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$\ $ \vskip 1.5in
\centerline{``On the First $n$ Strongly Compact Cardinals''}
\vskip .5in
\centerline{by}
\vskip .5in
\centerline{Arthur W. Apter${}^*$}
\centerline{Department of Mathematics}
\centerline{Baruch College of CUNY}
\centerline{New York, New York 10010}
\vskip 1.75in
\noindent Abstract: Using techniques of Kimchi and Magidor, we
generalize an earlier result and show that it is relatively
consistent for the first $n$ strongly compact cardinals
to be somewhat supercompact yet not fully
supercompact. \hb
\no ${}^*$The research for this paper was partially supported by
PSC-CUNY Grants 661371 and 662341 and by a salary grant from Tel
Aviv University. \hb\vfill\eject
\vskip 1in
\centerline{``On the First $n$ Strongly Compact Cardinals''}
\vskip .5in
\centerline{by}
\vskip .5in
\centerline{Arthur W. Apter}
\centerline{Department of Mathematics}
\centerline{Baruch College of CUNY}
\centerline{New York, New York 10010}
\vskip .75in
\indent The class of strongly compact cardinals is, without a
doubt, one of the most peculiar in the entire theory of large
cardinals. As is well know, the class of strongly compact
cardinals suffers from a severe identity crisis. Magidor's
fundamental result of [Ma] shows that it is consistent for
the least strongly compact cardinal to coincide with either
the least measurable or the least supercompact cardinal. The
result of [A1] shows that it is consistent for the least
strongly compact cardinal to be somewhat supercompact although
not fully supercompact. The result of Kimchi and Magidor [KM]
shows that it is consistent for the class of strongly compact
cardinals to coincide with the class of supercompact cardinals
(except at limit points where Menas' result [Me] shows that
such a coincidence may not occur) or for the first $n$ for
$n \in \omega$ strongly compact cardinals to coincide with the
first $n$ measurable cardinals. \hb
\indent The purpose of this paper is to show that matters can
be muddled still further. Specifically, we generalize the
result of [A1] using the methods of [KM] and prove the
following \hfil\break
\noindent Theorem: Let $V \models `` K = \{\k_1, \ldots,
\k_n\}$ with $\k_1 < \k_2 < \cdots < \k_n$ the first $n$
supercompact cardinals''. For
each $\k\in K$, let $\f_\k$ be a formula in the language of
set theory which defines an increasing $\Sigma_2$ function
from the ordinals to the ordinals which, in additiion, has
the following properties: \hfil\break
\noindent a) $\f_\k$ is preserved at and above $\k$ when
forcing with a cardinal preserving partial ordering of size
$\k$, i.e., for $P \in V$ a cardinal preserving partial
ordering, $\vert P \vert = \k$, $\a \ge \k$, $V \models ``\d
=\f_\k(\a)$'' iff $V^P \models ``\d=\f_\k(\a)$''. \hfil
\vskip .09in
\noindent b) If for any model $M$ of $ZFC$, $M \models ``\a
< \b$ and $\b$ is $\f_\k(\b)$ supercompact'', then
$\f_\k(\a) < \b$. \hfil\vskip .09in
\no c) If for any model $M$ of ZFC, $M \models ``\d$ is
$\f_{\k_i}(\d)$ supercompact, $\d$ is $\f_{\k_j}(\d)$
supercompact, and $i \le j$'', then $\f_{\k_i}(\d)
\ge \f_{\k_j}(\d)$. \hb
