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\date{Winter Meeting of the ASL\\
January 17-18, 2003\\Baltimore, Maryland}
\title{Some Results Concerning Strong
Compactness and Supercompactness}
\author{Arthur W$.$ Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010\\
awabb@cunyvm.cuny.edu\\
http://math.baruch.cuny.edu/$\sim$apter}
\begin{document}
\begin{slide}
\maketitle
Suppose $\gk \le \gl$ are infinite cardinals
with $\gk$ regular.
$\gk > \go$ is {\it measurable} iff
$\gk$ carries a $\gk$-{\it additive},
{\it nontrivial} ultrafilter ${\cal U}$.
The ultrafilter ${\cal U}$ is
$\gk$-{\it additive} iff given \break
$\la A_\ga : \ga < \gd < \gk \ra$ a sequence
of sets such that
$\forall \ga < \gd [A_\ga \in {\cal U}]$, then
$\bigcap_{\ga < \gd} A_\ga \in {\cal U}$.
The ultrafilter ${\cal U}$ is
{\it nontrivial} iff the only measure 1
sets (i.e., sets in the ultrafilter)
have cardinality $\gk$.
The ultrafilter ${\cal U}$ is
{\it normal} iff
for every function $f : \gk \to \gk$ such
that $f(\ga) < \ga$ for every
$\ga > 0$, $\ga < \gk$, then there
is an ordinal $\ga_0 < \gk$ such that
$\{\ga : f(\ga) = \ga_0\} \in {\cal U}$.
Assuming the Axiom of Dependent Choice
(a weak form of the Axiom of Choice),
every measurable cardinal carries a
normal measure.
$P_\gk(\gl) = \{ x \subseteq \gl : |x| < \gk\}$.
$\gk$ is $\gl$ {\it strongly compact} iff
$P_\gk(\gl)$ carries a $\gk$-additive,
{\it fine} ultrafilter ${\cal U}$.
The ultrafilter ${\cal U}$ is {\it fine} iff
for every $\ga < \gl$,
$\{p : \ga \in p\} \in {\cal U}$.
$\gk$ is $\gl$ {\it supercompact} iff
$\gk$ is $\gl$ strongly compact
and ${\cal U}$ is in
addition {\it normal}, i.e., if \break
$f : P_\gk(\gl) \to \gl$ is a function
such that
$f(p) \in p$ for every
$p \in P_\gk(\gl)$, then there is
$\ga_0 < \gl$ such that
$\{p : f(p) = \ga_0\} \in {\cal U}$.
$\gk$ is {\it strongly compact}
({\it supercompact}) iff
$\gk$ is $\gl$ strongly compact
($\gl$ supercompact)
for all $\gl \ge \gk$.
Equivalently, $\gk$ is $\gl$
supercompact iff
there is an elementary embedding
$j : V \to M$, $V$ the universe,
$M \subseteq V$ a transitive inner model
such that
$j \rest \gk = {\rm id}$,
$j(\gk) > \gl$, and $M^\gl \subseteq M$,
i.e., if
$f: \gl \to M$, $f \in V$, then $f \in M$.
The above equivalent form of $\gl$
supercompactness
can be used to show
that
(*) {\it If $\gk$ is
$2^\gk$ supercompact, then $\gk$ is
the $\gk^{\rm th}$ measurable cardinal.}
Strongly compact cardinals were introduced
by Keisler and Tarski in their 1964
{\it Fundamenta Mathematicae} paper
``From Accessible to Inaccessible Cardinals''.
Supercompact cardinals were introduced by
Solovay a few years later (ca$.$ 1967 or
1968).
Since every measurable cardinal
carries a normal measure, it was
initially conjectured by Solovay
that the notions of strongly compact
and supercompact cardinal were
the same, i.e., that
%every strongly compact measure
every $\gk$-additive, fine ultrafilter
over
$P_\gk(\gl)$ could be ``normalized''
in some way.
This was refuted by Solovay's
student Telis Menas in the
early 1970s, who showed the
following.
\begin{theorem}\label{t1}(Menas,
{\it Annals of Mathematical Logic 1974})
Suppose $\gk$ is a measurable
cardinal which is a limit of
strongly compact cardinals.
Then $\gk$ is strongly compact.
