1$.
\item For any inaccessible cardinal $\gd \le \gk$,
$\forces_{\FP_\gd} ``\dot \FQ_\gd$ is
${<} \gd$-strategically closed''.
%\item $\forces_{\FP_\gk} ``\dot \FQ_\gk$
%is $\gk$-directed closed''.
\item $G_0 \ast G_1$ is $V$-generic over $\FP \ast \dot \FQ$.
\item $\forces_{\FP} ``\card{\dot \FQ} \le \gl$ and
$\dot \FQ$ is $\gk$-directed closed''.
\item $j(\FP \ast \dot \FQ) =
\FP \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$.
\end{enumerate}
%Under these circumstances,
\noindent Since $V \models {\rm GCH}$,
$M[G_0][G_1] \models ``\card{\FR} = j(\gk)$'', and
$V \models ``\card{j(\gk^+)} = \card{j(2^\gk)} =
\card{\{f : f : P_\gk(\gl) \to \gk^+\}} =
\card{\{f : f : \gl \to \gk^+\}} =
\card{\{f : f : \gl \to \gl\}}$'',
$V[G_0][G_1] \models``$There are $\gl^+ = 2^\gl =
\card{j(\gk^+)} = \card{j(2^\gk)}$ many dense
open subsets of $\FR$ present in $M[G_0][G_1]$''.
Because $M[G_0][G_1]$
remains $\gl$-closed with respect to $V[G_0][G_1]$
and $\FR$ is $\gl$-strategically closed in both
$M[G_0][G_1]$ and $V[G_0][G_1]$, working in
$V[G_0][G_1]$, it is possible to build an
$M[G_0][G_1]$-generic object $G_2$ over $\FR$ such that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
Still working in $V[G_0][G_1]$, one then lifts $j$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$.
Since $M[G_0][G_1][G_2]$ remains $\gl$-closed
with respect to $V[G_0][G_1]$ and
$V[G_0] \models ``\card{\FQ} \le \gl$'', there is a
master condition $q \in V[G_0][G_1]$
for $\{j(p) : p \in G_1\}$.
%Using GCH in $V$,
Because $V \models ``\card{j(\gl^+)} = \card{j(2^\gl)} =
\card{\{f : f : P_\gk(\gl) \to \gl^+\}} =
\card{\{f : f : \gl \to \gl^+\}} = \gl^+$'' and
$M[G_0][G_1][G_2] \models ``\card{j(\FQ)} \le j(\gl)$'',
we may then build in $V[G_0][G_1]$ an
$M[G_0][G_1][G_2]$-generic object $G_3$ for
$j(\FQ)$ containing $q$.
It is then the case that
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3$,
so we may fully lift $j$ in $V[G_0][G_1]$ to a
$\gl$ supercompactness embedding
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
This argument remains valid (and in fact becomes
even simpler) if no forcing is done at
stage $\gk$ in $V$, i.e., if
$\dot \FQ$ is a term for trivial forcing.
%Finally, we mention that
%although readers need not be familiar
%with all of the details of \cite{AS97a} and
%\cite{AS97b},
%we strongly recommend that copies
%of these papers be kept
%close at hand to improve
%comprehensibility when reading this paper.
\section{Forcing Notions from \cite{AS97a}
and \cite{AS97b}}\label{s2}
In order to present in a meaningful way the iteration
to be used in the proof of Theorem \ref{t2},
we first recall the definitions and properties of the
fundamental building blocks
of this partial ordering.
In particular, we describe now
a specific form of
the partial orderings
$\FP^0_{\gd, \gl}$,
$\FP^1_{\gd, \gl}[S]$,
and
$\FP^2_{\gd, \gl}[S]$
of \cite[Section 4]{AS97b},
where the fixed but arbitrary regular cardinal
$\gg < \gd$ is replaced by the specific
regular cardinal $\go$.
%, but all other relevant definitions and proofs are the same
%as in \cite[Section 4]{AS97b}.
So that readers are not overly burdened, we
abbreviate our definitions and descriptions somewhat.
Full details may be found by consulting
\cite{AS97b}, along with
the relevant portions of \cite{AS97a}.
We quote nearly verbatim from \cite[Section 2]{A06}.
Fix $\gd < \gl$, $\gl > \gd^+$ regular cardinals in our
ground model $V$, with $\gd$ inaccessible and $\gl$
%either inaccessible or
the successor of a cardinal of cofinality
greater than $\gd$.
We assume GCH holds for all
cardinals $\eta \ge \gd$.
The first notion of forcing $\FP^0_{\gd, \gl}$ is just
the standard notion of forcing
for adding a nonreflecting stationary
set of ordinals $S$ of cofinality
$\go $ to $\gl$.
Next, work in
$V_1 = V^{\FP^0_{\gd, \gl}}$, letting $\dot S$
be a term always forced to denote $S$.
$\FP^2_{\gd, \gl}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ nonstationary).
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\ga : \ga \in S \ra$,
the existence of which is given by
\cite[Lemma 1]{AS97a} and \cite[Lemma 1]{AS97b}.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gd, \gl}
[S] $.
First, since each element of
$S$ has cofinality $\go$, the proofs of
\cite[Lemma 1]{AS97a} and \cite[Lemma 1]{AS97b}
show each $x \in X$ can be assumed to be
such that order-type$(x) = \go$. Then,
$\FP^1_{\gd, \gl}[ S]$ is defined as the set of all
4-tuples $\la w, \ga, \bar r, Z \ra$ satisfying the
following properties.
\begin{enumerate}
\item $w \in {[\gl]}^{< \gd}$.
\item $\ga < \gd$.
\item $ \bar r = \la r_i : i \in w \ra$ is a
sequence of functions from $\ga$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\ga$.
\item $Z \subseteq \{x_\gb : \gb \in S\}$
is a set such that if $z \in Z$, then for some
$y \in {[w]}^\go$, $y \subseteq z$ and $z - y$
is bounded in the $\gb$ such that $z = x_\gb$.
