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\title{Normal Measures and Strongly Compact Cardinals
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal, %strong cardinal,
$(\gk + 2)$-strong cardinal $\gk$,
indestructibility,
Magidor iteration of Prikry forcing.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants and a Visiting Fellowship
at the Isaac Newton Institute.}
\thanks{The author wishes to thank James Cummings and
Menachem Magidor for helpful
conversations on the subject matter of this paper.}
\thanks{This paper was written
while the author was a Visiting Fellow
at the Isaac Newton Institute for Mathematical Sciences in the programme
``Mathematical, Foundational and Computational Aspects of the
Higher Infinite (HIF)'' held from 19 August 2015 until 18 December 2015
and funded by EPSRC grant EP/K032208/1.
The author's participation %in this program
would not have been possible
without the generous support of Dean Jeffrey Peck of Baruch College's Weissman
School of Arts and Sciences and Professor Warren Gordon, Chair of the Baruch
College Mathematics Department, both of whom the author thanks.}
\thanks{The author wishes to thank the referee for helpful comments and
suggestions which have been incorporated into the current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{November 22, 2015}
%\date{November 25, 2015}
%\date{November 29, 2015}
%\date{November 30, 2015}
\date{December 1, 2015\\
(revised July 16, 2016 and June 18, 2017)}
%\date{\today}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
We prove four theorems concerning the number of normal measures
a non-$(\gk + 2)$-strong strongly compact cardinal $\gk$ can carry.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We consider in this paper the number of normal measures
a non-($\gk + 2$)-strong strongly compact cardinal $\gk$ can carry.
It follows from a theorem of Solovay \cite[Corollary 20.20(i)]{Je} that if $\gk$ is
($\gk + 2$)-strong, then $\gk$ is a measurable
limit of measurable cardinals with $2^{2^\gk}$ many normal measures over $\gk$,
the maximal number of normal measures a measurable cardinal can have.
It is known, however, that there can be strongly compact cardinals $\gk$ which
are not $(\gk + 2)$-strong. A result of Menas \cite[Theorem 2.21]{Me74}
shows that if $\gk$ is a measurable limit of strongly compact cardinals
(which might or might not also be supercompact), then $\gk$ itself must be
strongly compact. By the arguments of \cite[Theorem 2.22]{Me74},
the smallest such $\gk$ cannot be $(\gk + 2)$-strong.
In addition, Magidor's famous theorem of \cite{Ma76} establishes that
it is consistent, relative to the existence of a strongly compact cardinal, for
the least strongly compact cardinal $\gk$ to be the least measurable cardinal.
Under these circumstances, by the previously mentioned theorem of Solovay,
$\gk$ also cannot be $(\gk + 2)$-strong.
The work of Menas and Magidor therefore raises the following \bigskip
\noindent Question: Suppose $\gk$ is a strongly
compact cardinal which is not $(\gk + 2)$-strong.
How many normal measures is it
consistent for $\gk$ to carry?
\bigskip
In trying to provide answers to this question,
we will begin by
examining what occurs when the strongly compact cardinals being considered are
either the least measurable limit of supercompact cardinals or the least measurable cardinal.
%the non-($\gk + 2$)-strong strongly compact cardinals given by the aforementioned
%examples of Menas and Magidor.
Specifically, we will prove the following two theorems.
\begin{theorem}\label{t1}
Suppose $V$ is a model of ``ZFC + GCH'' in which
$\gk$ is the least measurable limit of supercompact cardinals
and $\gl \ge \gk^{++}$ is a regular cardinal. There is
then a partial ordering $\FP \subseteq V$ such that
$V^\FP$ is a model of ZFC in which the following hold:
\begin{itemize}
\item $\gk$ is the least measurable limit of supercompact cardinals.
\item $2^\gk = \gk^+$.
\item $2^{\gk^+} = 2^{2^\gk} = \gl$.
\item $\gk$ carries $2^{2^{\gk}}$ many normal measures.
\end{itemize}
%Suppose $V \models ``$ZFC + GCH + $\gk$ is
%the least measurable limit of supercompact cardinals +
%$\gl \ge \gk^{++}$ is a regular cardinal''. There is then a partial ordering
%$\FP \subseteq V$ such that
%$V^\FP \models ``$ZFC + $\gk$ is the least measurable limit of
%supercompact cardinals +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
%$\gk$ carries $2^{2^\gk}$ many normal measures''.
%Con(ZFC + GCH +
%$\gk$ is the least measurable limit of supercompact cardinals +
%\break $\gl \ge \gk^{++}$ is a regular cardinal) $\implies$
%Con(ZFC + $\gk$ is the least measurable limit of supercompact cardinals +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
%$\gk$ carries $2^{2^\gk}$ many normal measures).
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V$ is a model of ``ZFC + GCH'' in which
$\gk$ is the least supercompact cardinal
and $\gl \ge \gk^{++}$ is a regular cardinal. There is
then a partial ordering $\FP \subseteq V$ such that
$V^\FP$ is a model of ZFC in which the following hold:
\begin{itemize}
\item $\gk$ is both the least measurable and least strongly compact cardinal.
\item $2^\gk = \gk^+$.
\item $2^{\gk^+} = 2^{2^\gk} = \gl$.
\item $\gk$ carries $2^{2^{\gk}}$ many normal measures.
