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\title{An Equiconsistency for
Universal Indestructibility
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E45, 03E55.}
\thanks{Keywords: Universal indestructibility,
indestructibility, equiconsistency, measurable cardinal,
strong cardinal, hyperstrong cardinal, Woodin cardinal,
strongly compact cardinal, supercompact cardinal,
core model.}
\thanks{The authors owe a huge debt of
gratitude to both Joel Hamkins and
Ralf Schindler. Without their
help, encouragement, and many conversations
on the subject matter contained herein,
this paper would not have been possible.
The authors also wish to thank the referee,
for helpful comments, suggestions, and
corrections which have been incorporated into
the current version of the paper.}}
\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{June 30, 2008\\
(revised March 6, 2009)}
\begin{document}
\maketitle
\begin{abstract}
We obtain an equiconsistency
for a weak form of universal
indestructibility for strongness.
The equiconsistency is relative
to a cardinal weaker in consistency
strength than a Woodin cardinal,
Stewart Baldwin's notion of
{\em hyperstrong cardinal}.
We also briefly indicate
%We conclude by indicating
how our methods are applicable to
universal indestructibility for supercompactness
and strong compactness.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s0}
Ever since the first author and
Hamkins introduced the concept of
{\em universal indestructibility} in \cite{AH},
every theorem proven has used large cardinals
whose consistency strength is beyond
supercompactness. The reason for this is
that indestructibility of whatever
property the cardinal $\gk$ possesses
(measurability, full or partial
versions of strongness, strong compactness,
supercompactness, etc.) has been under
all ${<}\gk$-directed closed partial
orderings\footnote{In Laver's original
paper \cite{L}, these partial orderings
were referred to as being {\em $\gk$-directed
closed}.},
including the collapse forcings
${\rm Coll}(\gk, \gl)$ and ${\rm Coll}(\gk, {<} \gl)$
for $\gl > \gk$
an arbitrary cardinal.
Thus, core model considerations
(see, e.g., \cite{St} and
\cite{Ze}) indicate the necessity
of very large cardinals.
The purpose of this paper is to
show that by weakening what we consider,
%if the concept of universal indestructibility is weakened somewhat,
it is possible to
reduce the consistency strength used to
establish our form of universal indestructibility
below supercompactness and
actually obtain an
equiconsistency.
Before stating our main theorem, however,
we briefly digress to give the
relevant terminology and definitions.
We begin with a definition due to
Stewart Baldwin \cite{Ba}, that of a
{\em hyperstrong cardinal $\gk$}.\footnote{Although
our definition differs slightly from the one
found in \cite{Ba}, the two are equivalent.}
\begin{definition}\label{d1}
{\bf (S$.$ Baldwin)}
$\gk$ is 0 hyperstrong iff
$\gk$ is strong.
$\gk$ is $\ga + 1$ hyperstrong iff
for any ordinal $\gd > \gk$, there is an
elementary embedding $j : V \to M$
witnessing the $\gd$ strongness of $\gk$
(i.e., ${cp}(j) = \gk$,
$j(\gk) > \card{V_\gd}$, and
$V_\gd \subseteq M$)
generated by a $(\gk, \gd)$-extender
such that
$M \models ``\gk$ is $\ga$ hyperstrong''.
For $\gl$ a limit ordinal,
$\gk$ is $\gl$ hyperstrong iff
for any ordinal $\gd > \gk$, there is an
elementary embedding $j : V \to M$
witnessing the $\gd$ strongness of $\gk$
generated by a $(\gk, \gd)$-extender
such that
$M \models ``\gk$ is $\ga$ hyperstrong
for every $\ga < \gl$''.
$\gk$ is hyperstrong iff $\gk$ is $\ga$ hyperstrong
for every ordinal $\ga$.
\end{definition}
%Note that our definition of hyperstrong differs from Baldwin's
%only in that what we call $0$ hyperstrong, he calls $1$ hyperstrong.
Definition \ref{d1}
easily implies that
a hyperstrong cardinal is quite large
in size. In particular, a hyperstrong cardinal is a
limit of strong cardinals.
%(and much more).
%(and in general, that for any ordinal
%$\ga$, a hyperstrong cardinal is a
%limit of cardinals which are $\ga$ hyperstrong).
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
If $\FP \in V$ is a partial ordering
and $G \subseteq V$ is $V$-generic over $\FP$,
then we will abuse notation somewhat and use
$V[G]$ and $V^\FP$ interchangeably to denote
the generic extension.
If $\FP$ is a partial ordering
and $\gk$ is a cardinal,
%$\FP$ is {\it $\gk$-distributive}
%if for every sequence
%$\la D_\ga : \ga < \gk \ra$
%of dense open subsets of $\FP$,
%$\bigcap_{\ga < \gk} D_\ga$
%is dense open.
%$\FP$ is {\it $\gk$-closed}
%if every increasing chain of elements
%$\la p_\ga : \ga < \gd < \gk \ra$ of $\FP$
%has an upper bound $p \in \FP$.
%$\FP$ is {\it ${} \gk$-directed closed}
%if for every cardinal $\delta < \gk$
%and every directed
%set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$,
%(where $\langle p_\alpha : \alpha < \delta \rangle$
%is {\it directed} if every two elements
%$p_\rho$ and $p_\nu$ have a common upper bound of the form $p_\sigma$)
%there is an upper bound $p \in \FP$.
$\FP$ is {\it $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\it ${\prec}\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga < \gk\rangle$, 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 $\gd < \gk$.
$\FP$ is $(\gk, \infty)-${\em distributive}
if for every sequence
$\la D_\ga : \ga < \gk \ra$ of dense open
subsets of $\FP$, $\bigcap_{\ga < \gk} D_\ga$
is also a dense open subset of $\FP$.
