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\title{The Wholeness Axioms and the Class of
Supercompact Cardinals
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55, 03E65.}
\thanks{Keywords: Wholeness Axioms,
supercompact cardinal,
strongly compact cardinal, indestructibility,
level by level equivalence between strong
compactness and supercompactness,
%level by level inequivalence between strong
%compactness and supercompactness,
non-reflecting stationary set of ordinals.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}\\
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/apter\\
awapter@alum.mit.edu}
\date{November 1, 2010}
\begin{document}
\maketitle
\begin{abstract}
We show that certain relatively consistent
structural properties
of the class of supercompact cardinals
%known to be relatively consistent
are also
relatively consistent with the Wholeness Axioms.
%known to be relatively consistent with
%the class of supercompact cardinals are also
%relatively consistent with the class of supercompact
%cardinals and the Wholeness Axioms.
%We prove some structural theorems concerning
%the Wholeness Axioms and the class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
The Wholeness Axiom WA, introduced by Paul Corazza
in \cite{C00a} and \cite{C00b}, is intended as a weakening of
Kunen's inconsistency result of \cite{Ku71}
concerning the nonexistence of a nontrivial
elementary embedding from the universe to itself.
In addition, in \cite{H01}, Hamkins gave a
stratification of WA into countably many axioms
${\rm WA}_0, {\rm WA}_1, \ldots, {\rm WA}_\infty$,
with ${\rm WA}_\infty$ the original Wholeness Axiom WA.
More specifically, work in the language $\{\in, {\bf j}\}$
extending the usual language of set theory $\{\in\}$,
where {\bf j} is a unary function symbol representing
the elementary embedding.
For $n \in \{0, 1, 2, 3, \ldots, \infty\}$,
${\rm WA}_n$ is defined as:
\begin{enumerate}
\item (Elementarity) All instances of
$\varphi(x) \iff \varphi({\bf j}(x))$ for
$\varphi$ a formula in the language $\{\in\}$.
\item (Separation) All instances of the Separation Axiom
for $\Sigma_n$ formulae in the full language $\{\in, {\bf j}\}$.
\item (Nontriviality) The axiom $\exists x[{\bf j}(x) \neq x]$.
\end{enumerate}
The purpose of this paper is to augment the studies of
the Wholeness Axioms found in \cite{C00a}, \cite{C00b},
\cite{C03}, \cite{H01}, \cite{C06}, \cite{C07}, and \cite{AF}
and show that some relatively consistent structural
properties of the class of supercompact cardinals
are also relatively consistent with the Wholeness Axioms.
%and establish some structural theorems concerning the
%Wholeness Axioms and the class of supercompact cardinals.
Specifically, we prove the following.
\begin{theorem}\label{t1}
If the Wholeness Axiom ${\rm WA}_0$ is consistent,
then it is consistent with the following:
\begin{enumerate}
\item The classes of supercompact and strongly
compact cardinals coincide precisely, except
at measurable limit points.
\item Each supercompact cardinal $\gk$ is
Laver indestructible \cite{L} under
$\gk$-directed closed forcing.
\item Each non-supercompact strongly compact
cardinal $\gk$ has its strong compactness
indestructible under $\gk$-directed closed
forcing not changing $\wp(\gk)$.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{t2}
If the Wholeness Axiom ${\rm WA}_0$ is consistent,
then it is consistent with {\rm GCH} and level by level equivalence
between strong compactness and supercompactness.
\end{theorem}
\begin{theorem}\label{t3}
If the existence of an ${\rm I}_3$ cardinal
is consistent, then the (full) Wholeness Axiom {\rm WA}
is consistent with the following:
\begin{enumerate}
\item The classes of supercompact and strongly
compact cardinals coincide precisely, except
at measurable limit points.
\item Each supercompact cardinal $\gk$ is
Laver indestructible under
$\gk$-directed closed forcing.
\item Each non-supercompact strongly compact
cardinal $\gk$ has its strong compactness
indestructible under $\gk$-directed closed
forcing not changing $\wp(\gk)$.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{t4}
If the existence of an ${\rm I}_3$ cardinal
is consistent, then the (full) Wholeness Axiom {\rm WA}
is consistent with {\rm GCH} and level by level equivalence
between strong compactness and supercompactness.
\end{theorem}
A few remarks concerning the above theorems are now in order.
By the work of \cite{C00b} and \cite{C06},
ZFC + ${\rm WA}_0$ implies the existence of
a proper class of supercompact limits of supercompact
cardinals (and much more).
Further, by the work of Menas \cite{Me}, if $\ga < \gk$
and $\gk$ is the $\ga^{\rm th}$ measurable limit of
either supercompact or (non-supercompact) strongly
compact cardinals, then $\gk$ is strongly compact
but is not supercompact.
Thus, Theorems \ref{t1} -- \ref{t4}
are meaningful.
In addition, the indestructibility for
non-supercompact strongly compact cardinals found
in Theorems \ref{t1} and \ref{t3} was first
discussed in \cite{A98}.
Also, property (1) of Theorems \ref{t1} and \ref{t3}
was first introduced and established by Kimchi and
Magidor in \cite{KM}, and the property given in
%found to be consistent with the Wholeness Axioms in
Theorems \ref{t2} and \ref{t4} (``level by level equivalence
between strong compactness and supercompactness'', i.e.,
for $\gk < \gl$ regular cardinals, $\gk$ is $\gl$
strongly compact iff $\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact)
was first discussed and shown to
%be possible
be relatively consistent
by the author and Shelah in \cite{AS97a}.
Consequently, Theorems \ref{t1} -- \ref{t4}
demonstrate that certain interesting
structural properties that the class of
supercompact cardinals may possess
are also relatively consistent with the Wholeness Axioms.
Before beginning the proofs of Theorems \ref{t1} -- \ref{t4},
we very briefly mention some preliminary material
concerning notation and terminology.
For $\ga < \gb$ ordinals, $(\ga, \gb)$
is as in usual interval notation.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
If $G \subseteq \FP$ is $V$-generic, we will abuse
notation somewhat and use both $V[G]$ and
$V^\FP$ to denote the generic extension by $\FP$.
We will also occasionally abuse notation by writing
$x$ when we really mean $\check x$.
