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\title{A Note on Strong Compactness and Resurrectibility
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, indestructibility, resurrectibility}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{January 24, 2000\\
(revised April 26, 2000)}
\begin{document}
\maketitle
\begin{abstract}
We construct a model containing a proper
class of strongly compact cardinals in
which no strongly compact cardinal
$\gk$ is $\gk^+$ supercompact and in
which every strongly compact cardinal
has its strong compactness resurrectible.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Proof of the Main Theorem}\label{s1}
The subject of Laver indestructibility \cite{L}
in the context of strong compactness is one which
has now been the subject of several papers.
Readers may consult \cite{A98}, \cite{A99},
\cite{AG}, \cite{AH1}, and \cite{H4} for
further information on this topic.
However, in spite of the research that has
been done in this area, there are still
many unsolved problems.
For instance, in the list of open questions
at the end of \cite{A99}, it is asked
whether the first $\ga$ strongly compact
cardinals can be non-supercompact and still
exhibit some sort of indestructibility
properties, where $\ga > 2$.
(The case of $\ga = 2$ is partially discussed
in \cite{A99}, and the case of $\ga = 1$ is
discussed in \cite{AG}.)
The purpose of this note is to provide a
partial answer to this question by
constructing a model containing a proper
class of strongly compact cardinals in
which no strongly compact cardinal $\gk$
is even $\gk^+$ supercompact and in which
every strongly compact cardinal
has its strong compactness resurrectible.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + There is a proper class of
supercompact limits of supercompact cardinals +
There are no inaccessible limits of supercompact
limits of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC + There is a proper class of
strongly compact cardinals + No strongly compact
cardinal $\gk$ is $\gk^+$ supercompact +
Every strongly compact cardinal has its strong
compactness resurrectible''.
\end{theorem}
Note that we say a strongly compact
cardinal $\gk$ has its strong compactness
{\it resurrectible} if after forcing with an
arbitrary $\gk$-directed closed partial
ordering $\FP$, there is a $\gk$-distributive
partial ordering
$\FQ \in V^\FP$ so that
$V^{\FP \ast \dot \FQ} \models ``\gk$ is
strongly compact''.
Also, we say that the partial ordering
$\FQ$ is {\it $\gk$-distributive}
if for any ordinal $\ga < \gk$,
forcing with $\FQ$ adds no new
$\ga$ sequences of elements from the
ground model.
\begin{pf}
To prove Theorem \ref{t1}, let
$\la \gd_\ga : \ga \in {\rm Ord} \ra$
enumerate in $V$ the supercompact cardinals
together with their measurable limit points.
%We assume without loss of generality, by
%``cutting off'' the universe if necessary,
%that there is no inaccessible limit of
%supercompact limits of supercompact cardinals.
We assume, by doing a preliminary
forcing if necessary, that
$V \models {\rm GCH}$.
Let $\FP^0$ be the partial ordering used in the
proof of Theorem 3 of \cite{A99}.
For completeness, we give the definition of
$\FP^0$ below.
Let $\gg < \gd$ be so that $\gg$ is
regular and $\gd$ is supercompact.
By Lemma 13, pages 2028 - 2029 of \cite{AS97b}
(see also the proof of the Theorem of \cite{A98}),
there is a $\gg$-directed closed partial ordering
$\FP_{\gg, \gd} \in V$
of rank $\gd + 1$ with
$|\FP_{\gg, \gd}| = \gd$ so that
${V}^{\FP_{\gg, \gd}} \models
``$There are no strongly compact cardinals in
the interval $(\gg, \gd)$
since unboundedly many cardinals in
$(\gg, \gd)$ contain non-reflecting
stationary sets of ordinals of
cofinality $\gg$ + $\gd$ is a
fully indestructible supercompact cardinal''.