\noindent There is then a partial ordering
$P$ so that for each
$\k \in K$, $V^P \models ``\k$ is at least $\f_\k(\k)$
supercompact, $\k$ is not supercompact, and $\k$ is fully
strongly compact''; further, $V^P \models ``K$ consists of the
first $n$ strongly compact cardinals''. \hb
\indent Restrictions a) and b) above on each $\f_\k$ are made
for the same technical reasons as in [A1]. Note, however,
that many $\Sigma_2$ functions, e.g., $\d\mapsto\d^+$,
$\d \mapsto$The least inaccessible $>\d$,
$\d \mapsto$The least measurable $>\d$, etc., meet
restrictions a) and b). Restriction c) is needed to make the
arguments of [KM] go through and will be explained at the end
of the paper. \hb
\indent Our Theorem essentially says that it is possible for
the first $n$ strongly compact cardinals to be somewhat
supercompact yet not fully supercompact. This provides an
intermediate result to the results of [KM], which tell us
that the first $n$ strongly compact cardinals can be either
the first $n$ measurable cardinals or the first $n$
supercompact cardinals. \hb
\indent Before beginning the proof of our Theorem, we briefly
mention some preliminary material. Essentially, our notation
is standard, with $V_\a$ representing the universe through
stage $\a$, and for $\a < \b$ ordinals, $[\a, \b]$,
$[\a, \b)$, $(\a, \b]$, and $(\a, \b)$ as in standard interval
notation. Further, we assume complete familiarity with the
notions of strong compactness, supercompactness,
etc. Papers [A1], [A2], [KM], [L], [Ma], [Me], and [SRK] of
the References will provide sufficient details. Finally, we
will be assuming some familiarity with the techniques and
methods of [KM], although where necessary, appropriate
details will be provided. \hfil\break
\indent We turn now to the proof of our Theorem. \hfil\break
\indent Proof of Theorem: We will first define, for each
$\k \in K$, a partial ordering $P^\k$ so that
$V^{P^\k} \models ``\k$ is at least $\f_\k(\k)$
supercompact, $\k$ is not supercompact, $\k$ is fully
strongly compact, and there are no strongly compact cardinals
in the interval $[\d_\k, \k)$'', where $\d_{\k_1} = \aleph_1$,
and $\d_{\k_i} = \k^+_{i - 1}$ if $2 \le i \le n$. The
partial ordering $P$ then used to construct our final
model will be the product ordering $\prod_{1 \le i \le n}
P^{\k_i}$. \hb
\indent To define $P^\k$, we proceed inductively. $P^\k_0$ is
the trivial partial ordering, and if $\l < \k$ is a limit
ordinal, then $P^\k_\l =$ inverse limit($\langle
P^\k_\a : \a < \l \rangle$) if $\l$ is singular, and
$P^\k_\l =$ direct limit($ \langle P^\k_\a : \a <
\l \rangle$) if $\l$ is regular. To define $P^\k_{\a+1}$, let
$\g_\a < \k$ be the least ordinal so that \break
$\forces_{P^\k_\a} ``\g_\a$ is $\f_\k(\g_\a)$
supercompact''. $\g_\a$ will exist, since $\k$ is supercompact and
$\f_\k$ is $\Sigma_2$. $\dot{Q^\k_\a}$ is then a term for the
partial ordering $Q^\k_\a \in V^{P^\k_\a}$ which adds a
non~-~reflecting stationary set of ordinals of cofinality $\d_\k$
to $\g_\a$; specifically, $\dot{Q^\k_\a}$ is a term for a
non~-~reflecting bounded stationary
subset of $\{\b < \g_\a : \ $cof($\b) = \d_\k \}$,
ordered by $q$ extends $p$ iff $q \supseteq p$ and $q$ is an