\end{enumerate}
%\noindent As in Section 4 of \cite{AS97b}
%and Section 1 of \cite{AS97a}, the
%definition of $Z$ implies
%$|Z| < \gd$.
\noindent
The ordering on $\FP^1_{\gd, \gl}[S]$ is given by
$\la w^1, \ga^1, \bar r^1, Z^1 \ra \le
\la w^2, \ga^2, \bar r^2, Z^2 \ra$ iff the following hold.
\begin{enumerate}
\item $w^1 \subseteq w^2$.
\item $\ga^1 \le \ga^2$.
\item If $i \in w^1$, then $r^1_i
\subseteq r^2_i$.
\item $Z^1 \subseteq Z^2$.
\item If $z \in Z^1 \cap {[w^1]}^\go$ and
$\ga^1 \le \ga < \ga^2$, then $|\{i \in z :
r^2_i(\ga) = 0\}| = |\{i \in z : r^2_i(\ga) = 1\}| = \go$.
\end{enumerate}
The proof of \cite[Lemma 4]{AS97a} shows that
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$ is forcing equivalent to
${\hbox{\rm Add}}(\gl, 1) \ast \dot {\hbox{\rm Add}}(\gd, \gl)$.
The proofs of \cite[Lemmas 3 and 5]{AS97a}
and \cite[Lemma 6]{AS97b} show that
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ preserves cardinals
and cofinalities, is $\gl^+$-c.c.,
is ${<}\gd$-strategically closed, and is such that
$V^{\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]} \models
``2^\eta = \gl$ for every cardinal $\eta \in [\gd, \gl)$
and $\gd$ is nonmeasurable''.
%The arguments used are valid
%for any cofinality, and in particular, hold
These proofs are valid regardless of the
cofinality of the ordinals in $S$, and in particular, hold
when the fixed but
arbitrary regular cardinal $\gg < \gd$ found
in the definitions given in \cite[Section 4]{AS97b}
is replaced by the specific regular cardinal $\go$.
\section{The Proofs of Theorems \ref{t2} and \ref{t3}}\label{s3}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose $V$, $h$, and $\gk$ are as in
the hypotheses for Theorem \ref{t2}.
In particular, recall that by
condition (\ref{i3}) on $h$,
$\gr_\gd$ is the cardinal predecessor of $h(\gd)$, so
$h(\gd) = \gr^+_\gd$.
The partial ordering $\FP$ used in
the proof of Theorem \ref{t2}
is the Easton support iteration having
length $\gk + 1$ which begins by
forcing with $\add(\go, 1)$
and then does trivial forcing,
except at stages $\gd \le \gk$
which are inaccessible cardinals in $V$.
If such a $\gd$
is not $\gr_\gd$ supercompact, then
the forcing done at stage $\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}]$,
where $\dot S_{h(\gd)}$ is a term
for the nonreflecting stationary
set of ordinals of cofinality $\go$
introduced by $\FP^0_{\gd, h(\gd)}$.
If such a $\gd$ is $\gr_\gd$ supercompact,
then the forcing done at stage $\gd$ is
$\FP^0_{\gd, h(\gd)} \ast
(\FP^1_{\gd, h(\gd)}[\dot S_{h(\gd)}] \times
\FP^2_{\gd, h(\gd)}[\dot S_{h(\gd)}])$,
where $\dot S_{h(\gd)}$ is as in the
previous sentence.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is supercompact''.
\end{lemma}
\begin{proof}
We modify the proof of \cite[Lemma 3.1]{A06}.
Suppose $\gl > h(\gk)$ is any regular
cardinal, and $j : V \to M$
is any elementary embedding witnessing
the $\gl$ supercompactness of $\gk$
which is generated by a supercompact
ultrafilter over $P_\gk(\gl)$.
Since $V \models ``$No cardinal is
supercompact up to an inaccessible cardinal'',
$M \models ``$No cardinal in the half-open
interval $(\gk, \gl]$ is inaccessible''.
From this, it immediately follows that
$j(\FP) = \FP \ast \dot \FQ$,
where the first ordinal at which
$\dot \FQ$ is forced to act nontrivially
is well above $\gl$.
Since $V \models {\rm GCH}$,
the standard lifting arguments mentioned in Section \ref{s1}
%standard lifting arguments (as given,
%e.g., in the proof of
%\cite[Theorem 4]{A01})
%are then applicable and show that
%Standard arguments (see, e.g., Lemma ??? of ???)
now apply and show that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ was arbitrary,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal is supercompact
up to an inaccessible cardinal''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``$There exists a cardinal
which is supercompact up to an inaccessible cardinal''.
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
%By Hamkins' Gap Forcing Theorem of \cite{H2} and \cite{H3},
By Theorem \ref{gf}, this
factorization of $\FP$ and the fact that
forcing cannot create an inaccessible
cardinal indicate
that any $\gd$ which is supercompact
up to an inaccessible cardinal in
$V^\FP$ had to have been supercompact
up to an inaccessible cardinal in $V$.
Since $V \models ``$No cardinal is
supercompact up to an inaccessible cardinal'',
this is impossible.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
We modify the proof of \cite[Lemma 3.3]{A06}.
Suppose
$V^\FP \models ``\gd < \gl$ are
regular cardinals such that
$\gd$ is $\gl$ strongly compact''.
We begin by noting that
$V \models ``\gd$ is $\gl$ supercompact''.
To see this, by the definition of $\FP$,
it is easily established that
any subset of $\gd$ in
$V^\FP$ of size below $\gd$
has a name of size
below $\gd$ in $V$.
Therefore, by the factorization of
$\FP$ given in the proof of Lemma \ref{l2}
and Theorem \ref{gf},
%the results of \cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
Since $V \models ``$No cardinal is
supercompact up to an inaccessible
cardinal'', $V \models ``\gd$ is not a
measurable limit of cardinals $\gg$ which
are $\gl$ supercompact''. Thus, by
level by level equivalence between strong
compactness and supercompactness,
$V \models ``\gd$ is $\gl$ supercompact''.
