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\date{Set Theory, Analysis, and their
Neighbours (STAATN) Meeting\\
January 7, 1999\\London, England}
\title{Laver Indestrucibility and Strong Compactness}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York NY 10010\\
USA}
\begin{document}
\maketitle
We take as our convention that sc means supercompact and
strc means strongly compact.\\
In the mid '70s, Richard Laver proved the following:
\begin{theorem}\label{t1}(Laver, {\it Israel Journal of
Mathematics}, 1978):\\
Let $V \models ``$ZFC + $\gk$ is sc''.
There is then a p.o$.$ $\FP \in V$,
$|\FP| = \gk$, s.t$.$
$V^\FP \models ``\gk$ is sc and
$\gk$'s scness is indestructible under
$\gk$-directed closed forcing''.
This means that if
$\forces_{\FP} ``\dot \FQ$ is $\gk$-directed
closed'', then
$V^{\FP \ast \dot \FQ} \models ``\gk$ is sc''.
\end{theorem}
Remarks:\\
1$.$ A sc card$.$ $\gk$ as above is called
{\it Laver indestructible}.\\
2$.$ Laver indestructibility has been a fundamental
tool in large cardinals and forcing since its
discovery. Its first application was by Magidor,
who used it in his construction, starting from
a sc card$.$ with a huge card$.$ above it, of
a model in which GCH holds below $\ha_\omega$ and
$2^{\ha_\omega} = \ha_{\omega + 2}$.\\
We can ask the following question:\\
If $\gk$ is a non-sc strc card$.$, what sort of
indestructibility properties can $\gk$ exhibit?\\
There are two ways of approaching the above
question. We can either use properties of
scness, or we can see what can be proven
using an arbitrary strc card$.$\\
Some results using the first of these approaches
are as follows:
\begin{theorem}\label{t2}(A$.$, {\it JSL} '98):\\
Let $V \models ``$ZFC + $\gk$ is a measurable
limit of sc cards$.$ + $2^\gk = \gk^+$''.
There is then a p.o$.$ $\FP \in V$,
$|\FP| = \gk$ s.t$.$
$V^\FP \models ``\gk$ is a measurable limit of
sc cards.''
Further, if
$\forces_{\FP} ``\dot \FQ$ is a $\gk$-directed
closed p.o$.$ which doesn't add subsets to $\gk$'', then
$V^{\FP \ast \dot \FQ} \models ``\gk$ is a meas$.$ limit
of sc cards.''
\end{theorem}
Remarks:\\
1$.$ Menas showed that any measurable limit of strc
cards$.$ is strc. He also showed that many measurable
limits of strc cards$.$ aren't strc.\\
2$.$ Theorem \ref{t2} thus shows that certain
non-sc strc cards$.$ can exhibit a weak form of
indestructibility. Examples of p.o$.$s $\FQ$ as in
Theorem \ref{t2} include any $\gk^+$-directed closed
p.o$.$, the p.o$.$ adding a branch to a normal
Souslin tree on $\gk^+$, the p.o$.$ adding a
$\gk$-club to a $\gk$-stationary subset of
$\gk^+$, and the p.o$.$ which adds to any
regular card$.$ $\gl > \gk$ a
non-reflecting stationary set of ordinals of
cofinality $\gk$.\\
\begin{theorem}\label{t3}(A$.$ \& Gitik,
{\it JSL} '99):\\
Let $V \models ``$ZFC + $\gk$ is sc''.
There is then a p.o$.$
$\FP \in V$, $|\FP| = \gk$ s.t$.$
$V^\FP \models ``\gk$ is both the least
measurable and least strc card.''
Further, if
$\forces_{\FP} ``\dot \FQ$ is $\gk$-directed
closed'', then
$V^{\FP \ast \dot \FQ} \models ``\gk$ is both
the least measurable and least strc card.''
\end{theorem}
Remark: Theorem \ref{t3} thus shows,
since the least measurable card$.$
can't be sc, that it is consistent for a
non-sc strc card$.$ to be fully Laver
indestructible.
\begin{theorem}\label{t4}(Hamkins, Fall '97):\\
Let $V \models ``$ZFC + $\gk$ is a sc limit of
sc cards''. There is then a p.o$.$
$\FP \in V$, $|\FP| = \gk$ s.t$.$
$V^\FP \models ``\gk$ is the least measurable
limit of sc cards.'' Further, if
$\forces_{\FP} ``\dot \FQ$ is $\gk$-directed
closed'', then
$V^{\FP \ast \dot \FQ} \models ``\gk$ is
the least measurable limit of sc cards.''
\end{theorem}
Remark: Menas' work shows the least measurable limit
of sc (or strc) cards$.$ isn't sc.
Thus, Theorem \ref{t4} provides a model in which
this particular non-sc strc card$.$ is fully
Laver indestructible, whereas Theorem \ref{t2}
provides a model in which this particular
non-sc strc$.$ card$.$ has a weak form of
indestructibility.\\
Hamkins has studied what can be proven using the
second of the above approaches, i.e., what
indestructibility properties can be established
using an arbitrary non-sc strc card. He has shown
the following:
\begin{theorem}\label{t5}(Hamkins, to appear in
the {\it APAL}):\\
Let $V \models ``$ZFC + $\gk$ is strc''.
There is then a p.o$.$
$\FP \in V$, $|\FP| = \gk$ s.t$.$
$V^\FP \models ``\gk$ is strc and
$\gk$'s strcness is indestructible
under forcing with the p.o$.$ adding
one Cohen subset to $\gk$,
certain p.o$.$s which add clubs to $\gk$,
and certain p.o$.$s which add
``long Prikry sequences'' to $\gk$''.