Note that since forcing with a partial ordering
%which is both ${<}\gk$-strategically
%closed or both $\gk$-closed and
which is $(\gk, \infty)$-distributive
adds no new subsets of $\gk$,
the measurability of any measurable
cardinal $\gk$ (or equivalently,
its $\gk + 1$ strongness) is automatically
indestructible under such partial orderings.
As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra : \FP \in {\cal A}$
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.''
Recall 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. There are
exactly analogous definitions
of universal indestructibility for
measurability, strongness, or
strong compactness, for which we
refer readers to \cite{AH} for further details.
This motivates the definition that
{\em weak universal indestructibility for strongness}
holds in a model $V$ for ZFC if
every $V$-strong and partially strong
(including measurable) cardinal $\gd$
has its degree of strongness
indestructible under partial orderings
which are both ${<}\gd$-strategically closed
and $(\gd, \infty)$-distributive.
If some measurable cardinal $\gd$
has its degree of strongness
indestructible under partial orderings
which are both ${<}\gd$-strategically closed
and $(\gd, \infty)$-distributive,
then $\gd$'s degree of strongness will be
said to be {\em weakly indestructible}.
In addition, a partial ordering $\FP$
which is both ${<}\gd$-strategically closed
and $(\gd, \infty)$-distributive
will be said to be $\gd$-{\em suitable}.
We are now able to state our main theorem.
\begin{theorem}\label{t1}
Weak universal indestructibility
for strongness in the presence
of a strong cardinal is equiconsistent with
a hyperstrong cardinal.
\end{theorem}
Some discussion of the genesis of
this paper is perhaps in order here.
Apter had the idea that
the form of universal indestructibility
discussed might generate an
equiconsistency. He proposed
to Sargsyan that this
equiconsistency be established using a Woodin
cardinal, which by the work
of our earlier paper \cite{AS},
is sufficient for
the forcing portion of
Theorem \ref{t1}. Email correspondence
with Schindler then indicated that
weak universal indestructibility was
unlikely to yield an inner model
containing a Woodin cardinal.
Over a period of several months,
intense discussions (both in person
and by email) ensued among Apter, Hamkins,
Sargsyan, and Schindler.
Attempts were made to demonstrate the
equiconsistency by using two
cardinals $\gk_1 < \gk_2$ such that
$\gk_1$ had strongness properties
and $\gk_2$ had $\Sigma_2$ reflection properties.
This approach, however, was misleading in our
efforts to prove Theorem \ref{t1}.
In particular, %as Schindler observed, it
it could not result in an equiconsistency,
because the forcing portion of the proof
required only a limited amount of
$\Sigma_2$ reflection.
%because of the Woodin-like
%properties of the second cardinal.
%It then became clear that
%in fact universal
%indestructibility cannot be equiconsistent
%with any large cardinal (from
%which universal indestructibility could be forced)
%that also had $\Sigma_2$ reflection properties.
These observations forced us to reconsider
the consistency proofs for universal indestructibility,
which led Sargsyan to
%suggesting the inductive methods presented in this paper. This
coming up with the key techniques
used in the current proof of Theorem \ref{t1}. This
approach in turn led to the definition
of a large cardinal, which we initially called an
{\em enhanced strong cardinal},
that didn't have the ad hoc $\Sigma_2$
reflection properties but could still
be used for the forcing part of
our proof. Later it came to our attention that
S$.$ Baldwin had already
studied this notion in
\cite{Ba}, which prompted us to adopt
his notation and terminology for this paper.
%This prompted Sargsyan to come up with the idea of using a
%hyperstrong cardinal and the %essential structure of
%key techniques employed in the current proof of Theorem \ref{t1}.
Before turning our attention to the proof
of Theorem \ref{t1}, we note that
a hyperstrong
cardinal has consistency strength
strictly weaker than a Woodin cardinal.
More explicitly, we have the following theorem.
\begin{theorem}\label{t2}
ZFC + There is a Woodin cardinal $\vdash$
Con(ZFC + There is a proper
class of hyperstrong cardinals).
\end{theorem}
\begin{proof}
Suppose $\gk$ is a Woodin cardinal.
%Assume for the duration of the proof of
%Theorem \ref{t2} that all parameters used
%in the definition of enhanced strong cardinal
%are ordinals below $\gk$.
Assume towards a contradiction
that $\gd < \gk$ is such that
$V_\gk \models ``$No cardinal $\gl > \gd$
is a hyperstrong cardinal''.
Let $f : \gk \to \gk$ be defined by
$$f(\ga) = \rm The\ least\ \gb \ such \ that \
V_\gk \models ``\ga \ is\ not\ \gb\ hyperstrong".$$
Let $A$ code the graph of $f$.
Because $\gk$ is a Woodin cardinal,
by \cite[Theorem 26.14]{K},
for any $A \subseteq V_\gk$,
there is a proper class of $A$-strong
cardinals in $V_\gk$.
There thus must be some $\gl > \gd$, $\gl < \gk$
such that for any $\eta > \gl$, $\eta < \gk$,
there is an elementary embedding
$j_\eta : V \to M_\eta$ witnessing
the $\eta$ strongness of $\gl$
generated by a $(\gl, \eta)$-extender
${\cal E}_\eta \in V_\gk$ with
$A \cap V_\eta = j_\eta(A) \cap V_\eta$.
If $\gs \in A$ codes the pair $\la \gl, f(\gl) \ra$,
then for any $\eta > \max(f(\gl), \gs)$,
$\eta < \gk$,
%where $\gs \in A$ codes the pair $\la \gl, f(\gl) \ra$,
we have that
$j_\eta(f)(\gl) = f(\gl)$.