This has as a consequence that
${V}^{\FP_{\gg, \gd}} \models
``$Any partial ordering not adding bounded
subsets to $\gd$ preserves that there are
no strongly compact cardinals in the
interval $(\gg, \gd)$''. Further,
$\FP_{\gg, \gd}$ is defined as a modification
of Laver's indestructibility partial ordering
of \cite{L}, i.e., as an Easton support
iteration of length $\gd$
defined in the style of \cite{L} so that
every stage at which a non-trivial forcing
is done is a ground model measurable cardinal,
the least stage at which a non-trivial forcing
is done can be chosen to be an arbitrarily large
measurable cardinal in $(\gg, \gd)$,
and at a stage
$\ga$ when a non-trivial
forcing $\FQ$ is done,
$\FQ = \FQ^0 \ast \dot \FQ^1$
where $\FQ^0$ is $\ga$-directed closed
and $\dot \FQ^1$ is a term for the
forcing adding
a non-reflecting stationary set of
ordinals of cofinality $\gg$
to some cardinal $\gb > \ga$.
(The exact definition of $\FQ^1$ can be
found in, e.g., \cite{AS97b}.
We note only that $\FQ^1$ is $\gg$-directed closed.)
We define a proper class Easton support iteration
$\FP^0 = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga
\in {\rm Ord} \ra$ as follows:
\begin{enumerate}
\item\label{e1} $\FP_1 = \FP_0 \ast \dot \FQ_0$,
where $\FP_0$ is the partial ordering for
adding a Cohen real, and
$\dot \FQ_0$ is a term for
$\FP_{\ha_2, \gd_0}$.
\item\label{e2} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and
$\forces_{\FP_\ga} ``$There is a $\gd_\ga$-directed
closed partial ordering so that after forcing with it,
$\gd_\ga$ isn't $\zeta$ supercompact
for some $\zeta$'', then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for such
a partial ordering of minimal rank which
destroys the $\zeta$ supercompactness of
$\gd_\ga$ for the minimal possible $\zeta$.
By Hamkins' work of \cite{H1}, \cite{H2},
and \cite{H3}, $\zeta$ will be no greater
than the degree of supercompactness of
$\gd_\ga$ in $V$.
\item\label{e3} If $\gd_\ga$ is a measurable
limit of supercompact cardinals and case
\ref{e2} above doesn't hold
(which will mean that
$\forces_{\FP_\ga} ``\gd_\ga$ is a measurable
limit of supercompact cardinals whose
degree of supercompactness is fully
indestructible and whose strong
compactness is fully indestructible''), then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where $\dot \FQ_\ga$ is a term for the
trivial partial ordering
$\{\emptyset\}$.
\item\label{e4} If $\gd_\ga$ is not a
measurable limit of supercompact cardinals,
$\ga = \gb + 1$, $\gd_\gb$ is a measurable
limit of supercompact cardinals,
and case \ref{e2} above holds for $\gd_\gb$,
then inductively, since a direct limit must
be taken at stage $\gb$,
$|\FP_\gb| = \gd_\gb < \gd_{\gb + 1} = \gd_\ga$.
This means inductively $\FP_\gb$
has been defined so as
to have rank $< \gd_\ga$, so by Lemma 3.1 of \cite{A99}
and the succeeding remark, $\dot \FQ_\gb$ can be
chosen to have rank $< \gd_\ga$.
Also, by Lemma 3.1 of \cite{A99} and the succeeding remark,
$\zeta < \gd_\ga$ for $\zeta$ the least so that
${V}^{\FP_\gb \ast \dot \FQ_{\gb}} =
{V}^{\FP_\ga} \models ``\gd_\gb$
isn't $\zeta$ supercompact''. Let
$\dot \gg_\ga$ be so that
$\forces_{\FP_\ga} ``\dot \gg_\ga = \gd^+_\gb$'',
and let
$\sigma \in (\gd_\gb, \gd_\ga)$ be the least
measurable cardinal (in either $V$ or
$V^{\FP_\ga}$) so that
$\forces_{\FP_{\ga}} ``\sigma >
\max(\dot \gg_\ga, \dot \zeta,
{\rm rank}(\dot \FQ_\gb))$''. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$ defined so that
$\sigma$ is below the least stage at which,
in the definition of
$\FP_{\gg_\ga, \gd_\ga}$, a non-trivial forcing is done.
