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%
% ------------------------------------------------------------------------------
%
\title{Reducing the Consistency Strength
of an Indestructibility Theorem
% A Reduction in Consistency Strength
% of the Assumptions used to Establish
% an Indestructibility Theorem
% for an Indestructibility Theorem
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal,
enhanced supercompact cardinal, almost huge cardinal,
strong cardinal, indestructibility.}}
\author{Arthur W.~Apter
\thanks{The author's research was partially
supported by PSC-CUNY Grants and
CUNY Collaborative Incentive Grants.
In addition, the author wishes to
express his gratitude to the referee,
for many 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/apter\\
awapter@alum.mit.edu}
\date{March 4, 2007\\
(revised October 3, 2007)}
\begin{document}
\maketitle
\begin{abstract}
Using an idea of Sargsyan, we
show how to reduce the consistency
strength of the assumptions employed
to establish a theorem concerning
a uniform level of indestructibility
for both strong and supercompact cardinals.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s1}
In \cite{A02}, the following
three theorems were proven.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + $\gk$ is the
least supercompact cardinal''.
There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is
the least strong cardinal +
$\gk$ is not $2^\gk$ supercompact +
$\gk$'s strongness is indestructible under
forcing with partial orderings which are
$\gk$-directed closed''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + There is a
proper class of supercompact cardinals +
There is no supercompact limit of supercompact
cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + There is a
proper class of strong cardinals +
No strong cardinal $\gk$ is $2^\gk$
supercompact + Every strong cardinal
$\gk$ has its strongness indestructible under
$\gk$-directed closed forcing''.
\end{theorem}
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$
is almost huge''.
%Assume
%$j : V \to M$ is an elementary
%embedding witnessing $\gk$'s
%hugeness, with $j(\gk) = \gl$.
There is then a partial ordering
$\FP \in V$ such that for
$V^*$ the universe $V^\FP$
truncated at $\gk$,
$V^* \models ``$ZFC + There is
a proper class of supercompact
limits of supercompact cardinals +
Every strong cardinal $\gk$ has
its strongness indestructible under
$\gk$-directed closed forcing +
Every supercompact cardinal $\gk$ has
its supercompactness indestructible under
$\gk$-directed closed forcing''.
\end{theorem}
Whereas Hamkins' gap forcing
results of \cite{H2, H3}
suggest that the hypotheses used
to establish Theorems \ref{t1} and \ref{t2}
are close to optimal, the assumption
of an almost huge cardinal employed
in the proof of Theorem \ref{t3}
seems to be way too strong. Indeed,
\cite{A02} concludes by asking if
it is possible to weaken significantly
the hypotheses used to prove this theorem.
The purpose of this note is to provide
an affirmative answer to this question.
Before presenting our main theorem, however,
we first give a key definition,
which uses an idea due to Sargsyan,
a modification of which will appear
in \cite{AS07a}.
Say that the cardinal $\gk$ is an
{\em enhanced supercompact cardinal}
iff there is a strong cardinal
$\gk_0 > \gk$ such that
for every cardinal $\gl > \gk_0$,
there is an elementary embedding
$j : V \to M$
witnessing the $\gl$ supercompactness
of $\gk$ generated by a
supercompact ultrafilter over
$P_\gk(\gl)$ such that
$M \models ``\gk_0$ is strong''.
We observe that the existence of an
enhanced supercompact cardinal is
much stronger than the existence
of a supercompact cardinal.
To see this, let $\gk < \gk_0 < \gl$
be as above, and let
$j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$ such that
$M \models ``\gk_0$ is strong''.
It is then the case that
$M \models ``\gk$ is $\gg$ supercompact
for every $\gg < \gk_0$ and $\gk_0$
is strong'', so by \cite[Lemma 1.1]{A02a},
$M \models ``\gk$
is supercompact''. By reflection,
in $V$, $\gk$ must be a supercompact
limit of supercompact cardinals,
a supercompact limit of
supercompact cardinals which are
limits of supercompact cardinals, etc.
%(and much more as well).
We are now able to state our primary result.
\begin{theorem}\label{t4}
Suppose
$V \models ``$ZFC + There is
a proper class of enhanced
supercompact cardinals which
are limits of enhanced supercompact
cardinals''.