As a consequence, in $V_\gk$,
for all sufficiently large $\eta < \gk$,
the elementary embedding $k_\eta : V \to M^*_\eta$
generated by ${\cal E}_\eta$ is such that
$M^*_\eta \models ``\gl$ is $\gb$ hyperstrong
for every $\gb < f(\gl)$'', i.e.,
$V_\gk \models ``\gl$ is $f(\gl)$ hyperstrong''.
This contradiction means
we may therefore conclude that
$V_\gk \models ``$There is a proper class of
hyperstrong cardinals''.
This completes the proof of Theorem \ref{t2}.
\end{proof}
We turn our attention now to the
proof of Theorem \ref{t1}.
For clarity of exposition,
we split its presentation into
two distinct components. We begin
with our forcing construction, i.e.,
we first prove the following result.
\begin{theorem}\label{t3}
Con(ZFC + There is a hyperstrong cardinal) $\implies$
Con(ZFC + There is a strong cardinal + Weak
universal indestructibility for strongness holds).
\end{theorem}
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is a hyperstrong cardinal''.
By \cite[Theorem 3.12]{Ba},
it is also possible to assume that $V \models {\rm GCH}$.
Without loss of generality, by truncating
the universe if necessary, we assume in addition that
$V \models ``$No cardinal $\gl > \gk$ is
$(\gl + 2)$ strong''.
Our partial ordering
is a modification of the
one found in the proof of \cite[Theorem 5]{AH}.
Specifically, we define a reverse Easton
iteration having length at most $\gk$
which begins by adding a Cohen subset of $\go$.
All other
nontrivial stages of forcing can only occur at
a $\gd < \gk$ which is a Mahlo
cardinal in $V$. Suppose $\FP_\gd$
has been defined.
If $\gd$ is not $\gd + 2$ strong in
$V^{\FP_\gd}$, then we perform
trivial forcing. Otherwise,
if there is some
condition $p \in \FP_\gd$ forcing that
$\gd$ is $\gl$ strong for every $\gl < \eta$,
where $\eta$ is the least cardinal
in $V^{\FP_\gd}$ greater than $\gd$
which is $(\eta + 2)$ strong,
and these degrees of strongness are weakly
indestructible via any partial ordering
having rank less than $\gk$, then we
stop our construction and take
${(V_\eta)}^{V^{\FP_\gd}}$ as our
final desired model (assuming we have
forced above $p$).
%Otherwise, we continue the iteration.
Finally, if the preceding two
cases do not hold, then
in $V^{\FP_\gd}$, $\gd$ is
(at least) $\gd + 2$ strong, and
there is some minimal
$\ga < \eta$, $\ga \ge \gd + 2$ such that the $\ga$
strongness of $\gd$ is destroyed by some
$\gd$-suitable partial ordering $\FQ^*$
of rank less than $\gk$.
%which is either $\gd$-strategically closed or both
%$\gd$-closed and $(\gd, \infty)$-distributive.
By the proof of \cite[Lemma 4]{AH},
we may assume that forcing with $\FQ^*$ leaves
no cardinals $\gb$ which are
$\gb + 2$ strong in the half-open interval
$(\ga, \card{\FQ^*}]$.\footnote{If
necessary, instead of using $\FQ^*$ alone,
force with, e.g., $\FQ^* \ast \dot
{\rm Coll}(\gl, \card{\FQ^*})$, where
$\forces_{\FQ^*} ``\gl$ is the least inaccessible
cardinal greater than $\ga$''.}
We then let
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$,
where $\dot \FQ_\gd$ is a term for the lottery sum of
all $\gd$-suitable partial orderings $\FQ^*$
having rank below
the least $V$-strong cardinal above $\gd$ such that
forcing with $\FQ^*$ destroys the
$\ga$ strongness of $\gd$ and leaves
no cardinals $\gb$ which are $\gb + 2$ strong in
the half-open interval $(\ga, \card{\FQ^*}]$.
%which are either $\gd$-strategically closed or both
%$\gd$-closed and $(\gd, \infty)$-distributive and
%which leave no measurable cardinals in
%the half-open interval $(\ga, \card{\FQ^*}]$
%and which destroy
%the $\ga$ strongness of $\gd$, where
%$\ga$ is the smallest degree of strongness
%less than $\eta$
%that can be destroyed by a partial ordering
%$\FR$ having rank below $\gk$
%which is either $\gd$-strategically closed or both
%$\gd$-closed and $(\gd, \infty)$-distributive
%and leaves no
%measurable cardinals in the half-open interval
%$(\ga, \card{\FR}]$.
The proof of Theorem \ref{t3} is now
completed by the following three lemmas.
\begin{lemma}\label{l1}
Suppose that $\gd$ and $\eta$ are as in the
last case given above in the inductive
definition of $\FP$, and $\gr < \eta$
is such that
$V^{\FP_\gd} \models ``\gd$ is
$\gr$ strong, and there is a
$\gd$-suitable partial ordering
$\FQ^{**}$ such that forcing with
$\FQ^{**}$ destroys the $\gr$
strongness of $\gd$ and leaves no
cardinals $\gb$ which are $\gb + 2$
strong in the half-open interval
$(\gr, \card{\FQ^{**}}]$''. Then
%under these circumstances,
there is always a $\gd$-suitable
partial ordering $\FQ^* \in V^{\FP_\gd}$
having rank less than the least $V$-strong
cardinal $\gg$ greater than $\gd$ such that
forcing with $\FQ^*$ destroys the
$\gr$ strongness of $\gd$ and leaves
no cardinals $\gb$ which are $\gb + 2$ strong
in the half-open interval
$(\gr, \card{\FQ^*}]$.