\item\label{e5} If $\gd_\ga$ is not a measurable limit
of supercompact cardinals and case \ref{e4} doesn't hold,
then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where for
$\gg_\ga = {(\bigcup_{\gb < \ga} \gd_\gb)}^+$,
$\dot \FQ_\ga$ is a term for
$\FP_{\gg_\ga, \gd_\ga}$.
\end{enumerate}
By the proofs of Theorem 3 given in \cite{A99}
and Hamkins' Theorem 9 given in \cite{AH1} and
the fact $\FP^0$ is an iteration of partial
orderings satisfying the appropriate degree
of directed closure,
$V_1 = V^{\FP^0} \models ``$ZFC + The strongly compact
and supercompact cardinals coincide except at
measurable limit points +
Every supercompact cardinal
is fully Laver indestructible +
There is a proper class of measurable limits of
supercompact cardinals +
Every measurable limit $\gk$ of supercompact cardinals
has both its measurability and its strong compactness
fully indestructible under $\gk$-directed closed
forcing''.
Further, by the results in Section 5 of
\cite{H4}, the only measurable limits of
supercompact cardinals in $V_1$ were
supercompact limits of supercompact cardinals in $V$.
Therefore, since
$V \models ``$There are no inaccessible limits of
supercompact limits of supercompact cardinals'', if
$V_1 \models ``\gk$ is the $\ga^{\rm th}$
measurable limit of supercompact cardinals'', $\ga < \gk$.
As the definition of $\FP^0$ and the fact
$V \models {\rm GCH}$ ensure that
$V_1 \models ``2^\gk = \gk^+$ for any measurable limit
$\gk$ of supercompact cardinals
(which was a supercompact limit of supercompact
cardinals in $V$)'', a result of Menas from \cite{Me}
immediately yields that
$V_1 \models ``$No measurable limit of supercompact
cardinals is $2^\gk = \gk^+$ supercompact''.
Work now in $V_1$. Let
$\la \sigma_\ga : \ga \in {\rm Ord} \ra$
enumerate the measurable limits of supercompact cardinals.
For $\ga = 0$, let $\FP_\ga$ be the Easton support
iteration of partial orderings which add, for
every supercompact cardinal in the interval
$(0, \sigma_0)$, a non-reflecting stationary set of
ordinals of cofinality $\omega$.
For each ordinal $\ga > 0$, let
$\FP_\ga$ be the Easton support iteration of
partial orderings which add, for every
supercompact cardinal in the interval
$(\bigcup_{\gb < \ga} \sigma_\gb, \sigma_\ga)$,
a non-reflecting stationary set of ordinals of cofinality
${(\bigcup_{\gb < \ga} \sigma_\gb)}^+$.
(The precise definition of $\FP_\ga$ can be
found in \cite{A97}.)
$\FP^1$ is then defined as the Easton support product
$\prod_{\ga \in {\rm Ord}} \FP_\ga$.
Take $\FP = \FP^0 \ast \dot \FP^1$.
By the standard Easton arguments,
$V^\FP \models {\rm ZFC}$.
By Lemmas 1-6 of \cite{A97} and the succeeding remarks,
$V^\FP \models ``$There is a proper class
of strongly compact cardinals + No strongly
compact cardinal $\gk$ is $\gk^+$ supercompact''.
Thus, the proof of Theorem \ref{t1} will be
complete once we have shown the following.
\begin{lemma}\label{l1}
$V^\FP \models ``$Every strongly compact cardinal
is resurrectible''.
\end{lemma}
\begin{proof}
Let $\gk$ be strongly compact in
$V^\FP = V^{\FP^1}_1$.
