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\title{More Easton Theorems for Level
by Level Equivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
level by level equivalence between strong
compactness and supercompactness, Easton theorem.}}
\date{July 31, 2011\\(revised August 12, 2012)}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
Brent Cody for helpful conversations
on the subject matter of this paper.}
\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}
\begin{document}
\maketitle
\begin{abstract}
We establish
two new Easton theorems
for the least supercompact
cardinal that are consistent
with the level by level equivalence
between strong compactness and
supercompactness.
These theorems generalize \cite[Theorem 1]{A05}.
In both our ground model and the
model witnessing the conclusions
of our theorem, there are no
restrictions on the structure
of the class of
supercompact cardinals.
%We also briefly indicate how our
%methods of proof yield an
%Easton theorem that is consistent
%with the level by level equivalence
%between strong compactness and
%supercompactness in a universe
%with a restricted number of large cardinals.
%We conclude by posing some related
%open questions.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
%We also wish to emphasize
%what we mean when we say that
%level by level equivalence between
%strong compactness and supercompactness holds.
%Specifically, we will say
Say that a model of ZFC witnesses
{\em level by level equivalence between
strong compactness and supercompactness}
iff for every measurable cardinal
$\gk$ and every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except
if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$
supercompact.
Models containing supercompact
cardinals which also witness the
level by level equivalence between
strong compactness and supercompactness
and satisfy GCH
were first constructed in \cite{AS97a}.
Note that the results of \cite{AS97a} generalize
the fundamental work of Magidor \cite{Ma1} where
it is shown, relative to a supercompact cardinal,
that it is consistent for the least strongly compact
and supercompact cardinals to coincide precisely.\footnote{Magidor
also showed in \cite{Ma1} that it is consistent,
relative to the existence of a strongly compact
cardinal, for the least strongly compact and
measurable cardinals to coincide precisely.
In this situation, the least strongly compact cardinal
is not the least supercompact cardinal.}
These results also generalize the later work of
Kimchi and Magidor \cite{KM} who showed, relative
to the existence of a class of supercompact cardinals,
that it is consistent for the classes of strongly
compact and supercompact cardinals to coincide precisely,
except at measurable limit points.
We remark that the exceptions in the previous paragraph
are provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are either $\gl$ strongly
compact or $\gl$ supercompact,
then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
(Menas' results of \cite{Me} were also a precursor
to the later work of \cite{Ma1}, \cite{KM}, and \cite{AS97a}.)
When this situation occurs,
we will henceforth say that
{\em $\gk$ is a witness
to the Menas exception at $\gl$}.
If $\gk$ is measurable and
for every regular cardinal
$\gl > \gk$, $\gk$ is $\gl$
strongly compact iff $\gk$ is
$\gl$ supercompact, then we will
say that {\em $\gk$ is a witness to
level by level equivalence between
strong compactness and supercompactness}.
We continue now with the main narrative.
In \cite{A05}, the following theorem was proven.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is
either a strong cardinal or the
regular limit of strong cardinals$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i1a} $F(\gd) \in (\gd, \gd^*)$
is a cardinal, where
$\gd^*$ is the least strong
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2a} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3a} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F)(\gd) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''.
\end{theorem}
In Theorem \ref{t1},
it is unfortunately the case that
the Easton function $F$ is defined
only on a restricted set of
inaccessible cardinals at and below the least
supercompact cardinal $\gk$.
Furthermore, because of the restrictions
placed on $F$'s range by clause (\ref{i3a}),
if $\gd \in A$, then $2^\gd \le \gd^{++}$.
It therefore becomes desirable to see if
it is possible to remove these constraints.
The purpose of this paper is to show that
this is indeed the case, and to address further
the general question of what GCH patterns are
consistent with the level by level equivalence between
strong compactness and supercompactness.
Specifically, we prove the following two theorems.
\begin{theorem}\label{t2}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is a regular cardinal
which is not the successor of a singular cardinal and
$\neg \exists \gg < \gd[\gg$
is $\ga$ supercompact for every $\ga < \gd]\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i0b} If $\gd_1 < \gd_2$,
%and $\gd_1, \gd_2 \in A$,
then $F(\gd_1) \le F(\gd_2)$.
\item\label{i1b} $F(\gd) \in (\gd, \gd')$
is a cardinal, where
$\gd'$ is the least {Mahlo}
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2b} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3b} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F)(\gd) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''.
\end{theorem}
\begin{theorem}\label{t3}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is
either a strong cardinal or the
regular limit of strong cardinals$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i1c} $F(\gd) \in (\gd, \gd^*)$
is a cardinal, where
$\gd^*$ is the least strong
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2c} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3c} If $\gd \in A$ is
$\gl$ supercompact for $\gl \ge \gd^{+ 16}$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that
%either $j(F(\gd)) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{+ 17}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''.
\end{theorem}
We take this opportunity to make several
remarks concerning Theorems \ref{t2} and \ref{t3}.
Theorem \ref{t2} is a generalization of
Theorem \ref{t1} where $A$, the domain of the
Easton function $F$, has been expanded.
It includes not only those $\gd \le \gk$ which
are either strong cardinals or regular limits
of strong cardinals, but
also both certain successor cardinals and
additional regular limit cardinals which are neither strong
cardinals nor limits of strong cardinals.
%In some sense, $A$ represents a
%``maximal expansion below $\gk$''
%``significant expansion below $\gk$''
%of the domain of the Easton
%function in Theorem \ref{t1}.
More specifically, as the proof of
\cite[Lemma 2.4]{AC2} shows, if $\gd$ is
$\gg$ supercompact for every $\gg < \gl$
and $\gl$ is strong, then $\gd$ is
supercompact as well. From this,
since $\gk$ is the least supercompact
cardinal, it immediately
follows that $\gk \in A$ and
if $\gl < \gk$ is either a strong
cardinal or a limit of strong cardinals, then
for no cardinal $\gd < \gl$
can it be the case that $\gd$
is $\gg$ supercompact for every $\gg < \gl$.
Consequently, the domain of the Easton function
in Theorem \ref{t1} is a subset of the
domain of the Easton function in Theorem \ref{t2}.
We note that it is a proper subset.
To see this, suppose $\gd < \gk$ is
%either a strong cardinal or
a non-measurable limit of strong cardinals.
Suppose in addition that $\gr > \gd$ is, e.g.,
the least measurable cardinal above $\gd$,
the least measurable limit of
measurable cardinals above $\gd$,
the least cardinal above $\gd$ which is $\gr^{+ n}$
supercompact for every $n \in \go$, etc. If
$\gg \in (\gd, \gr]$ is an inaccessible cardinal, then
$\gg$ is a member of the domain of the Easton
function in Theorem \ref{t2}
but is not a member of the domain of the Easton
function in Theorem \ref{t1}.
Further, if $\gd$ is a non-measurable limit of strong cardinals,
$\gr$ is the least measurable cardinal above $\gd$, and
$\gg \in (\gd, \gr)$ is a successor cardinal which
is not the successor of a singular cardinal, then
$\gg$ is a member of the domain of the Easton
function in Theorem \ref{t2} as well.
The restrictions given by $\gd'$
on $F$'s range are to allow
%generalizations of the proofs of a proof
for the preservation
of level by level equivalence between strong
compactness and supercompactness while still
maintaining a great deal of freedom in the
values that $F$ may attain.
Other values of $\gd'$, e.g., the least
weakly compact cardinal above $\gd$,
are also possible.