\end{theorem}
Note that although Laver's forcing is
iterable so that we can fairly easily
make any class $\cal K$ of scs
simultaneously indestructible, the forcing of
Theorem \ref{t3}
(which makes the least strc be both the least
measurable and be fully indestructible) is
non-iterable. Thus, we can ask:\\
Is it possible for the first two strcs to be
non-sc and still exhibit some form of
indestructibility?\\
The answer is yes.
\begin{theorem}\label{t6}(A$.$, 8/98):\\
Let $V \models ``$ZFC + GCH +
$\gk_1 < \gk_2 < \gl$ are s.t$.$
$\gk_1$ and $\gk_2$ are both $\gl$
sc and $\gl$ is strongly inaccessible''.
There is then a p.o$.$
$\FP \in V$, $|\FP| = \gk_2$ s.t$.$
$V^{\FP}_\gl \models ``\gk_1$ and
$\gk_2$ are both the first two strcs
and first two measurables,
$\gk_1$'s strcness is fully indestructible, and
$\gk_2$'s measurability (but not necessarily its
strcness) is fully indestructible''.
\end{theorem}
Sketch of proof: Assume $\gk_1$, $\gk_2$, and
$\gl$ are all the least with their properties.
By first forcing over $V$ with the p.o$.$ from
Theorem \ref{t3}, we may assume that we have
constructed a model
$V_1 \models ``$ZFC + $\gk_1$ is the least
measurable and is $< \gl$ strc +
$\gk_1$'s $< \gl$ strcness is indestructible
under forcing with
$\gk_1$-directed closed p.o$.$s of rank $< \gl$ +
$\gk_2$ is $\gl$ sc + $\gl$ is the least
inaccessible above $\gk_2$''.\\
Working in $V_1$, we define an Easton
support iteration
$\FP^* = \la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk_2 \ra$. Let
$\la \gd_\ga : \ga < \gk_2 \ra$ enumerate the
measurables in the interval
$(\gk_1, \gk_2)$. Take
$\FP_0 = \{\emptyset\}$.
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$,
where if
$\forces_{\FP_\ga} ``$There is a $\gd_\ga$-directed
closed p.o$.$ of rank below the least inaccessible
above $\gd_\ga$ s.t$.$ forcing with it destroys
$\gd_\ga$'s measurability'', then
$\dot \FQ_\ga$ is a term for such a p.o.
If, however, this is not the case, i.e., if
$\forces_{\FP_\ga} ``$The measurability of
$\gd_\ga$ is indestructible under forcing
with $\gd_\ga$-directed closed p.o$.$s of
rank below the least inaccessible above
$\gd_\ga$'', then
$\dot \FQ_\ga$ is a term for the p.o$.$
adding a non-reflecting stationary set of
ordinals of cofinality $\gk_1$ to
$\gd_\ga$.\\
Why does $\FP^*$ work?
Its definition ensures it destroys all
$V_1$-measurables in $(\gk_1, \gk_2)$
and creates no new ones.
Since $\FP^*$ is $\gk_1$-directed closed,
$V^{\FP^*}_1 \models ``\gk_1$ is the
least measurable, is $< \gl$ strc, and the
$< \gl$ strcness of $\gk_1$ is indestructible
under forcing with $\gk_1$-directed closed
p.o$.$s of rank $< \gl$''.
Suppose $j : V_1 \to M$ is an elementary
embedding witnessing the $\gl$-scness of
$\gk_2$. If
$\forces_{\FP^*} ``\dot \FQ$ is
$\gk_2$-directed closed and
$|\dot \FQ| < \gl$'' and
$V^{\FP^* \ast \dot \FQ} \models
``\gk_2$ isn't measurable'',
then since $\gl$ is the least
inaccessible above $\gk_2$ in
both $V$ and $M$,
at stage $\gk_2$ in the
definition of $j(\FP^*)$,
$\dot \FQ_{\gk_2}$ is a term for a p.o$.$
of rank $< \gl$ destroying
$\gk_2$'s measurability. By
the $\gl$-closure of $M$ wrto $V_1$,
in $V_1$,
$\forces_{\FP^*} ``\dot \FQ_{\gk_2}$
destroys $\gk_2$'s measurability''.
We can then use the usual reverse Easton
argument to show
$j : V_1 \to M$ extends after forcing with
$\FP^* \ast \dot \FQ_{\gk_2}$, i.e.,
$V^{\FP^* \ast \dot \FQ_{\gk_2}} \models
``\gk_2$ is measurable''.
This contradiction allows us to infer
$V^{\FP^*}_1 \models ``$The measurability of
$\gk_2$ is indestructible when forcing with
p.o$.$s of rank $< \gl$ which are
$\gk_2$-directed closed''. We thus know that
$\dot \FQ_{\gk_2}$ is a term for the p.o$.$
adding a non-reflecting stationary set of
ordinals of cofinality $\gk_1$ to
$\gk_2$. This allows us to use an
argument of Magidor to show
$V^{\FP^*}_1 \models ``\gk_2$ is
$\gl$ strc''.\\
There are still many questions remaining
concerning strcness and indestructibility.
Some are:\\
1$.$ Can $\gk_2$ above have its strcness
fully indestructible?\\
2$.$ In general, can we have the first
$\ga$ strcs non-sc and fully indestructible?\\
3$.$ If we don't wish to assume scness, how
much indestructibility can be forced for an
arbitrary, strc card.?\\
Note that work of Hamkins shows that if an
arbitrary non-sc strc card$.$ is to be made
fully indestructible, the p.o$.$ used can't
in any way resemble Laver's p.o$.$, i.e.,
it can't be an Easton support iteration of
p.o$.$s which have the appropriate amount of
strategic closure.
\end{document}