end extension of $p$, i.e., $p=q \ \cap \ $sup($p$). $P^\k_{\a+1}$
is then defined as $P^\k_\a * \dot{Q^\k_\a}$, and
$P^\k = $ direct limit($\langle
P^\k_\a : \a < \k \rangle$). \hfil\break
\noindent Lemma 1: $V^{P^\k} \models ``\k$ is strongly
compact and there are no strongly compact cardinals in the
interval $[\d_\k, \k)$''. \hfil\break
\indent Proof of Lemma 1: As in [KM], forcing with $P^\k$
or any of its component partial orderings will create no
new measurable cardinals; in fact, the arguments of [KM] show
that forcing with $P^\k$ or any of its component partial
orderings will create no new cardinals $\d$ which are
$\f_\k(\d)$ supercompact. Thus, since the cardinals $\d$ in
the field of $P^\k$ are all forced by some component of
$P^\k$ to be $\f_\k(\d)$ supercompact, each cardinal $\d$ in
the field of $P^\k$ will be $\f_\k(\d)$ supercompact in
$V$. Thus, since restriction c) of the Theorem tells us that
for $i \le j$ and for any model $M$ of ZFC, if
$M \models ``\d$ is both $\f_{\k_i}(\d)$ and $\f_{\k_j}(\d)$
supercompact'', then $\f_{\k_i}(\d) \ge \f_{\k_j}(\d)$, the
arguments of [KM] show that $V^{P^\k} \models ``\k$ is
strongly compact''. (We will comment on this further at the
end of the paper.) \hb
\indent To show that $V^{P^\k} \models ``$There are no strongly
compact cardinals in the interval $[\d_\k, \k)$'', it suffices
to show that for unboundedly many $\g \in [\d_\k, \k)$,
$V^{P^\k} \models ``$There is a non~-~reflecting stationary
subset of $\g$ of ordinals of cofinality $\d_\k$. (This is
since a theorem of [SRK] states that if $\b$ contains a
non~-~reflecting stationary subset of ordinals of cofinality
$\a$, then there are no strongly compact cardinals in the
interval $(\a, \b)$. Here $\d_\k$, being a successor cardinal,
of course isn't strongly compact.) To see this, let $\g$ be
a cardinal in the field of $P^\k$, and let $P^\k =
Q_\g * \dot{Q^\g}$, where the field of $Q_\g$ consists of
all cardinals $\le \g$, and the field of $\dot{Q^\g}$
consists of all cardinals $> \g$. By the definition of
$P^\k$, $V^{Q_\g} \models ``\g$ contains a non~-~reflecting
stationary subset of ordinals of cofinality $\d_\k$''. The
definition of $P^\k$ ensures that for any $\a$ so that
$P^\k = P^\k_\a * \dot{Q^\k_\a}$ and $\forces_{P^\k_\a} ``$The
field of $\dot{Q^\k_\a}$ is a $V$ measurable cardinal
$> \g$'', $\forces_{P^\k_\a} ``\dot{Q^\k_\a}$ is
$< \sigma_\g$ strategically closed'' where $\sigma_\g$
is the least (measurable) cardinal in the field of
$\dot{Q^\g}$ and $< \sigma_\g$ strategically closed for a
partial ordering $Q$ means that for any fixed $\b < \sigma_\g$,
in the two person game in which the players construct a
sequence $\langle q_\a : \a < \b \rangle$ so that each
$q_{\a_0}$ extends $q_\a$ for $\a < \a_0$ and player I
plays odd stages and player II plays even and limit stages,
player II always has a winning strategy ensuring that the
game can be continued for any $\a < \b$. Since each cardinal
$\a$ in the field of $\dot{Q^\g}$ is a $V$ measurable cardinal
so that $\a \ge \sigma_\g > \g$, the definition of
$\dot{Q^\g}$ ensures that $\forces_{Q_\g} ``\dot{Q^\g}$ is
