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\title{A Reduction in Consistency Strength
for Universal Indestructibility
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Universal indestructibility,
indestructibility, measurable cardinal, Woodin cardinal,
strongly compact cardinal, supercompact cardinal,
cardinal Woodin for
supercompactness, high-jump cardinal, almost huge cardinal.}}
\author{Arthur W.~Apter\thanks{The
first author's research was
partially supported by
PSC-CUNY Grants and
CUNY Collaborative
Incentive 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\\
\\
Grigor Sargsyan\\
Group in Logic and the Methodology of Science\\
University of California\\
Berkeley, California 94720 USA\\
http://math.berkeley.edu/$\sim$grigor\\
grigor@math.berkeley.edu}
\date{September 4, 2006\\
(revised March 30, 2007)}
\begin{document}
\maketitle
\begin{abstract}
We show how to reduce the
assumptions in consistency
strength used to prove
several theorems on
universal indestructibility.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s0}
In \cite{AH}, the first author and
Hamkins introduced the concept of
{\em universal indestructibility} and
established several theorems concerning this
notion, most prominently the relative
consistency of universal indestructibility
for both supercompactness and strong compactness.
In \cite{A05}, the first author extended this
work and showed the relative consistency
of two strongly compact cardinals with
universal indestructibility for both
supercompactness and strong compactness.
All of these results were proven using a
{\em high-jump cardinal}, a very strong
notion reflected by almost hugeness.
The purpose of this paper is to
reduce the consistency strength used
to prove each of these theorems from
a high-jump cardinal to something
reflected by this notion which we
will call a {\em cardinal Woodin for supercompactness}.
Specifically, we prove the following two theorems.
\begin{theorem}\label{t1}
Universal indestructibility
for supercompactness
in the presence of a
supercompact cardinal is consistent
relative to the existence of a
cardinal Woodin for supercompactness.
\end{theorem}
\begin{theorem}\label{t2}
Universal indestructibility
for either supercompactness
or strong compactness in the
presence of two strongly
compact cardinals is consistent
relative to the existence of a
cardinal Woodin for supercompactness.
\end{theorem}
\noindent As we will also indicate
without proving explicitly,
other theorems from \cite{AH}
are consistent relative to
the existence of a cardinal Woodin
for supercompactness.
Before continuing, we take this
opportunity to remind
readers of some of the relevant definitions.
We say that
{\it universal indestructibility for
supercompactness}
holds in a model $V$ for ZFC if
every $V$-supercompact and
partially supercompact (including measurable)
cardinal $\gd$
has its degree of supercompactness
fully Laver indestructible
\cite{L} under $\gd$-directed
closed forcing. Analogously,
{\it universal indestructibility for
strong compactness}
holds in a model $V$ for ZFC if
every $V$-strongly compact and
partially strongly compact (including measurable)
cardinal $\gd$
has its degree of strong compactness
fully indestructible under $\gd$-directed
closed forcing.
We recall from \cite{AH}
that the cardinal $\gk$ is a
{\it high-jump cardinal} if
there is an elementary embedding
$j : V \to M$ having critical
point $\gk$ such that for
some $\theta$ we have that
$M^\theta \subseteq M$ and
$j(f)(\gk) < \theta$ for every
function $f : \gk \to \gk$.
As Lemma 2 of \cite{AH} indicates,
if $\gk$ is almost huge, then
$\gk$ is the $\gk^{\rm th}$
high-jump cardinal.
Our key new concept is that of
a cardinal $\gk$ being
{\em Woodin for supercompactness}.
This will hold if for every
$f : \gk \to \gk$ with
$f(\ga)$ a cardinal, there is some
$\gd < \gk$ with $f '' \gd \subseteq \gd$
%for which $\gd$ is supercompact up to $\gk$,
%$f '' \gd \subseteq \gd$, and there is
and an elementary embedding
$j : V \to M$ having critical
point $\gd$ generated by a
supercompact ultrafilter having
rank below $\gk$ such that
$M^{j(f)(\gd)} \subseteq M$.
Our terminology comes from the
usual definition of a Woodin cardinal.