For $\gk$ a regular cardinal and $\ga$ an arbitrary ordinal,
${\rm Add}(\gk, \ga)$ is the standard partial ordering
for adding $\ga$ many Cohen subsets of $\gk$.
The partial ordering $\FP$ is {\em $\gk$-directed closed}
if every directed subset of $\FP$
of size less than $\gk$ has an upper bound.
$\FP$ is {\it $\gk$-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 ${<}\gk$-{\em strategically closed}
if $\FP$ is $\gd$-strategically closed for
every cardinal $\gd < \gk$.
If $\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
is an Easton support iteration of length $\gk$
and $0 \le \gg < \gd < \gk$,
we will abuse notation by writing
$\FP_{\gg, \gd}$ for both the portion of the iteration
{\em strictly between $\gg$ and $\gd$}
(i.e., we will use this notation in the proofs of
Theorems \ref{t1}, \ref{t3}, and \ref{t4})
and the portion of
the iteration {\em between $\gg$ and $\gd$ but including $\gd$}
(particularly in the proof of Theorem \ref{t2},
where both usages will occur).
It will be clear from the context exactly which of the
two of these is meant.
As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra \mid \FP \in {\cal A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. The terminology ``lottery sum''
is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the expressions ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
Finally, we mention that we are assuming a reasonable
familiarity with standard concepts in large cardinals
and forcing, as found, e.g., in \cite{J} or \cite{K}.
We do note explicitly that we will say {\em $\gk$ is
${<} \gl$ supercompact} if $\gk$ is $\gd$ supercompact
for every $\gd < \gl$. In addition,
{\em an ${\rm I}_3$ cardinal $\gk$}
is a cardinal such that there exists an elementary
embedding $j : V_\gl \to V_\gl$ having critical point
$\gk$ with $\gl$ the supremum of
{\em the critical sequence associated with $j$}, i.e.,
$\gl = \bigcup_{i < \go} \gk_i$, where
$\gk = \gk_0 = {\rm cp}(j)$ and $\gk_{i + 1} = j(\gk_i)$.
Additional information on ${\rm I}_3$ cardinals
may be found in \cite{K}.
\section{The Proofs of Theorems \ref{t1} -- \ref{t4}}\label{s2}
We turn now to the proofs of our theorems.
We will provide full details for the proof
of Theorem \ref{t1} and indicate how the
proofs of our remaining results follow from
earlier work, which is found both in this paper and elsewhere.
In particular, the proofs of Theorems \ref{t3}
and \ref{t4} will really only be proof sketches.
We begin with the proof of Theorem \ref{t1}.
\begin{proof}
Assume that ${\rm WA}_0$ is consistent.
As Hamkins remarks in \cite{H01},
this means that there is a model
$\la V, \in j \ra$, where $\la V, \in \ra$
is a model of ZFC and $j : V \to V$ is a
nontrivial amenable elementary embedding.
We take the structure $\la V, \in j \ra$ as our ground model.
We also let $\la \gk_i \mid i < \go \ra$ be
the critical sequence associated with $j$.
%, i.e., $\gk = \gk_0 = {\rm cp}(j)$, and $\gk_{i + 1} = j(\gk_i)$.
As shown in \cite{H01},
\[V_{\gk_0} \prec V_{\gk_1} \prec \cdots \prec V\]
is an elementary chain of models, with
$V = \bigcup_{i < \go} V_{\gk_i}$.
In particular, elementarity implies that
$V_{\gk_0} \models ``$There is a proper
class of supercompact limits of supercompact cardinals''.
Note that by the work of \cite{C00b} and \cite{C06},
for each $i < \go$, $V \models ``\gk_i$ is a
supercompact limit of supercompact cardinals''.
Therefore, since $V_{\gk_j} \prec V$ for any
$j < \go$, for each $i < \go$ and all $j > i$,
$V_{\gk_j} \models ``\gk_i$ is a supercompact
limit of supercompact cardinals''.
In particular, for any $i < \go$, there is a cardinal
$\gk^*_i > \gk_i$ such that in both
$V_{\gk_{i + 1}}$ and $V$, $\gk^*_i$ is the least
supercompact cardinal greater than $\gk_i$.
We now follow the plan of attack found in the proof
of the Main Theorem of \cite{H01} by
first defining a class partial ordering $\FP_{\gk_0}$ in $V_{\gk_0}$
such that after forcing with $\FP_{\gk_0}$ over
$V_{\gk_0}$, the resulting model satisfies
properties (1) -- (3) of Theorem \ref{t1}.
We give the definition used in the proof of
\cite[Theorem 2]{A03}, as opposed to the original
one employed in the proof of the Theorem of \cite{A98}.
Specifically, working in $V_{\gk_0}$, let
${\cal K}$ be the class of supercompact cardinals.
Let
${\cal D} = \la \gd_\ga \mid \ga \in ({\rm Ord})^{V_{\gk_0}} \ra$
enumerate in increasing order all
regular limits of strong cardinals.
For any ordinal $\gd$, define $\gs_\gd$
as the successor of the smallest regular cardinal greater than
or equal to the supremum of the supercompact
cardinals below $\gd$, or $\go$ if there are
no supercompact cardinals below $\gd$.\footnote{In
\cite{A03}, $\gs_\gd$ is defined as
the smallest regular cardinal greater than
or equal to the supremum of the supercompact
cardinals below $\gd$, or $\go$ if there are
no supercompact cardinals below $\gd$. This
difference in definition, however, is inessential,
and does not affect any of the proofs given in \cite{A03}.}
The partial ordering $\FP_{\gk_0}$ with which we force
is the proper class Easton support iteration
$\la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd
\in ({\rm Ord})^{V_{\gk_0}} \ra$
%such that $\FP_0$ is trivial forcing and $\dot \FQ_0$
which begins by adding a Cohen subset of $\omega$
and then does trivial forcing except when
$\gd \in {\cal D}$. At such a stage,
%$\dot \FQ_\gd$ is a term for trivial forcing
%unless $\gd \in {\cal D}$. When $\gd \in {\cal D}$,
$\dot \FQ_\gd$ has the form
$\dot \FQ_{\gd, 1} \ast \dot \FQ_{\gd, 2}$, where
$\dot \FQ_{\gd, 1}$ is a term for the lottery sum
of all $\gd$-directed closed partial orderings
having rank below the least
$V_{\gk_0}$-strong cardinal $\gd'$ above
$\gd$, and $\dot \FQ_{\gd, 2}$ is a term for
the standard partial ordering (see \cite{AS97a})
which adds a
non-reflecting stationary set of ordinals of cofinality
$\sigma_{\gd'}$ to $\gd'$.\footnote{We refer readers
to \cite{AS97a} for the exact definition and properties
of the standard partial ordering for adding
a non-reflecting stationary set of ordinals of
cofinality $\gd$ to the regular cardinal $\gl$.