$< \sigma_\g$ directed closed'', i.e., that
$\forces_{Q_\g} ``$Forcing with $\dot{Q^\g}$ adds no new
subsets of $\g$''. This means that $V^{Q_\g * \dot{Q^\g}} =
V^{P^\k} \models ``\g$ contains a non~-~reflecting
stationary subset of ordinals of cofinality $\d_\k$''. Since
there are unboundedly many such $\g \in [\d_\k, \k)$ in the
field of $P^\k$, Lemma 1 is proven. \hfil\break
\finpf Lemma 1 \hfil\break
\noindent Lemma 2: $V^{P^\k} \models ``\k$ is not
supercompact''. \hfil\break
\indent Proof of Lemma 2: If $V^{P^\k} \models ``\k$ is
supercompact'', let $j$ be a supercompact embedding of
$V^{P^\k}$ into a sufficiently closed inner model $M$
with critical point $\k$ so that $M \models ``\k$ is
$\f_\k(\k)$ supercompact''. By reflection, $A = \{\g <
\k : \g$ is $\f_\k(\g)$ supercompact$\}$ is unbounded in
$\k$. It is also the case that $M \models ``\k$ is a
stage in the definition of $j(P^\k)$ at which a (direct)
limit is taken'', so reflection again allows us to assume
that for any $\g \in A$, $P^\k_\g =$ (direct) limit($
\langle P^\k_\a : \a < \g \rangle$). By the definition of
$P^\k$, $P^\k_{\g + 1} = P^\k_\g * \dot{Q^\k_\g}$ is so
that $\forces_{P^\k_{\g + 1}} ``\g$ contains a
non~-~reflecting stationary subset of ordinals of cofinality
$\d_\k$'', so the proof of Lemma 1 tells us that this fact
is also true in $V^{P^\k}$, i.e., that $V^{P^\k} \models ``\g$
is not $\f_\k(\g)$ supercompact''; in fact, $V^{P^\k} \models
``\g$ isn't weakly compact''. This contradiction proves
Lemma 2. \hfil\break\finpf Lemma 2 \hfil\break
\noindent Lemma 3: $V^{P^\k} \models ``\k$ is $\f_\k(\k)$
supercompact''. \hfil\break
\indent Proof of Lemma 3: Let $j:V \to M$ be a supercompact
embedding with critical point $\k$ so that $M$ is at least
$2^{{[\f_\k(\k)]}^{< \k}}$ closed and so that
${\f_\k(\k)}^M = {\f_\k(\k)}^V$. Property a) of
$\f_\k$ of the
hypotheses of the Theorem shows that $\f_\k(\k)$ has the
same meaning in either $V^{P^\k}$ or $M^{P^\k}$ as it did
in $V$ or $M$. We can therefore write $\f_\k(\k)$
unambiguously. \hfil\break
\indent If $V^{P^\k} \models ``\k$ is $\f_\k(\k)$
supercompact'', then we are done, so assume this is not the
case, i.e., $V^{P^\k} \models ``\k$ is not $\f_\k(\k)$
supercompact''. Work now in $M$. $P^\k$ is an initial
segment of $j(P^\k)$, and by the closure properties of
$M$, $M^{P^\k} \models ``\k$ is not $\f_\k(\k)$
supercompact''. Writing $j(P^\k)$ as $P^\k * \dot{Q}$,
property b) of $\f_\k$ of the hypotheses of the Theorem
allows us to conclude that for the least cardinal $\sigma_\k$
in the field of $\dot{Q}$, $M^{P^\k} \models
``\f_\k(\k) < \sigma_\k$''. \hfil\break
\indent If $G$ is $V$~-~generic over $P^\k$ and $H$ is
$M[G]$~-~generic over $Q$, define an embedding \break
$j^*:V[G] \to M[G*H]$ by $j^*(i_G(\tau)) = i_{G*H}(j(\tau))$
for any term $\tau$ denoting a set in $V[G]$. Since
$\f_\k(\k) < \sigma_\k$, the closure properties of $M$
and the Kunen~-~Paris arguments of [KP] show that $j^*$ is
a well defined elementary embedding with critical point $\k$
so that the ultrafilter ${\cal U}$ defined by
$x \in {\cal U}$ iff $\langle j(\a) : \a < \f_\k(\k) \rangle
\in j^*(x)$ is a supercompact ultrafilter over