\end{lemma}
\begin{proof}
Suppose there is
such a partial ordering $\FQ^{**}$ having
rank greater than $\gg$. Let $\gl > \gg$
be sufficiently large so that if
$j : V \to M$ is an elementary embedding
witnessing the $\gl$ strongness of $\gg$, then
$M \models ``$Forcing with the
$\gd$-suitable partial ordering $\FQ^{**}$ over
$M^{\FP_\gd}$ destroys the $\gr$
strongness of $\gd$ and leaves
no cardinals $\gb$ which are $\gb + 2$ strong in the
half-open interval $(\gr, \card{\FQ^{**}}]$,
where $\gr < \eta$ and
$\eta$ is the least cardinal greater than
$\gd$ which is $(\eta + 2)$ strong
in $M^{\FP_\gd}$''.
Since the critical point of $j$ is $\gg$ and
$\gd, \gr, \eta < \gg$, by reflection,
there is a $\gd$-suitable partial ordering
$\FQ^* \in V^{\FP_\gd}$ having rank less than
$\gg$ such that
$V \models ``$Forcing with the
$\gd$-suitable partial ordering $\FQ^{*}$ over
$V^{\FP_\gd}$ destroys the $\gr$
strongness of $\gd$ and leaves
no cardinals $\gb$ which are $\gb + 2$ strong in the
half-open interval $(\gr, \card{\FQ^{*}}]$,
where $\gr < \eta$ and
$\eta$ is the least cardinal greater than
$\gd$ which is $(\eta + 2)$ strong
in $V^{\FP_\gd}$''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
Lemma \ref{l1} allows us to infer that if
$\gd \le \gk$ is a strong cardinal, then
$\FP_\gd \subseteq V_\gd$.
\begin{lemma}\label{l2}
If the construction
of $\FP$ terminates at some stage $\gs < \gk$, then
${(V_\eta)}^{V^{\FP_{\gs}}}$ as described above
is a model for weak universal
indestructibility for strongness containing
a strong cardinal.
\end{lemma}
\begin{proof}
We assume throughout the course of the
proof of Lemma \ref{l2} that we are
forcing above the appropriate condition $p$.
By the inductive definition of $\FP$,
the same proof as given for \cite[Theorem 5]{AH}
yields that weak %universal
indestructibility for strongness
must hold in ${(V_\eta)}^{V^{\FP_{\gs}}}$
for any measurable cardinal
$\gd \in {(V_\eta)}^{V^{\FP_{\gs}}}$ such that
$\gd \le \gs$ and
$V^{\FP_\gd} \models ``\gd$ is (at least)
$\gd + 2$ strong''.\footnote{To see in
greater detail why this is the case,
we quote almost verbatim from the proof
of \cite[Theorem 5]{AH}, making the
minor necessary modifications where necessary.
Let $\ga \ge \gd + 2$, $\ga < \eta$ be the
least ordinal such that
$V^{\FP_{\gd + 1}} \models ``\gd$ isn't
$\ga$ strong''. Suppose
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gb$
strong''. Necessarily, $\gb < \ga$.
In addition, no $\gd$-suitable forcing
$\FQ^{**} \in V^{\FP_{\gd + 1}}$ can destroy
the $\gb$ strongness of $\gd$. This is since
if $\FQ^*$ is the forcing chosen by the lottery
at stage $\gd$, then $\FQ^* \ast \dot \FQ^{**}$
would have destroyed the $\gb$ strongness of
$\gd$ over $V^{\FP_\gd}$, contradicting the
minimality of $\ga$. In particular, the
$\gb$ strongness of $\gd$ is preserved by the
forcing $\FP_{\gd, \gg}$ which leads to any
of the later models $V^{\FP_\gg}$ for $\gg < \gk$.
Furthermore, since the next nontrivial stage of
forcing after $\gd$ takes place at an inaccessible
cardinal beyond $\ga$ and $\card{\FQ^*}$, the
$\ga$ strongness of $\gd$ is never resurrected by the
later stages of forcing. Consequently, in all
of the later models $V^{\FP_\gg}$ for $\gg < \gk$,
the partial strongness of $\gd$ is weakly indestructible
by all $\gd$-suitable partial orderings of
rank less than $\gk$.
Note that this proof requires no use of
Hamkins' Gap Forcing Theorem of \cite{H2} and \cite{H3}.}
The only problematic measurable cardinals
below $\gs$ are hence
%in the open interval $(\gd, \eta)$.
those measurable cardinals
$\gd \in {(V_\eta)}^{V^{\FP_{\gs}}}$ such that
$V^{\FP_\gd} \models ``\gd$ isn't $\gd + 2$ strong''.
However, by the definition of weak indestructibility,
$V^{\FP_\gd} \models ``\gd$'s degree of strongness,
namely measurability, is weakly indestructible''.
Since nontrivial stages of forcing above $\gd$
can only occur at ordinals $\gg$ such that
$V^{\FP_\gg} \models ``\gg$ is (at least)
$\gg + 2$ strong'', for $\dot \FP^*$ such that
$\FP_\gd \ast \dot \FP^* = \FP_\gs$,
%(where $\ga > \gd$, $\ga < \gk$) or
%$\FP_\gd \ast \dot \FP^* = \FP$,
it is the case that
$\forces_{\FP_\gd} ``$Forcing with
$\dot \FP^*$ adds no new subsets of the least
strong limit cardinal above $\gd$''. Thus,
$V^{\FP_\gd \ast \dot \FP^*} = V^{\FP_\gs} \models
``\gd$'s degree of strongness,
namely measurability, is weakly indestructible'',
i.e., weak %universal
indestructibility for
strongness holds in ${(V_\eta)}^{V^{\FP_{\gs}}}$
for any measurable cardinal $\gd \le \gs$. Since
${(V_\eta)}^{V^{\FP_\gs}} \models ``$Any
measurable cardinal $\gd > \gs$ isn't
$\gd + 2$ strong'', our remarks above
immediately yield that
${(V_\eta)}^{V^{\FP_\gs}} \models ``$Weak
indestructibility for strongness holds
for every measurable cardinal $\gd > \gs$''.
