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\date{Logic Colloquium '03\\
August 14-20, 2003\\Helsinki, Finland}
\title{Indestructibility and Strong
Compactness}
\author{Arthur W$.$ Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
awabb@cunyvm.cuny.edu\\
http://math.baruch.cuny.edu/$\sim$apter}
\begin{document}
\begin{slide}
\maketitle
This lecture will discuss
forcing indestructibility for
non-supercompact strongly
compact cardinals.
Throughout, we assume some
familiarity with the large
cardinal notions of
measurable, strong, strongly
compact, and supercompact cardinal,
along with related forcing techniques.
We begin with a brief discussion of
forcing indestructibility for
supercompact cardinals.
This was first done by
Richard Laver, who proved the
following theorem.
\begin{theorem}\label{t1}(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
some of what is known about forcing
indestructibility for a
non-supercompact strongly
compact cardinal.
If we start with a non-supercompact
strongly compact cardinal
in our ground model,
then Hamkins has shown the following.
\begin{theorem}\label{t1a}(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
${\rm Add}(\gk, 1)$
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}
Theorem \ref{t1a} only provides
a very restricted
\break amount of
indestructibility.
If we are willing to
use supercompactness in
order to force indestructibility
for non-supercompact
strongly compact cardinals,
then we can do much more.
For instance, we have the following.
\begin{theorem}\label{t2}(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}
Remark: The reason this theorem is
relevant is that Menas
({\it Annals of Mathematical Logic 1974})
has shown
that if $\gk$ is a measurable
cardinal which is a limit of
either supercompact or
non-supercompact
strongly compact cardinals,
then $\gk$ is strongly compact.
Further, if $\gk$ is the first
such cardinal,
the second such cardinal, etc.,
then $\gk$
isn't $2^\gk$ supercompact.
Thus, the cardinal for which
indestructibility has been
forced in Theorem \ref{t2}
is a non-supercompact strongly
compact cardinal.
If we increase the consistency
strength of our assumptions,
we can improve Theorem \ref{t2}
to the following.
\begin{theorem}\label{t3}(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}
Under the appropriate assumptions,
Theorem \ref{t3} can be
iterated to get models in which
every measurable limit $\gk$ of
supercompact cardinals has its
strong compactness indestructible
under $\gk$-directed closed forcing.
However, neither this nor
Theorems \ref{t2} and \ref{t3} tell us
whether it is relatively consistent
for a
non-supercompact strongly compact
cardinal which isn't a measurable
limit of supercompact cardinals
to be indestructible.
That this is possible is shown by
the following theorem.
\begin{theorem}\label{t4}(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}
Remark: Theorem \ref{t4}
generalizes a celebrated theorem
of Magidor ({\it Annals of
Mathematical Logic 1976}) which
states that relative to the
existence of a strongly compact
cardinal, it is consistent for
the least strongly compact cardinal $\gk$
to be the least measurable cardinal.
If this is the case, then
$\gk$ isn't supercompact.
Theorem \ref{t4} raises the following
Question: Is it possible for the
first two strongly compact cardinals
to be non-supercompact and to
exhibit significant indestructibility properties
for their strong compactness?
The answer to this question is yes and
forms the basis for the rest of this
lecture. Specifically, we have the
following theorem.
\vfill\eject
\begin{theorem}\label{t5}(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
exhibit indestructibility properties
for their strong compactness.
Specifically, the least strongly
compact cardinal $\gk_1$ has
its strong compactness indestructible
under arbitrary
$\gk_1$-directed closed forcing.
The second strongly compact cardinal
$\gk_2$ has its strong compactness
indestructible under $\gk_2$-directed
closed forcing that is either
trivial or
begins by adding a Cohen
subset to $\gk_2$, i.e.,
can be written in the form
${\rm Add}(\gk_2, 1) \ast \dot \FQ'$.
%where ${\rm Add}(\gk_2, 1)$ is the
%partial ordering for adding
%a Cohen subset to $\gk_2$.
\end{theorem}
Remark:
The forcing used to prove
Theorem \ref{t4} can't
be iterated above a
strongly compact cardinal
and preserve strong
compactness, since it
is a Gitik style iteration
of ``Prikry like'' forcings
which adds bona fide
Prikry sequences.
This means the forcing
used to prove Theorem \ref{t5}
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, as will be discussed later.
This is in sharp contrast
to Laver's forcing.
What follows is a sketch of the
proof of Theorem \ref{t5}.
We begin with the following definition:
Suppose $\mathfrak A$ is a
collection of partial orderings.