By Lemmas 1-6 of \cite{A97}, we know
that $\gk = \sigma_\ga$ for some $\ga$.
Work again in $V_1$. Write
$\FP^1 = \FQ^\ga \times \FQ$, where
$\FQ^\ga = \prod_{\gb > \ga} \FP_\gb$.
Let now $\dot \FR$ be a term for a
partial ordering in $V^\FP$ so that
$\forces_{\FP} ``\dot \FR$ is
$\gk$-directed closed'', i.e., so that
$\forces_{\FP^0 \ast (\dot \FQ^\ga \times \dot \FQ)}
``\dot \FR$ is $\gk$-directed closed''.
Abusing notation somewhat, we can assume
without loss of generality that in $V_1$,
$\forces_{\FQ^\ga \times \FQ} ``\dot \FR$ is
$\gk$-directed closed''.
Since
$(\FQ^\ga \times \FQ) \ast \dot \FR$
is a forcing iteration over $V_1$,
by the Term
Forcing Lemma of \cite{H5} applied in $V_1$
(see also \cite{C} for more information on term forcing),
there is a $\gk$-directed closed partial ordering
$\FR_{{\rm term}} \in V_1$ so that forcing with
$\FR_{{\rm term}} \times (\FQ^\ga \times \FQ)$
is equivalent to forcing with
$(\FQ^\ga \times \FQ) \ast \dot \FR \ast \dot \FS$,
where $\dot \FS$ is a term for a partial ordering.
Rewrite
$\FR_{{\rm term}} \times (\FQ^\ga \times \FQ)$ as
$(\FR_{{\rm term}} \times \FQ^\ga) \times \FQ$. Since
$\FQ^\ga$ is $\gk$-directed closed in $V_1$,
$(\FR_{{\rm term}} \times \FQ^\ga)$ is
$\gk$-directed closed in $V_1$
as well.
Thus, by the relevant indestructibility properties in $V_1$,
$V^{(\FR_{{\rm term}} \times \FQ^\ga)}_1 \models
``\gk$ is a measurable limit of supercompact cardinals''.
Therefore,
the definition of $\FQ$ allows Lemma 4
of \cite{A97} to be applied here to show
$V^{(\FR_{{\rm term}} \times \FQ^\ga) \times \FQ}_1 =
V^{\FR_{{\rm term}} \times (\FQ^\ga \times \FQ)}_1 =
V^{(\FQ^\ga \times \FQ) \ast \dot \FR \ast \dot \FS}_1 =
V^{\FP^1 \ast \dot \FR \ast \dot \FS}_1 =
V^{\FP \ast \dot \FR \ast \dot \FS} \models
``\gk$ is strongly compact''.
And, since
$\FR_{{\rm term}} \times \FQ^\ga$ is
$\gk$-directed closed,
any new $\gb$ sequences
for $\gb < \gk$ present after
forcing with the equivalent partial orderings
$(\FR_{{\rm term}} \times \FQ^\ga) \times \FQ$,
$\FR_{{\rm term}} \times (\FQ^\ga \times \FQ)$, or
$(\FQ^\ga \times \FQ) \times \FR_{{\rm term}}$
must have been added by $\FQ$.
Hence, as forcing with
$\FR_{{\rm term}} \times (\FQ^\ga \times \FQ)$
over $V_1$
is equivalent to forcing with
$(\FQ^\ga \times \FQ) \ast \dot \FR \ast \dot \FS$
over $V_1$,
for $\gb < \gk$, forcing with
$\FS$ can't add any new $\gb$ sequences.
This completes the proof of Lemma \ref{l1}.
\end{proof}
The proof of Lemma \ref{l1} completes the
proof of Theorem \ref{t1}.
\end{pf}
We conclude this section by making two remarks.
First, we observe that by the
L\'evy-Solovay results \cite{LS},
the definition of
$\FP^1$ can be redone so that we first add a
Cohen real and then define
$\FP^1$ as we did earlier.