Therefore, ${(V_\eta)}^{V^{\FP_\gs}} \models ``\gs$
is a strong cardinal, and weak universal
indestructibility for strongness holds''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
If the construction
of $\FP$ does not terminate before stage $\gk$, then
the arguments of
Lemma \ref{l2} show that weak %universal
indestructibility for strongness must
hold in $V^\FP$ for every measurable
cardinal $\gd \neq \gk$ .
The proof of Theorem \ref{t3}
is consequently completed by the following lemma.
\begin{lemma}\label{l3}
Suppose that the definition of $\FP$
does not terminate before stage $\gk$. Then
$V^\FP \models ``\gk$ is a strong cardinal
whose strongness is weakly indestructible''.
\end{lemma}
\begin{proof}
We proceed inductively,
taking as our inductive hypothesis that
if $\ga \ge 1$ is an ordinal and
$N \subseteq V$ is an arbitrary
transitive model of ZFC + GCH such that
$\FP \in N$, $V_\gk \in N$, and
%$N \models ``$GCH + $\gk$ is $\gb$ hyperstrong for
%every $\gb < \ga$'', then
$N \models ``\gk$ is $\ga$ hyperstrong'', then
$N^\FP \models ``$The $\gk + \ga$ strongness of $\gk$ is
weakly indestructible''.
For $\ga = 1$,
this amounts to showing that if $\FP \in N$,
$V_\gk \in N$, $N \subseteq V$,
%and $N \models ``$GCH + $\gk$ is $0$ hyperstrong'', i.e., if
and $N \models ``$GCH + $\gk$ is $1$ hyperstrong'',
then $N^\FP \models ``$The $\gk + 1$ strongness
of $\gk$, i.e., the measurability of $\gk$,
is weakly indestructible''.
To see that this is indeed the case, let
$\mu \in N$ be a normal measure over $\gk$
such that for $j_\mu : N \to M_\mu$ the
ultrapower embedding via $\mu$,
$M_\mu \models ``\gk$ isn't measurable''.
Note that
$N \models ``\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra :
\gd < \gk \ra$ is a reverse Easton iteration
such that at nontrivial stages $\gd$,
$\forces_{\FP_\gd} `\dot \FQ_\gd$ is
${<}\gd$-strategically closed' ''.
The preceding two sentences yield that
$j_\mu(\FP) = \FP \ast \dot \FQ'$, where the
first ordinal at which $\dot \FQ'$ is forced
to do nontrivial forcing is above $\gk^+$, and
$\forces_{\FP} ``\dot \FQ'$ is ${\prec} \gk^+$-strategically
closed''. Since $N \models {\rm GCH}$,
standard arguments
%as given, e.g., in \cite[Theorem 3.5]{H4},
yield that
$j_\mu$ lifts in $N$ to
$j_\mu : N^\FP \to M^{j_\mu(\FP)}_\mu$.\footnote{An
outline of these arguments is as follows.
Let $G$ be $N$-generic over $\FP$.
Since $\FP$ is $\gk$-c.c., $M_\mu[G]$ remains
$\gk$-closed with respect to $N[G]$.
Because $N \models {\rm GCH}$ and $M_\mu$
is given by an ultrapower embedding, we may let
$\la D_\ga : \ga < \gk^+ \ra \in N[G]$
enumerate the dense open subsets of $\FQ'$
present in $M_\mu[G]$. As in the
construction of the generic object
$H'$ given later in the proof
of this lemma, it is possible to use the
${\prec}\gk^+$-strategic closure of
$\FQ'$ in both $M_\mu[G]$ and $N[G]$ to
build in $N[G]$ an $M_\mu[G]$-generic
object $G'$ over $\FQ'$. Since
$j_\mu '' G \subseteq G \ast G'$,
$j_\mu$ lifts to $j_\mu : N[G] \to M_\mu[G][G']$.}
From this, it follows that
$N^\FP \models ``\gk$ is measurable''.
Since the measurability
of $\gk$ is weakly indestructible,
we have established the base case of our induction.
We now assume that $\ga > 1$ is an arbitrary
(successor or limit) ordinal.
If our inductive hypothesis is false at $\ga$, then
let $N$ and $\FQ \in N^\FP$ which is $\gk$-suitable
be such that
$N^{\FP \ast \dot \FQ} \models ``\gk$
isn't $\gk + \ga$ strong''.
For the sake of simplicity, we assume without
loss of generality that $N = V$. Choose
$\gl$ to be sufficiently large, e.g.,
suppose $\gl$ is the least strong limit cardinal above
$\max(\card{{\rm TC}(\FP \ast \dot \FQ)}, \gk + \ga)$
having cofinality $\gk$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ strongness of $\gk$
generated by a $(\gk, \gl)$-extender such that
$M \models ``\gk$ is $\gb$ hyperstrong
for every $\gb < \ga$''.
%By induction, for every $\gb < \ga$,
%in both $V^\FP$ and
%$M^\FP$, the $\gk + \gb$ strongness of
%$\gk$ is weakly indestructible. Further,
By the choice of $j$ and $M$, $\FQ \in M^\FP$.
Since $V \models ``$No cardinal
$\gb > \gk$ is $\gb + 2$ strong'' and
$V_\gl \subseteq M$,
$M \models ``$There are neither
strong cardinals nor cardinals $\gb$
which are $\gb + 2$ strong
in either the half-open interval $(\gk, \gl]$
or the half-open interval $(\gk, \card{\FQ}]$''.