${P_\k(\f_\k(\k))}^{V[G]}$ present in $V[G*H]$. In $M[G]$,
since $\f_\k(\k) < \sigma_\k$ and $\sigma_\k$ is inaccessible,
$2^{{[\f_\k(\k)]^{< \k}}} < \sigma_\k$, meaning that $Q$
will be so that forcing with $Q$ over $M[G]$ adds no new
subsets of $2^{{[\f_\k(\k)]^{< \k}}}$. The closure
properties of $M$ ensure that this property is true as well
when forcing with $Q$ over $V[G]$, i.e., that
${\cal U} \in V[G]$. Thus, assuming $\forces_{P^\k} ``\k$ is
not $\f_\k(\k)$ supercompact'' leads to the contradiction
$\forces_{P^\k} ``\k$ is $\f_\k(\k)$ supercompact'' as well,
meaning it must be the case that $\forces_{P^\k} ``\k$ is
$\f_\k(\k)$ supercompact''. This proves Lemma 3. \hb
\finpf Lemma 3 \hfil\break
\noindent Lemma 4: $V^P \models ``$If $\k \in K$, then $\k$
is strongly compact, at least $\f_\k(\k)$ supercompact, but
not fully supercompact''. \hfil\break
\indent Proof of Lemma 4: For each $\k \in K$, write
$P = Q_\k \times Q^\k$, where $Q_\k = \prod_{\{ \l \in K
: \l \le \k \}} P^\l$ and $Q^\k = \prod_{\{\l \in K : \l
> \k \}} P^\l$; further,
write $Q_\k$ as $Q_{< \k} \times P^\k$ for $Q_{< \k} =
\prod_{\{\l \in K :\l < \k\}} P^\l$. It is a
fundamental result of both [A2] and [KM] that we can assume
that a preliminary
forcing has been done to ensure that $V \models ``$If $\k$
is a supercompact cardinal, then $\k$ is Laver [L]
indestructible''. Thus, since each component partial ordering
$P^\l$ of $Q^\k$ has been defined so as to be at least
$\k$~-~directed closed, $Q^\k$ is $\k$~-~directed closed,
so $V^{Q^\k} \models ``\k$ is a supercompact
cardinal''. As the subsets of $\k$ are the same in either
$V$ or $V^{Q^\k}$, the definition of $P^\k$ is the same in
either $V$ or $V^{Q^\k}$, so by Lemmas 1~-~3,
$V^{Q^\k \times P^\k} \models ``\k$ is strongly compact, at
least $\f_\k(\k)$ supercompact, but not fully
supercompact''. Since $K$ is a finite set, the
fact that forcing with $P^\k$ over either $V$ or $V^{Q^\k}$
adds no new bounded subsets to $\k$ ensures that
$V^{Q^\k \times P^\k} \models ``\vert Q_{< \k} \vert < \k$'',
so the L\'evy~-~Solovay results [LS] ensure that
$V^{\bigp} = V^P \models ``\k$ is strongly compact, at least
$\f_\k(\k)$ supercompact, but not fully supercompact''. This
proves Lemma 4. \hfil\break
\finpf Lemma 4 \hfil\break
\noindent Lemma 5: $V^P \models ``K$ consists of the first
$n$ strongly compact cardinals''. \hb
\indent Proof of Lemma 5: Assume that $V^P \models ``\g$ is
the $i^{\rm th}$ strongly compact cardinal for $1 \le i \le n$
and isn't an element of $K$''. As Lemma 4 tells us each
$\k \in K$ remains strongly compact in $V^P$, it must be the
case that $\g < \k_n$. Thus, if $\k$ is the least
element of $K > \g$, then using the same notation as
before, $\g \in [\d_\k, \k)$. Writing $P = \bigp$ as in
the preceding lemma, the closure properties of $Q^\k$
already noted and Lemma 1 show that $V^{Q^\k \times P^\k}
\models ``\g$ is not strongly compact'', so the arguments
of [LS] show that $V^{\bigp} = V^P \models ``\g$ is not
strongly compact''. This proves Lemma 5. \hfil\break
\finpf Lemma 5 \hfil\break
\indent Lemmas 1~-~5 complete the proof of our Theorem.