Further, $V \models ``\gl$ is below the
least inaccessible cardinal $\gz$ above $\gd$''.
Continuing with the proof of Lemma \ref{l3},
because $V \models ``$No cardinal is
supercompact up to an inaccessible
cardinal and $\gk$ is supercompact'',
$V \models ``$No cardinal $\gr > \gk$
is inaccessible''.
Consequently, $V^\FP \models ``$No cardinal
$\gr > \gk$ is inaccessible'' as well.
From this, it immediately follows that $\gd \le \gk$.
By Lemma \ref{l1}, Lemma \ref{l3}
is true if $\gd = \gk$.
It therefore
suffices to prove Lemma \ref{l3}
when $\gd < \gk$, which we
assume for the duration of the
proof of this lemma.
Let $A = \{\gg \le \gd : \gg$
is an inaccessible cardinal$\}$. Write
$\FP = \FP_A \ast \dot \FQ$, where
$\FP_A$ is the portion of $\FP$
acting on ordinals at most $\gd$, and
$\dot \FQ$ is a term for the
rest of $\FP$, i.e., the portion
of $\FP$ acting on ordinals above $\gd$.
Since $\gl < \gz$ and
$\forces_{\FP_A} ``\dot \FQ$ is
$\gz$-strategically closed'',
to complete the proof of Lemma \ref{l3},
it hence suffices to show that
$V^{\FP_A} \models ``\gd$ is $\gl$
supercompact''.
Consider now the following two cases. \bigskip
\noindent Case 1: $\sup(A) = \gs < \gd$.
If this is true, then by the definition of $\FP$,
it must be the case that $\card{\FP_A} < \gd$.
Thus, by the L\'evy-Solovay results \cite{LS},
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''
as well. \bigskip
\noindent Case 2: $\sup(A) = \gd$.
It must be the case that
$V \models ``\gd$ is
$\gr_\gd$ supercompact'', because
otherwise, by the definition of $\FP$,
$V^{\FP_A} \models ``\gd$ is not
measurable''.
However, by the arguments found in
\cite[next to last paragraph on page 2033]{AS97b},
$V^{\FP_A} \models ``\gd$ is
$\gr_\gd$ supercompact''.
Hence, we may assume without loss of
generality that $\gl \ge h(\gd) = \gr^+_\gd$.
Consequently, let $j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gd$
satisfying condition (\ref{i4})
of Theorem \ref{t2} (so
$j$ is generated by a supercompact ultrafilter
over $P_\gd(\gl)$ and is such that
$j(h)(\gd) = h(\gd)$).
Since $\gl \ge h(\gd)$,
$\FP_A$ is forcing equivalent to
$\FP_\gd \ast \dot \FQ^*$, where
$\forces_{\FP_\gd} ``|\dot \FQ^*|
= h(\gd) \le \gl$ and
$\dot \FQ^*$ is $\gd$-directed closed''.
($\FQ^*$ is forcing equivalent to
$\add(h(\gd), 1) \ast
\dot \add(\gd, h(\gd))$.)
In addition, the same reasoning as
found in the proof of Lemma \ref{l1}
shows that
$M \models ``$No cardinal in the
half-open interval $(\gd, \gl]$ is inaccessible''.
Thus, $j(\FP_\gd \ast \dot \FQ^*)$ is
forcing equivalent to
$\FP_\gd \ast \dot \FQ^* \ast \dot \FR
\ast j(\dot \FQ^*)$, where the first
ordinal at which $\dot \FR$ is forced to act nontrivially
is well above $\gl$.
As in the proof of Lemma \ref{l1},
the standard lifting arguments mentioned in Section \ref{s1}
%the standard lifting arguments
%(as given, e.g., in the proof of Theorem 4 of \cite{A01})
are then once again
applicable and show that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Case 2 and Lemma \ref{l3}.
\end{proof}
By the remarks in the last
paragraph of Section \ref{s2}, the fact that
by condition (2) of Theorem \ref{t2},
for any inaccessible cardinal $\gd$,
$h(\gd)$ is below the least inaccessible
cardinal above $\gd$, and the
definition of $\FP$,
$V^\FP \models ``$For every $\gd \le \gk$
which is an inaccessible cardinal,
$2^\gg = h(\gd)$ for all cardinals $\gg \in [\gd, h(\gd))$''.
By the proof of Lemma \ref{l3},
$V^\FP \models ``\gd$ is $\gr_\gd$ supercompact
if $\gd$ is a measurable cardinal''.
These observations,
together with Lemmas \ref{l1} -- \ref{l3},
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}.
\begin{proof}
Let
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals''.
Without loss of generality, by first doing
a preliminary forcing as in \cite{AS97a} if necessary,
we may also assume
%that $V$ is as in Theorem \ref{t0}, i.e.,
that GCH and level by level equivalence between
strong compactness and supercompactness
hold in $V$.
This allows us to define in $V$ our partial
ordering $\FP$ as the Easton support iteration
which begins by forcing with $\add(\go, 1)$ and then does
nontrivial forcing only at stages $\gd$ which are
inaccessible cardinals in $V$. If
$V \models ``\gd$ is inaccessible but nonmeasurable'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for $\add(\gd, \gd^{++})$.
If $V \models ``\gd$ is measurable'', then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for $\add(\gd, \gd^{+})$.
Exactly the same arguments
as in the proof of \cite[Theorem 3]{A02a} (i.e.,
standard arguments in tandem with Theorem \ref{gf}) show
%the results of \cite{H2} and \cite{H3}) show
that cardinals and cofinalities are preserved
when forcing with $\FP$ and
$V^\FP \models ``$ZFC + $\K$ is the class of
supercompact cardinals''.