Theorem \ref{t3} is a generalization of
Theorem \ref{t1} which changes the Easton function $F$
so that the power set of
any ground model strong cardinal or
regular limit of strong cardinals $\gd$
can be larger than $\gd^{++}$.
In order to achieve this, we employ forcing
conditions first defined in \cite[Section 4]{AS97b},
as ordinary Cohen forcing seems to pose
certain technical challenges which are
difficult to overcome.
Also, the use of $\gd^{+ 16}$ and $\gd^{+ 17}$ in
the statement of Theorem \ref{t3}
is for ease of presentation and comprehensibility.
In essence, $\gd^{+ 17}$
should be seen as representing %virtually
any suitable regular cardinal $\gl$
for which it will be the case that $2^\gd = \gl$.
%and its successor $\gl^+$.
%It should be viewed
The cardinals $\gd^{+ 16}$ and $\gd^{+ 17}$ may be viewed
as a form of ``wild card'' standing in for the
more general possibilities, which are somewhat
technical in nature and are explicitly stated
in \cite[Section 4]{AS97b}.
We note that there are many
natural functions
%satisfying the conditions given in the
meeting the requirements of the
statements of Theorems \ref{t2} and \ref{t3}.
For instance, if we let
$B = \{\gd \in A \mid \gd$ is a successor cardinal$\}$ and
$C = A - B = \{\gd \le \gk \mid \gd$ is a limit cardinal and
$\neg \exists \gg < \gd[\gg$
is $\ga$ supercompact for every $\ga < \gd]\}$,
then
\[
F_0(\gd) = \left\{ \begin{array}{cc}
\gd^{+ 68} & \mbox{if
$\gd \in B$}\\
\gd^{++} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent and
\[
F_1(\gd) = \left\{ \begin{array}{lcl}
{\rm The \ least \ inaccessible \ cardinal
\ above \ } \gd & \mbox{if
$\gd \in B$}\\
\gd^{+} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent are candidates for the function $F$
mentioned in the statement of Theorem \ref{t2}.
If we let $B = \{\gd \in A \mid \gd$
is either a strong cardinal which
is not a limit of strong cardinals
or a non-measurable limit of
strong cardinals$\}$ and
$C = A - B = \{\gd \le \gk \mid \gd$
is a measurable limit of strong
cardinals$\}$,
then
\[
F_0(\gd) = \left\{ \begin{array}{cc}
\gd^{+ 95} & \mbox{if
$\gd \in B$}\\
\gd^{+ 17} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent and
\[
F_1(\gd) = \left\{ \begin{array}{lcl}
{\rm The \ least \ Mahlo \ cardinal
\ above \ } \gd & \mbox{if
$\gd \in B$}\\
\gd^{+ 17} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent are candidates for the function $F$
mentioned in the statement of Theorem \ref{t3}.
In fact, the Easton functions can essentially
take on arbitrary values for either
Theorem \ref{t2} or Theorem \ref{t3} when $\gd \in B$,
subject to the restrictions given above in
the statements of these theorems.
Before presenting the proofs
of our theorems, we briefly
state some preliminary information.
Our notation and terminology
will follow that given in \cite{A05} and \cite{A06}.
We do wish to mention a few
things explicitly, however.
When forcing, $q \ge p$ means
that $q$ is stronger than $p$.
For $\gk$ a regular cardinal and
$\ga$ an ordinal,
${\rm Add}(\gk, \ga)$ is the
standard Cohen partial ordering
for adding $\ga$ 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 $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
A partial ordering $\FP$ is
{\em $\gk$-directed closed} for
$\gk$ a cardinal 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
$\langle p_\ga \mid \ga \le\gk\rangle$, 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.
$ \FP$ is {\em ${<}\gk$-strategically closed}
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
In addition, if $\FP$ is $\gk$-directed closed, then
$\FP$ is ${<}\gk$-strategically closed.
%If $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga,
%\dot \FQ_\ga \ra : \ga < \gk + 1 \ra$
%is a reverse Easton iteration of length
%$\gk + 1$
%such that at stage $\ga$, a non-trivial forcing is done
%adding a subset of some ordinal
%$\gd_\ga$, then we will say that
%$\gd_\ga$ is in the field of $\FP$.
Finally, we mention that we are assuming
familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult
\cite{J} or \cite{K}
%, \cite{K}, or \cite{SRK}
for further details.
We do note, however, that
we will say $\gk$ is {\em supercompact (strongly compact)
up to the cardinal $\gl$} if
$\gk$ is $\gg$ supercompact ($\gg$ strongly compact) for every
$\gg < \gl$.
Also, if $\gk$ is $\gl$ supercompact and
$\gl$ is a cardinal, then
$\gk$ is supercompact up to $\gl^+$, i.e.,
$\gk$ is $\ga$
supercompact for every $\ga < \gl^+$.
%We note only that the cardinal
%$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
%supercompact for every cardinal $\gd < \gl$.
\section{The Proofs of Theorems \ref{t2} and \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let $V$, $A$, $F$, and $\gk$ be as in the hypotheses
for Theorem \ref{t2}.
Let $\la \gz_\ga \mid \ga < \gk \ra \in V$ enumerate in
increasing order
%the set $\{\go\} \cup
$\{\gz < \gk \mid \gz$ is either a {Mahlo}
cardinal or a limit of {Mahlo} cardinals$\}$.
We define three partial orderings
$\FP^0$, $\FP^1$, and $\FP^2$, where
$\FP^0 \in V$, $\FP^1 \in V^{\FP^0}$, and
$\FP^2 \in V^{\FP^0 \ast \dot \FP^1}$.
The partial ordering $\FP$ with which we force
to complete the proof of Theorem \ref{t2}
will then be defined as
$\FP = (\FP^0 \ast \dot \FP^1) \ast \dot \FP^2 =
\ov \FP \ast \dot \FP^2$. Specifically,
$\FP^0 = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra \in V$
is the reverse Easton iteration of length $\gk$
%(defined in $V$)
which begins by forcing with $\add(\go, 1)$
(so $\FP_0 = \add(\go, 1)$).
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the Easton support product
$\prod_{\gd \in A \cap [\gz_\ga, \gz_{\ga + 1})} \add(\gd, F(\gd))$.
$\FP^1 \in V^{\FP^0}$ is $(\add(\gk, F(\gk)))^{V^{\FP^0}}$, and
$\FP^2 \in V^{\ov \FP}$ is
$\prod_{\gd \in A \cap [\go, \gz_0)} \add(\gd, F(\gd))$.
A few explanatory remarks are perhaps now
in order concerning the above
definition of $\FP$.
Note that it is possible to write
$\ov \FP = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_2$-directed closed''.
In Hamkins' terminology of \cite{H2, H3},
``$\ov \FP$ admits a gap at $\ha_1$''.
$\ov \FP$ has been defined in this manner
so that the results of \cite{H2, H3} may be
applied and allow us to infer that the model $V^{\ov \FP}$
satisfies level by level equivalence
between strong compactness and supercompactness.
In particular, the gap at $\ha_1$
ensures that any cardinal $\gd$
which is $\gl$ supercompact in $V^{\ov \FP}$ had to have been
$\gl$ supercompact in $V$.