Since by its definition,
a cardinal Woodin for supercompactness
is also a Woodin cardinal, it
follows that if $\gk$ is
Woodin for supercompactness, then
$\gk$ is both regular and a limit of
measurable cardinals (and as
Lemma \ref{l3} will show, much more).
In addition, essentially the same proof
used with Woodin cardinals shows that
the least cardinal Woodin for
supercompactness isn't weakly compact.
The following lemma is central
to establishing our results.
\begin{lemma}\label{l1}
If $\gk$ is a high-jump cardinal, then
$\gk$ carries a normal measure
concentrating on
$A = \{\gd < \gk \mid \gd$ is Woodin for
supercompactness$\}$.
\end{lemma}
\begin{proof}
Our proof is reminiscent of
the usual proof that
a superstrong cardinal
has a normal measure concentrating
on Woodin cardinals
(see Proposition 26.12 of \cite{K}).
Suppose $j : V \to M$ is an elementary
embedding witnessing that $\gk$ is
a high-jump cardinal, with
$\gth$ such that
$M^\gth \subseteq M$ and
$j(f)(\gk) < \gth$ for every
$f : \gk \to \gk$. Fix $f: \gk \to \gk$
such that $f(\ga)$ is a cardinal,
and let $\gl = j(f)(\gk)$.
Take $\U$ and $\U'$
as the $\gl$ supercompactness
measure and normal measure
over $\gk$ derived from $j$, i.e.,
$X \in \U$ iff
$\la j(\ga) \mid \ga < \gl \ra \in j(X)$, and
$Y \in \U'$ iff $\gk \in j(Y)$.
Then both $\U$ and
$\U'$ are elements of $M$.
In addition, for
$i : V \to {\rm Ult}(V, \U)$ and
$k : M \to {\rm Ult}(M, \U) = M^*$,
the closure properties of $M$ imply that
$i(f)(\gk) = k(f)(\gk)$. Now, for
$\ell : {\rm Ult}(V, \U) \to M$
the factor embedding, i.e., the
elementary embedding such that
$\ell \circ i = j$, we know that
$\ell \rest \gl = {\rm id}$. Also,
by the definitions of
$\ell$ and $\gl$,
$\ell(i(f)(\gk)) = j(f)(\gk) = \gl$.
Further, if $i(f)(\gk) < \gl$,
there must be some $\nu < \gl$
(namely $i(f)(\gk)$) such that
$\ell(\nu) = \gl$. However, since
$\ell \rest \gl = {\rm id}$, $\ell(\nu)
= \nu$. This means that $i(f)(\gk) =
\gl = j(f)(\gk) = k(f)(\gk)$.
And, because ${\rm cp}(j) = \gk$,
$M \models ``f : \gk \to \gk$ and
$j(f) : j(\gk) \to j(\gk)$ are
functions which agree below $\gk$''.
By elementarity,
$M^* \models ``k(f)$ and $k(j(f))$
agree below $k(\gk)$'', which
immediately yields that
$k(f)(\gk) = k(j(f))(\gk)$.
Putting all of the preceding
together allows us to infer that
$M \models ``j(f) : j(\gk) \to j(\gk)$,
$j(f) '' \gk \subseteq \gk$, and
there is a $\gd < j(\gk)$ (namely $\gk$) and
an elementary embedding
$k : M \to M^*$ with
critical point $\gk$ generated
by a supercompact ultrafilter
having rank below $j(\gk)$ such that
${(M^*)}^{k(j(f))(\gk)} =
{(M^*)}^{k(f)(\gk)} \subseteq M^*$''.
By reflection,
$V \models ``$There is a $\gd < \gk$ and
an elementary embedding
$k^* : V \to N$ with critical point
$\gd$ generated by a supercompact
ultrafilter having rank below $\gk$
such that $f '' \gd \subseteq \gd$ and
$N^{k^*(f)(\gd)} \subseteq N$''. Hence,
$V \models ``\gk$ is Woodin for supercompactness'',
so since $M^\gth \subseteq M$,
$M \models ``\gk$ is Woodin for
supercompactness'' as well. Consequently,
$\gk \in j(A)$, which means $A \in \U'$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{pf}
Having completed the proof of
Lemma \ref{l1}, we turn our
attention now to the proof
of Theorem \ref{t1}.