\end{itemize}
%Suppose $V \models ``$ZFC + GCH +
%$\gk$ is supercompact + $\gl \ge \gk^{++}$ is a regular cardinal''.
%There is then a partial ordering $\FP \subseteq V$ such that
%$V^\FP \models ``$ZFC + $\gk$ is both the least measurable and least strongly compact
%cardinal +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
%$\gk$ carries $2^{2^\gk}$ many normal measures''.
%Con(ZFC + GCH +
%$\gk$ is supercompact + $\gl \ge \gk^{++}$ is a regular cardinal)
%$\implies$
%Con(ZFC + $\gk$ is both the least measurable and least strongly compact
%cardinal +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
%$\gk$ carries $2^{2^\gk}$ many normal measures).
\end{theorem}
%Both of the above theorems concern what happens when
%for the strongly compact cardinal $\gk$ under examination,
%$2^\gk = \gk^+$. We may also consider what happens when
%$2^\gk > \gk^+$, Under these circumstances, we have the
%following two theorems.
Theorems \ref{t1} and \ref{t2} handle the case where
the non-$(\gk + 2)$-strong
strongly compact cardinal $\gk$ in question carries
$2^{2^\gk}$ many normal measures and $2^\gk = \gk^+$.
We may also ask if it is possible to have $2^\gk > \gk^+$.
The next two theorems take care of this situation for certain
non-$(\gk + 2)$-strong strongly compact cardinals $\gk$.
Specifically, we will also prove the following two theorems.
\begin{theorem}\label{t3}
Suppose $V$ is a model of ZFC in which there is a
supercompact limit of supercompact cardinals.
There is then a model of ZFC containing a strongly compact
cardinal $\gk$ such that $\gk$ satisfies the following properties:
\begin{itemize}
\item $\gk$ is a measurable limit of strongly compact cardinals
but is not the least measurable limit of strongly compact cardinals.
%\item $\gk$ is not the least measurable limit of strongly compact cardinals.
\item $\gk$ is not $(\gk + 2)$-strong.
\item $2^\gk = \gk^{+ 17}$.
\item $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$.
\item $\gk$ carries $2^{2^\gk}$ many normal measures.
\end{itemize}
%Con(ZFC + There is a supercompact limit of supercompact cardinals)
%$\implies$
%Con(ZFC + There is a strongly compact cardinal
%measurable limit of supercompact cardinals
%$\gk$ which is not the least measurable
%limit of supercompact cardinals but is both a measurable limit of
%supercompact cardinals and is
%not $(\gk + 2)$-strong + $2^\gk = \gk^{+ 17}$ +
%$2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$ + $\gk$ carries
%$2^{2^\gk}$ many normal measures).
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V$ is a model of ZFC in which there is a
supercompact limit of supercompact cardinals.
There is then a model of ZFC containing a strongly compact
cardinal $\gk$ such that $\gk$ satisfies the following properties:
\begin{itemize}
\item $\gk$ is both the least strongly compact cardinal
and a measurable limit of strongly compact cardinals.
%\item $\gk$ is not the least measurable limit of strongly compact cardinals.
\item $\gk$ is not $(\gk + 2)$-strong.
\item $2^\gk = \gk^{+ 17}$.
\item $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$.
\item $\gk$ carries $2^{2^\gk}$ many normal measures.
\end{itemize}
%Con(ZFC + There is a supercompact limit of supercompact cardinals)
%$\implies$
%Con(ZFC + The least strongly compact cardinal
%$\gk$ is a limit of measurable cardinals but is not $(\gk + 2)$-strong + $2^\gk = \gk^{+ 17}$ +
%$2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$ + $\gk$ carries
%$2^{2^\gk}$ many normal measures).
\end{theorem}
In Theorems \ref{t3} and \ref{t4}, there is nothing special about
``$\gk^{+ 17}$'' and ``$\gk^{+ 95}$''.
The values for both $2^\gk$ and $2^{2^\gk}$ can be produced by
an Easton function $F$
as in \cite[Theorem, Section 18, pages 83--88]{Me76}. %, i.e.,
%any values of $2^\gk$ and $2^{2^\gk}$
%defined only at inaccessible cardinals which are given by any function $F$ having domain
In particular, in order to be able both to preserve all
supercompact cardinals and control the size of each of
$2^\gd$ and $2^{2^\gd}$ for $\gd$ an inaccessible
cardinal, we require $F$ to have the following properties:
\begin{itemize}
\item $F$'s domain is the class of regular cardinals.
\item $F$ is definable over $V$ by a $\Delta_2$ formula.
\item For all regular cardinals $\gd_1 \le \gd_2$, $F(\gd_1) \le F(\gd_2)$.
\item For every regular cardinal $\gd$, %$\gd \in \dom(F)$,
${\rm cof}(F(\gd)) > \gd$. In fact:
\item For every regular cardinal $\gd$, %$\gd \in \dom(F)$,
$F(\gd)$ is regular (something not required in \cite{Me76},
but necessary for our purposes, since we need to be able to
control the value of $2^{F(\gd)}$).
\end{itemize}
%domain the class of regular cardinals, be
%definable over $V$ by a $\Delta_2$ formula, be such that
%$\gd_1 \le \gd_2$ implies that $F(\gd_1) \le F(\gd_2)$ and
%${\rm cof}(F(\gd)) > \gd$, %so long as $F$ has the %together with the
%and also have the
%additional property that for any (regular) cardinal $\gd$, $F(\gd)$ is a regular cardinal.