We use an iteration
of products in the definition of $\ov \FP$ in order to allow
the usual supercompactness lifting arguments to be
applied.\footnote{The author wishes to thank Brent Cody
for suggesting this approach.} In addition, as readers may
verify for themselves, the standard Easton arguments for products
and iterations (see \cite{J}) show that in $V^\FP$,
cardinals and cofinalities are preserved and
$2^\gd = F(\gd)$ for every $\gd \in A$.
Since in $V^{\ov \FP}$, $\FP^2$ has cardinality
the least Mahlo cardinal,
the L\'evy-Solovay results \cite{LS} ensure that forcing
with $\FP^2$ over $V^{\ov \FP}$ will not destroy
any relevant properties true in $V^{\ov \FP}$,
e.g., level by level equivalence between
strong compactness and supercompactness.
%, and that any cardinal $\gd$
%which is $\gl$ supercompact in $V^{\ov \FP}$ had to have been
%$\gl$ supercompact in both $V^{\ov \FP}$ and $V$.
Finally, the forcing $\FP^2$ is performed at the end of
the construction, and not the beginning, so that a
gap at $\ha_1$ may be introduced.
\begin{lemma}\label{l1}
$V^{\ov \FP} \models ``\gk$ is the least
supercompact cardinal''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} is a somewhat
more complicated version of the proof of
\cite[Lemma 2.1]{A05}.
By our remarks in the preceding
paragraph, since forcing with $\ov \FP$
creates no new supercompact cardinals,
it suffices to show that
$V^{\ov \FP} \models ``\gk$ is supercompact''.
%To do this, we prove that
%$V^{\ov \FP} \models ``\gk$ is supercompact'',
%and then use the fact that
%$V^{\ov \FP} \models ``\card{\FP^2} < \gk$''
%in conjunction with the results of \cite{LS} to demonstrate that
%$V^\FP \models ``\gk$ is supercompact''.
Towards this end,
let $\gl > \gk^+ = 2^\gk$ be
the successor of a regular cardinal,
and let $j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$ with
$F(\gk) = j(F)(\gk)$.
%By the usual arguments,\footnote{More specifically, ....}
%we can also assume that
%$M \models ``\gk$ is not $\gl$ supercompact''.
Let $\theta = F(\gk) = j(F)(\gk)$ (so
$\theta$ is either $\gk^+$ or
$\gk^{++}$).
Since $V \models {\rm GCH}$,
$M \models ``\gk$ is $\ga$ supercompact
for every $\ga < \gl$''.
In addition, since
$V \models ``$No cardinal is supercompact
up to $\gk$'',
$M \models ``$No cardinal is supercompact
up to $\gk$'', i.e.,
$M \models ``\gk \in j(A)$''. Further,
$M \models ``\gz_\gk = \gk$''.
The definition of $\ov \FP$ therefore implies that we can write
$j(\ov \FP) = \FP_\gk \ast
((\dot \add(\gk, \theta) \times \dot \FQ^0) \ast \dot \FQ^1)
\ast \dot \add(j(\gk), j(\theta)) =
\FP_\gk \ast \dot \FQ
\ast \dot \add(j(\gk), j(\theta))$, where
$\dot \add(\gk, \theta) \times \dot \FQ^0$ is a term
for the Easton support product
$\prod_{\gd \in j(A) \cap [\gz_\gk, \gz_{\gk + 1})} \add(\gd, F(\gd))$,
$\dot \FQ^1$ is a term for the portion of $j(\ov \FP)$ defined between
$\gz_{\gk + 1}$ and $j(\gk)$,
and the first
ordinal at which $\dot \FQ^0$ is forced to do nontrivial forcing
is greater than or equal to $\gl^+$.
%We now proceed as in \cite[Lemma 2.1]{A05}.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be
$V[G]$-generic over
$(\add(\gk, \theta))^{V[G]}$.
Since $\FP_\gk$ is $\gk$-c.c.,
standard arguments show that
$M[G]$ remains $\gl$ closed
with respect to $V[G]$,
and that $\FQ^0$ is
$\gl^+$-directed closed in both
$M[G]$ and $V[G]$.
Since
$M[G] \models ``|\FQ^0| < j(\gk)$'' and
$j(\gk)$ is inaccessible in both $M$ and $M[G]$,
there are $j(\gk)$
dense open subsets of $\FQ^0$
present in $M[G]$. However, since
$|j(\gk)| =
|\{f \mid f : P_\gk(\gl) \to \gk$
is a function$\}| = \gl^+$ by GCH,
we can use the usual diagonalization
arguments
(as given, e.g., in the construction
of the generic object $G_1$ in
\cite[Lemma 2.4]{AC2})
to construct in
$V[G]$ an $M[G]$-generic
object $H'$ over $\FQ^0$. (An
outline of this argument is as follows.
%By GCH and
%the fact that $j$ is given by an
%ultrapower embedding, we may let
Let $\la D_\ga \mid \ga < \gl^+ \ra$ enumerate in
$V[G]$ the dense open subsets of $\FQ^0$ present
in $M[G]$. Because $M[G]$ remains
$\gl$ closed with respect to $V[G]$,
by the $\gl^+$-directed
closure of $\FQ^0$ in both $M[G]$ and $V[G]$,
we may work in $V[G]$ and
meet each $D_\ga$ in order to construct $H'$.)
%Since $j '' G_0 \subseteq G_0 \ast G_1$,
%we may lift $j$ in $V[G_0]$ to
%$j : V[G_0] \to M[G_0][G_1]$.}
%and lift $j$ to $j : V[G] \to M[G][H][H']$ in $V[G][H]$.
%Note that since $M^\gl \subseteq M$,
%it is the case that
%in both $V$ and $M$,
%$\forces_{\FP_\gk}
%\ast \dot \add(\gk, \theta)}
%``\dot \FQ^0$ is
%$\gl^+$-directed closed''.
Since $\FQ^0$ is $\gl^+$-directed closed
in both $M[G]$ and $V[G]$,
standard arguments again show that
$M[G][H']$ remains $\gl$ closed
with respect to $V[G][H'] = V[G]$.
In addition, since $H$ is $V[G]$-generic over
$\add(\gk, \theta)$ and $M[G][H'] \subseteq V[G]$,
$H$ is $M[G][H']$-generic over $\add(\gk, \theta)$.
Consequently, since $\add(\gk, \theta)$ is
$\gk^+$-c.c$.$ in both $V[G]$ and $M[G][H']$,
$M[G][H'][H] = M[G][H][H']$ remains $\gl$ closed
with respect to $V[G][H]$.
Further, since
$M[G][H][H'] \models ``|\FQ^1| = j(\gk)$'' and
GCH holds in both $V$ and $M$,
there are $2^{j(\gk)} = j(\gk^+)$
dense open subsets of $\FQ^1$
present in $M[G][H][H']$. However, since
$|j(\gk^+)| =
|\{f \mid f : P_\gk(\gl) \to \gk^+$
is a function$\}| = \gl^+$ by GCH,
we can use the same diagonalization
arguments as in the construction of $H'$
to construct in
$V[G][H]$ an $M[G][H][H']$-generic
object $H''$ over $\FQ$ and lift $j$ to
$j : V[G] \to M[G][H][H'][H'']$ in $V[G][H]$.
Note that $M[G][H][H'][H'']$ remains
$\gl$ closed with respect to
$V[G][H][H'][H''] = V[G][H]$, since
$\FQ^1$ is $\gl^+$-directed closed in
$M[G][H][H']$ and $M[G][H][H']$ is
$\gl$ closed with respect to $V[G][H]$.