Further, if $\gk$ is the least
such cardinal, then $\gk$
isn't $2^\gk$ supercompact.
\end{theorem}
Thus, if there are large
enough cardinals in the universe,
there is a strongly compact
cardinal which isn't supercompact.
In fact, via a forcing argument,
Menas was able to show the following.
%\vfill\eject
\begin{theorem}\label{t2}(Menas,
{\it Annals of Mathematical Logic 1974})
Relative to the existence of
a measurable cardinal which is
a limit of supercompact cardinals,
it is consistent for the least
strongly compact cardinal not
to be supercompact.
\end{theorem}
(Theorem \ref{t2} was later extended
by Jacques Stern, who showed in
unpublished work that it was
relatively consistent for the
first two strongly compact
cardinals not to be supercompact.)
Menas' work began the area of
set theory colloquially known
as the study of identity crises,
i.e., the study of strong
compactness, supercompactness,
and their possible interrelationships.
The fundamental breakthrough results
in this area were proven by
Menachem Magidor in the mid 1970s.
Inspired by Menas' work, Magidor
proved the following theorem.
%\vfill\eject
\begin{theorem}\label{t3}(Magidor,
{\it Annals of Mathematical Logic 1976})
Relative to the existence of a
strongly compact cardinal, it is
consistent for the least strongly
compact cardinal to be the
least measurable cardinal.
However, relative to the existence
of a supercompact cardinal, it
is consistent for the least
strongly compact cardinal to
be the least supercompact cardinal.
\end{theorem}
Thus, in Magidor's words, the
least strongly compact cardinal
suffers from a severe identity
crisis.
There is a universe in which
it is the least supercompact
cardinal, in which case, by
(*) mentioned earlier, it
is not the least measurable cardinal.
However, there is also a universe
in which it is the least measurable
cardinal, in which case,
by (*) mentioned earlier, it is
not supercompact.
Although the aforementioned theorems
show that the notions of strongly
compact and supercompact cardinal
need not coincide, they do
not answer the following fundamental
Question 1: Is the theory
``ZFC + There is a strongly compact
cardinal'' equiconsistent with the
theory ``ZFC + There is a supercompact
cardinal''?
This question, in fact, remains open,
with no immediate hope of resolution.
Magidor's Theorem \ref{t3} itself
raises further questions about what
can happen with the second, third,
etc$.$ strongly compact and
supercompact cardinals.
In the mid 1980s, in work
unpublished by them, Magidor and
his student Yechiel Kimchi extended
Theorem \ref{t3} by proving the following.
\begin{theorem}\label{t4}(Kimchi and Magidor,
mid 1980s)
Relative to the existence of a
class $\mathfrak K$ of supercompact
cardinals, it is consistent for
$\mathfrak K$ to be the class
of supercompact cardinals and for
the classes of strongly compact
and supercompact cardinals to
coincide precisely, except at
measurable limit points.
Further, relative to the existence
of $n \in \go$ supercompact cardinals,
it is consistent for the
first $n$ strongly compact
cardinals to be the first $n$
measurable cardinals.
\end{theorem}
(By Menas' Theorem \ref{t1},
if there are large
\break enough cardinals
in the universe, there can never be
a complete coincidence between
the classes of strongly compact
and supercompact cardinals.)
Thus, as one would expect,
Theorem \ref{t3} extends to show
that, relative to the appropriate
assumptions, it is consistent
for the classes of strongly
compact and supercompact cardinals
to coincide whenever possible.
However, Theorem \ref{t4} still
leaves open the following
Question 2: Is it relatively
consistent for the first
$\go$ strongly compact
cardinals to be the first
$\go$ measurable cardinals?
A positive answer to Question 2
is known in a model in which
the Axiom of Replacement fails.
Although Theorem \ref{t4}
gives two distinct
examples of
models containing different
characterizations of the
strongly compact cardinals,
there are other models
in which the strongly
compact cardinals can be
characterized in additional ways.
For instance, there are models
(A$.$, {\it APAL 1997},
A$.$ and Hamkins,
{\it JSL to appear})
in which a fixed ground model function
$f : {\rm Ord} \to 2$ can be
used to characterize exactly
which strongly compact cardinals
are supercompact and which ones aren't.