We proceed in analogy to the
proof of Theorem 5 given in \cite{AH},
using the same definition for
our forcing conditions as
found there. Suppose
$V \models ``$ZFC + $\gk$ is
Woodin for supercompactness''.
We define a reverse Easton iteration having
length $\gk$ which
%begins by adding a Cohen subset of $\go$ and then
does nontrivial forcing
only at those stages $\gd < \gk$ which are
measurable cardinals in $V$. If at such
a $\gd$, some condition $p \in \FP_\gd$ forces that
$\gd$ is ${<} \gg$
supercompact in $V^{\FP_\gd}$ for $\gg$
the next measurable cardinal above $\gd$ and the
${<} \gg$ supercompactness of $\gd$ is indestructible
under $\gd$-directed closed forcing having rank
less than $\gk$, then we stop the construction
and use as our final model
${(V_\gg)}^{V^{\FP_\gd}}$,
assuming we have forced above $p$.
Otherwise, we continue the iteration.
In this case, there is some minimal
$\ga < \gg$ such that the $\ga$
supercompactness of $\gd$
is destroyed by some $\gd$-directed closed
partial ordering $\FQ$ of rank below $\gk$.
By the work of \cite{AH}, we may assume
that forcing with $\FQ$ leaves no measurable
cardinals in the half-open interval
$(\ga, \card{\FQ}]$.
We then let
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ$,
where $\dot \FQ$ is a term for such a
$\FQ$ of smallest possible rank.
By the same arguments as in \cite{AH},
if there is a stage of forcing
$\gd < \gk$ at which we can stop the
construction and use
${(V_\gg)}^{V^{\FP_\gd}}$ as our
final model, then we have obtained
a model for universal indestructibility
for supercompactness containing
a supercompact cardinal.
Thus, it suffices to show that this
is indeed what occurs.
If this is not the case, then
let $f : \gk \to \gk$
be defined inductively by
$f(\gd) = 0$ if
$\gd$ isn't a measurable
cardinal, but for $\gd$
a measurable cardinal,
$f(\gd)$ is the least inaccessible
cardinal above $\max(\ga, \gb, \gg)$,
where $V^{\FP_\gd} \models
``\gg$ is the least measurable
cardinal above $\gd$ and
$\ga$ is the smallest degree
of supercompactness of $\gd$
below $\gg$
%the least measurable cardinal above $\gd$
that can be destroyed
by some $\gd$-directed closed
forcing $\FQ$ which leaves no
measurable cardinals in the
half-open interval $(\ga, \card{\FQ}]$'', and
$\gb$ is the
%maximum of the cardinality of $\FQ$ and the
smallest rank below $\gk$ of such a $\FQ$.
%and $V^{\FP_\gd} \models ``\gg$
%is the least measurable cardinal above $\gd$''.
By the fact $\gk$ is Woodin for
supercompactness, let $\gd < \gk$
be such that $f '' \gd \subseteq \gd$
and there is an elementary embedding
$j : V \to M$ having critical point $\gd$ with
$M^{j(f)(\gd)} \subseteq M$. Write
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ_\gd
\ast \dot \FR$. By the definition of
$f$, the closure properties of $M$,
and the fact that $j(\gk) \ge \gk$
(in actuality, $j(\gk) = \gk$),
we then have that in both
$V^{\FP_\gd}$ and $M^{\FP_\gd}$,
forcing with $\FQ_\gd$ destroys
the $\ga$ supercompactness of $\gd$, where
$\ga$ is minimal below the least measurable
cardinal above $\gd$ (which is the same in both
$V^{\FP_\gd}$ and $M^{\FP_\gd}$), forcing with
$\FQ_\gd$ leaves no measurable cardinals in
the half-open interval $(\ga, \card{\FQ_\gd}]$,
and $\FQ_\gd$ has smallest possible rank below
$\gk = j(\gk)$.