Then the {\it lottery sum of
$\mathfrak A$} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$,
ordered with $0$
weaker than everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
The terminology (although not the definition)
of the lottery sum of a collection of
partial orderings is due to Hamkins.
Intuitively, if $G$ is $V$-generic over
$\oplus \mathfrak A$, then $G$ first selects an
element of $\mathfrak A$
(or, as Hamkins puts it,
``holds a lottery among the posets in
$\mathfrak A$'') and then forces with it.
Let now
$V^* \models ``$ZFC + $\gk_1$ and
$\gk_2$ are the first two supercompact
cardinals''.
Without loss of generality,
by first forcing with the
partial ordering of
Theorem \ref{t4} (which
can be defined so as to
have cardinality $\gk_1$)
followed by the $\gk_1$-directed
closed partial ordering which
forces GCH for all cardinals
greater than or equal to
$\gk_1$, we assume that
$V^*$ has been generically
extended to a model
$V$ such that
$V \models ``\gk_1$
is both the least strongly
compact and least
measurable cardinal +
$\gk_1$'s strong compactness
is indestructible under
$\gk_1$-directed closed forcing +
GCH holds for all cardinals
greater than or equal to $\gk_1$ +
$\gk_2$ is supercompact''.
The partial ordering $\FP$
to be used in the proof of
Theorem \ref{t5} is now
defined as follows.
For any ordinal $\gd$, let
$\gd'$ be the least
$V$-strong cardinal above
$\gd$.
$\FP$ begins by adding a
Cohen subset of $\gk_1$.
The remainder of $\FP$ is the
reverse Easton iteration
having length $\gk_2$
which does a non-trivial forcing
only at those cardinals in
the open interval
$(\gk_1, \gk_2)$
%$\gd \in (\gk_1, \gk_2)$
which are in $V$ measurable
limits of strong cardinals.
At such a stage $\gd$, the
forcing done is the lottery
sum of all $\gd$-directed
closed partial orderings
having rank below $\gd'$
which begin by
adding a Cohen subset to $\gd$,
followed by the partial ordering
which adds a non-reflecting
stationary set of ordinals of
cofinality $\gk_1$ to $\gd'$.
%the least $V$-strong cardinal above $\gd$.
The intuition behind the
above definition of
$\FP$ is as follows.
The non-reflecting
stationary set of ordinals
of cofinality $\gk_1$ added
at each non-trivial stage of
forcing is used to destroy
all strongly compact cardinals
in the open interval
$(\gk_1, \gk_2)$.
The lottery sum employed at each
non-trivial stage of forcing
is used to force indestructibility
for $\gk_2$.
The Cohen subset added at each
non-trivial stage of forcing is
used to ensure that $\gk_2$
becomes a non-supercompact
strongly compact cardinal.
The entire iteration $\FP$
is defined in a way so as to
be $\gk_1$-directed closed,
which means that after
forcing with $\FP$,
$\gk_1$ remains as both the
least strongly compact and
least measurable cardinal and
retains the indestructibility
of its strong compactness
under $\gk_1$-directed closed forcing.
The following lemmas show that
$\FP$ is as desired.
%\begin{lemma}\label{l1}
Lemma 1: In $V^\FP$, there are no strongly
compact cardinals in the open
interval $(\gk_1, \gk_2)$.
%\end{lemma}
%\begin{proof}
{\bf Proof:} By the supercompactness
of $\gk_2$ in $V$,
there are unboundedly many in $\gk_2$
$V$-measurable limits
of $V$-strong cardinals in
the open interval $(\gk_1, \gk_2)$.
Therefore, by its definition, after forcing
with $\FP$, unboundedly many in $\gk_2$
cardinals in the open interval $(\gk_1, \gk_2)$
will contain non-reflecting
stationary sets of ordinals of
cofinality $\gk_1$.
By a theorem of Solovay, this
means that in $V^\FP$,
no cardinal $\gd \in (\gk_1, \gk_2)$
is strongly compact.
This completes the proof of Lemma 1.
\hfill$\square$
%\end{proof}
Lemma 2: In $V^\FP$, $\gk_2$ is
strongly compact.
{\bf Proof Sketch:} Let $\gl > 2^{\gk_2}$
be any regular cardinal.
%of cofinality
%greater than $\gk_2$ at which GCH holds
%(such as an appropriate
%singular strong limit
%cardinal or a regular cardinal).
Let $j : V \to M$ be an
elementary embedding witnessing the
$\gl$ supercompactness of $\gk_2$
such that $M \models ``\gk_2$
isn't $\gl$ supercompact''.