By the definition of $\FP$, it is further the case that
the inaccessible cardinals of $V$ and
$V^\FP$ are precisely the same and
$V^\FP \models ``2^\gd = \gd^{++}$ if
$\gd$ is inaccessible but nonmeasurable in $V$ +
$2^\gd = \gd^+$ if $\gd$ is measurable in $V$''.
Thus, the proof of Theorem \ref{t3} will be
complete once we have established the following
three lemmas.
\begin{lemma}\label{l4}
If
$V \models ``\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact and $\gl$
is a successor cardinal'', then
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 3.1]{A02a},
quoting almost verbatim when appropriate.
If $\gk$ and $\gl$ are as in the hypotheses of Lemma \ref{l4},
then we consider the following two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: Either $\gl$ is not the successor of an
inaccessible cardinal or $\gl$ is the
successor of a measurable cardinal. Write
$\FP = \FP_\gl \ast \dot \FP^\gl$, where
$\FP_\gl$ acts nontrivially on
ordinals below $\gl$, and $\FP^\gl$
consists of the rest of $\FP$.
By the choice of $\gl$, $|\FP_\gl| \le \gl$.
Suppose $j : V \to M$ is an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk$
which is generated by a supercompact ultrafilter
over $P_\gk(\gl)$ and that $\gl = \gd^+$.
Note that since $2^\gd = \gd^+ = \gl$ and $M^\gl \subseteq M$,
$V \models ``\gl$ is the successor of a measurable cardinal''
iff
$M \models ``\gl$ is the successor of a measurable cardinal''.
Hence, by the definition of $\FP_\gl$,
no matter which of the two clauses in Case 1 holds,
%if $j : V \to M$ is an embedding witnessing the
%$\gl$ supercompactness of $\gk$,
$\FP_\gl$ is an initial segment of
$j(\FP_\gl)$.
Therefore,
the standard lifting arguments mentioned in Section \ref{s1}
%the standard lifting arguments
%mentioned in the proof of Lemma \ref{l1}
once again show that
%reverse Easton arguments show
$V^{\FP_\gl} \models ``\gk$ is $\gl$ supercompact''.
Since
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
${(2^{[\gl]^{< \gk}})}^{+}$-directed closed'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models ``\gk$ is
$\gl$ supercompact''.
\bigskip
Case 2: $\gl$ is the successor of a
nonmeasurable inaccessible cardinal.
Once again, write
$\FP = \FP_\gl \ast \dot \FP^\gl$, where
$\FP_\gl$ acts nontrivially on
ordinals below $\gl$, and $\FP^\gl$ is the rest of
$\FP$.
%i.e., the ordinals on which
%$\FP^\gl$ is forced to act nontrivially are above $\gl$.
In this instance, it is not the case that
$|\FP_\gl| \le \gl$, since for the $\gd$ such that
$\gl = \gd^+$,
$|\FP_\gl| = \gd^{++} = \gl^+ > \gl$.
However, arguments originally due to
Magidor \cite{Ma2}, which are also given in
both \cite[pages 119--120]{AS97a} and
\cite[Case 2 of Lemma 3.1]{A02a} and are found
other places in the literature as well,
will yield that
$V^{\FP_\gl} \models ``\gk$ is $\gl = \gd^+$
supercompact''.
For the convenience of readers, we present these
arguments below.
\setlength{\parindent}{1.5em}
Write
$\FP_\gl = \FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++})$,
where $\FQ_0$ acts nontrivially on
ordinals below $\gk$, and
$\dot \FQ_1$ is forced to act nontrivially on
all remaining ordinals
in the interval $[\gk, \gd)$. Let $G$ be $V$-generic
over $\FP_\gl$, with
$G_0 \ast G_1 \ast G_2$ the corresponding
factorization of $G$. Fix
$j : V \to M$ an elementary embedding witnessing
the $\gl = \gd^+ = 2^\gd$ supercompactness of
$\gk$ which is generated by a supercompact
ultrafilter $\U$ over $P_\gk(\gl)$.
Since $M \models ``\gl$ is the successor of a
nonmeasurable inaccessible cardinal'', we then have
$j(\FP_\gl) = \FQ_0 \ast \dot \FQ_1
\ast \dot \add(\gd, \gd^{++}) \ast \dot \FR_0
\ast \dot \FR_1$, where $\dot \FR_1$ is a term for
$\add(j(\gd), j(\gd^{++}))$ as computed in
$M^{\FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++}) \ast \dot \FR_0}$.
Therefore, as in \cite[Case 2 of Lemma 3.1]{A02a},
%by using the argument given in Lemma \ref{l1},
since $M[G_0][G_1][G_2]$ remains $\gl$-closed with respect to
$V[G_0][G_1][G_2]$ and $V \models {\rm GCH}$,
it is possible working in $V[G_0][G_1][G_2]$
to construct an $M[G_0][G_1][G_2]$-generic object
$G_3$ over $\FR_0$ and lift $j$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
It is then the case that
$M[G_0][G_1][G_2][G_3]$ remains $\gl$-closed
with respect to $V[G_0][G_1][G_2]$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_2 \rest \ga = \{p \rest \ga : p \in G_2\}$. Clearly,
$V[G_0][G_1][G_2] \models ``|G_2 \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2][G_3]}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) : p \in G_2 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2][G_3]}$.
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
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_2 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_2 \rest \gb$ and $G_2 \rest \ga$, for
$p \rest \gb = q \in G_2 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2][G_3] \models ``$GCH holds
for all cardinals greater than or equal to $j(\gd)$'',
\break $M[G_0][G_1][G_2][G_3] \models ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2][G_3]$ is a
maximal antichain of $\add(j(\gd), j(\gd^{++}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{++}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a supercompact ultrafilter over
$P_\gk(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra \in
V[G_0][G_1][G_2]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G_0][G_1][G_2][G_3]$.