%both $2^\gk$ and $2^{2^\gk}$ are also regular cardinals.
%They stand for any
%``reasonably definable cardinals appropriate for reverse Easton
%iterations'' (i.e., values of an Easton function
%defined only at inaccessible cardinals
%defined as in \cite[Theorem, Section 18, pages 83--88]{Me76}).
\noindent However, for comprehensibility and ease of presentation,
Theorems \ref{t3} and \ref{t4} have been stated as written.
Also, Theorems \ref{t3} and \ref{t4} are in some ways
``weaker'' than Theorems \ref{t1} and \ref{t2}.
This is in the sense that in Theorem \ref{t3}, unlike in
Theorem \ref{t1}, $\gk$ is not the least measurable
limit of supercompact cardinals.
In Theorem \ref{t4}, unlike in Theorem \ref{t2},
$\gk$ is not the least measurable cardinal.
We will discuss this further towards the end of the paper.
Before beginning the proofs of our theorems, we
briefly discuss some preliminary information. Essentially, our
notation and terminology are standard.
When exceptions occur, these will be clearly noted.
In particular, when forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}. If $\FP$ is a notion of forcing for the ground
model $V$ and $G$ is $V$-generic over $\FP$,
then we will abuse notation somewhat by using both $V[G]$ and $V^\FP$
to denote the generic extension when forcing with $\FP$.
We will also occasionally abuse notation by writing $x$ when we
actually mean $\dot x$ or $\check x$.
Suppose $\gk$ is a regular cardinal.
%The partial ordering $\add(\gk, \gl)$ is defined as
As in \cite{L}, we will say that $\FP$ is {\em $\gk$-directed closed}
if every directed subset of $\FP$ of size less than $\gk$ has
an upper bound. %a common extension.
For $\gl$ any ordinal, the standard partial ordering
for adding $\gl$ many Cohen subsets of $\gk$ will be written as
$\add(\gk, \gl)$. It is defined as $\{f \mid f : \gk \times \gl \to \{0, 1\}$
is a function such that
$\card{\dom(f)} < \gk\}$, ordered by
$q \ge p$ iff $q \supseteq p$.
%the standard partial ordering for adding $\gl$ many Cohen subsets of $\gk$,
Note that $\add(\gk, \gl)$ is $\gk$-directed closed.
%Define $\add(\gk, \gl)$ and state its (directed closure) properties.
%Definition of strong cardinal, abuse notation with $V[G]$ and $V^\FP$.
We presume a basic knowledge and understanding of large cardinals
and forcing, as found in, e.g., \cite{Je}
(see also \cite{Ka} for additional material on
large cardinals), to
which we refer readers for further details. We do mention that the cardinal
%$\gk$ is {\em $\gl$-strong} for any ordinal $\gl$ if there is an elementary
%embedding $j : V \to M$ having critical point $\gk$ such that $V_\gl \subseteq M$.
%In particular,
$\gk$ is {\em $\gl$-strongly compact} for $\gl \ge \gk$ a cardinal if
$P_\gk(\gl) = \{x \subseteq \gl \mid \card{x} < \gk\}$ carries a $\gk$-additive,
{\em fine ultrafilter $\U$} (where $\U$ being {\em fine} means that for every $\ga < \gl$,
$\{p \in P_\gk(\gl) \mid \ga \in p\} \in \U$).
If $\U$ is in addition {\em normal} (i.e., if every $f : P_\gk(\gl) \to \gl$ is constant
on a $\U$ measure $1$ set), then $\gk$ is {\em $\gl$-supercompact}.
An equivalent definition for $\gk$ being $\gl$-strongly compact is that there is an
elementary embedding $j : V \to M$ having critical point $\gk$ such that
for any $x \subseteq M$, $x \in V$ with $\card{x} \le \gl$, there is some $y \in M$
having the properties that $x \subseteq y$ and
$M \models ``\card{y} < j(\gk)$''.
An equivalent definition for $\gk$ being $\gl$-supercompact is that there
is an elementary embedding $j : V \to M$ having critical point $\gk$ such that
$j(\gk) > \gl$ and
$M^\gl \subseteq M$ (i.e., every $f : \gl \to M$ with $f \in V$ is such that $f \in M$).
$\gk$ is {\em strongly compact} ({\em supercompact})
if $\gk$ is $\gl$-strongly compact ($\gl$-supercompact)
for every cardinal $\gl \ge \gk$.
Also, $\gk$ is {\em $(\gk + 2)$-strong} if there is an elementary embedding
$j : V \to M$ having critical point $\gk$ such that $V_{\gk + 2} \subseteq M$.
It is the case that if $\gk$ is supercompact, then $\gk$ is $(\gk + 2)$-strong
(and much more).
%$\gk$ is {\em strong} if $\gk$ is $\gl$-strong for every ordinal $\gl$.
Because Magidor's notion of iterated Prikry forcing from \cite{Ma76}
will be used in the proofs of Theorems \ref{t2} and \ref{t4},
we take this opportunity to briefly review its definition and some of its properties.
We follow the conventions of \cite[pages 39 -- 40]{Ma76} and take the liberty to quote nearly
verbatim from \cite{Ma76} when appropriate.