We note only that this partial ordering is both
$\gd$-directed closed and ${<}\gl$-strategically closed.}
We continue by following the proof of the Main Theorem
of \cite{H01}, taking the liberty to quote verbatim
when appropriate. We also refer readers to \cite{H01}
for any missing or unexplained details.
We begin by letting $G_{\gk_0} \subseteq \FP_{\gk_0}$
be $V$-generic. Next, we consider the partial ordering
$j(\FP_{\gk_0}) = \FP_{\gk_1}$. Observe that if
$\gd < \gk$ and $\varphi(x)$ is the formula
in the language of set theory which says either
``$x$ is a strong cardinal'' or ``$x$ is
a supercompact cardinal'', then by the fact that
$V_{\gk_0} \prec V_{\gk_1}$, for $\gd < \gk_0$,
$V_{\gk_0} \models \varphi(\gd)$ iff
$V_{\gk_1} \models \varphi(\gd)$.
%To see this, by the fact $\gk = {\rm cp}(j)$, if
%$V_\gk \models \varphi(\gd)$, then
This means we may write $j(\FP_{\gk_0})$ as
$\FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk_1}$.
Further, since
$V_{\gk_1} \models ``\gk_0$ is supercompact'',
by forcing above a condition opting
for trivial forcing in the lottery sum held
at stage $\gk_0$,
we may assume that
$V[G_{\gk_0}] \models ``\FP_{\gk_0, \gk_1}$ is
$\gk^+_0$-directed closed''.
Force now to obtain a $V[G_{\gk_0}]$-generic object
$G_{\gk_0, \gk_1} \subseteq \FP_{\gk_0, \gk_1}$, which
also provides a $V$-generic object $G_{\gk_1} =
G_{\gk_0} \ast G_{\gk_0, \gk_1} \subseteq \FP_{\gk_1}$.
Since $j '' G_{\gk_0} = G_{\gk_0} \subseteq G_{\gk_1}$,
as usual, $j$ lifts (in $V[G_{\gk_1}]$)
to $j : V[G_{\gk_0}] \to V[G_{\gk_1}]$
with $j(G_{\gk_0}) = G_{\gk_1}$. In addition,
as in \cite{H01},
$V_{\gk_0}[G_{\gk_0}] \prec V_{\gk_1}[G_{\gk_1}]$.
Next, consider the partial ordering
$j(\FP_{\gk_1}) = \FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk_2} =
\FP_{\gk_1} \ast \dot \FP_{\gk_1, \gk_2}$. As before,
by forcing above a condition opting for trivial forcing
in the lottery sum held at stage $\gk_1$, since
$V_{\gk_2} \models ``\gk_1$ is supercompact'',
we may assume that
$V[G_{\gk_1}] \models ``\FP_{\gk_1, \gk_2}$ is
$\gk^+_1$-directed closed''. Therefore, since
$j '' G_{\gk_0, \gk_1}$ is a directed subset of
$\FP_{\gk_1, \gk_2}$ in $V[G_{\gk_1}]$ having size $\gk_1$,
by the directed closure of $\FP_{\gk_1, \gk_2}$ in
$V[G_{\gk_1}]$, there is a master condition $q_1$ for
$j '' G_{\gk_0, \gk_1}$. Let $G_{\gk_1, \gk_2}
\subseteq \FP_{\gk_1, \gk_2}$ be a $V[G_{\gk_1}]$-generic
object containing $q_1$. We now have a
$V$-generic object $G_{\gk_2} = G_{\gk_1} \ast G_{\gk_1, \gk_2}
\subseteq \FP_{\gk_2}$ such that
$j '' G_{\gk_1} \subseteq G_{\gk_2}$.
By continuing inductively in this manner for $\go$ many steps,
we obtain $V$-generic objects $G_{\gk_n} \subseteq
\FP_{\gk_n}$ for every $n \in \go - \{0\}$ and master
conditions $q_n$ for $j''G_{\gk_{n - 1}, \gk_n}$ such that
$q_n \in G_{\gk_{n}, \gk_{n + 1}}$ and
$j '' G_{\gk_n} \subseteq G_{\gk_{n + 1}}$.
This means that working in $V[G_{\gk_{n + 1}}]$, it is always
possible to lift $j$ to
$j : V[G_{\gk_n}] \to V[G_{\gk_{n + 1}}]$ and have that
$j(G_{\gk_n}) = G_{\gk_{n + 1}}$ and
$V_{\gk_n}[G_{\gk_n}] \prec V_{\gk_{n + 1}}[G_{\gk_{n + 1}}]$.
This produces the elementary chain of models of length $\go$
$$V_{\gk_0}[G_{\gk_0}] \prec V_{\gk_1}[G_{\gk_1}]
\prec \cdots \prec V_{\gk_n}[G_{\gk_n}] \prec \cdots .$$
Let $\ov V = \bigcup_{n \in \go} V_{\gk_n}[G_{\gk_n}]$.
Since $\ov V$ is the union of an elementary chain of
models, the theory of $\ov V$ is the theory of each
$V_{\gk_n}[G_{\gk_n}]$. In particular, the theory of
$\ov V$ is the same as the theory of $V_{\gk_0}[G_{\gk_0}]$,
so $\ov V$ is a model of ZFC satisfying properties
(1) -- (3) of Theorem \ref{t1}.
Also, because for every $n \in \go$
we have already lifted $j$ to
$j : V[G_{\gk_n}] \to V[G_{\gk_{n + 1}}]$,
we have defined a map $j : \ov V \to \ov V$.