\hfil\break
\finpf Theorem \hfil\break
\indent We note that each $\f_\k$ may imply additional
assumptions on $\k$, e.g., if $\f_\k$ is the function
$\d \mapsto \ $The least inaccessible cardinal $>\d$,
then there is assumed to be (if it can't already be
shown to exist) an inaccessible cardinal $> \k$. Also,
in general, for the same reasons as in [A1], $\f_\k$
can't be assumed to be $\Sigma_3$. Further, as in [A1],
under certain circumstances we can get a precise bound
on the non~-~supercompactness of some $\k \in K$. If,
for instance, $\f_\k$ is the function $\d \mapsto\d^+$,
then the argument of [L] or [KM] allows us to assume
without loss of generality that $2^\k = \k^+$ and
$2^{\k^+} = \k^{++}$. After applying the arguments of
Lemmas 2~-~4, $\k$ will be $\k^+$ supercompact but not
$\k^{++}$ supercompact. (Even if we have no knowledge of
the size of power sets of cardinals, for many $\f_\k$
the preceding arguments show that $\k$ isn't
$2^{{[\f_\k(\k)]^{< \k}}}$ supercompact.) \hb
We remark that in the initial version of this paper, our
Theorem was stated for a class of strongly compact cardinals
and not just for a finite set. Unfortunately, as was pointed
out by the referee, there is a gap in the original
Kimchi~-~Magidor proof so that it is now only known how, from
the existence of $n \in \omega$ supercompact cardinals, to
force and obtain the consistency of the coincidence of the
first $n$ strongly compact cardinals with the first $n$
measurable cardinals. To outline the original
Kimchi~-~Magidor argument and highlight the problem, let
$K = \{ \k_1, \ldots, \k_n \}$ be as in the statement of our
Theorem, and for each $\k \in K$, let $A_\k \subseteq \k
- \{ \d \in K : \d < \k \}$ be a set of measurable
cardinals. Let $P^\k$ be defined as before, only this time
adding non~-~reflecting stationary sets to the elements of
$A_\k$, and let again $P = \prod_{1 \le i \le n}
P^{\k_i}$. \hb
To show $V^P \models ``\k$ is strongly compact for $\k \in
K$'', as in Lemma 4, it will suffice to show $V^{P^\k} \models
``\k$ is strongly compact''. To do this, let $j:V \to M$ be
an elementary embedding witnessing $\k$ is $\l$ strongly
compact. If $G$ is $V$-generic over $P^\k$, then as usual,
we'd like to find $H \subseteq j(P^\k)$, $H \in V[G]$ so
that $G*H$ is $M$-generic over $j(P^\k)$ and so that $j$
extends to a strongly compact embedding $j' : V[G] \to
M[G * H]$. If $\l$ is sufficiently large, then if $j$ is
a $\l$ supercompact embedding and $\k_{i+1}, \ldots,
\k_n \in j(A_\k)$ (as would certainly be the case in the
original Kimchi~-~Magidor situation, i.e., when all ground model
measurable cardinals are being destroyed), by elementariness,
we would necessarily have $M[G * H] \models ``\k_{i+1},
\ldots, \k_n$ are no longer measurable as they contain
non~-~reflecting stationary sets''. By the closure properties of
$M$, this would also have to be true in $V[G * H']$ for
$H'$ some ``initial segment'' of $H$, and we'd actually have to
extend $V$ by $G * H'$ and have $j'$ be so that
$j' : V[G * H'] \to M[G * H]$. This would mean that $\k_{i+1},
\ldots, \k_n$ could no longer be strongly compact in
$V[G * H']$. \hb
To avoid this difficulty, Kimchi and Magidor construct a
strongly compact embedding $j : V \to M$ so that $M \models
``\k_{i+1}, \ldots, \k_n$ are non-measurable''. They do
this inductively as follows. Let $\l$ be sufficiently large,
and let $j_i : V \to M_{i+1}$ be an embedding witnessing the
$\l$ supercompactness of $\k$ so that $\k_{i+1}, \ldots,
\k_n \in j_i(A_\k)$. If ${\cal U}_{i+1} \in M_{i+1}$ is a
normal measure over $\k_{i+1}$ so that $M_{i+2}$ =
transitive collapse($M_{i+1}^{\k_{i+1}} / {\cal U}_{i+1}$)
$\models `` \k_{i+1}$ is non~-~measurable'' (${\cal U}_{i+1}$
exists since $j_i(A_\k) \in M_{i+1}$ is a set of measurable
cardinals), then let $j_{i+1} : M_{i+1} \to M_{i+2}$ be the