In $M$, $\gk_2$ is a
measurable limit of
strong cardinals, meaning
by the definition of $\FP$
it is possible to opt for
%the forcing adding a Cohen subset of $\gk_2$ in
${\rm Add}(\gk_2, 1)$ in
the stage $\gk_2$ lottery
held in $M$ in the
definition of $j(\FP)$.
Further, since
$M \models ``$No cardinal
$\gd \in (\gk_2, \gl]$ is
strong''
(otherwise, $\gk_2$
is in $M$ supercompact
up to a strong cardinal
and hence fully
supercompact), the next non-trivial
forcing in the
definition of $j(\FP)$ takes place
well above $\gl$.
Since $\gl$ has been chosen
large enough, we can let
$\mu \in M$ be a normal measure
over $\gk_2$ minimal in the
Mitchell ordering and
$k : M \to N$ be the
elementary embedding generated
by the ultrapower via $\mu$.
$i = k \circ j$ is therefore an
elementary embedding with
${\rm cp}(i) = \gk_2$ such that
$i : V \to N$ witnesses the
$\gl$ strong compactness of
$\gk_2$ and
$N \models ``\gk_2$ isn't a
measurable cardinal''.
Thus, only trivial forcing
takes place at stage $\gk_2$
in $N$ in the definition of
$i(\FP)$.
We can therefore
now apply an argument
of Magidor to show that
after forcing with $\FP$,
$i$ lifts to an elementary embedding
witnessing the $\gl$ strong
compactness of $\gk_2$.
Since $\gl$ was arbitrary, this
completes the proof sketch of Lemma 2.
\hfill$\square$
Thus, the key idea in the
proof of Lemma 2 is to
choose a sufficiently
large $\gl$ and associated
supercompactness embedding
$j : V \to M$ such that
at stage $\gk_2$ in the
definition of the forcing in $M$,
we are able to opt for Cohen forcing.
This allows us then to run an
argument of Magidor for the
preservation of strong compactness.
Lemma 3: In $V^\FP$, $\gk_2$
isn't supercompact.
In fact, in $V^\FP$, $\gk_2$
has trivial Mitchell rank, i.e.,
there is no normal measure $\mu$
over $\gk_2$ in $V^\FP$
such that for
$j : V^\FP \to M^{j(\FP)}$ the elementary
embedding generated by the
ultrapower via $\mu$,
$M^{j(\FP)} \models ``\gk_2$ is measurable''.
{\bf Proof Sketch:}
The proof uses an argument
due to Hamkins.
Let $G$ be $V$-generic over $\FP$. If $V[G]
\models ``{\gk_2}$ does not have trivial Mitchell
rank'', then let $j : V[G] \to M[j(G)]$ be an
embedding generated by a normal measure over
${\gk_2}$ in $V[G]$ witnessing this fact.
By Hamkins' Gap Forcing Theorem,
$j$ must lift an embedding $j : V \to
M$ that is definable in $V$.
Also by Hamkins' Gap Forcing Theorem,
${\gk_2}$ is measurable in $M$. Therefore,
since ${\gk_2}$ is a measurable
limit of strong cardinals
in $M$, it follows that there is a non-trivial
forcing done at stage ${\gk_2}$ in $M$.
This means
$j(G) = G \ast S \ast H$, where $S$ is a
Cohen subset of ${\gk_2}$ added
by forcing over $M[G]$ with
${{\rm Add}({\gk_2}, 1)}^{M[G]}$
at stage ${\gk_2}$ in $M[G]$,
and $H$ is
$M[G][S]$-generic for the rest of the forcing
$j(\FP)$. Since
$V_{{\gk_2}+1}^V \subseteq M \subseteq
V$, it follows that $V_{{\gk_2} + 1}^{V[G]} = V_{{\gk_2}
+ 1}^{M[G]}$. From this it follows that
${{\rm Add}({\gk_2}, 1)}^{M[G]} =
{{\rm Add}({\gk_2}, 1)}^{V[G]}$,
and the dense open subsets of
what we can now unambiguously write as
${\rm Add}({\gk_2}, 1)$ are the same in both $M[G]$ and
$V[G]$. Thus, the set $S$, which is an element of
$V[G]$, is $V[G]$-generic over ${\rm Add}({\gk_2}, 1)$,
a contradiction. This completes
the proof sketch of Lemma 3.
\hfill$\square$
Thus, the key idea in the proof of
Lemma 3 is that if $\gk_2$ has
non-trivial Mitchell rank after
forcing with $\FP$, then there
must be a Cohen generic subset
of $\gk_2$ which couldn't possibly
have been added by $\FP$.
Since a supercompact cardinal
has non-trivial Mitchell rank,
$\gk_2$ isn't supercompact.