From this, we may immediately infer that
$M^\FP \models ``\FQ$ has rank below the
least $M$-strong cardinal above $\gk$
and forcing with $\FQ$ destroys the
$\gk + \ga$ strongness of $\gk$''.
In addition, it is the case that
forcing with $\FQ$
leaves no cardinals
$\gb$ which are $\gb + 2$ strong
in the half-open interval
$(\gk, \card{\FQ}]$.
To see this, note that it is
possible (in either $V$ or $M$) to write
$\FP \ast \dot \FQ = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is (at least)
$\ha_1$-strategically closed''.
By Hamkins' Gap Forcing Theorem of
\cite{H2} and \cite{H3}, this means that any
cardinal $\gb \in (\gk, \card{\FQ}]$ which
is $\gb + 2$ strong in $M^{\FP \ast \dot \FQ}$
had to have been $\gb + 2$ strong in $M$.
As we have observed above, there can be
no such cardinals.
Further, by the choice of $j$ and $M$,
%because $M \models ``\gk$ is $\ga$ hyperstrong'',
%$M \models ``\gk$ is $\gb$ hyperstrong for every $\gb < \ga$''.
it is inductively the case that
$M^\FP \models ``$The $\gk + \gb$
strongness of $\gk$ is weakly
indestructible for every $\gb < \ga$''.
Consequently, $\FQ$ is an allowable choice
in the stage $\gk$ lottery held in
$M^\FP$ in the definition of $j(\FP)$, so
above the appropriate condition $q$,
$j(\FP \ast \dot \FQ)$ is forcing
equivalent to
$\FP \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$,
where the first nontrivial stage
in $\dot \FR$ is forced to occur
well above $\gl$. For the remainder
of the proof of Lemma \ref{l3},
we assume without loss of generality
that we are forcing above $q$.%such a condition.
%Standard arguments then show that
%$V^{\FP \ast \dot \FQ} \models ``\gk$ is $\gk + \ga$ strong''.
We now show that the embedding $j$
lifts in $V^{\FP \ast \dot \FQ}$ to
$j : V^{\FP \ast \dot \FQ} \to M^{j(\FP \ast \dot \FQ)}$.
%that the embedding $j$ lifts
The methods for doing this
are quite similar to those given in the proof of
\cite[Theorem 4.10]{H4} (as well as elsewhere).
%or the argument given in the
%proof of Lemma 4.2 of \cite{A03}.
For the benefit of readers, we give
the argument here as well, taking the
liberty to quote freely from \cite[Theorem 4.10]{H4}.
%We may assume that
Because $j$ is an extender embedding, we have that
$M = \{j(f)(a) : a \in {[\gl]}^{< \omega}$,
$f \in V$, and $\dom(f) = {[\gk]}^{|a|}\}$.
%and $\rge(f) \subseteq V\}$.
Since we may write
$j(\FP)$ as $\FP \ast \dot \FQ \ast \dot \FR$,
as we have already observed,
we know that the first ordinal at which
$\dot \FR$ is forced to do a
lottery sum is above $\gl$.
Since $\gl$ has been chosen to have cofinality $\gk$,
we may assume that $M^\gk \subseteq M$.
This means that if
$G$ is $V$-generic over $\FP$ and
$H$ is $V[G]$-generic over $\FQ$,
$\FR$ is ${\prec} \gk^+$-strategically closed in both
$V[G][H]$ and $M[G][H]$, and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$.
As in \cite{H4},
by using a suitable coding that allows us
to identify finite subsets of $\gl$
with elements of $\gl$,
by the definition of $M$,
there must be some
$\ga_0 < \gl$ and function $g$ such that
$\dot \FQ = j(g)(\ga_0)$.
%(assuming that $\dot \FQ$ has been chosen reasonably).
Let $N^* = \{i_{G \ast H}(\dot z) : \dot z =
j(f)(\gk, \ga_0, \gl)$ for some function $f \in V\}$.
It is easy to verify that
$N^* \prec M[G][H]$, that $N^*$ is closed under
$\gk$ sequences in $V[G][H]$, and that
$\gk$, $\ga_0$, $\gl$, $\FQ$, and $\FR$ are all
elements of $N^*$.
Further, since
%$\FP \subseteq V_\gk$ and
$\FR$ is $j(\gk)$-c.c$.$ in $M[G][H]$ and
there are only $2^\gk = \gk^+$ many functions
$f : [\gk]^3 \to V_\gk$ in $V$, there are at most
$\gk^+$ many dense open subsets of $\FR$ in $N^*$.
Therefore, since $\FR$ is
${\prec} \gk^+$-strategically closed in both
$M[G][H]$ and $V[G][H]$, we can
%use the argument for the construction of the
%generic object $G_1$ to be given in Lemma \ref{l5} to
build $H' \subseteq \FR$
in $V[G][H]$ as follows. Let
%an $N$-generic object $H'$ over $\FR$ as follows. Let
$\la D_\gs : \gs < \gk^+ \ra$ enumerate in
$V[G][H]$ the dense open subsets of
$\FR$ present in $N^*$ so that
every dense open subset of $\FR$
occurring in $N^*$ appears at an
odd stage at least once in the
enumeration.
If $\gs$ is an odd ordinal,
$\gs = \tau + 1$ for some $\tau$.
Player I picks
$p_\gs \in D_\gs$ extending $q_\tau$
%$\sup(\la q_\gb : \gb < \gs \ra)$
(initially, $q_{0}$ is the empty condition),
and player II responds by picking
$q_\gs \ge p_\gs$
according to a fixed strategy ${\cal S}$
(so $q_\gs \in D_\gs$).