Suppose that $A$ is a set of measurable cardinals.
Let $\Omega = \sup(A)$.
For each $\gk \in A$, let $\U_\gk$ be a normal measure over $\gk$
giving measure $0$ to the set of measurable cardinals below $\gk$.
For $\gk \in A$ or $\gk = \Omega$, we will define inductively a notion of
forcing $\FP_\gk$ which changes the cofinality of every member of $A \cap \gk$ to $\go$.
If $\gk_0$ is the smallest member of $A$, $\FP_{\gk_0}$ is trivial forcing, and
$\dot \U_{\gk_0}$ is a term in the forcing language with respect to $\FP_{\gk_0}$
denoting $\U_{\gk_0}$.
Then, for $\gk > \gk_0$, $\gk \in A$ or $\gk = \Omega$,
$\FP_\gk$ is defined as the set of all sequences of the form
$\la p_\ga, \dot B_\ga \ra_{\ga \in A \cap \gk}$, where $p_\ga$ is a finite increasing
sequence of members of $\ga$, $p_\ga$ is different from the empty sequence
for only finitely many $\ga$ s, and $\dot \U_\ga$ and $\dot B_\ga$ are terms with %such that
$\forces_{\FP_\ga} ``\dot \U_\ga$ is a normal measure over $\ga$ such that
$\dot \U_\ga \supseteq \U_\ga$, $\dot B_\ga \in \dot \U_\ga$, and
$\sup(p_\ga) < \inf(\dot B_\ga)$''.
Let $E = A \cap \gk$.
The ordering on $\FP_\gk$ is %given by
%$\la p_\ga, C_\ga \ra_{\ga \in A \cap \gk} \ge \la q_\ga, B_\ga \ra_{\ga \in A \cap \gk}$ iff
$\la q_\ga, \dot C_\ga \ra_{\ga \in E} \ge \la p_\ga, \dot B_\ga \ra_{\ga \in E}$ iff
$q_\ga$ extends $p_\ga$ as a finite sequence,
$\forces_{\FP_\ga} ``\dot C_\ga \subseteq \dot B_\ga$'', and if
$\gb \in q_\ga - p_\ga$, then $\la q_\gg, \dot C_\gg \ra_{\gg \in E \cap \ga}
\forces_{\FP_\ga} ``\gb \in \dot B_\ga$''.
Intuitively, $\FP_\gk$ may be thought of as the iteration of
Prikry forcing which has finite support in the stems, full support in
the measure $1$ sets, and changes the cofinality of every member of $A$ to $\go$.
The work of \cite{Ma76} shows that $\FP_\gk$ is well-defined
(which is certainly not obvious from the definitions given above).
Also, if $\gk < \sup(A)$, the preceding definition of $\FP_\gk$ makes sense
even if $\gk$ is a measurable cardinal and $\gk \not\in A$.
Assume now that $\gk$ is a measurable cardinal which
is a limit of measurable cardinals.
Assume also that in the above definition, $A$ is composed of
an unbounded subset of the measurable cardinals below $\gk$.
By \cite[Lemma 4.4(i)]{Ma76}, forcing with $\FP_\gk$ preserves all cardinals.
In addition, by the definition of $\FP_\gk$ just given,
$\FP_\gk$ is $\gk^+$-c.c., and $|\FP_\gk| = 2^\gk$.
It therefore follows that forcing with $\FP_\gk$
preserves the value of $\card{2^\gd}$ %the power set function
for all cardinals $\gd \ge \gk$.
Further, \cite[Theorem 2.5]{Ma76} tells us that if
$A \not\in \U$ for some normal measure $\U$ over $\gk$, then
$\U$ extends to a normal measure $\ov \U$ after forcing with $\FP_\gk$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorem \ref{t1}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
%$\gd$-strategically closed''.
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
%In Hamkins' terminology of
%\cite{H2, H3},
%$\FP$ is {\em mild
%with respect to a cardinal $\gk$}
%iff every set of ordinals $x$ in
%$V^\FP$ of size below $\gk$ has
%a ``nice'' name $\tau$
%in $V$ of size below $\gk$,
%i.e., there is a set $y$ in $V$,
%$|y| <\gk$, such that any ordinal
%forced by a condition in $\FP$
%to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is {\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any $A \in \ov V$.
%$j : V \to M$ is {\em amenable to $V$} when $j \rest A \in V$ for any $A \in V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$
that admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an elementary embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
%Finally, if $\FP$ is mild with
%respect to $\gk$ and $\gk$ is
%$\gl$ strongly compact in $V[G]$
%for any $\gl \ge \gk$, then
%$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\noindent A consequence of Theorem \ref{tgf} is that if $\FP$
admits a gap at some regular $\gd < \gk$ and
$\gk$ is either supercompact or measurable in $V^\FP$,
%$V^\FP \models ``\gk$ is supercompact'' or $V^\FP \models ``\gk$ is measurable'',
then $\gk$ is supercompact or measurable in $V$ as well.
%$V \models ``\gk$ is supercompact'' as well. $V^\FP \models ``\gk$ is measurable'', then
\section{The Proofs of Theorems \ref{t1} -- \ref{t4}}\label{s2}
We turn now to the proofs of our theorems.
\begin{proof}
To prove Theorem \ref{t1}, let $V$ be a model
of ``ZFC + GCH'' in which $\gk$ is the least measurable limit of supercompact cardinals.