Also, there is a model
(A$.$ and Cummings,
{\it Arch$.$ Math$.$ Logic 2001})
in which the strongly compact
cardinals are precisely the
strong cardinals,
and a model
(A$.$, {\it MLQ to appear})
in which the strongly compact
cardinals are precisely the measurable
limits of strong cardinals.
(A {\it strong cardinal} is
a cardinal in which the embedding
characterization of supercompactness
is weakened to require not full
closure of the transitive inner
model $M$, but only containment
of larger and larger $V_\gl$.)
Theorem \ref{t4} also leaves open
other possibilities.
For instance, although there can
be a global
coincidence between strong
compactness and supercompactness,
what about a local coincidence?
In other words,
%is it possible to get a model
is there a way of obtaining a model
containing
supercompact cardinals in which,
for every pair of regular cardinals
$\gk \le \gl$,
$\gk$ is $\gl$ strongly compact
iff $\gk$ is $\gl$ supercompact?
This question is answered in
the affirmative by the following.
\begin{theorem}\label{t5}(A$.$ and
Shelah,
{\it Transactions AMS 1997})
Let $V \models ``$ZFC +
$\mathfrak K$ is the class of
supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH +
$\mathfrak K$ is
the class of supercompact cardinals +
For every pair of regular cardinals
$\gk \le \gl$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a limit
of cardinals which are $\gl$ supercompact''.
\end{theorem}
A model witnessing the
conclusions of Theorem \ref{t5}
is called a model for
{\it level by level equivalence
between strong compactness and
supercompactness}.
In such a model, the
Kimchi-Magidor property holds,
i.e., the strongly compact
and supercompact cardinals
coincide precisely,
except at measurable limit points.
By a theorem of Magidor,
if there are supercompact cardinals
in the universe,
there are always cardinals $\gk < \gl$
for which $\gl$ is singular,
$\gk$ is $\gl$ strongly compact, yet
$\gk$ isn't $\gl$ supercompact.
Also, the exception in Theorem \ref{t5}
is a local version of the
``Menas exception'' given by
Theorem \ref{t1}.
Another tack one can take when
studying universes in which
strongly compact and supercompact
cardinals exist is to examine
indestructibility phenomena.
The basic result along these
lines was established by
Richard Laver, who proved the
following theorem.
\begin{theorem}\label{t6}(Laver,
{\it Israel J$.$ Math$.$ 1978})
Let $V \models ``$ZFC +
$\gk$ is supercompact''.
There is then a partial ordering
$\FP \in V$, $|\FP| = \gk$ such that
$V^\FP \models ``\gk$ is
supercompact''.
Further, if $\FQ \in V^\FP$ is
$\gk$-directed closed, then
$V^{\FP \ast \dot \FQ} \models ``\gk$
is supercompact''.
\end{theorem}
Note that a partial ordering
$\FQ$ is $\gk$-{\it directed closed}
iff every compatible set of
conditions of size less than $\gk$
has a common extension.
A supercompact cardinal such as the
above $\gk$ in $V^\FP$ is called
{\it Laver indestructible} or
simply {\it indestructible}.
The terminology comes from the
fact that $\gk$'s supercompactness
is preserved whenever any
$\gk$-directed closed forcing
is done.
Laver's forcing easily iterates,
and it is possible to create
a universe in which each supercompact
cardinal is Laver indestructible.
Laver indestructibility is one of
the most powerful tools used in
large cardinals and forcing.
Its first application was given
by Magidor, who used it to
construct a model in which,
for every $n \in \go$,
$2^{\ha_n} = \ha_{n + 1}$, yet
$2^{\ha_\go} = \ha_{\go + 2}$.
Notice that Laver's result
says nothing about whether
%it is possible to force
one can force
indestructibility for a
non-supercompact strongly
compact cardinal.
It is therefore reasonable
to explore this possibility.
The following results highlight
what is known about
indestructibility for a
non-supercompact strongly
compact cardinal.
\begin{theorem}\label{t7}(A$.$,
{\it JSL 1998})
It is consistent, relative to
GCH and
the existence of a measurable
limit of supercompact cardinals,
for the least measurable limit
$\gk$ of supercompact cardinals
to have its strong compactness
indestructible under
$\gk^+$-directed \break closed forcing.