%Assume
%$j : V \to M$ is an elementary
%embedding witnessing $\gk$'s
%hugeness, with $j(\gk) = \gl$.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + There is
a proper class of supercompact
limits of supercompact cardinals +
Every strong cardinal $\gk$ has
its strongness indestructible under
$\gk$-directed closed forcing +
Every supercompact cardinal $\gk$ has
its supercompactness indestructible under
$\gk$-directed closed forcing''.
\end{theorem}
Before continuing, we mention that
we presume some degree
of familiarity with the results
and methods of \cite{A02}.
Also, note that when forcing,
$q \ge p$ means that $q$
{\em is stronger than} $p$.
We abuse notation slightly and
take both $V[G]$ and $V^\FP$
as being the generic extension
of $V$ by $\FP$ (assuming of
course that $G$ is $V$-generic over $\FP$).
The partial ordering $\FP$ is
{\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has an upper bound.
In addition, an {\em indestructibly
supercompact cardinal $\gk$} is one
as first given by Laver in \cite{L}, i.e.,
one which is supercompact and remains
supercompact after arbitrary
$\gk$-directed closed forcing.
Returning to the discussion at hand,
to see that Theorem \ref{t4} is meaningful,
we begin with the following preliminary result.
\begin{theorem}\label{t5}
Suppose
$V \models ``$ZFC + $\gk$ is almost huge''. Then
$V_\gk \models ``$ZFC + There is a proper
class of enhanced supercompact cardinals
which are limits of enhanced supercompact cardinals''.
\end{theorem}
\begin{proof}
Suppose $j : V \to M$ is an elementary
embedding witnessing the almost hugeness of $\gk$,
with $\gl^* = j(\gk)$. Let $\gk_0 > \gk$
be the least cardinal such that the statement
$``\gk_0$ is $\gg$ strong
for every $\gg < \gl^*$'' is true in
both $V$ and $M$.
Since $M^{< \gl^*} \subseteq M$, $\gk_0$ exists,
and in addition, the statement
$``\gk$ is $\gg$ strong
for every $\gg < \gl^*$'' is also true in
both $V$ and $M$.
Consequently, by elementarity,
$M \models ``j(\gk)$ is $\gg$ strong for
every $\gg < j(\gl^*)$'', i.e.,
$M \models ``\gl^*$ is $\gg$ strong for
every $\gg < j(\gl^*)$''. This
means by \cite[Lemma 1.1]{A02a} that
$M \models ``\gk_0$ is $\gg$ strong for
every $\gg < j(\gl^*)$''.
Fix $\gl < \gl^*$, $\gl > \gk_0$, and let
$j_\gl : V \to M_\gl$ be the $\gl$ supercompactness
embedding generated by $j$, i.e., $j_\gl$
is generated by the ultrafilter ${\cal U}_\gl$
defined by $x \in {\cal U}_\gl$ iff
$\la j(\ga) \mid \ga < \gl \ra \in j(x)$. Take
$i : M_\gl \to M$ as the canonical elementary
embedding such that $i \circ j_\gl = j$.
Suppose $\gd < \gl^*$ is such that
$M_\gl \models ``\gk_0$ is not $\gd$ strong''.
By elementarity, $M \models ``i(\gk_0)$ is not
$i(\gd)$ strong'', so since
${\rm cp}(i) > \gk_0$,
$M \models ``\gk_0$ is not $i(\gd)$ strong''.
However, $i(\gd) \le i(j_\gl(\gd)) = j(\gd) < j(\gl^*)$,
so this contradicts the last sentence of
the previous paragraph.
It must therefore be the case that
$M_\gl \models ``\gk_0$ is $\gg$ strong for every
$\gg < \gl^*$'', from which we may now infer that
$M \models ``$For $\gk_0$ the least
cardinal greater than $\gk$
which is $\gg$ strong for every
$\gg < \gl^*$, and for every $\gl < \gl^*$,
$\gl > \gk_0$, there is a supercompact ultrafilter
${\cal U}_\gl$ over $P_\gk(\gl)$ with associated
supercompactness embedding $j_\gl : M \to M_\gl'$
such that $M_\gl' \models `\gk_0$ is $\gg$ strong
for every $\gg < \gl^*$' ''.
We abbreviate this by
$M \models \varphi(\gk, \gl^*)$.
It thus follows by reflection
that for unboundedly in $\gk$ many
$\gd < \gk$, $V \models \varphi(\gd, \gk)$.
For each such $\gd$, by elementarity
and the fact that $\gd < \gk = {\rm cp}(j)$,
$M \models \varphi(\gd, \gl^*)$.