Lemma 4: Suppose $\FQ \in V^\FP$ is a
partial ordering which is $\gk_2$-directed
closed and can be written in the form
${\rm Add}(\gk_2, 1) \ast \dot \FQ'$. Then
%adds a Cohen subset to $\gk_2$. Then
$V^{\FP \ast \dot \FQ} \models ``\gk_2$
is strongly compact''.
In fact,
$V^{\FP \ast \dot \FQ} \models ``\gk_2$
is supercompact''.
{\bf Proof Sketch:} Suppose $\FQ \in V^\FP$
is such a partial ordering. Let
$\gl > \max(|{\rm TC}(\dot \FQ)|, 2^{\gk_2})$
be an arbitrary regular cardinal, and let
$j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness
of $\gk_2$ such that
$M \models ``\gk_2$ isn't $\gl$ supercompact''.
By the choice of $\gl$, it is
possible to opt for $\FQ$ in the stage
$\gk_2$ lottery held in $M$ in the
definition of $j(\FP)$.
Further, as in Lemma 2, since
$M \models ``$No cardinal
$\gd \in (\gk_2, \gl]$ is strong'',
the next non-trivial
forcing in the definition of
$j(\FP)$ takes place well above $\gl$.
Thus, above the appropriate condition,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FR$.
This means it is now possible to use
the standard reverse Easton arguments
to lift $j$ in $V^{\FP \ast \dot \FQ}$
to an elementary embedding
$j : V^{\FP \ast \dot \FQ} \to
M^{j(\FP \ast \dot \FQ)}$ witnessing
the $\gl$ supercompactness of $\gk_2$.
Since $\gl$ was arbitrary,
this completes the proof sketch of Lemma 4.
\hfill$\square$
Thus, the key idea in the proof of Lemma 4
is to choose a sufficiently large $\gl$
and associated supercompactness embedding
$j : V \to M$ such that at stage
$\gk_2$ in the definition of the
forcing in $M$, we are able to
opt for $\FQ$ as our partial ordering.
Since no additional non-trivial
forcing in $M$ takes place until
well after $\gl$,
this allows us then to run the
standard reverse Easton arguments
to show that $j$ lifts to a $\gl$
supercompactness embedding after
forcing with $\FP \ast \dot \FQ$.
Theorem \ref{t5} now follows from
Lemmas 1-4.
By Lemma 1, there are no strongly
compact cardinals in $V^\FP$ in
the interval $(\gk_1, \gk_2)$,
so by Lemma 2, in $V^\FP$,
$\gk_2$ is the second strongly
compact cardinal.
By Lemma 3, $\gk_2$ isn't supercompact.
By Lemma 4, $\gk_2$ remains strongly
compact after forcing with a partial
ordering $\FQ$ that can be written in
the form
${\rm Add}(\gk_2, 1) \ast \dot \FQ'$,
and $\gk_2$ is clearly strongly
compact after doing a trivial forcing
over $V^\FP$.
This completes the proof sketch
of Theorem \ref{t5}.
\hfill$\square$
In some sense, Theorem \ref{t5}
raises more questions than it answers.
We close by listing some of them.
In particular:
1. Is it possible to prove
Theorem \ref{t5} without the
requirement that indestructibility
for $\gk_2$ for non-trivial forcing
be under partial orderings which
begin by adding a Cohen subset of $\gk_2$?
(With the current proof of Theorem \ref{t5},
if we drop this requirement in the
definition of $\FP$, then
$\gk_2$ remains a supercompact
cardinal after forcing with $\FP$.)
2. Is it possible to prove
Theorem \ref{t5} such that after
doing a non-trivial
$\gk_2$-directed closed
forcing, $\gk_2$'s supercompactness
isn't resurrected, i.e., such that after
doing a non-trivial
$\gk_2$-directed closed
forcing, $\gk_2$ remains a
non-\break supercompact
strongly compact cardinal?
3. In general, for $\ga > 2$,
is it consistent,
relative to anything, for the
first $\ga$ strongly compact
cardinals $\gk_\ga$
all to be non-supercompact and
to have their strong compactness
indestructible under (some version
of) $\gk_\ga$-directed closed forcing?
Note that
as the proof of Theorem \ref{t5}
currently stands, the forcing $\FP$
can't be iterated, in the sense that
if we define a similar type of
partial ordering above $\gk_2$,
forcing with it resurrects the
supercompactness of $\gk_2$.
Thus, the techniques used in proving
Theorem \ref{t5} are inadequate for
answering Question 3.
\vfill\eject
\end{slide}
\end{document}
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.