Working in $V[G_0][G_1][G_2]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$ of
elements of $\add(j(\gd), j(\gd^{++}))$ such that
$\forall \ga \in (\gd, \gd^{++}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_4 = \{p \in \add(j(\gd), j(\gd^{++})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2][G_3]$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2][G_3]$ is
$\gd^+$-closed with respect to
$V[G_0][G_1][G_2]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, 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
$M[G_0][G_1][G_2][G_3]$ is $\gd^+$-closed
%under $\gd^+$ sequences
with respect to
$V[G_0][G_1][G_2]$, 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
$M[G_0][G_1][G_2][G_3]$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gk(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gk(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gk(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{++})$ be such that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{++})$ is such that for some
$\ga \in (\gd, \gd^{++})$, $p = p \rest \ga$,
$G_4$ is such that if
$p \in G_2$, $j(p) \in G_4$.
Thus, working in $V[G_0][G_1][G_2]$,
we have shown that $j$ lifts to
$j : V[G_0][G_1][G_2] \to M[G_0][G_1][G_2][G_3][G_4]$,
i.e.,
$V[G_0][G_1][G_2] \models ``\gk$ is $\gl = \gd^+$
supercompact''.
Since as in Case 1,
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
${(2^{[\gl]^{< \gk}})}^{+}$-directed closed'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models
``\gk$ is $\gl$ supercompact''.
This completes the proof of Case 2 and Lemma
\ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``2^\gd = \gd^+$ if $\gd$
is a measurable cardinal''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd$ is a measurable cardinal''.
As in the proof of Lemma \ref{l2}, write
$\FP = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
Again by Theorem \ref{gf}, this
%Hamkins' Gap Forcing Theorem of \cite{H2} and \cite{H3}, this
factorization of $\FP$ indicates
that $\gd$ is measurable in $V$.
As we have already observed, the measurability
of $\gd$ in $V$ implies that
$V^\FP \models ``2^\gd = \gd^+$''.
Thus, %by our remarks in the preceding sentence,
the proof of Lemma \ref{l5} will be
complete once we have shown that
$V^\FP \models ``\gd$ is a measurable cardinal''.
However, since $\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$,
where $\FP_{\gd + 1}$ acts nontrivially on
ordinals less than or equal to $\gd$ and
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ is
$(2^\gd)^+$-directed closed'',
it will suffice to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is a measurable cardinal''.
To do this, we combine
the standard lifting arguments mentioned in Section \ref{s1}
%the standard lifting techniques
with Magidor's argument found in the proof of
Case 2 of Lemma \ref{l4} above
and an idea of Levinski found in \cite{Le}.
Suppose $G \ast H$ is $V$-generic over
$\FP_{\gd + 1} = \FP_\gd \ast \dot \add(\gd, \gd^+)$.
Let $j : V \to M$ be an elementary embedding
witnessing $\gd$'s measurability
generated by a normal measure over $\gd$ such that
$M \models ``\gd$ is nonmeasurable''.
Write $j(\FP_{\gd + 1}) = j(\FP_\gd \ast \dot \add(\gd, \gd^+))
= \FP_\gd \ast \dot \FQ_\gd \ast \dot \FR \ast \dot \add(j(\gd),
j(\gd^+))$, where $\dot \FQ_\gd$ is a term for the stage $\gd$
forcing done in $M^{\FP_\gd}$ and $\dot \FR$ is a term for the
forcing done in $M^{\FP_\gd \ast \dot \FQ_\gd} = M^{\FP_{\gd + 1}}$
(strictly) between stages $\gd$ and $j(\gd)$. Because
$M \models ``\gd$ is nonmeasurable'',
$\dot \FQ_\gd$ is a term for $(\add(\gd, \gd^{++}))^{M^{\FP_\gd}}$.
We use now Levinski's ideas of \cite{Le} to show that
it is possible to rearrange $H$ to form an
$M[G]$-generic object $H'$ over $\FQ_\gd$ in $V[G][H]$.
Since $V \models {\rm GCH}$ and
$j$ is generated by an ultrafilter over $\gd$,
$V \models ``\card{(\gd^{++})^M} = \gd^+$''.
In addition, since $\FP$ is an Easton support iteration,
$\FP_\gd$ is $\gd$-c.c., which means that
cardinals at and above $\gd$ are preserved from
$V$ to $V[G]$ and $M$ to $M[G]$. Hence,
$(\gd^{++})^{M[G]} = (\gd^{++})^M$,
$(\gd^+)^{V[G]} = (\gd^+)^V$, and
$V[G] \models ``\card{(\gd^{++})^{M[G]}} = \gd^+$''.
Let $(\gd^{++})^{M[G]} = \gr$.
%From this, it follows that in $V[G]$, $(\add(\gd, \gd^{++})^{M[G]}$
%has the form $\add(\gd, \gr)$, where
%$V[G] \models ``\card{\gr} = \gd^+$''.
Working in $V[G]$, %(or $V$),
we may therefore let $f : \gd^+ \to \gr$
be a bijection. For any
$p \in \add(\gd, \gd^+)$, $g(p) = \{\la \la \ga, f(\gb) \ra, \gg \ra
: \la \la \ga, \gb \ra, \gg \ra \in p\} \in (\add(\gd, \gr))^{M[G]}$.
As can be easily checked (see \cite{Le}),
$H' = \{g(p) : p \in H\}$ is an $M[G]$-generic object over
$(\add(\gd, \gr))^{M[G]}$.
We continue with the lifting argument.
Since $M$ is $\gd$-closed with respect to $V$,
$\FP_\gd \ast \dot \add(\gd, \gd^+)$ is
$\gd^+$-c.c$.$ in $V$, and $\FP_\gd \ast \dot \FQ_\gd$
is $\gd^+$-c.c$.$ in $M$,
$M[G][H']$ remains $\gd$-closed with respect to $V[G][H]$.