This means that
$M \models ``\varphi(\gk, \gl^*)$, and $\gk$ is
a limit of cardinals $\gd$ such that $\varphi(\gd, \gl^*)$''.
Once again by reflection, for unboundedly
in $\gk$ many $\gd < \gk$,
$V \models ``\varphi(\gd, \gk)$, and $\gd$ is
a limit of cardinals $\gg$ such that $\varphi(\gg, \gk)$''.
This means that
$V_\gk \models ``$ZFC + There is a proper
class of enhanced supercompact cardinals
which are limits of enhanced supercompact cardinals''.
This completes the proof of Theorem \ref{t5}.
\end{proof}
Now that we have completed the proof of
Theorem \ref{t5}, we turn our attention
to the proof of Theorem \ref{t4}.
\begin{proof}
Suppose
$V \models ``$ZFC + There is a proper class
of enhanced supercompact cardinals which are limits
of enhanced supercompact cardinals''.
We begin by recalling for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\it lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
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
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{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 names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
We are now ready to present the
definition of our partial ordering $\FP$
used in the proof of Theorem \ref{t4}, which
is a modification of the
partial ordering defined in the proof
of \cite[Theorem 3]{A02}.
$\FP$ is a proper class reverse Easton iteration
which begins by adding
a Cohen subset of $\go$ and then
does nontrivial
forcing only at those stages $\gd$
which are $V$-strong cardinals.
At such a stage, the forcing done
is the lottery sum of all partial
orderings $\FR$ having rank below the least
$V$-strong cardinal $\eta$ above $\gd$ which
are $\gd$-directed closed
and which have the
additional property that
forcing with $\FR$ destroys the
$\gg$ strongness of $\gd$ for some $\gg < \eta$,
assuming this sort of $\FR$ exists.
If this sort of $\FR$ does not exist, then
the forcing done at stage $\gd$ is
the lottery sum of all partial orderings
$\FS$ having rank below
the least $V$-strong cardinal
$\eta$ above $\gd$ which are
$\gd$-directed closed and which have the
additional property that forcing with
$\FS$ destroys the $\gg$ supercompactness
of $\gd$ for some $\gg < \eta$, again
assuming this kind of $\FS$ exists.
If this kind of $\FS$
does not exist, then the
forcing done at stage $\gd$ is the
trivial forcing $\{\emptyset\}$.
Standard arguments show that $V^\FP \models {\rm ZFC}$.
In addition,
the Gap Forcing Theorem of \cite{H2, H3}
allows us to infer that
any $V$-strong cardinal $\gk$ which is not
supercompact in $V$ is no longer strong in $V^\FP$
(since any collapse of a cardinal greater
than $\gk$'s degree of supercompactness to $\gk$
forces $\gk$ to be non-measurable, which means
that there is indeed a partial ordering
destroying the $\gg$ strongness of
$\gk$ for some $\gg$).
We may also infer that in $V^\FP$, every
$V$-supercompact cardinal $\gk$
is a strong cardinal whose strongness is
indestructible under $\gk$-directed closed forcing
(since any $\gk$-directed closed
partial ordering in $V^\FP$
allegedly destroying some degree $\gg$ of
$\gk$'s strongness can actually be shown
by a standard lifting argument to preserve
a degree of $\gk$'s supercompactness implying
that $\gk$ is $\gg$ strong).
Therefore,
the same arguments as given in
\cite{A02} show that
$V^\FP \models ``$There is a proper class of
strong cardinals + Every strong cardinal
$\gk$ has its strongness indestructible
under $\gk$-directed closed forcing + Every
$V$-supercompact cardinal $\gk$ has become
a strong cardinal whose strongness is
indestructible under $\gk$-directed closed forcing''.
In particular, as in \cite[Lemma 3.2]{A02}, if
$V \models ``\gk$ is supercompact'', then
$\forces_{\FP_\gk} ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
\begin{lemma}\label{l1}
In $V^\FP$, every $V$-enhanced supercompact
cardinal $\gk$ is indestrucibly
supercompact.
%has its supercompactness
%indestructible under $\gk$-directed closed forcing.
\end{lemma}
\begin{proof}
%We use ideas found in the proof of Lemma 3.3 of \cite{A02}.
Suppose
$V \models ``\gk$ is an enhanced supercompact cardinal''.
Write $\FP = \FP_\gk \ast \dot \FP^\gk$. Since
$\forces_{\FP_\gk} ``\dot \FP^\gk$ is
$\gk$-directed closed'', it suffices to show that
$V^{\FP_\gk} \models ``\gk$ is an
indestructibly supercompact cardinal''.