Then, as the number of dense open subsets
of $\add(j(\gk), j(\theta))$ in
$M[G][H][H'][H'']$ is either
$j(\gk^{++})$ (if $F(\gk) =
\gk^+$) or $j(\gk^{+++})$ (if
$F(\gk) = \gk^{++}$), which
by GCH and the fact that
$\gl \ge \gk^{++} \ge \theta$
has size $\gl^+$ in
$V[G][H]$, and as $\add(j(\gk), j(\theta))$
is $\gl^+$-directed closed in both
$M[G][H][H'][H'']$ and $V[G][H]$ and
$V[G][H] \models ``\card{j''H} \le \gl$'',
we can once again
use the standard diagonalization arguments
to construct in $V[G][H]$ an
$M[G][H][H'][H'']$-generic object $H'''$ containing
a master condition for $j''H$.
We can now fully lift $j$ to
$j : V[G][H] \to M[G][H][H'][H''][H''']$ in
$V[G][H]$, thereby showing that
$V[G][H] \models ``\gk$ is $\gl$ supercompact''.
Since $\gl \ge \gk^{++}$ was an arbitrary
successor of a regular cardinal,
this completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $\gd \in A$
%is a nontrivial stage of forcing and
and $\gl > \gd$ is a regular
cardinal such that
$V \models
``\gd$ is $\gl$ supercompact'', then
$V^{\ov \FP} \models ``\gd$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} is once
again along the lines of the proof of
\cite[Lemma 2.2]{A05}, only somewhat more complicated.
In analogy to Lemma \ref{l1}, let
$\gth = F(\gd)$, i.e., $\gth$ is either
$\gd^+$ or $\gd^{++}$.
Write
$\ov \FP = \FP_{\gd + 1} \ast \dot \FQ$.
By the definition of $\ov \FP$,
since $V \models ``\gd$ is $\ga$ supercompact
for every $\ga < \gl^+$'',
$\forces_{\FP_{\gd + 1}} ``\dot \FQ$ is $\gg$-directed closed
where $\gg > \gl^+$ is Mahlo''.
Hence, to show that
$V^{\ov \FP} \models ``\gd$ is $\gl$ supercompact'',
it suffices to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$
supercompact''.
In addition, the definition of $\ov \FP$ implies that
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for a partial ordering
having the form
$\add(\gd, \gth) \times \FR$. Here,
since $V \models ``\gd$ is $\ga$ supercompact
for every $\ga < \gl^+$'',
$\FR$ is an Easton product of
the Cohen forcings %partial orderings of the form
$\add(\gg, \eta)$, where by GCH at and above $\gd$
in $V$ and $V^{\FP_\gd}$,
$\gg \ge \gl^{++} = (2^{[\gl]^{< \gd}})^+ = (2^\gl)^+$.
Consequently, in $V^{\FP_\gd}$, $\FR$ is
$(2^{[\gl]^{< \gd}})^+$-directed closed.
%by the definition of $\ov \FP$ and GCH in $V$,
%it must be the case that
%$\forces_{\FP_{\gd + 1}} ``\dot \FQ$ is
%$\gl^{++} = (2^{[\gl]^{< \gd}})^+ =
%(2^\gl)^+$-directed closed''.
Therefore, since $\gth \le \gd^{++} < \gl^{++}$,
to show that
$V^{\FP_{\gd + 1}} = V^{\FP_\gd \ast \dot \FQ_\gd} =
V^{\FP_\gd \ast (\dot \add(\gd, \gth) \times \dot \FR)} \models
``\gd$ is $\gl$ supercompact'', it suffices to show that
$V^{\FP_{\gd} \ast \dot \add(\gd, \gth)} \models
``\gd$ is $\gl$ supercompact''.
With a severe abuse of notation,
for the remainder of the proof of Lemma \ref{l2},
we denote $\FP_\gd \ast \dot \add(\gd, \gth)$ by
$\FP_{\gd + 1}$.
%In addition, by the definition of
%$\ov \FP$ and $\FP_{\gd + 1}$,
%because $F(\gd) = \theta$,
%we may consequently infer that
%$|\FP_{\gd + 1}| = \theta$.
Suppose now that $\gl \ge \theta$.
By the definition of $F$,
we may choose $j : V \to M$
as an elementary embedding
witnessing the $\gl$ supercompactness
of $\gd$ generated by a supercompact ultrafilter
over $P_\gd(\gl)$ such that
$j(F)(\gd) = F(\gd) = \theta$.
If $V \models ``\gl$ is the successor of a
singular cardinal'', then since
$M^\gl \subseteq M$,
$M \models ``\gl$ is the successor of a
singular cardinal'' as well.
If not, i.e., if
$V \models ``\gl$ is either a regular
limit cardinal or the successor of
a regular cardinal'',
then by GCH in $V$ and $M$,
$M \models ``\gd$ is $\ga$ supercompact
for every $\ga < \gl$''.
In either case, as in Lemma \ref{l1},
$j(\FP_{\gd + 1}) = \FP_\gd \ast
((\dot \add(\gd, \theta) \times \dot \FQ^0) \ast \dot \FQ^1)
\ast \dot \add(j(\gd), j(\theta))$,
%= \FP_\gk \ast \dot \FQ
%\ast \dot \add(j(\gk), j(\theta))$, where
%$\dot \add(\gk, \theta) \times \dot \FQ^0$ is a term
%for the Easton support product
%$\prod_{\gd \in A \cap [\gz_\gk, \gz_{\gk + 1})} \add(\gd, F(\gd))$,
%$\dot \FQ^1$ is a term for the portion of $j(\ov \FP)$ defined between
%$\gz_{\gk + 1}$ and $j(\gk)$, and
where the first
ordinal at which $\dot \FQ^0$ is forced to do nontrivial forcing
is greater than or equal to $\gl^+$.
The argument given in the proof of
Lemma \ref{l1} may therefore now
be used to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
Thus, we assume that
$\gl < \theta$. Since $\gl > \gd$, this means that
$\gl = \gd^+$.
The remainder of the proof of Lemma \ref{l2}
is along the lines of \cite[Lemma 2.2]{A05}, from which
for the relevant portions we quote almost verbatim.
If $j(F)(\gd) =
F(\gd) = \gd^+$, then the
argument given in the preceding
paragraph shows that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
We consequently assume that
$\gl = \gd^+$ and $j(F)(\gd) =
F(\gd) = \gd^{++}$. Under these circumstances,
the argument given on
\cite[pages 119--120]{AS97a},
\cite[pages 88--90]{A02},
\cite[pages 832--833]{AH4}, or
\cite[pages 591--592]{A03}
(which is originally due to Magidor
and is also found earlier in
\cite{JMMP}, \cite{JW}, and \cite{Ma2})
suitably modified to take into account
the definition of $\ov \FP$
can be used to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
%For the convenience of readers, we give this
%argument here as well.
Getting specific, write
$\FP_{\gd + 1} = \FP_\gd \ast \dot \add(\gd, \gd^{++})$.
%Let $G = G_0 \ast G_1$ be $V$-generic over $\FP_{\gd + 1}$.
Let $G$ be $V$-generic over $\FP_\gd$, and let $H$ be
$V[G]$-generic over $(\add(\gd, \gd^{++}))^{V[G]}$.
Fix $j : V \to M$ an elementary embedding
witnessing the $\gl = \gd^+$
supercompactness of $\gd$ generated
by a supercompact ultrafilter
${\cal U}$ over $P_\gd(\gl)$ such that
$j(F)(\gd) = F(\gd) = \gd^{++}$.