Therefore, since $j$ is generated by an ultrafilter over $\gd$
and $V \models {\rm GCH}$,
the standard arguments mentioned in Section \ref{s1}
%the proof of Lemma \ref{l1}
%(as found once again in the proof of \cite[Theorem 4]{A01})
show that it is possible to construct in $V[G][H]$ an
$M[G][H']$-generic object $H''$ over $\FR$ and lift
$j$ to $j : V[G] \to M[G][H'][H'']$.
Because the first ordinal at which $\FR$ does
nontrivial forcing is above $(\gd^{++})^{M[G]}$,
$M[G][H'][H'']$ remains $\gd$-closed with respect to
$V[G][H]$.
It remains to lift $j$ in $V[G][H]$
through the stage $\gd$ forcing
$\add(\gd, \gd^+)$.
However, Magidor's argument as given in
the proof of Case 2 of Lemma \ref{l4} above
for the construction of the generic object $G_4$,
replacing the use of a normal measure over
$P_\gk(\gl)$ with a normal measure over $\gd$,
allows us working in $V[G][H]$ to construct
an $M[G][H'][H'']$-generic object $H'''$
for $\add(j(\gd), j(\gd^+))$
such that if $p \in H$, $j(p) \in H'''$.
Thus, working in $V[G][H]$,
we have shown that $j$ lifts to
$j : V[G][H] \to M[G][H'][H''][H''']$,
i.e.,
$V[G][H] \models ``\gd$ is a measurable cardinal''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$For every pair of regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except possibly if
$\gk$ is a measurable limit of cardinals $\gd$
which are $\gl$ supercompact, or $\gl$ is
a nonmeasurable inaccessible cardinal''.
\end{lemma}
\begin{proof}
We significantly modify
the proof of \cite[Lemma 3.2]{A02a}.
Suppose
$V^\FP \models ``\gk < \gl$ are regular,
$\gl$ is either a successor or measurable cardinal,
$\gk$ is $\gl$ strongly compact, and
$\gk$ is not a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact''.
By its definition,
forcing with $\FP$ preserves all cardinals and cofinalities.
In addition, by the proof of Lemma \ref{l5},
$V \models ``\gl$ is a measurable cardinal'' iff
$V^\FP \models ``\gl$ is a measurable cardinal''.
Consequently, $V \models ``\gl$ is either a
successor or measurable cardinal''.
Consider now the following two cases.
\bigskip\noindent Case 1: $\gl$ is a successor
cardinal in both $V^\FP$ and $V$.
By the definition of $\FP$,
any subset of $\gk$ in $V^\FP$ of size below $\gk$
has a name of size below $\gk$ in $V$.
Thus, by the factorization of $\FP$ given in
the second sentence of the proof of
Lemma \ref{l5} and Theorem \ref{gf},
%the results of \cite{H2} and \cite{H3},
$V \models ``\gk$ is $\gl$ strongly compact''.
By Lemma \ref{l4}, any cardinal $\gd$ such that
$V \models ``\gd$ is $\gl$ supercompact''
remains $\gl$ supercompact in $V^\FP$.
This means
$V \models ``\gk$ is not a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact''.
Hence, by level by level equivalence between
strong compactness and supercompactness in $V$,
$V \models ``\gk$ is $\gl$ supercompact'',
so another application of Lemma \ref{l4}
implies that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\bigskip\noindent Case 2: $\gl$ is a measurable
cardinal in both $V^\FP$ and $V$.
As in Case 1, $V \models ``\gk$ is $\gl$ strongly compact''.
It is in addition true that
$V \models ``\gk$ is not a measurable limit of
cardinals $\gd$ which are $\gl$ supercompact''.
To see this, assume not, and let $\gd < \gk$ be such that
$V \models ``\gd$ is $\gl$ supercompact''.
It is then true that
$V \models ``\gd$ is $\gg$ supercompact for
every successor cardinal $\gg < \gl$'', so by
Lemma \ref{l4} and the fact that forcing
with $\FP$ preserves cardinals and cofinalities,
$V^\FP \models ``\gd$ is $\gg$ supercompact for
every successor cardinal $\gg < \gl$''.
By an application of the alternate
proof sketched in \cite[Exercise 22.9]{K},
since $V^\FP \models ``\gl$ is a measurable cardinal'',
$V^\FP \models ``\gd$ is $\gl$ supercompact''. Thus, if
$V \models ``\gk$ is a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact'', then
$V^\FP \models ``\gk$ is a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact'', a contradiction.
Therefore, by level by level equivalence between
strong compactness and supercompactness in $V$,
$V \models ``\gk$ is $\gl$ supercompact''.
The argument just given then shows that
$V^\FP \models ``\gk$ is $\gl$ supercompact'' as well.
This completes the proof of Case 2 and Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l4} -- \ref{l6} complete the
proof of Theorem \ref{t3}.
\end{proof}
%In conclusion to Section \ref{s3},
We note that the definition of the
partial ordering $\FP$ used in the proof
of Theorem \ref{t3} shows that
$V^\FP \models ``2^\gd = \gd^+$ if
$\gd$ is a successor or singular cardinal''. In addition,
any cardinal $\gk$ in $V^\FP$
which is a measurable limit of cardinals
$\gd$ which are $\gl$ strongly compact
where $\gl > \gk$ is regular and is
either a successor or measurable cardinal
must be in $V^\FP$ a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
This is since Theorem \ref{gf},
%the results of \cite{H2} and \cite{H3},
which tells us that there are no
new instances of measurability, strong compactness,
or supercompactness in $V^\FP$, implies
that $\gk$ must be in $V$ a measurable limit
of cardinals $\gd$ which are $\gl$ strongly compact.
$\gk$ can then be written in $V$ as a
measurable limit of cardinals $\gd$ which are
$\gl$ strongly compact where each such $\gd$
is not itself a measurable limit of cardinals
$\gg$ which are $\gl$ strongly compact.
By level by level equivalence between strong
compactness and supercompactness in $V$,
each such cardinal $\gd$ must be
$\gl$ supercompact in $V$.