The argument that $j$ is elementary, and consequently, that
$\la \ov V, \in j \ra$ is a model of ${\rm WA}_0$, is now
the same as in \cite{H01}. Specifically, if
$\ov V \models \varphi(x)$, then because we have an
elementary chain of models,
$V_{\gk_n}[G_{\gk_n}] \models \varphi(x)$
for sufficiently large $n$.
An application of $j$ now yields that
$V_{\gk_{n + 1}}[G_{\gk_{n + 1}}] \models
\varphi(j(x))$ for such $n$,
so once again, the fact that we have an elementary
chain of models yields that $\ov V \models \varphi(j(x))$.
In addition, since $j \rest V[G_{\gk_n}]$ was defined in
$V[G_{\gk_{n + 1}}]$, it follows that
$j \rest V_{\gk_n}[G_{\gk_n}] \in \ov V$.
Hence, $j : \ov V \to \ov V$ is amenable,
so $\la \ov V, \in, j \ra$ is a model of ${\rm WA}_0$.
This completes the proof of Theorem \ref{t1}.
\end{proof}
\begin{pf}
Turning now to the proof of Theorem \ref{t2},
let $\la V, \in, j \ra$ be a model for ${\rm WA}_0$.
As in \cite{H01}, we may assume that
$V \models {\rm GCH}$ as well. In addition,
as in the proof of Theorem \ref{t1}, let
$\la \gk_i \mid i < \go \ra$ be the critical
sequence generated by $j$.
As in the proof of Theorem \ref{t1},
we first define a class partial ordering
$\FP_{\gk_0}$ such that after forcing over
$V_{\gk_0}$ with $\FP_{\gk_0}$, the resulting model
satisfies GCH and level by level equivalence
between strong compactness and supercompactness.
%we begin with a description of the partial ordering
%$\FP$ used in the proof of the general case of
%the Theorem of \cite{AS97a}.
We start by defining
the partial orderings
$\FP^0_{\gd, \gl}$,
$\FP^1_{\gd, \gl}[S]$,
and
$\FP^2_{\gd, \gl}[S]$
of \cite[Section 1]{AS97a}
in an arbitrary ground model $\ov V \models {\rm ZFC}$.
So that readers are not overly burdened, we
abbreviate our definitions and descriptions somewhat.
Full details may be found by consulting
\cite{AS97a}.
Fix $\gd < \gl$, $\gl \ge \ha_1$ regular cardinals in our
ground model $\ov V$.
%We assume in addition that $V \models {\rm GCH}$.
The first notion of forcing $\FP^0_{\gd, \gl}$ is once again
the standard partial ordering
for adding a non-reflecting stationary
set of ordinals $S$ of cofinality
$\gd $ to $\gl^+$.
%Note that $\FP^0_{\gd, \gl}$ is $\gd$-directed closed.
Next, work in
$V_1 = {\ov V}^{\FP^0_{\gd, \gl}}$, letting $\dot S$
be a term always forced to denote $S$.
$\FP^2_{\gd, \gl}[S]$ is the usual partial ordering
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ non-stationary).
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
\cite[Lemma 1]{AS97a}.
We are ready to define in $V_1$
the partial ordering $\FP^1_{\gd, \gl}
[S] $.
First, since each element of
$S$ has cofinality $\gd$, the proof of \cite[Lemma 1]{AS97a}
shows each $x \in X$ can be assumed to be
such that order type$(x) = \gd$. 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 $w \in {[\gl^+]}^{< \gl}$.
\item $\ga < \gl$.
\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
$y \in {[w]}^\gd$, $y \subseteq z$ 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 $z \in Z^1 \cap {[w^1]}^\gd$ and
$\ga^1 \le \ga < \ga^2$, then $|\{i \in z \mid
r^2_i(\ga) = 0\}| = |\{i \in z \mid r^2_i(\ga) = 1\}| = \gd$.
\end{enumerate}
By \cite[Lemma 4]{AS97a},
$\FP^0_{\gd, \gl} \ast (\FP^1_{\gd, \gl}[\dot S] \times
\FP^2_{\gd, \gl}[\dot S])$ is equivalent to
${\hbox{\rm Add}}(\gl^+, 1) \ast \dot {\hbox{\rm Add}}(\gl, \gl^+)$
and so is $\gl$-directed closed.
In particular,
$\FP^0_{\gd, \gl} \ast \FP^2_{\gd, \gl}[\dot S]$
is equivalent to ${\hbox{\rm Add}}(\gl^+, 1)$, and
after forcing with
$\FP^0_{\gd, \gl} \ast \FP^2_{\gd, \gl}[\dot S]$,
$\FP^1_{\gd, \gl}[\dot S]$ is equivalent to
${\hbox{\rm Add}}(\gl, \gl^+)$.
By the remark on \cite[middle of page 108]{AS97a}
and the fact that $\FP^0_{\gd, \gl}$ is $\gd$-directed closed,
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ is
$\gd$-directed closed.
By \cite[Lemma 5]{AS97a},
%and Lemma 6 of \cite{AS97b}
$\FP^0_{\gd, \gl} \ast \FP^1_{\gd, \gl}[\dot S]$ preserves
GCH, cardinals and cofinalities, is $\gl^{++}$-c.c., and
is ${<}\gl$-strategically closed.
We are now in a position to define $\FP_{\gk_0}$.
We quote verbatim from \cite[page 121]{AS97a} when appropriate.
Work in $V_{\gk_0}$.
Let $\langle \delta_i\mid i \in ({\rm Ord})^{V_{\gk_0}} \rangle $
enumerate the
inaccessible cardinals,
and let $\lambda_i > \delta_i$ be the least regular
cardinal such that $V_{\gk_0} \models
``\delta_i$ isn't $\lambda_i$ supercompact''
if such a $\lambda_i$ exists.
If no such $\lambda_i$ exists, i.e., if $\delta_i$ is supercompact,
then let $\lambda_i = \Omega$, where we think of $\Omega$ as some
giant ``ordinal'' larger than any $\alpha \in
({\rm Ord})^{V_{\gk_0}}$.