%$\gk$-directed
%closed forcing that preserves
%$\gk$'s measurability.
\end{theorem}
\begin{theorem}\label{t8}(Hamkins,
{\it Kobe J$.$ Math$.$ 1999})
It is consistent, relative to
the existence of a supercompact
limit of supercompact cardinals,
for the least measurable limit
$\gk$ of supercompact cardinals
to have its strong compactness
indestructible under $\gk$-directed
closed forcing.
\end{theorem}
\begin{theorem}\label{t9}(A$.$ and
Gitik, {\it JSL 1998})
It is consistent, relative to
a supercompact cardinal, for
the least strongly compact
cardinal $\gk$ to be the
least measurable cardinal,
and to have its strong compactness
indestructible under
$\gk$-directed closed forcing.
\end{theorem}
\begin{theorem}\label{t10}(A$.$,
August 2002)
It is consistent, relative to
the existence of two supercompact
cardinals, for the first two
strongly compact cardinals to
be non-\break supercompact and to
satisfy indestructibility properties.
Specifically, the least strongly
compact cardinal $\gk_1$ has
its strong compactness indestructible
under $\gk_1$-directed closed forcing,
and the second strongly compact cardinal
$\gk_2$ has its strong compactness
indestructible under $\gk_2$-directed
closed forcing that also adds a
Cohen subset to $\gk_2$.
\end{theorem}
Remark:
The forcing used to prove
Theorem \ref{t9} can't
be iterated above a
strongly compact cardinal
and preserve strong
compactness.
This means the forcing
used to prove Theorem \ref{t10}
requires something different
in order to obtain the
indestructibility properties
for the second strongly
compact cardinal.
In fact, for different reasons,
this forcing doesn't iterate
either.
This is in sharp contrast
to Laver's forcing.
Notice that the preceding four
theorems all use supercompactness
to obtain indestructibility for
the relevant non-supercompact
strongly compact cardinals.
One can also study what sort
of indestructibility properties
can be established using an
arbitrary non-supercompact
strongly compact cardinal.
Hamkins has done this, and
has established the following.
\begin{theorem}\label{t11}(Hamkins,
{\it APAL 2000})
Let $V \models ``$ZFC +
$\gk$ is strongly compact''.
There is then a partial ordering
$\FP \in V$, $|\FP| = \gk$ such that
$V^\FP \models ``$ZFC + $\gk$ is
strongly compact + $\gk$'s strong
compactness is indestructible under
forcing with the partial ordering
that adds a Cohen subset to $\gk$,
certain partial orderings which add
closed, unbounded sets to $\gk$,
and certain partial orderings which
add `long Prikry sequences' to $\gk$''.
\end{theorem}
Hamkins' partial ordering $\FP$ of
Theorem \ref{t11} is defined in
a way that is
fundamentally similar
%roughly analogous
to the definition of Laver's
partial ordering of Theorem \ref{t6}.
It is therefore not surprising that
only a very limited amount of
indestructibility can be forced for
an arbitrary non-supercompact strongly
compact cardinal, in light of the
following theorem.
\begin{theorem}\label{t12}(Hamkins,
{\it APAL 2000})
The following theories are
equiconsistent:
a) ``ZFC + $\gk$ is supercompact''
b) ``ZFC + $\gk$ is a strongly compact
cardinal whose strong compactness can
be made indestructible by any partial
ordering even naively resembling
Laver's forcing''.
\end{theorem}
A sketch of the proof of Theorem
\ref{t12} is as follows:
The forward direction is just
Theorem \ref{t6}.
For the reverse direction,
if $\gk$'s strong compactness can
be made indestructible by a partial
ordering such as Laver's forcing,
then in particular, for any cardinal
$\gl > \gk$, $\gk$ remains measurable
(and in fact, $\gk$ remains strongly
compact) after having forced with
the partial ordering which collapses
$\gl$ to have cardinality $\gk$.
Since $|\gl| = \gk$, this means that
$\gk$ is $\gl$ supercompact.
By Hamkins' Gap Forcing Theorem,
$\gk$ had to have been $\gl$ supercompact
in the ground model.