If $\gs$ is a limit ordinal, player II
uses ${\cal S}$ to pick
$q_\gs$ extending each
$q \in \la q_\gg : \gg < \gs \ra$.
%$\sup(\la q_\gb : \gb < \gs \ra)$.
By the ${\prec} \gk^+$-strategic closure of
$\FR$ in $V[G][H]$,
player II's strategy can be assumed to be
a winning one, so
$\la q_\gs : \gs < \gk^+ \ra$ can be taken
as an increasing sequence of conditions with
$q_\gs \in D_\gs$ for $\gs < \gk^+$.
Let
$H' = \{p \in \FR : \exists \gs <
\gk^+ [q_\gs \ge p]\}$.
%is an $N$-generic object over $\FR$.
We show now that $H'$ is actually
$M[G][H]$-generic over $\FR$.
If $D$ is a dense open subset of
$\FR$ in $M[G][H]$, then
$D = i_{G \ast H}(\dot D)$ for some name
$\dot D \in M$. Consequently,
$\dot D = j(f)(\gk, \gk_1, \ldots, \gk_n)$
for some function $f \in V$ and
$\gk < \gk_1 < \cdots < \gk_n < \gl$. Let
$\ov D$ be a name for the intersection of all
$i_{G \ast H}(j(f)(\gk, \ga_1, \ldots, \ga_n))$,
where $\gk < \ga_1 < \cdots < \ga_n < \gl$ is
such that
$j(f)(\gk, \ga_1, \ldots, \ga_n)$
yields a name for a dense open subset of
$\FR$.
Since this name can be given in $M$ and
$\FR$ is $\gl$-strategically closed in
$M[G][H]$ and therefore $(\gl, \infty)$-distributive in
$M[G][H]$, $\ov D$ is a name for a dense open
subset of $\FR$ which is
definable without the parameters
$\gk_1, \ldots, \gk_n$.
Hence, by its definition,
$i_{G \ast H}(\ov D) \in N^*$.
Thus, since $H'$ meets every dense open
subset of $\FR$ present in $N^*$,
$H' \cap i_{G \ast H}(\ov D) \neq \emptyset$,
so since $\ov D$ is forced to be a subset of
$\dot D$,
$H' \cap i_{G \ast H}(\dot D) \neq \emptyset$.
This means $H'$ is
$M[G][H]$-generic over $\FR$, so in
$V[G][H]$, as
$j''G \subseteq G \ast H \ast H'$,
$j$ lifts to
$j : V[G] \to M[G][H][H']$ via the definition
$j(i_G(\tau)) = i_{G \ast H \ast H'}(j(\tau))$.
It remains to lift $j$ through the forcing $\FQ$
while working in $V[G][H]$.
To do this, it suffices to show that
$j''H \subseteq j(\FQ)$ generates an
$M[G][H][H']$-generic object $H''$ over $j(\FQ)$.
Given a dense open subset
$D \subseteq j(\FQ)$, $D \in M[G][H][H']$,
$D = i_{G \ast H \ast H'}(\dot D)$ for some name
$\dot D = j(\vec D)(a)$, where
$a \in {[\gl]}^{< \go}$ and
$\vec D = \la D_\gs : \gs \in {[\gk]}^{|a|} \ra$
is a function.
%{[\gk]}^{< \go} \ra$.
We may assume that every $D_\sigma$ is a
dense open subset of $\FQ$.
Since $\FQ$ is $(\gk, \infty)$-distributive, it follows that
%$D' = \bigcap_{\gs \in {[\gk]}^{< \go}} D_\gs$ is
$D' = \bigcap_{\gs \in {[\gk]}^{|a|}} D_\gs$ is
also a dense open subset of $\FQ$. As
$j(D') \subseteq D$ and $H \cap D' \neq \emptyset$,
$j''H \cap D \neq \emptyset$. Thus,
$H'' = \{p \in j(\FQ) : \exists q \in j''H
[q \ge p]\}$ is our desired generic object, and
$j$ lifts in $V[G][H]$ to
$j : V[G][H] \to M[G][H][H'][H'']$.
This final lifted version of $j$ is
$\gl$ strong since $V_\gl \subseteq M$, meaning
${(V_\gl)}^{V[G][H]} \subseteq M[G][H] \subseteq
M[G][H][H'][H'']$.
Therefore, since
$V[G][H] \models ``\gl > \gk + \ga$ is a
strong limit cardinal'',
$V[G][H] \models ``\gk$ is $\gk + \ga$ strong''.
This contradiction completes our induction
and the proof of Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l1} -- \ref{l3} complete the
proof of Theorem \ref{t3}.
\end{proof}
Now that we have finished with the proof
of Theorem \ref{t3}, we turn our attention to
the inner model portion of our argument.
Specifically, we establish the following result.
\begin{theorem}\label{t4}
Con(ZFC + There is a strong cardinal + Weak
universal indestructibility for strongness holds) $\implies$
Con(ZFC + There is a hyperstrong cardinal).
\end{theorem}
\begin{proof}
We argue using standard core model techniques
exposited in \cite{St} and \cite{Ze}.
We are done if there is an inner model with a
hyperstrong cardinal, so we assume without
loss of generality that this is not the case.
%In particular, we begin by assuming without
%loss of generality that there is no inner
%model containing a Woodin cardinal,
%an assumption which is harmless by Theorem \ref{t2}.
%which Theorem \ref{t2} allows us to do.
Suppose that
$V \models ``$ZFC + $\gk$ is a strong cardinal +
Weak universal indestructibility for strongness holds''.