If possible, choose $\theta_i < \delta_i$ as the least
regular cardinal
such that $\theta_i < \delta_k <
\delta_i$
implies $\lambda_k < \delta_i$ (whenever $k < i$).
Note that $\theta_i$ is undefined for $\delta_i$ iff $\delta_i$
is a limit of cardinals which are ${<} \delta_i$ supercompact
because for $k < i$, if $\delta_k$ is ${<} \delta_i$
supercompact, then $\l_k \ge \delta_i$.
We define now a class Easton support iteration
$\FP_{\gk_0} = \langle \langle \FP_\a, \dot
\FQ_\a \rangle \mid \a \in ({\rm Ord})^{V_{\gk_0}} \rangle $
as follows.
\begin{enumerate}
\item $\FP_0$ is trivial.
\item Assuming $\FP_\a$ has been defined, $\FP_{\a + 1} = \FP_\a \ast
\dot \FQ_\a$, where
$ \dot \FQ_\a$ is a term for the trivial
partial ordering unless $\a$ is regular
and for some inaccessible $\delta = \delta_i < \alpha$
with $\theta_i$ defined,
either $\delta_i$ is $\a$ supercompact or $\a = \lambda_i$.
Under these circumstances, $\dot \FQ_\a$ is a term for
$$(\prod_{
\{ i < \a\ \mid \ \delta_i \ {\rm is} \ \ga \ {\rm supercompact}\}}
%{\hbox{\rm is }}\; \a \; {\hbox{\rm supercompact}} \}}
(\FP^0_{\theta_i, \a} \ast
\FP^2_{\theta_i, \a} [\dot S_{\theta_i, \a} ]) \ast
\prod_{
\{ i < \a\ \mid \ \delta_i \ {\rm is} \ \ga \ {\rm supercompact}\}}
%\underset \{ i < \a\mid \delta_i \; \; \hbox{\rm is} \; \; \a \; \;
%\hbox{\rm supercompact} \} \to{\Pi}
\FP^1_{\theta_i, \a} [\dot S_{\theta_i, \alpha}])$$
$$\times$$
$$(\prod_{\{ i < \a\ \mid \ \a = \lambda_i \}}
\FP^0_{\theta_i, \alpha}
\ast
\prod_{\{ i < \a\ \mid \ \a = \lambda_i \}}
\FP^1_{\theta_i, \alpha} [\dot S_{\theta_i, \alpha}])$$
$ =
(\dot \FP^0_\a
\ast \dot \FP^1_\a) \times (\dot \FP^2_\a \ast \dot \FP^3_\a),$
with the proviso that elements of $\dot \FP^0_\a$ and $\dot \FP^2_\a$
will have
full
support, and elements of $\dot \FP^1_\a$ and $\dot \FP^3_\a$ will have
support less than $\ga$.
\end{enumerate}
%\noindent Note that unless $| \{ i < \a \mid \delta_i$
%is ${<} \a$ supercompact$\} | = \a$,
% the elements of $\dot \FP^i_\a$
% will have full support for $i = 0, 1, 2, 3.$
%To prove Theorem \ref{t2}, we begin by noting that
Observe that
the definition of $\FP_{\gk_0}$ easily implies that
$\FP_{\gk_0}$ is an initial segment of
$j(\FP_{\gk_0}) = \FP_{\gk_1}$.
%$j(\FP_{\gk_0}) = \FP_{\gk_0} \ast \dot \FQ_{\gk_0}
%\ast \dot \FP_{\gk_0, \gk_1}$.
We consider now the partial ordering $\FP_{\gk_1}$.
It is a folklore fact
that if $\gd$ is ${<}\gg$ supercompact and $\gg$
is measurable, then $\gd$ is $\gg$ supercompact.\footnote{If
$\la \mu_\ga \mid \ga < \gg \ra$ is a sequence of normal
measures over $P_\gd(\ga)$ and $\nu$ is a normal
measure over $\gg$, then $\mu = \{x \subseteq
P_\gd(\gg) \mid \{\ga < \gg \mid x \cap P_\gd(\a) \in
\mu_\ga\} \in \nu\}$ is easily verified as
being a normal measure over $P_\gd(\gg)$.}
Thus, for any $\gd_i$ with $i < \gk_0$,
since $\gk_0$ is a measurable
cardinal in both $V$ and $V_{\gk_1}$,
it is impossible to have
$\gl_i = \gk_0$. By the definition of $\FP_{\gk_0}$,
this means that $\dot \FQ_{\gk_0}$ has the form
$\dot \FP^0_{\gk_0} \ast \dot \FP^1_{\gk_0}$, so by
%\cite[Lemma 4]{AS97a} and
our remarks on directed closure in the paragraph immediately
following the definition of the ordering on
$\FP^1_{\gd, \gl}[S]$,
$\forces_{\FP_{\gk_0}} ``\dot \FP^0_{\gk_0} \ast \dot \FP^1_{\gk_0}$
is equivalent to a $\gk_0$-directed closed partial ordering
having size $\gk^+_0$''.
Further, by the definition of $\gth_i$ and the fact that
$V \models ``\gk_0$ is a supercompact limit of
supercompact cardinals'' (which as we have already observed
means that
$V_{\gk_1} \models ``\gk_0$ is a supercompact
limit of supercompact cardinals''),
if $\gd_i > \gk_0$ is an inaccessible cardinal
for which $\gth_i$ is defined,
then $\gth_i \ge \gk_0$.\footnote{Note that
by the fact $\gk_0$ is a limit
of cardinals which are ${<}\gk_0$ supercompact,
$\gth_{\gk_0}$ is undefined.}
In addition, by the definition of $\gth_i$ and
the fact that $V_{\gk_1} \models ``\gk^*_0$
is supercompact'', if $\gd_i > \gk^*_0$ is an
inaccessible cardinal for which $\gth_i$ is defined,
then $\gth_i \ge \gk^*_0$.