We then have as above that
$j(\FP_{\gd + 1}) = \FP_\gd \ast
((\dot \add(\gd, \theta) \times \dot \FQ^0) \ast \dot \FQ^1)
\ast \dot \add(j(\gd), j(\theta))$,
where the first
ordinal at which $\dot \FQ^0$ is forced to do nontrivial forcing
is greater than or equal to $\gl^+ = \gd^{++}$.
Therefore, the arguments used in the proof of
Lemma \ref{l1} allow us to construct in $V[G][H]$
generic objects $H'$ and $H''$ over $\FQ^0$ and
$\FQ^1$ respectively and lift $j$ in $V[G][H]$ to
$j : V[G] \to M[G][H][H'][H'']$, where
$M[G][H][H'][H'']$ remains $\gd^+$ closed
with respect to $V[G][H]$.
We construct now in $V[G][H]$ an
$M[G][H][H'][H'']$-generic object over
$\add(j(\gd), j(\gd^{++}))$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p \mid
\gs < \ga\}$ and
$H \rest \ga = \{p \rest \ga \mid p \in H\}$. Clearly,
$V[G][H] \models ``|H \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${(\add(j(\gd), j(\gd^{++}))}^{M[G][H][H'][H'']}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) \mid p \in H \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G][H][H'][H'']}$.
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 $M[G][H][H'][H''] \models ``$GCH holds
for all cardinals at or above $j(\gd)$'',
$M[G][H][H'][H''] \models ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G][H][H'][H'']$ 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 supercompact ultrafilter over
$P_\gd(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga \mid \ga \in (\gd, \gd^{++}) \ra \in
V[G][H]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G][H][H'][H'']$.
Working in $V[G][H]$, we define
now an increasing sequence
$\la r_\ga \mid \ga \in (\gd, \gd^{++}) \ra$ 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
$\forall {\cal A} \in
\la {\cal A}_\ga \mid \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$H''' = \{p \in \add(j(\gd), j(\gd^{++})) \mid
\exists r \in \la r_\ga \mid \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G][H][H'][H'']$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga \mid \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb \mid \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G][H][H'][H'']$ 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
$\la {\cal B}_\gb \mid \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb \mid \gb \le \ga + 1 \ra$
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
$M[G][H][H'][H'']$ is closed under
$\gd^+$ sequences with respect to
$V[G][H]$, define by induction
an increasing sequence
$\la s_\gb \mid \gb < \eta \ra$ 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 $H'''$ is
$M[G][H][H'][H'']$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga \mid \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) \mid \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gd(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gd(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gd(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) \ge \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga \mid \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
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
$p \in H$, $j(p) \in H'''$.
Thus, working in $V[G][H]$,
we have shown that $j$ lifts to
$j : V[G][H] \to M[G][H][H'][H''][H''']$,
i.e.,
$V[G][H] \models ``\gd$ is $\gl = \gd^+$
supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^{\ov \FP} \models ``$Every measurable cardinal
$\gd$ either is a witness to level by
level equivalence between strong
compactness and supercompactness or is a
witness to the Menas exception at $\gl$
for some $\gl > \gd$'', i.e.,
$V^{\ov \FP} \models ``$Level by level equivalence
between strong compactness and
supercompactness holds''.
\end{lemma}
\begin{proof}
We modify the proof of \cite[Lemma 2.3]{A05}.
Since ${\ov \FP}$ may be defined
so that $\card{{\ov \FP}} \le \gk^{++}$,
by the results of \cite{LS},
Lemma \ref{l3} is true for any measurable cardinal
$\gd > \gk$. By Lemma \ref{l1}, Lemma \ref{l3}
is true for $\gd = \gk$. It thus
suffices to show that Lemma \ref{l3} holds
for any measurable cardinal $\gd < \gk$.
To establish this last fact,
we consider the following two cases.
\bigskip
\noindent Case 1: $\gd \in A$.
%is a nontrivial stage of forcing.
Suppose $\gl > \gd$ is such that
$V^{\ov \FP} \models ``\gd$ is $\gl$ strongly compact''.
Recall from the second paragraph of the
proof of Theorem \ref{t2} that ${\ov \FP}$
may be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$,
$\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is
$\ha_2$-directed closed''.
Further, it is easily seen that
any subset $x$ of $\gd$ in $V^{\ov \FP}$
of size below $\gd$
has a ``nice'' name $\gt$
of size below $\gd$ in $V$, i.e.,
there is a set $y$ in $V$,
$\card{y} < \gd$, such that any
ordinal forced by a condition in
${\ov \FP}$ to be in $\gt$ is an element
of $y$.
Therefore, in the terminology of
\cite{H2, H3}, ${\ov \FP}$ is a
``mild forcing with respect to $\gd$
admitting a gap at $\ha_1$'', so by
the results of \cite{H2, H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
Note now that $\gd$ cannot be a witness in $V$
to the Menas exception at $\gl$, i.e.,
$\gd$ is not in $V$ a limit of cardinals
which are $\gl$ supercompact.
This follows since otherwise, there are
$\gg < \gd$ such that $\gg$ is $\ga$
supercompact for every $\ga < \gd$,
an immediate contradiction to
the fact that $\gd \in A$.
%where nontrivial stages of forcing occur.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$\gd$ is $\gl$ supercompact in $V$.
We may therefore apply Lemma \ref{l2}
to infer that
$V^{\ov \FP} \models ``\gd$ is $\gl$ supercompact''.
\bigskip
\noindent Case 2: $\gd \not\in A$.
%is a trivial stage of forcing.
As before, suppose
$\gl > \gd$ is such that
$V^{\ov \FP} \models ``\gd$ is $\gl$ strongly compact''.
Let $S = \{\gr \in A \mid \gr < \gd\}$,
%be the set of nontrivial stages of forcing below $\gd$,
with $\gg$ either
the largest member of $S$ (if it exists),
or the supremum of the members of $S$ otherwise.
In the former situation, it must be true that
$F(\gg) < \gd$,
since $F(\gg)$ has size below the least
$V$-Mahlo cardinal above it.
Write $\ov \FP = \FP_S \ast \dot \FP^S$, where
$\FP_S$ is the portion of $\ov \FP$ acting
nontrivially on members of $S$, and
$\dot \FP^S$ is a term for the rest of $\ov \FP$.
It is therefore also true by the definition
of ${\ov \FP}$ that $\card{\FP_S} < \gd$.
The factorizations of ${\ov \FP}$ given in Case 1
and the one just presented consequently yield that
we once more have
in the terminology of \cite{H2, H3}
that ${\ov \FP}$ is a ``mild forcing
with respect to $\gd$ admitting a gap
below $\gd$''.
Hence, again by the results of \cite{H2, H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
By the level by level equivalence between
strong compactness and supercompactness in $V$,
this means that $\gd$ is either $\gl$
supercompact in $V$ or is a witness
to the Menas exception at $\gl$ in $V$.
%i.e., $\gd$ is a limit in $V$ of cardinals which are
%$\gl$ supercompact.
Regardless of which of these situations holds,
it must be the case that there is some cardinal
$\gr \le \gd$ such that $\gr$ is $\gl$
supercompact in $V$.