However, as in the proof of
Theorem 5 of \cite{AH},
the usual reverse Easton arguments show that
if $G_0$ is $V$-generic over $\FP_\gd$,
$G_1$ is $V[G_0]$-generic over $\FQ_\gd$, and
$G_2$ is $V[G_0][G_1]$-generic over $\FR$, then
$j$ lifts in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$. We may then
find a master condition $q$ for $j '' G_1$
in $V[G_0][G_1][G_2]$
with respect to the partial ordering $j(\FQ_\gd)$,
take $G_3$ as a $V[G_0][G_1][G_2]$-generic
object containing $q$, and working in
$V[G_0][G_1][G_2][G_3]$, lift $j$ further to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
As usual, $\U$ given by
$X \in \U$ iff $\la j(\gb) \mid \gb < \ga \ra
\in j(X)$
is a supercompactness measure over
${(P_\gd(\ga))}^{V[G_0][G_1]}$ which is
present in $V[G_0][G_1][G_2][G_3]$.
However, by the closure properties of
$\FR \ast \j(\dot \FQ_\gd)$ in both
$M[G_0][G_1]$ and $V[G_0][G_1]$,
$\U \in V[G_0][G_1]$.
This contradicts that forcing with
$\FQ_\gd$ over $V^{\FP_\gd}$ destroys
the $\ga$ supercompactness of $\gd$
and therefore completes the proof of
Theorem \ref{t1}.
\end{pf}
Theorems 7 and 8 from \cite{AH},
in which models for universal
indestructibility for strong
compactness and universal
indestructibility for strongness
are constructed, also remain valid
when forcing with the same partial orderings
as in \cite{AH}, using a
cardinal Woodin for
supercompactness and the method
of proof given in Theorem \ref{t1} above.
In addition, Theorem 6 of \cite{AH},
where a model for universal
indestructibility for supercompactness
is constructed
in which every Ramsey and weakly
compact cardinal also satisfies the
appropriate form of universal
indestructibility, can be proven
as well using a cardinal Woodin
for supercompactness and the same
partial ordering as in \cite{AH}.
All of these models contain either
a supercompact, strongly compact,
or strong cardinal.
In order to prove Theorem \ref{t2},
we need the following lemma, which is
the analogue of Lemma 1.1 of \cite{A05}.
It shows that the results of
\cite{LS} are true for cardinals
Woodin for supercompactness.
\begin{lemma}\label{l2}
Suppose
$V \models ``$ZFC + $\gk$ is
Woodin for supercompactness +
$\FP$ is a partial ordering such that
$\card{\FP} < \gk$''. Then
$V^\FP \models ``\gk$ is Woodin
for supercompactness''.
\end{lemma}
\begin{proof}
Suppose $p \in \FP$ and $\dot f$
are such that
$p \forces ``\dot f : \gk \to \gk$
is a function with $\dot f(\ga)$
a cardinal''. Define in $V$ a function
$g$ by $g(\ga) = {\card{\FP}}^+$ if
$\ga \le \card{\FP}$, and
$g(\ga) =$
The least inaccessible cardinal above
$\sup(\{\gb < \gk \mid$
For some $q$ extending $p$,
$q \forces ``\dot f(\ga) = \gb$''$\})$ if
$\ga > \card{\FP}$.
Since $\card{\FP} < \gk$ and $\gk$ is
a regular limit of measurable
cardinals, $g$ is a well-defined
function whose values are always
cardinals. It is then the case that
$p \forces ``$For every $\ga < \gk$,
$\dot f(\ga) < g(\ga)$''.
By the definitions of $g$ and
Woodin for supercompactness,
there is some $\gd < \gk$,
$\gd > \card{\FP}$ and elementary
embedding $j : V \to M$ having
critical point $\gd$ such that
$g '' \gd \subseteq \gd$ and
$M^{j(g)(\gd)} \subseteq M$.
By the results of \cite{LS},
since $\card{\FP} < \gd$, $j$ lifts in
$V^\FP$ to $j : V^\FP \to M^{j(\FP)}$.
We then have that
$p \forces ``$There is $\gd < \gk$
and an elementary embedding
$j : V^\FP \to M^{j(\FP)}$ having
critical point $\gd$ such that
$\dot f '' \gd \subseteq \gd$ and
${(M^{j(\FP)})}^{j(\dot f)(\gd)} \subseteq M^{j(\FP)}$''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{pf}
We are just about ready to begin
the proof of Theorem \ref{t2}.