associated elementary embedding. Note that as $j_{i+1}
(\k_l) = \k_l$ for $i+2 \le l \le n$, $M_{i+2} \models ``
\k_{i+1}$ is non~-~measurable and $j_{i+1}(\k_l) = \k_l$ is
measurable for $i+2 \le l \le n$''. We can thus inductively for
$i+2 \le l \le n$ find a normal measure ${\cal U}_l \in M_l$
so that $M_{l+1}$ = transitive collapse($M_l^{\k_l} /
{\cal U}_l) \models ``\k_{i+1}, \ldots, \k_l$ are non~-~measurable
and $\k_{l+1}, \ldots, \k_n$ are measurable''. This is since
the associated elementary embedding $j_l : M_l \to M_{l+1}$ is
so that $j_l(\k_m) = \k_m$ for $i+1 \le m < l$ as $\k_l$ is
the critical point of $j_l$, inductively, $M_l \models
``\k_{i+1}, \ldots, \k_{l-1}$ are non~-~measurable'', and
${\cal U}_l$ has been chosen so that $M_{l+1} \models
``\k_l$ is non~-~measurable''; also, as above, $j_l(\k_m) =
\k_m$ for $l+1 \le m \le n$, and we can inductively assume
$M_l \models ``\k_{l+1}, \ldots, \k_n$ are
measurable''. If
we now define $j = j_n \circ j_{n-1} \circ \cdots
\circ j_{i+1} \circ j_i$ and $M = M_{n+1}$, then it can be
verified that $j:V \to M$ is an elementary embedding witnessing
the $\l$ strong compactness of $\k$. By construction,
$M \models ``\k_{i+1}, \ldots, \k_n$ are non~-~measurable'', so
there is therefore no problem (as the forcing creates no new
measurable cardinals) if $G$ is $V$-generic over $P^\k$ in
finding $H \subseteq j(P^\k)$, $H \in V[G]$ so that $j$
extends to $j' : V[G] \to M[G * H]$ which is a strongly compact
embedding. \hb
Unfortunately, however, the above iteration breaks down at stage
$\omega$, as we have to consider new $\omega$ sequences, and
our control over $j(P^\k)$ may be lost. To this point in
time, no way is yet known around this difficulty, and the
Kimchi~-~Magidor method is applicable only in situations
dealing with a finite number of cardinals. In particular,
in our situation, we of course do not destroy all measurable
cardinals, only those cardinals $\d$ forced to be
$\f_\k(\d)$ supercompact, which as we have previously
observed, are those cardinals $\d$ actually
$\f_\k(\d)$ supercompact in $V$. The argument we use to show
$\k$
remains strongly compact is then essentially the same as the
one just given. The only real difference is that in order to
preserve the strong compactness and $\f_\k(\k_l)$ supercompactness
of $\k_l$ for $l = i+1, \ldots, n$, we choose for $i+1 \le
l \le n$ the measures ${\cal U}_l$ to be supercompact measures
over the appropriate version of $P_{\k_l}(\f_\k(\k_l))$ with
corresponding elementary embeddings $j_l : M_l \to M_{l+1}$
so that $M_{l+1} \models ``\k_m$ for $i+1 \le m \le l$
isn't $\f_\k(\k_m)$ supercompact and $\k_m$ for $l+1 \le
m \le n$ is $\f_\k(\k_m)$ supercompact'' and then define $j$
as before. Unfortunately, this doesn't eliminate the difficulty
at stage $\omega$, and we can still only handle a finite
number of cardinals. \hb
The above paragraph illustrates why restriction c) of the
statement of the Theorem is necessary, i.e., why we must have
$\f_{\k_i}(\d) \ge \f_{\k_j}(\d)$ if $i \le j$ and $\d$ is
both $\f_{\k_i}(\d)$ and $\f_{\k_j}(\d)$ supercompact in
$V$. If, e.g., $\f_{\k_i}$ were the function $\d \mapsto
\d^+$, $\f_{\k_{i+1}}$ were the function $\d \mapsto
\d^{++}$, and $V \models ``2^{\k_{i+1}^+} = \k_{i+1}^{++}$'',
then in $V^{Q_{\k_i}}$ (we use the notation of Lemma 4) we
would have unboundedly many cardinals $\d$ in the interval
$(\k_i, \k_{i+1})$ which were $\d^+$ supercompact. As was
just noted, when forcing with $P^{\k_i}$, we can only currently
preserve the $\d^+$ supercompactness of finitely many such
$\d$ if $\k_i$ is to remain strongly compact. If we destroy
the $\d^+$ supercompactness of all but finitely many of these
$\d$, then $\k_{i+1}$ will no longer be $\k_{i+1}^{++}$
supercompact. Of course, if $\f_{\k_i}$ were the function