We assume towards a contradiction
that this is not the case.
Let therefore $\gg$ and
$\FQ \in V^{\FP_\gk}$ be such that
$V^{{\FP_\gk} \ast \dot \FQ} \models ``\gk$
is not $\gg$ supercompact''.
By the same arguments as in \cite[Lemma 3.3]{A02},
both $\gg$ and the rank of $\FQ$ may be
chosen to be less than $\eta$,
the least $V$-strong cardinal above $\gk$.
Let $\gk_0 > \gk$ be a strong
cardinal as in the
definition of enhanced supercompact cardinal, and
let $\gl > \gk_0$ be such that
for some elementary embedding witnessing the
$\gl$ supercompactness of $\gk$,
$M \models ``\gk_0$ is strong (and
consequently, $\gk$ is supercompact)''.
We observe that $\eta \le \gk_0$ is the least
strong cardinal above $\gk$ in
$M$ as well as $V$. To see this,
by its closure properties,
$M \models ``\eta$ is $\ga$ strong
for every $\ga < \gk_0$'', so since
$M \models ``\gk_0$ is strong'',
again by \cite[Lemma 1.1]{A02a},
$M \models ``\eta$ is strong''. If
$M \models ``\eta' < \eta$ is strong'', then
another reference to $M$'s closure
properties tells us that
$V \models ``\eta'$ is $\ga$ strong for every
$\ga < \eta$ and $\eta$ is strong''.
Thus, once more by \cite[Lemma 1.1]{A02a},
$V \models ``\eta' < \eta$ is strong''.
This contradicts that
$V \models ``\eta$ is the least strong cardinal''.
By the choice of $\gl$,
$M^{{\FP_\gk} \ast \dot \FQ}$ and $V^{{\FP_\gk} \ast \dot \FQ}$
each satisfy the statement
``$\gk$ is not $\gg$ supercompact''.
In addition, by the remarks in the paragraph
immediately preceding the statement of
Lemma \ref{l1}, elementarity,
and the fact that
$M \models ``\gk$ is supercompact'', in $M$,
$\forces_{{\FP_\gk}} ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
Therefore, since $\eta$ is the least
strong cardinal above $\gk$
in both $V$ and $M$ and
$\FQ$'s rank is less than $\eta$,
the definition of $\FP$ tells us that
it is possible to opt for $\FQ$
in the stage $\gk$ lottery held
in $M^{j(\FP_\gk)}$.
%in the definition of $j(\FP)$.
By forcing above the appropriate condition,
we thus have that
$j({\FP_\gk} \ast \dot \FQ)$ is
forcing equivalent to
${\FP_\gk} \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$,
where the first nontrivial stage in $\dot \FR$
is well above $\gg$
(and in fact, occurs precisely at $\eta$).
Standard arguments, as given, e.g., in \cite{L}
now show that
$V^{{\FP_\gk} \ast \dot \FQ} \models ``\gk$
is $\gg$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be $V$-generic over
$\FP_\gk \ast \dot \FQ \ast \dot \FR$,
lift $j$ in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
$V[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ)$ containing $p$, lift $j$
again in $V[G_0][G_1][G_2][G_3]$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\eta$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gg))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0][G_1]$.)
This contradiction completes the
proof of Lemma \ref{l1}.
\end{proof}
As the referee has pointed out,
in the proof of Lemma \ref{l1},
we may assume that $\eta = \gk_0$.
This is since any strong cardinal of
$M$ below $\gk_0$ is fully strong
in $V$, and therefore witnesses the
enhanced supercompactness of $\gk$ via $j$.
\begin{lemma}\label{l2}
In $V^\FP$, every supercompact
cardinal is indestructibly supercompact.
\end{lemma}
\begin{proof}
We mimic to a certain extent the proof of
\cite[Lemma 3.4]{A02}.
Suppose
$V^\FP \models ``\gk$ is supercompact''. Write
$\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_2$-directed closed''.
Since $\FP$ admits this factorization,
once again
by the
Gap Forcing Theorem,
% of \cite{H2, H3},
$V \models ``\gk$ is supercompact''.
Therefore, by the factorization of $\FP$
given in Lemma \ref{l1}, it suffices to show that
$V^{\FP_\gk} \models ``\gk$ is indestructibly supercompact''.