Before doing so, however, we
prove the following lemma.
\begin{lemma}\label{l3}
If $\gk$ is Woodin for supercompactness, then
$\{\gd < \gk \mid \gd$ is ${<} \gk$
supercompact$\}$ is unbounded in $\gk$.
\end{lemma}
\begin{proof}
Assume towards a contradiction that
$\{\gd < \gk \mid \gd$ is ${<} \gk$
supercompact$\}$ is bounded in $\gk$.
Let therefore $\ga_0$
be such that for every $\ga \ge \ga_0$,
$\gth_\ga$ is the least cardinal
below $\gk$ with
the property that $\ga$ isn't $\gth_\ga$
supercompact. Define $f : \ga \to \ga$ by
$f(\ga) = \ga^+_0$ if $\ga < \ga_0$, and
$f(\ga) = $ The least
inaccessible cardinal above
$\gth_\ga$ if $\ga \ge \ga_0$.
By the fact $\gk$ is Woodin for
supercompactness, we may find $\gd < \gk$,
$\gd > \ga^+_0$
and an elementary
embedding $j : V \to M$
with critical point $\gd$ such that
$f '' \gd \subseteq \gd$ and
$M^{j(f)(\gd)} \subseteq M$.
By the closure properties of
$M$ and the definition of $f$,
it then immediately follows that
$\gd$ is $\gth_\gd$ supercompact
in both $V$ and $M$, a contradiction.
This completes the proof of
Lemma \ref{l3}.
\end{proof}
%Given what we just established,
We are now ready to prove Theorem \ref{t2}.
Suppose once again that
$V \models ``$ZFC + $\gk$ is
Woodin for supercompactness''.
By Lemma \ref{l3}, let
$\gd < \gk$ be the smallest cardinal such that
$V \models ``\gd$ is ${<} \gk$ supercompact''.
Force with the partial ordering
$\FP$ of Theorem 1 of \cite{AG}
defined with respect to $\gd$.
By the results of \cite{AG},
$V^\FP \models ``\gd$ is the
least measurable cardinal, $\gd$
is ${<} \gk$ strongly compact, and the
${<} \gk$ strong compactness of $\gd$
is indestructible under forcing with
$\gd$-directed closed partial orderings
having rank below $\gk$''.
Since $\FP$ may be defined so that
$\card{\FP} = \gd < \gk$, by Lemma \ref{l2},
$V^\FP \models ``\gk$ is Woodin for supercompactness''.
If we then let $\FQ$ be the partial ordering
of either Theorem 5 or Theorem 6 of \cite{AH}
(both of which force universal indestructibility
for supercompactness) or Theorem 7 of
\cite{AH} (which forces universal indestructibility
for strong compactness),
with the first nontrivial stage of forcing
taking place at or above the least
weakly compact cardinal in $V^\FP$ above $\gd$,
then the arguments given in the proof of
Theorem \ref{t1} of this paper show that
the construction of $\FQ$ terminates
at some stage $\gg < \gk$.
Let $\gs < \gk$, $\gs > \gg$ be
the least weakly compact cardinal
above $\gg$ in $V^{\FP \ast \dot \FQ}$.
By the same arguments as in \cite{A05},
${(V_\gs)}^{V^{\FP \ast \dot \FQ}}$
is our model for either universal
indestructibility for strong compactness
or universal indestructibility for
supercompactness (depending upon the
exact definition of $\FQ$) containing
two strongly compact cardinals.
This completes the proof of Theorem \ref{t2}.
\end{pf}
Since our constructions
require implict applications of Hamkins'
Gap Forcing Theorem of
\cite{H2} and \cite{H3},
our proofs are going to
require at the minimum
as a hypothesis a supercompact
limit of supercompact cardinals. Readers
are urged to consult
\cite{AH} for the explicit details.
The exact consistency strength
of universal indestructibility
as discussed in this paper
therefore remains unknown.
\bigskip\noindent Added in proof: It is possible
to reduce the consistency strength of
the assumptions used to establish
Theorems \ref{t1} and \ref{t2} still further.
Details can be found in our forthcoming paper
``An Equiconsistency for Universal Indestructibility''.
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\end{document}