Note that since GCH holds in $V$,
$\card{\FP_S} < \gd$, and
$V \models ``\gr$ is $\ga$ supercompact for every $\ga < \gl^+$'',
$\forces_{\FP_S} ``\dot \FP^S$
is $\gl^{++} = (2^\gl)^+ = (2^{[\gl]^{< \gr}})^+$-directed closed''.
This and the results of \cite{LS} then yield
that in both $V^{\FP_{S}}$ and
$V^{\FP_{S} \ast \dot \FP^{S}} = V^{\ov \FP}$,
$\gd$ is either $\gl$ supercompact or is
a witness to the Menas exception at $\gl$.
If $\gg = \sup(S)$,
then by the definition of ${\ov \FP}$,
$\gg$ has to be singular. Hence,
it must be possible as before to write
${\ov \FP} = \FP_{S} \ast \dot \FP^{S}$, where
$\card{\FP_{S}} < \gd$.
The analysis given in the preceding paragraph
thus once again applies to show that in $V^{\ov \FP}$,
$\gd$ is either $\gl$ supercompact or is a
witness to the Menas exception at $\gl$.
Therefore, regardless if we are in Case 1 or
Case 2, if
$V^{\ov \FP} \models ``\gd$ is $\gl$ strongly
compact'', $\gd$ is either $\gl$
supercompact in $V^{\ov \FP}$ or is a witness
to the Menas exception at $\gl$ in
$V^{\ov \FP}$.
This just means that in $V^{\ov \FP}$,
level by level equivalence between
strong compactness and supercompactness holds.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^{\ov \FP} \models ``\cal K$ is
the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
By Lemma \ref{l1},
$V^{\ov \FP} \models ``\gk$ is the
least supercompact cardinal''.
Thus, to prove Lemma \ref{l4},
it suffices to show that in
$V^{\ov \FP}$, the class of supercompact
cardinals above $\gk$ is the same
as in $V$. However, since as
we observed in the proof of
Lemma \ref{l3}, ${\ov \FP}$ may be
defined so that
$\card{{\ov \FP}} \le \gk^{++}$, this
follows by the results of \cite{LS}.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Since $\FP^2$ has cardinality the least
$V$-Mahlo cardinal,
the results of \cite{LS} therefore imply that
the conclusions of Lemmas \ref{l1} -- \ref{l4}
remain true in $V^{\ov \FP \ast \dot \FP^2} = V^\FP$.
This observation, together with
Lemmas \ref{l1} -- \ref{l4} and the remarks
made prior to the proof of Lemma \ref{l1}, 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 $V$, $A$, $F$, and $\gk$ be as in the
hypotheses for Theorem \ref{t3}.
In order to present in a meaningful way the iteration
to be used in the proof of Theorem \ref{t3},
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}.
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}.
Note that our presentation is excerpted
almost verbatim from \cite[Section 2]{A06}.
Fix regular cardinals $\gd < \gl$, $\gl > \gd^+$ 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 non-reflecting 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
$X = \la x_\ga \mid \ga \in S \ra$,
the existence of which is given by
Lemma 1 of \cite{AS97a} and \cite{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 proof of Lemma 1
of \cite{AS97a} and \cite{AS97b}
shows each $x \in X$ can be assumed to be
such that the order type of $x$ is $\go$. 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 \mid i \in w \ra$ is a
sequence of functions from $\ga$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\ga$.
\item $Z \subseteq \{x_\gb \mid \gb \in S\}$
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 $|\{i \in z \mid
r^2_i(\ga) = 0\}| = |\{i \in z \mid r^2_i(\ga) = 1\}| = \go$.
\end{enumerate}
The intuition behind the above definition of $\FP^1_{\gd, \gl}[S]$
is described in the first complete paragraph on
\cite[page 2033]{AS97b}, from which we quote.
We wish to be able simultaneously to make
$2^\gd = \gl$, destroy the measurability of $\gd$, and be able
to resurrect the ${<} \gl$ supercompactness of $\gd$
if necessary. By its design, $\FP^1_{\gd, \gl}[S]$
allows us to accomplish these tasks. Specifically,
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 non-measurable''.
%Having briefly presented the definitions of
%the partial orderings
%$\FP^0_{\gd, \gl}$,
%$\FP^1_{\gd, \gl}[S]$,
%and
%$\FP^2_{\gd, \gl}[S]$
%of \cite[Section 4]{AS97b},
%we are now in a position to define the
%partial ordering used in the proof of Theorem \ref{t3}.
We return now to the definition of the
partial ordering $\FP$ used in the proof of Theorem \ref{t3}.
$\FP$ will be the reverse Easton iteration
$\la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd \le \gk \ra$
of length
$\gk + 1$ which begins by adding a Cohen subset of $\go$
and then does nontrivial forcing only at
members of $A$.
If $\gd \in A$ is $\gg$ supercompact for
$\gg \ge \gd^{+ 16}$, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for
$\FP^0_{\gd, F(\gd)} \ast (\FP^1_{\gd, F(\gd)}[\dot S] \times
\FP^2_{\gd, F(\gd)}[\dot S])$ as described above.
For all other $\gd \in A$,
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for
$\FP^0_{\gd, F(\gd)} \ast \FP^1_{\gd, F(\gd)}[\dot S]$.
We explicitly note that if $\gd \in A$ is $\gg$
supercompact for $\gg \ge \gd^{+ 16}$, then by our restrictions
on $F$, $F(\gd) = \gd^{+ 17}$.
The standard Easton arguments (see, e.g., \cite{J})
in combination with the properties of
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$
and
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$
described in
the next to last paragraph then show that
forcing with $\FP$ preserves all cardinals and cofinalities
and that in $V^\FP$,
$2^\gd = F(\gd)$ for every $\gd \in A$.
In addition, note that it is possible to write
$\FP = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $\ha_1$-strategically closed''.
Therefore, by the results of \cite{H2, H3}, any cardinal $\gd$
which is $\gl$ supercompact in $V^\FP$ had to have been
$\gl$ supercompact in $V$, and any cardinal $\gd$
which is either a strong cardinal or
an inaccessible limit of strong cardinals in $V^\FP$ had
to have been in $V$ either a strong cardinal or
an inaccessible limit of strong cardinals.
\begin{lemma}\label{l5}
$V^\FP \models ``\gk$ is the least supercompact cardinal''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l5} is a simplified version of
the proof of Lemma \ref{l1}.
By our remarks in the preceding
paragraph, since forcing with $\FP$
creates no new supercompact cardinals,
it suffices to show that
$V^\FP \models ``\gk$ is supercompact''.
Towards this end,
let $\gl \ge \gk^{+ 17} > 2^\gk = \gk^+$ be a regular cardinal,
and let $j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$ with
$F(\gk) = j(F)(\gk) = \gk^{+ 17}$.
Let $\theta = \gk^{+ 17}$.
Note that since $V \models ``\gk$ is the
least supercompact cardinal'',
$M \models ``j(\gk) > \gk$ is the least
supercompact cardinal'', i.e.,
$M \models ``\gk$ is not supercompact''.
By %the remarks immediately following
\cite[Lemma 2.1]{AC2}, $M \models ``\gk$ is a
regular cardinal which is a limit of strong cardinals''.
Also,
%as in \cite[Lemma 2.4]{AC2},
$M \models ``$No cardinal $\gd \in (\gk, \gl]$ is strong'',
since otherwise, $\gk$ is supercompact up to a
strong cardinal and hence is fully supercompact in $M$.