Lemmas \ref{l4} -- \ref{l6} then imply that each of these
cardinals remains $\gl$ supercompact in
$V^\FP$.
\begin{pf}
We briefly indicate how Theorem \ref{t4} follows
as a corollary of (the proof of) Theorem \ref{t3}.
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal is supercompact up to an inaccessible cardinal +
Level by level equivalence between strong compactness
and supercompactness holds''.
Let $\FP$ be defined as in the proof of Theorem \ref{t3},
except that each use of $\add(\gd, \gd^{++})$ is
replaced by a use of $\add(\gd, \gd^{+ 19})$.
An analogous argument to the one found in
the proof of Lemma \ref{l2} shows that
$V^\FP \models ``$No cardinal is supercompact
up to an inaccessible cardinal''.
In addition, Levinski's ideas of \cite{Le}
used in the proof of Lemma \ref{l5} remain
valid if $\add(\gd, \gd^{++})$ is replaced by
$\add(\gd, \gd^{+ 19})$.
These key observations then allow us to infer
as in the proof of Theorem \ref{t3} that
$V^\FP \models ``\gk$ is supercompact + Level by
level equivalence between strong compactness and
supercompactness holds + For every inaccessible
cardinal which is not also measurable, $2^\gd = \gd^{+ 19}$ +
For every measurable cardinal $\gd$, $2^\gd = \gd^+$''.
This completes our discussion of the proof of Theorem \ref{t4}.
\end{pf}
Suppose $\gl$ is a measurable cardinal and
$j : V \to M$ is an elementary embedding
%witnessing $\gl$'s measurability
having critical point $\gl$
which is generated by a normal measure over $\gl$.
We remark that our application of
the ideas of \cite{Le} only requires
the existence of a function $f : \gl \to \gl$ such that
%for any inaccessible cardinal $\gg < \gl$,
$V \models ``{\rm cof}(f(\gg)) > \gg$ if
$\gg < \gl$ is inaccessible and
$\card{j(f)(\gl)} = \gl^+$''.
This allows for great flexibility in the proof of
Theorem \ref{t4} when determining the possible values
for $2^\gd$ if $\gd < \gk$ is a nonmeasurable inaccessible cardinal.
It is reasonable to hope that the partial orderings
described in Section \ref{s2} which are used in the
proof of Theorem \ref{t2} can also be employed to
prove versions of Theorems \ref{t1a} and \ref{t3}
containing different violations of GCH for the
relevant inaccessible cardinals.
%in which for the relevant inaccessible cardinals
%$\gl$, $2^\gl > \gl^{++}$.
However, because the definition of
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$
implies that in
$V^{\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]}$,
$\gd$ is nonmeasurable,
%$\gd$ is not $\gl$ supercompact
%(see \cite{AS97a} and \cite{AS97b}),
it will not be possible to force with these partial
orderings and end up with a universe containing
more than one supercompact cardinal.
We therefore conclude by asking whether it is
possible to establish alternate forms %prove versions
of Theorems \ref{t1a} and \ref{t3}
in which for the relevant inaccessible cardinals
$\gl$, $2^\gl > \gl^{++}$.
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Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
%\bibitem{S} R.~Solovay, ``Strongly Compact Cardinals
%and the GCH'', in: {\it Proceedings of the Tarski
%Symposium}, {\bf Proceedings of Symposia in Pure
%Mathematics 25}, American Mathematical Society,
%Providence, 1974, 365--372.
%\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
%``Strong Axioms of Infinity and Elementary Embeddings'',
%{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}
In addition, Theorem \ref{t3}
provides a generalization of
Theorem 2 of \cite{A01a} and
Theorem 4 of \cite{A01}, but
in the context of a universe
in which the structure of
the class of supercompact cardinals
can be {\em arbitrary}, and not limited
as in either of these two aforementioned
theorems.
Further, Theorem \ref{t3} also lifts
Theorem 2 of \cite{A01a} to a
model in which level by level equivalence
between strong compactness and
supercompactness holds.
%which is not necessarily true in \cite{A01a}.
Finally, and perhaps
most importantly, in Theorem \ref{t3}, it
can be the case that $2^\gk > \gk^{++}$.
This contrasts sharply with the
results found in
\cite{A03} and \cite{A05},
where the structure of the class
of supercompact cardinals can
be arbitrary,
level by level equivalence between
strong compactness and supercompactness holds,
yet if GCH fails at the least supercompact
cardinal $\gk$,
$2^\gk = \gk^{++}$.
\begin{lemma}\label{l2}
$V^\FP \models ``$For every
$\gd \le \gk$ which is either
a strong cardinal or an inaccessible
limit of strong cardinals,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$,
and $\gd$ is
$\gr_\gd$ supercompact
if $\gd$ is a measurable cardinal''.
\end{lemma}
\begin{proof}
Write
$\FP = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
By Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}, this
factorization of $\FP$ indicates
that any $\gd \le \gk$ which
is either in $V^\FP$ a strong
cardinal or an inaccessible limit of
strong cardinals had to have been in $V$
either a strong cardinal or an inaccessible
limit of strong cardinals.
Further, suppose $\gd < \gk$ is such that
$V \models ``\gd$ is a strong cardinal
which is not a limit of strong cardinals''.
Since $V \models {\rm GCH}$, Lemma 3.1 of
\cite{A01a} tells us that
$V \models ``\gd$ is not $2^\gd = \gd^+$
supercompact''. Consequently, by the
definition of $\FP$,
$V^\FP \models ``\gd$ is not a
measurable cardinal''. We may therefore
infer that any cardinal $\gd < \gk$
which is a strong cardinal in $V^\FP$
had to have been in $V$ a strong cardinal
which is a limit of strong cardinals.