%$V \models ``$ZFC + GCH + $\gk$ is the least measurable limit of supercompact cardinals''.
Without loss of generality, by
doing a preliminary forcing as in \cite{A98} if necessary, we assume in
addition that in $V$, every supercompact cardinal $\gd < \gk$ has its
supercompactness indestructible under $\gd$-directed closed forcing,
and GCH holds at and above $\gk$.
%$V \models ``\gk = \bigcup_{\ga < \gk} \gk_\ga$, where each $\gk_\ga$ is a
%Laver indestructible supercompact cardinal \cite{L} +
%supercompact cardinal whose supercompactness is indestructible under
%$\gk_\ga$-directed closed forcing +
%$V \models ``$Every supercompact cardinal $\gd < \gk$ has its supercompactness
%ndestructible under $\gd$-directed closed forcing +
%GCH holds at and above $\gk$''.
Let $V_1 = V^{\add(\gk^+, \gl)}$.
%where $\add(\gk^+, \gl)$ is the standard partial ordering
%for adding $\gl$ many Cohen subsets of $\gk^+$.
Because $\add(\gk^+, \gl)$ is $\gk^+$-directed closed,
$V_1$ and $V$ contain the same subsets of $\gk$.
Thus, $V_1 \models ``\gk$ is measurable''.
In addition, standard arguments show that in $V_1$,
$2^\gk = \gk^+$, and $2^{\gk^+} = 2^{2^\gk} = \gl$.
%$V_1 \models ``2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$''.
%Further, since for $\ga < \gk$, $V \models ``\gk_\ga < \gk$ has its
%supercompactness indestructible under $\gk_\ga$-directed closed
%forcing'', $V_1 \models ``\gk_\ga$ is supercompact''.
Further, if $V \models ``\gd < \gk$ is supercompact'', then because
$V \models ``\gd$ has its supercompactness indestructible under
$\gd$-directed closed forcing'', $V_1 \models ``\gd$ is supercompact''.
We may consequently
infer that $V_1 \models ``\gk$ is a measurable limit of supercompact cardinals''.
To show $\gk$ is in fact the least measurable limit of supercompact cardinals
in $V_1$, observe that
the closure properties of $\add(\gk^+, \gl)$ tell us forcing with
$\add(\gk^+, \gl)$ creates no new measurable cardinals below $\gk$.
To see that forcing with $\add(\gk^+, \gl)$ creates no new supercompact
cardinals below $\gk$, let $\gd < \gk$ be such that $V \models ``\gd$ is
not supercompact''. % but $\gg$ is supercompact''.
Let $\gr > \gk$ be large enough so that
$V \models ``\gd$ is not $\gr$-supercompact''.
If $V_1 \models ``\gd$ is
supercompact'', then again by the closure properties of $\add(\gk^+, \gl)$,
it must be the case that $V \models ``\gd$ is $\eta$-supercompact for
every $\eta < \gk$''. Since $V \models ``\gk$ is supercompact'',
we may now argue in analogy to
%the argument found in
\cite[page 31, paragraph 4]{AC2} to see that
$V \models ``\gd$ is $\gr$-supercompact''. In particular,
if $\ell : V \to M$ is an elementary
embedding witnessing the $\gr'$-supercompactness
of $\gk$ for some strong limit cardinal
$\gr' > \gr > \gk$, then as
$V \models ``\gd$ is $\eta$-supercompact for every $\eta < \gk$'',
$M \models ``\ell(\gd) = \gd$ is
$\eta$-supercompact for every $\eta < \ell(\gk)$''. Since
$\ell(\gk) > \gr'$ and $\gr'$ is a strong limit cardinal,
in both $M$ and $V$, $\gd$ is $\gr$-supercompact, a contradiction.
%It is therefore the case
We now know that in $V_1$,
$\gk$ is the least measurable limit of supercompact cardinals,
$2^\gk = \gk^+$, and $2^{\gk^+} = 2^{2^\gk} = \gl$.
%$V_1 \models ``$ZFC + $\gk$ is the least measurable limit of
%supercompact cardinals + $2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$''.
Working in $V_1$, let $\FP^*$ be the (possibly proper class)
reverse Easton iteration which
%begins by %adding a single Cohen subset of $\go$
%forcing with $\add(\go, 1)$ and then does nontrivial forcing
forces nontrivially
only at inaccessible cardinals $\gd$ which are not limits of inaccessible cardinals,
%where a single Cohen subset is added by $\FP^*$.
where the forcing done is $\add(\gd, 1)$.\footnote{$\FP^*$ is a proper class
if there are class many inaccessible cardinals, but is a set otherwise.
The standard Easton arguments show that $V_2 = V^{\FP^*}_1 \models {\rm ZFC}$
if $\FP^*$ is a proper class.}
%Standard arguments (see, e.g., the proof of \cite[Theorem]{A98}) show that every
A variant of Laver's original argument from \cite{L} shows that every
$V_1$-supercompact cardinal is preserved to $V_2 = V_1^{\FP^*}$.
Specifically, suppose $V_1 \models ``\gd$ is supercompact''.
Let $\gl > \gd$ be a fixed but arbitrary regular cardinal, with
$\gg = 2^{[\gl]^{< \gd}}$.