Therefore, by our remarks on directed closure
in the paragraph immediately
following the definition of the ordering on
$\FP^1_{\gd, \gl}[S]$ and the definition of $\FP_{\gk_0}$,
we may write
$\FP_{\gk_1} = \FP_{\gk_0} \ast
\dot \FP_{\gk_0, \gk^*_0} \ast
\dot \FP_{\gk^*_0, \gk_1}$,
where $\forces_{\FP_{\gk_0}} ``\dot \FP_{\gk_0, \gk^*_0}$
is equivalent to a $\gk_0$-directed closed partial ordering
and has size $\gg < \gk_1$'' and
$\forces_{\FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk^*_0}}
``\dot \FP_{\gk^*_0, \gk_1}$ is equivalent to a
$\gk^*_0$-directed closed partial ordering
and has size $\gk_1$''.
In particular, since $\gk^*_0 >\gk^+_0$,
$\forces_{\FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk^*_0}}
``\dot \FP_{\gk^*_0, \gk_1}$ is equivalent to a
$\gk^+_0$-directed closed partial ordering
and has size $\gk_1$''.
%$\forces_{\FP_{\gk_0} \ast \dot \FQ_{\gk_0}} ``\dot \FP_{\gk_0, \gk_1}$
%is $\gk^+_0$-directed closed and has size $\gk_1$''.
Again as in the proof of Theorem \ref{t1},
force to obtain a $V$-generic object
$G_{\gk_1} \subseteq \FP_{\gk_1}$. However, this time, factor
$G_{\gk_1}$ as $G_{\gk_0} \ast G_{\gk_0, \gk^*_0} \ast G_{\gk^*_0, \gk_1}$,
where $G_{\gk_0}$ is $V$-generic over $\FP_{\gk_0}$,
$G_{\gk_0, \gk^*_0}$ is $V[G_{\gk_0}]$-generic
over $\FP_{\gk_0, \gk^*_0}$, and
$G_{\gk^*_0, \gk_1}$ is
$V[G_{\gk_0}][G_{\gk_0, \gk^*_0}]$-generic over
$\FP_{\gk^*_0, \gk_1}$. In exactly the same manner as in
the proof of Theorem \ref{t1}, since $j '' G_{\gk_0}
= G_{\gk_0} \subseteq G_{\gk_1}$, $j$ lifts in
$V[G_{\gk_1}]$ to $j : V[G_{\gk_0}] \to
V[G_{\gk_1}]$, with $j(G_{\gk_0}) = G_{\gk_1}$ and
$V_{\gk_0}[G_{\gk_0}] \prec V_{\gk_1}[G_{\gk_1}]$.
Now, in analogy to the proof of Theorem \ref{t1},
consider the partial ordering
$j(\FP_{\gk_1}) = \FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk^*_0}
\ast \dot \FP_{\gk^*_0, \gk_1} \ast \dot \FP_{\gk_1, \gk^*_1}
\ast \dot \FP_{\gk^*_1, \gk_2}
= \FP_{\gk_1} \ast \dot \FP_{\gk_1, \gk^*_1} \ast \dot \FP_{\gk^*_1, \gk_2}$.
In $V[G_{\gk_1}]$, $j '' G_{\gk_0, \gk^*_0}$ is a
directed subset of $\FP_{\gk_1, \gk^*_1}$ having size $\gg$. Since
$V[G_{\gk_1}] \models ``\FP_{\gk_1, \gk^*_1}$ is equivalent to a
$\gk_1$-directed closed partial ordering'' and
$\gk_1 > \gg$, we can find a master condition
$q_{0, 0}$ for $j '' G_{\gk_0, \gk^*_0}$. Let $G_{\gk_1, \gk^*_1}$ be a
$V[G_{\gk_1}]$-generic object over $\FP_{\gk_1, \gk^*_1}$
containing $q_{0, 0}$. This means that
working in $V[G_{\gk_1}][G_{\gk_1, \gk^*_1}]$, we may now lift $j$ to
$j : V[G_{\gk_0}][G_{\gk_0, \gk^*_0}] \to V[G_{\gk_1}][G_{\gk_1, \gk^*_1}]$.
In $V[G_{\gk_1}][G_{\gk_1, \gk^*_1}]$, $j '' G_{\gk^*_0, \gk_1}$ is a
directed subset of $\FP_{\gk^*_1, \gk_2}$ having size $\gk_1$.
Since $V[G_{\gk_1}][G_{\gk_1, \gk^*_1}] \models ``\FP_{\gk^*_1, \gk_2}$ is
$\gk^+_1$-directed closed'', we can find a master condition
$q_{0, 1}$ for $j '' G_{\gk^*_0, \gk_1}$. Let
$G_{\gk^*_1, \gk_2}$ be a $V[G_{\gk_1}][G_{\gk_1, \gk^*_1}]$-generic
object over $\FP_{\gk^*_1, \gk_2}$ containing $q_{0, 1}$. Working in
$V[G_{\gk_1}][G_{\gk_1, \gk^*_1}][G_{\gk^*_1, \gk_2}] = V[G_{\gk_2}]$, lift
$j$ to $j : V[G_{\gk_0}][G_{\gk_0, \gk^*_0}][G_{\gk^*_0, \gk_1}] \to
V[G_{\gk_1}][G_{\gk_1, \gk^*_1}][G_{\gk^*_1, \gk_2}] $, i.e.,
working in $V[G_{\gk_2}]$, $j$ lifts to
$j : V[G_{\gk_1}] \to V[G_{\gk_2}]$. We may now continue
inductively for $\go$ many steps, building an
$\go$ sequence of generic objects and
master conditions, and thereby complete the
proof of Theorem \ref{t2} as in the proof of Theorem \ref{t1}.
\end{pf}
\begin{pf}
To prove Theorems \ref{t3} and \ref{t4}, let
$j : V_\gl \to V_\gl$ be an elementary embedding
in our ground model $V$ which
has critical point $\gk$ and witnesses that
$j$ is an ${\rm I}_3$ embedding.
As before, let
$\la \gk_i \mid i < \go \ra$ be the critical
sequence associated with $j$, with
$\gl = \bigcup_{i < \go} \gk_i$.
Suppose $\FP_\gl \in V$ and $\FP_\gl \subseteq V_{\gl}$, with
$\dot G_\gl$ a $\FP_\gl$-name for a
$V$-generic filter over $\FP_\gl$ and $\dot G_{\gk_n}$ a
$\FP_{\gk_n}$-name for $\dot G_\gl \rest \gk_n$.