This last sentence now tells us that the
argument of Lemma 4.2 of \cite{A01a}
goes through unchanged to show that
$V^\FP \models ``$For every
$\gd \le \gk$ which is either
a strong cardinal or an inaccessible
limit of strong cardinals,
$2^\gg = h(\gd)$ for all cardinals
$\gg \in [\gd, h(\gd))$,
and $\gd$ is
$\gr_\gd$ supercompact
if $\gd$ is a measurable cardinal''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We construct models for the level by level
equivalence between strong compactness and
supercompactness in which for each inaccessible
cardinal $\gd$, $2^\gd > \gd^{++}$.
In these models, no cardinal is supercompact up
to an inaccessible cardinal. This generalizes a
corollary of \cite[Theorem 3]{A02a}.
We construct models
for the level by level equivalence between
strong compactness and supercompactness
containing failures of GCH at inaccessible cardinals.
In particular, we
provide examples of such models in which
no cardinal is
supercompact up to an inaccessible cardinal, and
for each inaccessible cardinal
$\gd$, $2^\gd > \gd^{++}$.
%This generalizes a corollary of \cite[Theorem 3]{A02a}.
We also construct a model in which
level by level equivalence between strong
compactness and supercompactness holds
except possibly at inaccessible cardinals
in which GCH is true at every measurable cardinal
yet fails at every nonmeasurable inaccessible cardinal.
This provide examples of
models for level by level equivalence between
strong compactness and supercompactness
in which no cardinal is supercompact
up to an inaccessible cardinal, GCH is true at each
measurable cardinal, and GCH fails at every nonmeasurable
inaccessible cardinal.
These results lift and generalize \cite[Theorem 3]{A02a}.
%This lifts and generalizes
%Theorem 2 of \cite{A01a} and Theorem 4
%of \cite{A01}.
To do this,
we use arguments originally due to
Magidor \cite{Ma2}, which are also given
on pages 119--120 of \cite{AS97a}
(and are found other places throughout
the literature as well). Specifically,
for $\ga \in (\gd, \gd^{+})$ and
$p \in \add(\gd, \gd^{+})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$H \rest \ga = \{p \rest \ga : p \in H\}$. Clearly,
$V[G][H] \models ``|H \rest \ga| \le \gd$
for all $\ga \in (\gd, \gd^{+})$''. Thus, since
${\add(j(\gd), j(\gd^{+}))}^{M[G][H'][H'']}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{+}$,
$q_\ga = \bigcup\{j(p) : p \in H \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{+}))}^{M[G][H'][H'']}$.
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
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in H \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$H \rest \gb$ and $H \rest \ga$, for
$p \rest \gb = q \in H \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G][H'][H''] \models ``$GCH holds
for all cardinals at and above $j(\gd)$'',
$M[G][H'][H''] \models ``\add(j(\gd),
j(\gd^{+}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{+})$ many maximal antichains''.
This means that if
${\cal A} \in M[G][H'][H'']$ is a
maximal antichain of $\add(j(\gd), j(\gd^{+}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{+}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a normal measure over $\gd$
imply that
$V \models ``|j(\gd^{+})| = \gd^{+}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{+}) \ra \in
V[G][H]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{+}))$
present in
$M[G][H'][H'']$.
Working in $V[G][H]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{+}) \ra$ of
elements of $\add(j(\gd), j(\gd^{+}))$ such that
$\forall \ga \in (\gd, \gd^{+}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{+}) \ra
\exists \gb \in (\gd, \gd^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$H''' = \{p \in \add(j(\gd), j(\gd^{+})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{+}) \ra
[r \ge p]$ is an
$M[G][H'][H'']$-generic object over
$\add(j(\gd), j(\gd^{+}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{+}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G][H'][H'']$ is
$\gd$-closed with respect to
$V[G][H]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, 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
$M[G][H'][H'']$ is closed under
$\gd$ sequences with respect to
$V[G][H]$, 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
$s_{\gb + 1}$ lifts 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 $H'''$ is
$M[G][H'][H'']$-generic over
$\add(j(\gd), j(\gd^{+}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{+}) \ra
\exists \gb \in (\gd, \gd^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{+} \ra$ is
unbounded in $j(\gd^{+})$. To see this, if
$\gb < j(\gd^{+})$ is an ordinal, then for some
$f : \gd \to M$ representing $\gb$,
we can assume that for $\gs < \gd$,
$f(\gs) < \gd^{+}$. Thus, by the regularity of
$\gd^{+}$ in $V$,
$\gb_0 = \bigcup_{\gs < \gd} f(\gs) <
\gd^{+}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{+} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{+})$ be such that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{+})$ is such that for some
$\ga \in (\gd, \gd^{+})$, $p = p \rest \ga$,
$H'''$ is such that if
$p \in H$, $j(p) \in H'''$.
We observe that the proof of Lemma \ref{l4}
may break down if $\gl$ is inaccessible.
This is since the inner model $M$ will not
necessarily have enough closure to allow either of the
proofs given in Cases 1 and 2 above to remain
valid.
Suppose $V \models ``\gd$ is a measurable cardinal''.
As we have already observed,
$V^\FP \models ``2^\gd = \gd^+$''.
As in the proof of Lemma \ref{l2}, write
$\FP = \FP_0 \ast \dot \FQ$, where
$\card{\FP_0} = \go$, $\FP_0$ is
nontrivial, and
$\forces_{\FP_0} ``\dot \FQ$ is
$\ha_1$-strategically closed''.
Again by Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}, this
factorization of $\FP$ indicates
that any $\gg$ which is measurable in
$V^\FP$ had to have been measurable in $V$.
Thus, by our remarks in the preceding sentence,
the proof of Lemma \ref{l5} will be
complete once we have shown that
$V^\FP \models ``\gd$ is a measurable cardinal''.
However, since $\FP = \FP_{\gd + 1} \ast \dot \FP^{\gd + 1}$,
where $\FP_{\gd + 1}$ acts nontrivially on
ordinals less than or equal to $\gd$ and
$\forces_{\FP_{\gd + 1}} ``\dot \FP^{\gd + 1}$ is
$(2^\gd)^+$-directed closed '',
it will suffice to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is a measurable cardinal''.