By the forcing equivalence of
$\FP^0_{\gk, \gk^{+ 17}} \ast (\FP^1_{\gk, \gk^{+ 17}}[\dot S] \times
\FP^2_{\gk, \gk^{+ 17}}[\dot S])$ with
$\add(\gk^{+ 17}, 1) \ast \dot \add(\gk, \gk^{+ 17})$
and the fact that
$M \models ``\gk$ is $\gk^{+ 16}$ supercompact'',
$j(\FP)$ is forcing equivalent to
$\FP_\gk \ast \dot \add(\gk^{+ 17}, 1)
\ast \dot \add(\gk, \gk^{+ 17}) \ast \dot \FQ \ast \dot
\add(j(\gk)^{+ 17}, 1) \ast \dot \add(j(\gk), j(\gk^{+ 17}))$.
This in turn is forcing equivalent to $\FP \ast \dot \FQ \ast \dot
\add(j(\gk)^{+ 17}, 1) \ast \dot \add(j(\gk), j(\gk^{+ 17}))$.
Let $G$ be $V$-generic over $\FP_\gk$, and let
$H$ be $V[G]$-generic over
${(\add(\gk^{+ 17}, 1) \ast \dot \add (\gk, \gk^{+ 17}))}^{V[G]}$.
%$\add(j(\gk)^{+ 17}, 1) \ast \dot \add(j(\gk), j(\gk^{+ 17}))$.
The arguments mentioned in the proof of Lemma \ref{l1}
for the construction of the generic objects
$H'$ and $H''$ may now be used to construct an
$M[G][H]$-generic object $H'$ over $\FQ$.
This allows us to lift $j$ in $V[G][H]$ to
$j : V[G] \to M[G][H][H']$, construct in $V[G][H]$
an $M[G][H][H']$-generic object $H''$ containing a
master condition for $j '' H$, and then fully lift $j$ to
$j : V[G][H] \to M[G][H][H'][H'']$.
This shows that
$V[G][H] \models ``\gk$ is $\gl$ supercompact''.
Since $\gl \ge \gk^{+ 17}$ was an arbitrary
regular cardinal, this completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$For every
$\gd \le \gk$ which is
a nontrivial stage of forcing,
%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 $\gd^{+ 16}$ supercompact
if $\gd$ is a measurable cardinal''.
\end{lemma}
\begin{proof}
Let $\gd \le \gk$ be a nontrivial stage of forcing.
By the definition of $\FP$, it must be the case that
in $V$, $\gd$ is either a strong cardinal or a regular
limit of strong cardinals.
We follow now the proof of \cite[Lemma 3.2]{A06}.
%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''.
%This factorization of $\FP$ indicates
%that the results of \cite{H2, H3}
%may be applied to show that
As we have already observed,
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
\cite[Lemma 2.1]{AC2} tells us that
$V \models ``\gd$ is not $2^\gd = \gd^+$
supercompact'', since otherwise, it would
have to be the case that
$V \models ``\gd$ is a limit of strong
cardinals''. Consequently, by the
definition of $\FP$,
$V^\FP \models ``\gd$ is not a
measurable cardinal''.
Further, by the definition of $\FP$, if
$V \models ``\gd$ is a regular limit of
strong cardinals which is not $\gd^{+ 16}$
supercompact'', then $V^\FP \models ``\gd$ 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 both (at least)
$\gd^{+ 16}$ supercompact and a limit of strong cardinals.
These last two sentences now tell us that the
argument of \cite[Lemma 4.2]{A01a}
goes through unchanged to show that
$V^\FP \models ``$For every
$\gd \le \gk$ which is a nontrivial
stage of forcing,
$\gd$ is $\gd^{+ 16}$ supercompact if
$\gd$ is a measurable cardinal''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
\begin{lemma}\label{l7}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 3.3]{A06}.
Suppose
$V^\FP \models ``\gd < \gl$ are
regular cardinals such that
$\gd$ is $\gl$ strongly compact and
$\gd$ is not a measurable limit of
cardinals $\gg$ which are
$\gl$ supercompact''.
Assume first $\gd > \gk$.
Since $\FP$ may
be defined so that $\card{\FP}= \gk^{+ 17}$,
by the results of \cite{LS},
Lemma \ref{l7} is true if $\gd > \gk$.
By Lemma \ref{l5}, Lemma \ref{l7}
is true if $\gd = \gk$.
It therefore
suffices to prove Lemma \ref{l7}
when $\gd < \gk$, which we
assume for the duration of the
proof of this lemma.
Let $A = \{\gg \le \gd \mid \gg$
is a strong cardinal or
an inaccessible limit of
strong cardinals$\}$. Write
$\FP = \FP_A \ast \dot \FQ$, where
$\FP_A$ is the portion of $\FP$
which does nontrivial forcing at
%whose field is composed of
ordinals at most $\gd$, and
$\dot \FQ$ is a term for the
rest of $\FP$, i.e., the portion
of $\FP$ doing nontrivial forcing
%whose field contains
at ordinals above $\gd$. We claim that
since $\gd < \gk$, it follows
that $\gl$ is below the
least $V$-strong cardinal $\gr$ above
$\gd$. This is because otherwise,
$V \models ``\gd$ is strongly compact
up to $\gr$ and $\gr$ is a strong cardinal'',
so by \cite[Lemma 1.1]{A02a},
$V \models ``\gd$ is strongly compact'',
a contradiction to the fact that $\gd < \gk$ and
$\gk$ is both the least $V$-supercompact
and least $V$-strongly compact cardinal.
As $\forces_{\FP_A} ``\dot \FQ$ is
$\gr$-strategically closed and
$\gr$ is inaccessible'',
$V^{\FP_A} \models ``\gd$
is $\gl$ strongly compact and $\gd$ is not
a measurable limit of cardinals
$\gg$ which are $\gl$ supercompact''.
Further, to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact'',
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
$\card{\FP_A} < \gd$.
%This is
%since if $\card{\FP_A} \ge \gd$,
%by the definition of $\FP$,
%the order type of $A$ is a successor ordinal,
%and $\gs$ must be the largest member of $A$.
%We may then infer that
%$V^{\FP_A} \models ``2^\gs \ge \gd$'', which
%contradicts the measurability of $\gd$ in
%$V^{\FP_A}$.
%Thus, since $\card{\FP_A} < \gd$,
Thus, by the results of \cite{LS},
$V \models ``\gd$ is $\gl$ strongly compact
and $\gd$ is not a measurable limit of cardinals
$\gg$ which are $\gl$ supercompact''.
Hence, by the level by level equivalence
between strong compactness and supercompactness
in $V$, $V \models ``\gd$ is $\gl$ supercompact'',
so again by the results of \cite{LS},
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''
as well. \bigskip
\noindent Case 2: $\sup(A) = \gd$.
As before,
we can write
$\FP_A = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is
nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is
$\ha_1$-strategically closed''.
Further, it is easily seen that
any subset of $\gd$ in
$V^{\FP_A}$ of size below $\gd$
has a name of size
below $\gd$ in $V$.
Therefore, as in Lemma \ref{l3}, by the results of
\cite{H2, H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
In addition, as in Case 1 above, it is
the case that
$V \models ``\gd$ is not a measurable
limit of cardinals $\gg$ which are
$\gl$ supercompact''.
This is since otherwise,
as $V \models ``\gd$ is
a limit of strong cardinals'',
some cardinal $\gg < \gd < \gk$
must be supercompact up to
a strong cardinal in $V$.