Suppose further that
$q = \la \dot q_i \mid i < \gl \ra \in \FP_\gl$
is a condition obtained inductively
satisfying the following three conditions.
\begin{enumerate}
\item $q \rest \gk_1$ is trivial.
\item For each $n \ge 1$, $\dot q_{\gk_n} \in \dot \FQ_{\gk_n}$
is a name for a master condition for
$\{j(p_{\gk_{n}}) \mid p \in \dot G_{\gk_n}\}$.
\item For each $n \ge 1$, $\dot q \rest (\gk_n, \gk_{n + 1}) \in
\dot \FP_{\gk_n, \gk_{n + 1}}$ is a name for a master condition for
$\{j(p) \rest (\gk_n, \gk_{n + 1}) \mid p \in \dot G_{\gk_n}\}$.
\end{enumerate}
The proof of \cite[Theorem 1.2]{C07} shows that
if $G_\gl$ is a $V$-generic filter containing $q$, then
$j$ lifts to
$j : (V_\gl)^{V[G_\gl]} \to (V_\gl)^{V[G_\gl]}$
witnessing that $j$ is an ${\rm I}_3$ embedding
with critical point $\gk$.
However, the inductive construction of the
master conditions mentioned in the proofs of
Theorems \ref{t1} and \ref{t2}
(specifically, the sequence $q = \la q_n \mid n < \go \ra$
of master conditions built at the $n^{\rm th}$
stage of the induction in the proof of Theorem \ref{t1}
or the sequence
$q = \la \la q_{n, 0}, q_{n, 1} \ra \mid n < \go \ra$
of master conditions built at the $n^{\rm th}$
stage of the induction in the proof of
Theorem \ref{t2}) provides us
with a way of constructing the requisite $q$
required in the proofs of Theorems \ref{t3} and \ref{t4},
where for $\FP_{\gk_n}$ as defined in either
Theorem \ref{t1} or Theorem \ref{t2}, $\FP_\gl$
is taken as the inverse limit of
$\la \FP_{\gk_n} \mid n < \go \ra$.
Therefore, by the work of \cite{C00b},
$\la (V_\gl)^{V[G_\gl]}, \in, j \ra$ is a model for WA.
By construction, depending upon how $\FP_\gl$ is defined,
$(V_\gl)^{V[G_\gl]}$ is a model for the conclusions of
either Theorem \ref{t3} or Theorem \ref{t4}.
This completes our sketch of the proofs of
Theorems \ref{t3} and \ref{t4}.
\end{pf}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we make several remarks.
First, as Corazza has shown in \cite{C00b},
if $\gk$ is the critical point of an elementary
embedding witnessing WA, then $\gk$ is super-$n$-huge
for every $n \in \go$. In addition, as Hamkins has
observed in \cite{H01}, the same fact follows if
$\gk$ is the critical point of an elementary embedding
witnessing ${\rm WA}_0$. Thus, the proofs of
Theorems \ref{t2} and \ref{t4} establish the
consistency (relative to very strong hypotheses)
of GCH and level by level equivalence between
strong compactness and supercompactness with the
existence of super-$n$-huge cardinals for every
$n \in \go$. We conjecture that the consistency
of these hypotheses with the existence of a
super-$n$-huge cardinal for a specific
$n \in \go$ can be established relative
to the existence of that kind of
super-$n$-huge cardinal alone,
and that the consistency of these hypotheses
relative to any particular form of huge cardinal
can be established relative to exactly that form
of huge cardinal.
In addition, we note that our methods of proof
are amenable to the establishment of the relative consistency
of other properties known to be consistent with the
class of supercompact cardinals with ${\rm WA}_0$
and WA. For example, in \cite{A01}, relative to ZFC and the
existence of a class ${\cal K}$ of supercompact cardinals,
the consistency of the theory ${\rm T} =
``$ZFC + ${\cal K}$ is the class of
supercompact cardinals + The classes of supercompact and
strongly compact cardinals coincide precisely, except
at measurable limit points + Every measurable cardinal
$\gk$ is $\gk^+$ supercompact'' was shown.
The partial ordering $\FP$ used to establish
this theorem is an Easton support iteration which
can be fit into the rubric of the partial orderings
discussed in this paper. Consequently,
by forcing over a model witnessing the conclusions
of either Theorem \ref{t2} or Theorem \ref{t4},
it is possible to establish
the relative consistency of T with ${\rm WA}_0$ and WA
in an analogous manner to what was just done.
For further details on the definition of $\FP$,
which is somewhat complicated, we refer readers to \cite{A01}.
Finally, for the same reasons as in \cite{H01} and \cite{C07},
it is unknown for
$1 \le n \le \infty$ whether the consistency of just ${\rm WA}_n$
implies the consistency of ${\rm WA}_n$ with
the conclusions of Theorems \ref{t1} and \ref{t2}.
We finish by asking if this is indeed the case.
\begin{thebibliography}{99}
%\bibitem{A00} A.~Apter, ``A New Proof of a
%Theorem of Magidor'', {\it Archive for Mathematical
%Logic 39}, 2000, 209--211.
\bibitem{A98} A.~Apter, ``Laver Indestructibility
and the Class of Compact Cardinals'', {\it Journal
of Symbolic Logic 63}, 1998, 149--157.
%\bibitem{A02} A.~Apter, ``On Level by Level Equivalence
%and Inequivalence between Strong Compactness and
%Supercompactness'', {\it Fundamenta Mathematicae 171},
%2002, 77--92.
\bibitem{A03} A.~Apter, ``Some Remarks on Indestructibility
and Hamkins' Lottery Preparation'', {\it Archive for
Mathematical Logic 42}, 2003, 717--735.
%\bibitem{A83} A.~Apter, ``Some Results on Consecutive
%Large Cardinals'', {\it Annals of Pure and Applied Logic 25},
%1983, 1--17.
\bibitem{A01} A.~Apter, ``Some Structural Results
Concerning Supercompact Cardinals'', {\it Journal of
Symbolic Logic 66}, 2001, 1919--1927.
%\bibitem{A10} A.~Apter, ``Tallness and Level by Level
%Equivalence and Inequivalence'', {\it Mathematical
%Logic \FQuarterly 56}, 2010, 4--12.