$\d \mapsto \d^{++}$, $\f_{\k_{i+1}}$ were the function
$\d \mapsto \d^+$, and $V \models ``2^{\k_{i+1}^+} =
\k_{i+1}^{++}$'', then in $V^{Q_{\k_i}}$, there would be no
cardinals $\d$ which were $\d^+$ or $\d^{++}$ supercompact in
the interval $(\k_i, \k_{i+1})$, thereby eliminating this
difficulty. \hb
As the referee has pointed out, there is an interesting
comparison the reader can make between the result of this
paper and the result of [KM]. Although the arguments we give
in general follow the proof of the result that from $n \in
\omega$ supercompact cardinals, it is possible to force and
obtain the coincidence of the first $n$ strongly compact
cardinals with the first $n$ measurable cardinals, the forcing
notion itself is similar to the one used in the proof of the
result that from a class of supercompact cardinals, it is
possible to force and obtain the coincidence of the classes of
strongly compact and supercompact cardinals (except at limit
points). \hb
In conclusion, we would like to contrast the result of this
paper and the corresponding result of [KM] with the result of
[AS], which says that from $n \in \omega$ supercompact cardinals,
it is possible to force so that not only do the first $n$
strongly compact and measurable cardinals coincide, but each such
cardinal $\k$ is also $\k^+$ supercompact. Of course, under
these circumstances, GCH must fail rather badly, and in general,
since each $\f_\k$ can define a cardinal so that $\f_\k(\d)
\ge 2^\d$, in the present situation, a coincidence between the
first $n$ strongly compact and measurable cardinals may be
impossible. (We can have GCH hold in the result of this
paper if we start with a model for GCH and the preliminary
Laver indestructibility has been done so as to make the $n$
supercompacts indestructible only under forcings which do not
destroy GCH, since adding non~-~reflecting stationary subsets
to cardinals won't destroy GCH.) For further details, the
interested reader is urged to consult [AS]. \hb
\no Acknowledgement: The author wishes to acknowledge helpful
conversations with Yechiel Kimchi, Menachem Magidor, and
Saharon Shelah on the subject matter of this paper and on
related topics. In addition, the author wishes to acknowledge
gratefully the referee for pointing out the gap
in the proofs of both the original version of this paper and
the corresponding result of [KM] and for making a number of
extremely useful comments. Many of the referee's remarks have
been incorporated almost verbatim into the concluding comments
above.
\hfil\break\vfill\eject
\frenchspacing\vskip 1in
\centerline{References}
\vskip 1.25in
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\item{[A2]} A. Apter, ``Some Results on Consecutive Large
Cardinals'', {\it Annals of Pure and Applied Logic 25},
1983, 1-17. \hb
\item{[AS]} A. Apter, S. Shelah, ``On Strongly Compact versus
Supercompact and Measurable'', in preparation. \hb
\item{[KM]} Y. Kimchi, M. Magidor, ``The Independence
Between the Concepts of Strong Compactness and
Supercompactness'', in preparation. \hfil\break
\item{[KP]} K. Kunen, J. Paris, ``Boolean Extensions and
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\item{[L]} R. Laver, ``Making the Supercompactness of $\k$
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\item{[LS]} A. L\'evy, R. Solovay, ``Measurable Cardinals and
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\item{[Ma]} M. Magidor, ``How Large is the First Strongly
Compact Cardinal?'', {\it Annals of Math. Logic 10}, 1976,
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\item{[Me]} T. Menas, ``On Strong Compactness and
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\item{[SRK]} R. Solovay, W. Reinhardt, A. Kanamori,
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{\it Annals of Math. Logic 13}, 1978, 73-116. \hfil\break
\vfill\eject\end