Take $j : V_1 \to M$ as an elementary embedding witnessing the
$\gg$-supercompactness of $\gd$.
Let $\dot \FQ^*$ be a term
in the forcing language with respect to $\FP^*_\gd$
for the portion of $\FP^*$ acting
on ordinals in the open interval $(\gd, \gg)$.
By the definition of $\FP^*$, %it is possible to write
$j(\FP^*_\gd \ast \dot \FQ^*) =
\FP^*_\gd \ast \dot \FQ^* \ast \dot \FR^* \ast j(\dot \FQ^*)$, where
$\dot \FR^*$ is a term for the portion of $j(\FP^*_\gd \ast \dot \FQ^*)$
acting on %forced to act on
ordinals in the open interval $(\gg, j(\gd))$.
Let $G_0 \ast G_1
\ast G_2$ be $V_1$-generic over
$\FP^*_\gd \ast \dot \FQ^* \ast \dot \FR^*$.
As in \cite{L}, we may
lift $j$ in $V_1[G_0][G_1][G_2]$ to
$j : V_1[G_0] \to M[G_0][G_1][G_2]$,
take a master condition $p$
for $j '' G_1$ and a
$V_1[G_0][G_1][G_2]$-generic object
$G_3$ over $j(\FQ^*)$ containing $p$, lift $j$
again in $V_1[G_0][G_1][G_2][G_3]$ to
$j : V_1[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gg^+$-directed closure of
$\FR^* \ast j(\dot \FQ^*)$
in both $M[G_0][G_1]$ and $V_1[G_0][G_1]$
that the supercompactness
measure over ${(P_\gd(\gl))}^{V_1[G_0][G_1]}$
generated by $j$ is actually a member of
$V_1[G_0][G_1]$.
Write $\FP^* = \FP^*_\gd \ast \dot \FQ^* \ast \dot \FS$.
Since $\gl$ was arbitrary, and since
$\forces_{\FP^*_\gd \ast \dot \FQ^*} ``\dot \FS$ is $\gg^+$-directed closed'',
$V_2 = V_1^{\FP^*} \models ``\gd$ is supercompact''.
Since forcing with $\FP^*$ does not change either cofinalities or
%the power function,
the size of power sets, in $V_2$,
$2^\gk = \gk^+$, and $2^{\gk^+} = 2^{2^\gk} = \gl$.
%$V_2 \models ``2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$''.
In addition, if we write $\FP^* = \FP_{\gk} \ast \dot \FQ$, it is the case that
$\forces_{\FP_{\gk }} ``\dot \FQ$ is (at least) $(2^\gk)^+$-directed closed''.
This means that the set of normal measures $\gk$ carries,
and hence $\gk$'s total number of normal measures, is the same in both
$V_2 = V_1^{\FP^*} = V_1^{\FP_{\gk} \ast \dot \FQ}$ and $V_1^{\FP_{\gk}}$.
We can now argue as in \cite[Lemma 1.1]{A01} to
infer that in both $V_1^{\FP_\gk}$ and $V_2$,
$\gk$ is a measurable cardinal
carrying $2^{2^\gk}$ many normal measures.
%not concentrating on measurable cardinals.
%We first argue as in \cite[Lemma 1.1]{A01} that
%$\gk$ carries $2^{2^\gk}$ many normal measures, and then
%use Theorem \ref{tgf} to show that there are $2^{2^\gk}$ many
%normal measures not concentrating on measurable cardinals.
%To begin,
Explicitly, let $j : V_1 \to M$ be an elementary embedding
generated by a normal measure over $\gk$ present in $V_1$ such that
$M^\gk \subseteq M$ and $M \models ``\gk$ is not measurable''.
Let $G$ be $V_1$-generic over $\FP_\gk$, and
write $j(\FP_\gk) = \FP_\gk \ast \dot \FR$.
Note that since $V_1 \models ``|\FP_\gk| = \gk$'',
by elementarity, we also have that in $M$,
$\forces_{\FP_\gk} ``|\dot \FR| = j(\gk)$''.
Therefore, in $M$,
$\forces_{\FP_\gk} ``$The number of dense open subsets
of $\dot \FR$ is at most $2^{j(\gk)}$''. % = j(\gk^+)$''.
In addition, since $V_1[G] \models ``2^\gk = \gk^+$'',
%$|j(\gk^+)| =
$V_1[G] \models ``|2^{j(\gk)}| = |j(2^\gk)| = |\{f \mid f : \gk \to 2^\gk$
is a function$\}| = |{[2^\gk]}^\gk| =
2^\gk = \gk^+$''. This means we can let
$\la D_\ga \mid \ga < \gk^+ \ra$ be
an enumeration in $V_1[G]$ of the
dense open subsets of $\FR$ present in $M[G]$.
%Hence, as $|j(\gk^+)| = \gk^+$, we can let
%$\la D_\ga : \ga < \gk^+ \ra$ be an enumeration in $V[G]$ of the dense opens
%subsets of $i_G(\dot \FQ)$ present in $M[G]$.
By the fact $\FP_\gk$ is $\gk$-c.c.,
${M[G]}^\gk \subseteq M[G]$.
Further,
%$M \models ``$The number of dense open subsets
%of $\FR$ is $2^{j(\gk)}$''.
%Since $M^\gk \subseteq M$, and
by the definition of $\FP_\gk$,
the first ordinal on which $\FR$ %is forced to act nontrivially is above $\gk$,
acts in $M[G]$ is above $\gk$.