%\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness'',
%{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
%\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
%and Strong Compactness II: Strong Cardinals'',
%{\it Archive for Mathematical Logic 40}, 2001, 25--38.
\bibitem{AF} A.~Apter, Sh.~Friedman, ``Coding into
HOD via Normal Measures with Some Applications'',
to appear in the {\it Mathematical Logic Quarterly}.
%\bibitem{AGH} A.~Apter, V.~Gitman, J.~D.~Hamkins,
%``Inner Models with Large Cardinal Features Usually
%Obtained by Forcing'', in preparation.
%submitted for publication to the {\it Journal of Symbolic Logic}.
%\bibitem{AH02} 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{AH03} A.~Apter, J.~D.~Hamkins,
%``Exactly Controlling the Non-Supercompact Strongly
%Compact Cardinals'', {\it Journal of Symbolic Logic 68},
%2003, 669--688.
\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{Bu} J$.$ Burgess, ``Forcing'', in:
%J$.$ Barwise, ed., {\it Handbook of
%Mathematical Logic}, North-Holland,
%Amsterdam, 1977, 403--452.
\bibitem{C00a} P.~Corazza, ``Consistency of
$V = {\rm HOD}$ with the Wholeness Axiom'',
{\it Archive for Mathematical Logic 39}, 2000, 219--226.
\bibitem{C07} P.~Corazza, ``Lifting Elementary
Embeddings $j : V_\gl \to V_\gl$'',
{\it Archive for Mathematical Logic 46}, 2007, 61--72.
\bibitem{C03} P.~Corazza, ``The Gap between ${\rm I}_3$
and the Wholeness Axiom'', {\it Fundamenta Mathematicae 179},
2003, 43--60.
\bibitem{C06} P.~Corazza, ``The Spectrum of Elementary
Embeddings $j : V \to V$'', {\it Annals of Pure and
Applied Logic 139}, 2006, 327--399.
\bibitem{C00b} P.~Corazza, ``The Wholeness Axiom and
Laver Sequences'', {\it Annals of Pure and Applied
Logic 105}, 2000, 157--260.
%\bibitem{C} J.~Cummings, ``A Model in which GCH
%Holds at Successors but Fails at Limits'',
%{\it Transactions of the American Mathematical Society 329},
%1992, 1--39.
%\bibitem{CFM} J.~Cummings, M.~Foreman,
%M.~Magidor, ``Squares, Scales, and Stationary
%Reflection'', {\it Journal of Mathematical
%Logic 1}, 2001, 35--98.
%\bibitem{DH} C.~ Di Prisco, J.~Henle, ``On the Compactness
%of $\ha_1$ and $\ha_2$'', {\it Journal of Symbolic
%Logic 43}, 1978, 394--401.
%\bibitem{F} M.~Foreman, ``More Saturated Ideals'', in:
%{\it Cabal Seminar 79-81}, {\bf Lecture Notes in
%Mathematics 1019}, Springer-Verlag, Berlin and New York,
%1983, 1--27.
%\bibitem{GS} M.~Gitik, S.~Shelah,
%``On Certain Indestructibility of
%Strong Cardinals and a \FQuestion of
%Hajnal'', {\it Archive for Mathematical
%Logic 28}, 1989, 35--42.
%\bibitem{H1} J.~D.~Hamkins,
%``Destruction or Preservation As You
%Like It'',
%{\it Annals of Pure and Applied Logic 91},
%1998, 191--229.
%\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{H} J.~D.~Hamkins, {\it Lifting and
%Extending Measures; Fragile Measurability},
%Doctoral Dissertation, University of California,
%Berkeley, 1994.
%\bibitem{H5} J.~D.~Hamkins, ``Tall Cardinals'',
%{\it Mathematical Logic \FQuarterly 55}, 2009, 68--86.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
{\it Annals of Pure and Applied Logic 101},
2000, 103--146.
\bibitem{H01} J.~D.~ Hamkins, ``The Wholeness Axioms
and $V = {\rm HOD}$'', {\it Archive for Mathematical
Logic 40}, 2001, 1--8.
\bibitem{J} T.~Jech, {\it Set Theory:
The Third Millennium Edition,
Revised and Expanded}, Springer-Verlag,
Berlin and New York, 2003.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite. Large Cardinals in Set
Theory from Their Beginnings}, second edition,
Springer-Verlag, Berlin and New York, 2003.
%\bibitem{Ke} J.~Ketonen, ``Strong Compactness and
%Other Cardinal Sins'', {\it Annals of Mathematical
%Logic 5}, 1972, 47--76.
\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
between the Concepts of Compactness and Supercompactness'',
circulated manuscript.
\bibitem{Ku71} K.~Kunen, ``Elementary Embeddings and
Infinitary Combinatorics'', {\it Journal of
Symbolic Logic 36}, 1971, 407--413.
%\bibitem{Ku} K.~Kunen, {\it Set Theory:
%An Introduction to Independence Proofs},
%{\bf Studies in Logic and the Foundations
%of Mathematics 102}, North-Holland,
%Amsterdam and New York, 1980.
\bibitem{L} R.~Laver, ``Making the
Supercompactness of $\gk$ Indestructible
under $\gk$-Directed Closed Forcing'',
{\it Israel Journal of Mathematics 29},
1978, 385--388.
%\bibitem{LS} A.~L\'evy, R.~Solovay,
%``Measurable Cardinals and the Continuum Hypothesis'',
%{\it Israel Journal of Mathematics 5}, 1967, 234--248.
%\bibitem{Ma} M.~Magidor, ``How Large is the First
%Strongly Compact Cardinal?'', {\it Annals of
%Mathematical Logic 10}, 1976, 33--57.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
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}
it suffices to show that
$j(\FP_{\gk_0}) = \FP_{\gk_0} \ast \dot \FP_{\gk_0, \gk_1}$, where
$\forces_{\FP_{\gk_0}} ``\dot \FP_{\gk_0, \gk_1}$ is
$\gk^+_1$-directed closed''.
This is since under these circumstances, the same argument
as given in the proof of Theorem \ref{t1} will remain valid.