Since $\gl$ was arbitrary, $\gk$ was
supercompact in the ground model.
\vfill\eject
We conclude by giving some
possible directions for future research
into the nature of different
universes in which strongly compact
and supercompact cardinals
can exist.
\begin{itemize}
\item Resolve the equiconsistency
question, and perhaps along the
way develop an inner model
theory for strong compactness
and supercompactness.
Failing this, try to establish
further ``inner model theorems
via forcing''
(Woodin's phrase), such as
Theorem \ref{t5}.
\item Find universes in which
there are different characterizations
of the strongly compact cardinals
in terms of other large cardinals
(such as, e.g., Woodin cardinals).
\item Explore the different kinds
of indestructibility properties
non-supercompact strongly compact
cardinals can exhibit.
\end{itemize}
\vfill\eject
\end{slide}
\end{document}
\setlength{\parindent}{0in}
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$\gk$ being a strong cardinal is a weakening
of $\gk$ being a supercompact cardinal.
In fact, we have the following diagram of
consistency \break strengths:
%\setlength{\parindent}{1.5in}
\begin{center}
$\gk$ is measurable $<$
$\gk$ is strong $<$
$\gk$ is strongly compact $\le$
$\gk$ is supercompact.
\end{center}
%\setlength{\parindent}{0in}
It is unknown if the last inequality is strict.
Assume now that $\gk \le \gl$ are both regular.
The partial ordering $\FP$ is
$\gk$-{\it directed closed} iff
whenever
$\{p_\ga : \ga < \gd < \gk\}$ is
a set of compatible conditions in
$\FP$, there is some
$p \in \FP$ extending each $p_\ga$.
The partial ordering $\FP$ is
$\gk$-{\it strategically closed} iff in
the two person game in which the players
construct an increasing sequence
$\la p_\ga : \ga \le \gk \ra$ of
elements of $\FP$,
where player I plays odd stages and
player II plays even and limit
stages (choosing the trivial
condition at stage 0), then player II
has a strategy ensuring that the game
can always be continued.
The partial ordering $\FP$ is
$\gk$-{\it distributive} iff
for any ordinal $\ga < \gk$,
forcing with $\FP$ adds no new
$\ga$ sequences of elements from
the ground model.
Some examples illustrating the
above definitions:
\begin{enumerate}
\item The partial ordering adding one
Cohen subset of $\gk$ is
$\gk$-directed closed,
$\gd$-strategically closed for
every $\gd < \gk$, and
$\gk$-distributive.
\item The partial ordering adding $\gl$
Cohen subsets of $\gk$ is
$\gk$-directed closed,
$\gd$-strategically closed for
every $\gd < \gk$, and
$\gk$-distributive.
\item The partial ordering adding
a non-reflecting stationary set of
ordinals of cofinality $\gk$ to $\gl$ is
$\gk$-directed closed,
$\gd$-strategically closed for
every $\gd < \gl$, and
$\gl$-distributive.
\item The L\'evy collapse of
$\gl$ to $\gk$ is
$\gk$-directed closed,
$\gd$-strategically closed for
every $\gd < \gk$, and
$\gk$-distributive.
\end{enumerate}
Suppose now that $\gk$ is regular and
$\gl \ge \gk$ is an arbitrary ordinal.
$\gk$ is $\gl$ {\it strong} iff
there is an elementary embedding
$j : V \to M$, $V$ the universe,
$M \subseteq V$ a transitive inner model
such that
$j \rest \gk = {\rm id}$,
$j(\gk) > |V_\gl|$, and $V_\gl \subseteq M$.
$\gk$ is {\it strong} iff $\gk$ is $\gl$
strong for every $\gl \ge \gk$.
Notice that $\gk$ being a strong cardinal is
a weakening of $\gk$ being a
supercompact cardinal.
Let $V \models ``$ZFC + $\K$
is the class of supercompact
cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\K$ is
the class of supercompact cardinals
+ The strongly compact and
supercompact cardinals coincide
precisely, except at measurable
limit points + Every supercompact
cardinal is Laver indestructible +
Every measurable limit of supercompact
cardinals $\gk$ has its strong compactness
indestructible under $\gk$-directed
closed forcing which preserves
$\gk$'s measurability''.