To see this, we first note that once again by the remarks
in the paragraph immediately preceding the
statement of Lemma \ref{l1},
$\forces_{\FP_\gk} ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''. Therefore, if
$V^{\FP_\gk} \models ``\gk$ is not indestructibly
supercompact'', by the definition of $\FP$,
the forcing done at stage $\gk$ is
equivalent to a
partial ordering $\FQ$ of rank
less than the least $V$-strong cardinal above $\gk$ which
destroys some degree of supercompactness of $\gk$.
Thus, $V^{\FP_\gk \ast \dot \FQ} = V^{\FP_{\gk + 1}} \models
``\gk$ is not $\gl$ supercompact'', where
$\gl$ is the smallest degree of
supercompactness $\FQ$ destroys.
As before, $\gl$
is also less than the least $V$-strong
cardinal above $\gk$.
Write $\FP = \FP_{\gk + 1} \ast \dot \FP^{\gk + 1}$.
Since by the definition of $\FP$,
$\forces_{\FP_{\gk + 1}} ``\dot \FP^{\gk + 1}$
is $\gg$-directed closed for $\gg$ the least
strong limit cardinal above $\gl$'',
$V^{\FP_{\gk + 1} \ast \dot \FP^{\gk + 1}} = V^\FP \models
``\gk$ is not $\gl$ supercompact''.
This contradiction to our assumption that
$V^\FP \models ``\gk$ is supercompact'' completes
the proof of Lemma \ref{l2}.
\end{proof}
Since
$V \models ``$There is a proper class of
enhanced supercompact cardinals which are
limits of enhanced supercompact cardinals'',
Lemma \ref{l1} implies that
$V^\FP \models ``$There is a proper
class of supercompact limits of supercompact cardinals''.
Lemmas \ref{l1} and \ref{l2} therefore complete
the proof of Theorem \ref{t4}.
\end{proof}
As the referee has pointed out,
the essence of the proof of Lemma \ref{l2} is that
if a supercompact cardinal remains supercompact
in $V^\FP$, then it must be indestructible, since
otherwise, the smallest degree of its supercompactness
which could have been destroyed would have
been destroyed earlier. This is the key idea of
the ``trial-by-fire method'' of \cite{AH99}.
We conclude by asking exactly what the
consistency strength is of
the theory ``ZFC + There is a proper
class of supercompact limits of supercompact cardinals +
Every strong cardinal $\gk$ has its strongness
indestructible under $\gk$-directed closed forcing +
Every supercompact cardinal $\gk$ has its
supercompactness indestructible under $\gk$-directed
closed forcing''.
To this point, the inner model theory
being developed by Woodin
for supercompactness is not yet advanced enough
to begin considering this question.
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{\it Fundamenta Mathematicae 174}, 2002, 87--96.
\bibitem{A02} A.~Apter, ``Strong Cardinals
can be Fully Laver Indestructible'',
{\it Mathematical Logic Quarterly 48}, 2002, 499--507.
\bibitem{AH99} A.~Apter, J.~D.~Hamkins,
``Universal Indestructibility'', {\it Kobe
Journal of Mathematics 16}, 1999, 119--130.
\bibitem{AS07a} A.~Apter, J.~D.~Hamkins,
G.~Sargsyan, R.~Schindler,
``An Equiconsistency for Universal
Indestructibility'', in preparation.
%\bibitem{GS} M.~Gitik, S.~Shelah,
%``On Certain Indestructibility of Strong
%Cardinals and a Question of Hajnal'',
%{\it Archive for Mathematical Logic 28},
%1989, 35--42.
\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{H5} J.~D.~Hamkins, ``Small Forcing makes any
%Cardinal Superdestructible'',
%{\it Journal of Symbolic Logic 63}, 1998, 51--58.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
{\it Annals of Pure and Applied Logic 101}, 2000, 103--146.
%\bibitem{J} T.~Jech, {\it Set Theory},
%Academic Press, New York and San
%Francisco, 1978.
%\bibitem{K} A.~Kanamori, {\it The
%Higher Infinite}, Springer-Verlag,
%Berlin and New York, 1994.
\bibitem{L} R.~Laver,
``Making the Supercompactness of $\gk$
Indestructible under $\gk$-Directed
Closed Forcing'',
{\it Israel Journal of Mathematics 29},
1978, 385--388.
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%``Measurable Cardinals and the Continuum Hypothesis'',
%{\it Israel Journal of Mathematics 5}, 1967, 234--248.