This means that $\FR$ is $\gk^+$-directed closed in both $M[G]$ and $V_1[G]$.
Thus, we can build in $V_1[G]$ a tree
$\cal T$ of height $\gk^+$ such that:
\begin{enumerate}
\item\label{k1} The root of $\cal T$ is the
empty condition.
\item\label{k2} If $p$ is an element at level
$\ga < \gk^+$ of $\cal T$, then the
successors of $p$ at level $\ga + 1$ are
a maximal incompatible subset of $D_\ga$
extending $p$.
By the definition of %both $\FP_\gk$ and
$\FR$, there will be
at least two successors of $p$ at level $\ga + 1$.
\item\label{k3} If $\gl < \gk^+$ is a limit
ordinal, then the elements of $\cal T$ at
height $\gl$ are upper bounds to any
path through $\cal T$ of height $\gl$.
\end{enumerate}
\noindent We observe that any path $H$
of height $\gk^+$ through $\cal T$ generates
an $M[G]$-generic object over $\FR$.
Therefore, since there are
$2^{\gk^+}$ many paths of height
$\gk^+$ through $\cal T$, there are
$2^{\gk^+} = 2^{2^\gk}$ many different
$M[G]$-generic objects over
$\FR$.
For $H$ an
$M[G]$-generic object over
$\FR$ generated as above,
since $j''G \subseteq G \ast H$,
$j : V_1 \to M$ lifts to
$j : V_1[G] \to M[G][H]$. This means that if
$j_1$ is the lift generated by $H_1$ and
$j_2$ is the lift generated by $H_2$,
$j_1(G) = \la G, H_1 \ra$ and
$j_2(G) = \la G, H_2 \ra$, i.e., there are
$2^{2^\gk}$ many different lifts of $j$
after forcing with $\FP$.
Since \cite[Lemma 1]{C93} tells us that any
$k : V_1[G] \to M[G][H]$ is generated by the
normal measure over $\gk$ given by
${\cal U} = \{x \subseteq \gk \mid
\gk \in k(x)\}$, there are
$2^{2^\gk}$ many different normal measures over
$\gk$ in $V_1[G]$ and $V_2$ as well.
%Therefore, since $\FP_\gk \subseteq V_\gk$, $\card{\FP_\gk} = \gk$, and
%every $p \in \FP_\gk$ has at least two incompatible extensions, by \cite[Lemma 1.1]{A01},
The proof of Theorem \ref{t1} will consequently be finished if we can show that
in $V_2$, $\gk$ is the least measurable limit of supercompact cardinals.
To do this, note that
%since the smallest cardinal to which $\FP^*$ adds a
by its definition, $\FP^*$ is $\gd_0$-directed closed, where $\gd_0$
is the least inaccessible cardinal (in either $V$ or $V_1$). Thus,
$\gd_0$ is the least inaccessible cardinal in $V_2$ as well,
so that there are no measurable or supercompact
cardinals below $\gd_0$ in either $V$, $V_1$, or $V_2$.
In addition, it is possible to write
$\FP^* = \add(\gd_0, 1) \ast \dot \FS$, where
$\forces_{\add(\gd_0, 1)} ``\dot \FS$ is (at least) $\gd^{++}_0$-directed closed''.
By Theorem \ref{tgf}, this means that forcing with $\FP^*$ over $V_1$ creates no new
measurable or supercompact cardinals greater than $\gd_0$,
and hence creates no new measurable or supercompact cardinals.
%Since the above work of the preceding paragraph
Since our earlier work
yields that $V_2 \models ``\gk$ is a measurable limit of
supercompact cardinals'' and $V_1 \models ``\gk$ is the least measurable
limit of supercompact cardinals'', it consequently follows that
$V_2 \models ``\gk$ is the least measurable limit of supercompact cardinals'' as well.
%$\FP = \FP_0 \ast \dot \FS$, where $\card{\FP_0} = \go$, $
%In our final model $V_2 = V^\FP_1$,
%because $V_1 \models ``2^\gk = \gk^+$'',
%t is the case that
%$V_2 \models ``$ZFC + $\gk$ is the least measurable limit of
%supercompact cardinals + $2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
%$\gk$ carries $2^{2^\gk}$ many normal measures''.
%This completes the proof of Theorem \ref{t1}.
By taking $\FP = \add(\gk^+, \gl) \ast \dot \FP^*$,
the proof of Theorem \ref{t1} has been completed.
\end{proof}
It is actually the case that the %proof just given
above shows that in
%both $V_1^{\FP_\gk}$ and
$V_2$, $\gk$ carries $2^{2^\gk}$ many
normal measures not concentrating on measurable cardinals.
To see this, observe that by the proof just given,
each of the $2^{2^\gk}$ many normal measures corresponding to the $2^{2^\gk}$
many lifted embeddings must concentrate on the set
$A = \{\gd < \gk \mid \gd$ is non-measurable in $V_1\}$.
Since as we have just seen, Theorem \ref{tgf} implies that no
$\gd \in A$ is measurable in $V_2$, $A$ consists of non-measurable
cardinals in $V_2$ as well.