%By Lemma 1.1 of \cite{A02},
As we have already observed, $\gg$
%in the proof of Lemma \ref{l5}, $\gg$
is then supercompact in $V$, which
contradicts that $V \models ``\gk$ is
the least supercompact cardinal''.
Thus, by the level by level
equivalence between strong compactness
and supercompactness in $V$,
$V \models ``\gd$ is $\gl$ supercompact''.
Note that since $\gd$ is in $V$
a regular limit of strong cardinals,
$\gd$ is a nontrivial stage of forcing.
It must therefore be the case that
$V \models ``\gd$ is
$\gd^{+ 16}$ supercompact'', because
otherwise, by the definition of $\FP$,
$V^{\FP_A} \models ``\gd$ is not
measurable''.
Since by Lemma \ref{l6},
$V^\FP \models ``\gd$ is
$\gd^{+ 16}$ supercompact'',
which means by the definition of
$\FP$ that
$V^{\FP_A} \models ``\gd$ is
$\gd^{+ 16}$ supercompact'' as well,
we may assume without loss of
generality that $\gl \ge \gd^{+ 17}$.
Consequently, let $j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter
over $P_\gd(\gl)$ such that
$j(F)(\gd) = F(\gd) = \gd^{+ 17}$.
Since $\gl \ge \gd^{+ 17}$,
$\FP_A$ is forcing equivalent to
$\FP_\gd \ast \dot \FQ^*$, where
$\forces_{\FP_\gd} ``|\dot \FQ^*|
= \gd^{+ 17} \le \gl$ and
$\dot \FQ^*$ is $\gd$-directed closed''.
($\FQ^*$ is forcing equivalent to
$\add(\gd^{+ 17}, 1) \ast
\dot \add(\gd, \gd^{+ 17})$.)
In addition, the same reasoning as
found in the proof of Lemma \ref{l5}
shows that
$M \models ``$No cardinal in the
half-open interval $(\gd, \gl]$ is strong''.
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 do nontrivial forcing is well above $\gl$.
The same argument as given in the
proof of Lemma \ref{l5} then shows that
$V^{\FP_A} \models ``\gd$ is $\gl$ supercompact''.\bigskip
Cases 1 and 2 now complete
the proof of Lemma \ref{l7}.
\end{proof}
Since $\FP$ may be defined so that
$\card{\FP} = \gk^{+ 17}$, the proof of
Lemma \ref{l4} shows that
$V^\FP \models ``{\cal K}$ is the class
of supercompact cardinals''.
Lemmas \ref{l5} -- \ref{l7}
and the remarks made prior to the proof of
Lemma \ref{l5} therefore complete the proof
of Theorem \ref{t3}.
\end{proof}
As in \cite{A05}, we ask
%conclude by asking
what other types of Easton theorems are
consistent with the level by level equivalence
between strong compactness and supercompactness.
In particular, is it possible to include
regular cardinals in the domain of the Easton
function $F$ which are above the least supercompact
cardinal $\gk$?
This is precluded by the restrictions
on $A$ given in Theorem \ref{t2}.
Is it possible for $F$'s domain
%the domain of the Easton function $F$
to be all regular cardinals,
as in Easton's original result?
Which cardinals may be included in $F$'s range?
Finally, as the referee has asked,
are there any GCH patterns which are incompatible
with the level by level equivalence between strong
compactness and supercompactness?
\begin{thebibliography}{99}
\bibitem{A05} A.~Apter,
``An Easton Theorem for Level by Level Equivalence'',
{\it Mathematical Logic Quarterly 51}, 2005, 247--253.
\bibitem{A03} A.~Apter,
``Failures of GCH and the Level by Level
Equivalence between Strong Compactness and
Supercompactness'', {\it Mathematical
Logic Quarterly 49}, 2003, 587--597.
\bibitem{A02} A.~Apter,
``On Level by Level Equivalence and
Inequivalence between Strong
Compactness and Supercompactness'',
{\it Fundamenta Mathematicae 171},
2002, 77--92.
\bibitem{A02a} A.~Apter, ``On the Non-Extendibility of
Strongness and Supercompactness through Strong
Compactness'', {\it Fundamenta Mathematicae 174},
2002, 87--96.
%\bibitem{A01} A.~Apter, ``Some Structural Results
%Concerning Supercompact Cardinals'',
%{\it Journal of Symbolic Logic 66}, 2001, 1919--1927.
\bibitem{A01a} A.~Apter, ``Supercompactness and
Measurable Limits of Strong Cardinals'',
{\it Journal of Symbolic Logic 66}, 2001, 629--639.
\bibitem{A06} A.~Apter, ``Supercompactness and
Measurable Limits of Strong Cardinals II:
Applications to Level by Level Equivalence'',
{\it Mathematical Logic Quarterly 52}, 2006, 457--463.
\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness II: Strong Cardinals'',
{\it Archive for Mathematical Logic 40}, 2001, 25--38.
\bibitem{AH4} A.~Apter, J.~D.~Hamkins,
``Indestructibility and the Level-by-Level Agreement
between Strong Compactness and Supercompactness'',
{\it Journal of Symbolic Logic 67}, 2002, 820--840.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong
Equality between Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349},
1997, 103--128.
\bibitem{AS97b} A.~Apter, S.~Shelah, ``Menas'
Result is Best Possible'',
{\it Transactions of the American Mathematical Society 349},
1997, 2007--2034.
%\bibitem{H5} J.~D.~Hamkins, ``Extensions with
%the Approximation and Cover Properties
%Have No New Large Cardinals'',
%to appear in
%{\it Fundamenta Mathematicae}.
\bibitem{H2} J.~D.~Hamkins, ``Gap Forcing'',
{\it Israel Journal of Mathematics 125}, 2001, 237--252.
\bibitem{H3} J.~D.~Hamkins, ``Gap Forcing:
Generalizing the L\'evy-Solovay Theorem'',
{\it Bulletin of Symbolic Logic 5}, 1999, 264--272.
%\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
%{\it Annals of Pure and Applied Logic 101},
%2000, 103--146.
%\bibitem{H5} J.~D.~Hamkins, ``Small Forcing Makes
%Any Cardinal Superdestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 51--58.
\bibitem{J} T.~Jech, {\it Set Theory:
The Third Millennium Edition,
Revised and Expanded},
%$3^{\rm rd}$ Millennium Edition,
Springer-Verlag, Berlin and New York, 2003.
\bibitem{JMMP} T.~Jech, M.~Magidor,
W.~Mitchell, K.~Prikry, ``Precipitous Ideals'',
{\it Journal of Symbolic Logic 45}, 1980, 1--8.
\bibitem{JW} T.~Jech, W.H.~Woodin,
``Saturation of the Closed Unbounded Filter on the
Set of Regular Cardinals'',
{\it Transactions of the American Mathematical
Society 292}, 1985, 345--356.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite}, Springer-Verlag,
Berlin and New York, 1994.
\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
between the Concepts of Compactness and Supercompactness'',
circulated manuscript.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
\bibitem{Ma1} M.~Magidor, ``How Large is the
First Strongly Compact Cardinal? or A Study on
Identity Crises'', {\it Annals of Mathematical Logic 10},
1976, 33--57.
\bibitem{Ma2} M.~Magidor, ``On the Existence of
Nonregular Ultrafilters and the Cardinality of
Ultrapowers'', {\it Transactions of the American
Mathematical Society 249}, 1979, 97--111.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
%\bibitem{So} 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}