This changes $\FP^1$ and
$\FP^1 \ast \dot \FR \ast \dot \FS$,
which is equivalent to
$(\FQ^\ga \times \FQ) \times \FR_{\rm term}$, into a
``gap forcing with a gap at $\ha_1$'' in
Hamkins' sense of \cite{H1}, \cite{H2},
and \cite{H3}, which, by the results of
\cite{H1}, \cite{H2}, and \cite{H3}, allows
us to infer that after forcing over
$V^\FP = V^{\FP^1}_1$ with $\FR \ast \dot \FS$,
$\gk$ hasn't become $\gk^+$ supercompact.
Second, if we desire each strongly compact
cardinal $\gl$ in our final model to satisfy a weaker
form of resurrectibility, i.e., to be so that
after forcing with a
$\gl$-directed closed partial ordering $\FR$
that doesn't add subsets to $\gl$,
there is a further partial ordering $\FS$
which doesn't add any new $\gl$ sequences such that
forcing with $\FS$ resurrects the strong compactness
of $\gl$, then this is possible to do starting
from a model
$V \models ``$ZFC + There is a proper class of
measurable limits of supercompact cardinals +
There are no inaccessible limits of measurable
limits of supercompact cardinals''.
%the hypotheses of a proper class of
%measurable limits of supercompact cardinals.
We simply use the techniques found in \cite{A98}
to construct $V_1$,
beginning by adding a Cohen real,
force over $V_1$ using the same definition of
$\FP^1$ as used in the proof of Theorem \ref{t1},
and then apply
the methods of Lemma \ref{l1} to infer the
desired form of resurrectibility in our
final model.
As before, if we start the definition of
$\FP^1$ by adding a Cohen real,
since no measurable limit of
strongly compact cardinals
$\gl \in V_1$ will be $\gl^+$ supercompact,
we can use the results of
\cite{H1}, \cite{H2}, and \cite{H3} to infer
that forcing with
$\FR \ast \dot \FS$ keeps every strongly
compact cardinal $\gl$ as a
non-$\gl^+$ supercompact cardinal.
\section{Concluding Remarks}\label{s2}
In conclusion to this note, we ask if
any of the strongly compact cardinals
in our final model $V^\FP$ is fully
indestructible, or indeed, if there
is any way of constructing a model
containing a proper class of strongly
compact cardinals in which no strongly
compact cardinal $\gk$ is $\gk^+$
supercompact yet all strongly compact
cardinals $\gk$ have their strong
compactness indestructible in some
variation of the usual sense of
indestructibility.
We conjecture that the answer to the
latter of these two questions is yes.
\begin{thebibliography}{99}
\bibitem{A99} A.~Apter, ``Aspects of Strong
Compactness, Measurability, and Indestructibility'',
submitted for publication to the
{\it Archive for Mathematical Logic}.
\bibitem{A98} A.~Apter,
``Laver Indestructibility and the Class of
Compact Cardinals'', {\it Journal of
Symbolic Logic 63}, 1998, 149--157.
%\bibitem{A} A.~Apter, ``On Measurable Limits
%of Compact Cardinals'', to appear in the
%{\it Journal of Symbolic Logic}.
%\bibitem{A81} A.~Apter,
%``Measurability and Degrees of Strong Compactness'',
%{\it Journal of Symbolic Logic 46}, 1981, 180--185.
%\bibitem{A80} A.~Apter, ``On the Least Strongly
%Compact Cardinal'', {\it Israel Journal of Mathematics 35},
%1980, 225--233.
\bibitem{A97} A.~Apter, ``Patterns of Compact Cardinals'',
{\it Annals of Pure and Applied Logic 89}, 1997, 101--115.
%\bibitem{A99} A.~Apter, ``Strong Compactness and the
%Class of Supercompact Cardinals'', submitted for
%publication to {\it Fundamenta Mathematicae}.
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\end{thebibliography}
\end{document}