Let $\gl > \gk$
%, e.g., $\gl = \gk^+ + \go$,
be an arbitrary ordinal which is
sufficiently large, e.g.,
$\gl$ is of size at least
$\gk + \go$. We now have
that if $j : V \to M$ is an elementary
embedding witnessing the $\gl$ strongness of
$\gk$ generated by a
$(\gk, \gl)$-extender ${\cal E}$, then
$M \models ``\gk$ is $(\gk + 2)$ strong''.
By elementarity,
$M \models ``$The $(\gk + 2)$ strongness of
$\gk$ is weakly indestructible''.
Consequently, for $\gd > \gk^+$ an arbitrary cardinal
and $\FP_\gd = ({\rm Coll}(\gk^+, \gd))^M$,
$M^{\FP_\gd} \models ``\gk$ is $(\gk + 2)$ strong''.
Since any subset of $\gd$ may now be coded
by a subset of $\gk^+$,
this means that %every subset of $\gd$ is present in
there is actually an extender
${\cal F} \in M^{\FP_\gd}$ witnessing
enough strongness to show that all subsets
of $\gd$ are captured in the relevant target model.
By downwards absoluteness to the core model
${(K)}^{M^{\FP_\gd}}$, this last fact is true in
${(K)}^{M^{\FP_\gd}}$ as well, i.e., in
${(K)}^{M^{\FP_\gd}}$,
there is some strongness extender
${\cal F}^* = {\cal F} \rest {(K)}^{M^{\FP_\gd}}$
witnessing that all subsets of
$\gd$ are captured in the relevant target model.
By the absoluteness of the core model under set forcing,
in the core model ${(K)}^M = {(K)}^{M^{\FP_\gd}}$,
%there is some strongness extender
%${\cal F}^{**} = {\cal F}^* \rest {(K)}^M$
${\cal F}^*$ is a strongness extender
witnessing that all subsets of
$\gd$ are captured in the relevant target model.
Since $\gd > \gk$ was arbitrary, this just means that
${(K)}^M \models ``\gk$ is a strong cardinal''.
Let $K = {(K)}^V$.
We show that $K \models ``\gk$
is $1$ hyperstrong''. To do this,
take once again $\gl$, ${\cal E}$, $j$, and $M$
as in the preceding paragraph.
Let ${\cal E}^* = {\cal E} \rest K$, with
$i : K \to N$ the $\gl$ strongness
embedding generated by ${\cal E}^*$ and
$\ell : N \to {(K)}^M$ the associated factor
elementary embedding whose critical point
is greater than $\gk$.
It is then the case that
$N \models ``\gk$ is a strong cardinal'', since
by elementarity,
$N \models ``\gk$ is a strong cardinal'' iff
${(K)}^M \models ``\ell(\gk)$ is a strong cardinal'',
i.e., iff
${(K)}^M \models ``\gk$ is a strong cardinal''.
Thus, for any $\gl$ which is sufficiently
large, there is an
elementary embedding witnessing the
$\gl$ strongness of $\gk$ in $K$ generated by a
$(\gk, \gl)$-extender such that in
the target model, $\gk$ is a strong cardinal.
Now that we know that
$K \models ``\gk$ is $1$ hyperstrong'',
we are able to proceed inductively.
Specifically, we assume that for
$\gl$ having been chosen to be sufficiently large,
$j : V \to M$ an elementary
embedding witnessing the $\gl$ strongness of
$\gk$ generated by a
$(\gk, \gl)$-extender ${\cal E}$, and $\gg$ either
a successor or limit ordinal,
${(K)}^M \models ``\gk$ is $\ga$ hyperstrong
for every $\ga < \gg$''.
The proof given in the preceding
paragraph, with
``$\gk$ is $\ga$
hyperstrong for every $\ga < \gg$''
replacing ``$\gk$ is a strong cardinal''
then shows that
$K \models ``\gk$ is $\gg$ hyperstrong''.
Since $\gg$ was arbitrary,
this completes the proof of Theorem \ref{t4}.
\end{proof}
\begin{pf}
The proofs of Theorems \ref{t3} and \ref{t4}
complete the proof of Theorem \ref{t1}.
\end{pf}
We note that
\cite[Theorems 1 and 2]{AS} may be
established using a supercompact
cardinal having analogous properties
to those given in Definition \ref{d1}.
More specifically,
these theorems may be proven using a
cardinal $\gk$ in which in Definition \ref{d1},
strong is replaced by supercompact, and
instead of elementary embeddings witnessing
strongness generated by extenders, we use
elementary embeddings witnessing supercompactness
generated by ultrafilters.
In particular, models containing a
supercompact cardinal which also witness
universal indestructibility for supercompactness
may be so constructed, as well as
models containing strongly compact cardinals
which witness various forms
of universal indestructibility
(including universal indestructibility
for strong compactness).
Since these theorems were originally
proven using a strengthened version of Woodin
cardinal in which strongness-like elementary
embeddings are replaced by supercompactness-like
elementary embeddings (see \cite{AS}
for the exact definition),
in analogy to Theorem \ref{t2} of
this paper, this
weakens the hypotheses used to obtain these theorems.
(The work of \cite{AS} also weakened the
hypotheses used to establish universal
indestructibility from the original
assumptions of \cite{AH}.)
Of course, since inner model theory for
supercompactness is still in its infancy,
getting an equiconsistency for
universal indestructibility for
either strong compactness or supercompactness
relative to a model satisfying supercompactness
assumptions currently remains out of reach.
We conclude by asking whether it is
possible to prove a version of
Theorem \ref{t1} in which the equiconsistency
%of universal indestructibility obtained
is for the type of indestructibility
Gitik and Shelah first described in \cite{GS}.
This would seem to require a version of the
Gap Forcing Theorem for Prikry iterations
relative to strongness,
something which has yet to be established.
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\end{document}