A sketch of the proof of Theorem
\ref{t7} is as follows:
The forward direction is just
Theorem \ref{t1}.
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.
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}
Notice that the preceding
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.
Hamkins' partial ordering $\FP$ of
Theorem \ref{t6} is defined in
a way that is
fundamentally similar
%roughly analogous
to the definition of Laver's
partial ordering of Theorem \ref{t1}.
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{t7}(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}
Remark: Because the forcing used to
prove Theorem \ref{t4} is
a Prikry iteration,
strongly compact''.\footnote{An
outline of the argument is
as follows. Let
$\ov V = V^{\FP_n \times
\cdots \times \FP_{i + 1}}$.
Fix $k : M \to N$ an elementary
embedding generated by a normal
ultrafilter over $\gk_i$
in $N$ having trivial
Mitchell rank, and let
$h : \ov V \to N$ be defined by
$h = k \circ j$. Working in
$\ov V$, $h(\FP_i) =
\FP_i \ast \dot \FQ \ast \dot \FR$,
where the field of $\dot \FQ$
is composed of ordinals in the interval
$(\gk_i, k(\gk_i)]$, and the
field of $\dot \FR$ is composed
of ordinals in the interval
$(k(\gk_i), h(\gk_i))$.
If $G_0$ is $\ov V$-generic over
$\FP_i$, since
$N \models ``\gk_i$ isn't measurable''
and GCH holds at $\gk_i$ in $\ov V$, it is
possible to use the standard diagonalization
techniques to construct in $\ov V[G_0]$ an
$N[G_0]$-generic object $G_1$ over $\FQ$.
Since GCH holds at $\gk_{i + 1}$
in $\ov V$ and no cardinal
$\gd \in [\gk_i, \gk_{i + 1}]$
is in the field of $\dot \FT$,
we can again use the standard
diagonalization techniques to
construct in $\ov V[G_0]$ an $M$-generic
object for the term forcing
partial ordering associated with
$\dot \FT$ defined with respect to
$\FP_i \ast \dot \FS$, transfer it
using $k$, and realize the transferred
generic using $G_0 \ast G_1$ to
obtain an $N[G_0][G_1]$-generic
object $G_2$ for $\FR$.
$h$ then lifts to
$h : \ov V[G_0] \to N[G_0][G_1][G_2]$,
which witnesses the $\gk_{i + 1}$
strong compactness of $\gk_i$ in
$\ov V[G_0]$, meaning that
$\gk_i$ is $\gk_{i + 1}$ strongly
compact in
$V^{\FP_n \times \cdots \times \FP_i}$.}
Let $G$ be $V$-generic over $\FP$. If $V[G]
\models ``\gk$ does not have trivial Mitchell
rank'', then let $j : V[G] \to M[j(G)]$ be an
embedding generated by a normal measure over
$\gk$ in $V[G]$ witnessing this fact. In the
terminology of \cite{H1}, \cite{H2}, and
\cite{H3}, $\FP$ admits a gap below $\gk$, and so
by the Gap Forcing Theorem of \cite{H2} and
\cite{H3}, $j$ must lift an embedding $j : V \to
M$ that is definable in $V$. Since $j(\FP)$ also
admits a gap below $\kappa$ in $M$ and $\kappa$
is measurable in $M[j(G)]$, we similarly conclude
that $\kappa$ is measurable in $M$. Therefore,
since $\kappa$ is a limit of strong cardinals (we
have already noted that any supercompact cardinal
is a limit of strong cardinals), it follows that
$\gk$ is in the field of $j(\FP)$. Thus, there is
nontrivial forcing at stage $\kappa$, and so
$j(G) = G \ast S \ast H$, where $S$ is a
non-reflecting stationary set of ordinals added
by forcing over $M[G]$ with ${(\FP_{\eta,
\gk})}^{M[G]}$ at stage $\kappa$ and $H$ is
$M[G][S]$-generic for the rest of the forcing
$j(\FP)$. Since $V_{\kappa+1}\subseteq M\subseteq
V$, it follows that $V_{\gk + 1}^{V[G]} = V_{\gk
+ 1}^{M[G]}$. From this it follows that
${{\rm Add}(\gk, 1)}^{M[G]} = {(\FP_{\eta,
\gk})}^{V[G]}$, and the dense open subsets of
what we can now unambiguously write as
${\rm Add}(\gk, 1)$ are the same in both $M[G]$ and
$V[G]$. Thus, the set $S$, which is an element of
$V[G]$, is $V[G]$-generic over ${\rm Add}(\gk, 1)$,
a contradiction. This proves Lemma \ref{l2}.