%\bibitem{MSS} W.~Mitchell, E.~Schimmerling,
%J.~Steel, ``The Covering Lemma up to a
%Woodin Cardinal'', {\it Annals of Pure
%and Applied Logic 84}, 1997, 219--255.
%\bibitem{So} R.~Solovay, ``Strongly Compact Cardinals
%and the GCH'', in:
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%{\bf Proceedings of Symposia in Pure Mathematics 25},
%American Mathematical Society, Providence,
%1974, 365--372.
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\end{thebibliography}
\end{document}
We are now ready to present the
definition of our partial ordering ${\FP_\gk}$
used in the proof of Theorem \ref{t4}, which
is a slight
modification of the
partial ordering defined in the proof
of Theorem 3 of \cite{A02}.
${\FP_\gk}$ is a proper class reverse Easton iteration
which begins by adding
a Cohen subset of $\go$ and then
does nontrivial
forcing only at those stages $\gd$
which are $V$-strong cardinals.
At such a stage, the forcing done
is the lottery sum of all partial
orderings $\FR$ having rank below the least
$V$-strong cardinal $\eta$ above $\gd$ which
are $\gd$-directed closed
and which have the
additional property that
forcing with $\FR$ destroys the
$\gg$ strongness of $\gd$ for some $\gg < \eta$,
assuming this sort of $\FR$ exists.
If this sort of $\FR$ does not exist, then the
forcing done at stage $\gd$ is the
lottery sum of all partial orderings
$\FS$ having rank below the least
$V$-strong cardinal $\eta$ above $\gd$
which are $\gd$-directed closed and
which have the additional property that
forcing with $\FS$ destroys the $\gg$
supercompactness of $\gd$ for some
$\gg < \eta$, again assuming this kind of
$\FS$ exists.
If this kind of $\FS$ does not exist, then the
forcing done at stage $\gd$ is the
trivial forcing $\{\emptyset\}$.
Standard arguments show that $V^{\FP_\gk} \models {\rm ZFC}$.
In addition, the same arguments as given in
\cite{A02} show that
$V^{\FP_\gk} \models ``$There is a proper class of
strong cardinals + Every strong cardinal
$\gk$ has its strongness indestructible
under $\gk$-directed closed forcing + Every
$V$-supercompact cardinal $\gk$ has become
a strong cardinal whose strongness is
indestructible under $\gk$-directed closed forcing''.
In particular, as in \cite[Lemma 3.2]{A02}, if
$V \models ``\gk$ is supercompact'', then
$\forces_{{\FP_\gk}_\gk} ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
\begin{lemma}\label{l1}
In $V^{\FP_\gk}$, every $V$-enhanced supercompact
cardinal $\gk$ is an indestrucible
supercompact cardinal.
%has its supercompactness
%indestructible under $\gk$-directed closed forcing.
\end{lemma}
\begin{proof}
%We use ideas found in the proof of Lemma 3.3 of \cite{A02}.
Suppose
$V \models ``\gk$ is an enhanced supercompact cardinal''.
Write ${\FP_\gk} = {\FP_\gk}_\gk \ast \dot {\FP_\gk}^\gk$. Since
$\forces_{{\FP_\gk}_\gk} ``\dot {\FP_\gk}^\gk$ is
$\gk$-directed closed'', it suffices to show that
$V^{{\FP_\gk}_\gk} \models ``\gk$ is an
indestructible supercompact cardinal''.
To do this, let
$\FQ \in V^{\FP_\gk}$ and $\gg$ be such that
$V^{{\FP_\gk} \ast \dot \FQ} \models ``\gk$
is not $\gg$ supercompact''. Let
$\gl > \max(\gg, \card{{\rm TC}(\dot \FQ)}$
be sufficiently large, e.g., a strong limit cardinal.
Since
$V \models ``\gk$ is an enhanced supercompact cardinal'',
we may take $j : V \to M$ as an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$
such that $M \models ``\gk$ is supercompact''.
Note that by elementarity, in $M$,
$\forces_{{\FP_\gk}_\gk} ``\gk$ is a strong cardinal
whose strongness is indestructible under
$\gk$-directed closed forcing''.
Therefore, by the definition of ${\FP_\gk}$,
$\FQ$ is an allowable choice in the
stage $\gk$ lottery sum performed in
$M^{{\FP_\gk}_\gk}$ in the definition of $j({\FP_\gk})$.
\end{proof}
\end{proof}