Although this fact is not important for the proof of Theorem \ref{t1},
it will be key in the proof of Theorem \ref{t2}.
\begin{pf}
We can now turn our attention to the proof of
Theorem \ref{t2}, which is proven similarly. Let
$V$ be a model of ``ZFC + GCH'' in which $\gk$ is supercompact.
%$V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Without loss of generality, by first doing the forcing of \cite{L}, we assume in
addition that in $V$, $\gk$'s supercompactness is indestructible under
$\gk$-directed closed forcing, and GCH holds at and above $\gk$.
%$V \models ``\gk$'s supercompactness is indestructible under
%$\gk$-directed closed forcing + GCH holds at and above $\gk$''.
If we as before let $V_1 = V^{\add(\gk^+, \gl)}$,
as in the proof of Theorem \ref{t1},
it is then the case that in $V_1$,
$\gk$ is supercompact,
$2^\gk = \gk^+$, and $2^{\gk^+} = 2^{2^\gk} = \gl$.
%$V_1 \models ``$ZFC + $\gk$ is supercompact +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$''.
Working in $V_1$, again as in the proof of Theorem \ref{t1},
let $\FP^*$ be the (possibly proper class)
reverse Easton iteration which
%begins by %adding a single Cohen subset of $\go$
%forcing with $\add(\go, 1)$ and then does nontrivial forcing
forces nontrivially
only at inaccessible cardinals $\gd$ which are not limits of inaccessible cardinals,
%where a single Cohen subset is added by $\FP^*$.
where the forcing done is $\add(\gd, 1)$.
%let $\FP^*$ be the (possibly proper class)
%reverse Easton iteration which begins by adding a single
%Cohen subset of $\go$ and then does nontrivial forcing
%only at inaccessible cardinals which are not limits of
%inaccessible cardinals, where a single Cohen subset is added by $\FP^*$.
%reverse Easton iteration which begins by %adding a single Cohen subset of $\go$
%forcing with $\add(\go, 1)$
%and then does nontrivial forcing
%only at inaccessible cardinals $\gd$ which are not limits of inaccessible cardinals,
%where a single Cohen subset is added by $\FP^*$.
%where the forcing done is $\add(\gd, 1)$.\footnote{Strictly speaking,
%unlike the proof of Theorem \ref{t1}, the proof of Theorem \ref{t2}
%does not require that $\FP^*$ be defined by starting by forcing
%with $\add(\go, 1)$. This is since no use of Theorem \ref{tgf} is
%made, so there is no need to introduce a gap at $\ha_1$.
%However, for uniformity in presentation, the same definition of $\FP^*$ is used.}
By the same arguments as in the proof of Theorem \ref{t1} and
the paragraph immediately following,
in $V_2 = V^{\FP^*}_1$,
%because $V_1 \models ``2^\gk = \gk^+$'',
it is the case that %$V_2 \models ``$ZFC +
$\gk$ is supercompact,
$2^\gk = \gk^+$, $2^{\gk^+} = 2^{2^\gk} = \gl$, and
$\gk$ carries $2^{2^\gk}$ many normal
measures not concentrating on measurable cardinals.
In addition, %the standard Easton arguments show that
as in the proof of Theorem \ref{t1}, if $\FP^*$ is a proper class,
then $V_2 \models {\rm ZFC}$.
Now, working in $V_2$, let $\FQ$ be the Magidor iteration of
Prikry forcing \cite{Ma76} which adds a cofinal $\go$ sequence to
each measurable cardinal below $\gk$,
with $V_3 = V^\FQ_2$.
As in \cite{Ma76}, $V_3 \models ``\gk$ is both the least
strongly compact and least measurable cardinal''.
Since as we have noted in Section \ref{s1}, %by the proof of \cite[Lemma 4.4]{Ma76},
forcing with $\FQ$ neither collapses
any cardinals nor %changes the size of any power sets of cardinals
changes the value of $\card{2^\gd}$ for cardinals $\gd \ge \gk$,
%power set function for cardinals greater than or equal to $\gk$,
%(see \cite[last paragraph of the proof of Theorem 3]{A12} for a discussion of these facts),
in $V_3$, $2^\gk = \gk^+$, and $2^{\gk^+} = 2^{2^\gk} = \gl$.
%$V_3 \models ``2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$''.
We have just mentioned that $V_2 \models
``\gk$ carries $2^{2^\gk}$ many normal
measures not concentrating on measurable cardinals''.
%by the proof of
Therefore, since as we observed in Section \ref{s1}
in the paragraph immediately prior to the discussion
of Theorem \ref{tgf},
%because \cite[Theorem 2.5]{Ma76} tells us that
any normal measure over $\gk$ not concentrating on $A =
\{\gd < \gk \mid \gd$ is a measurable cardinal$\}$
extends after forcing with $\FQ$
and forcing with $\FQ$ collapses no cardinals,
%it is the case that
$V_3 \models %``$ZFC + $\gk$ is both the least strongly compact and
%least measurable cardinal +
%$2^\gk = \gk^+$ + $2^{\gk^+} = 2^{2^\gk} = \gl$ +
``\gk$ carries $2^{2^\gk}$ many normal measures'' as well.
%Since
$V_3$ is thus as desired.
By taking $\FP = \add(\gk^+, \gl) \ast \dot \FP^* \ast \dot \FQ$,
the proof of Theorem \ref{t2} has been completed.
\end{pf}
%Key to the proofs of both Theorems \ref{t1} and \ref{t2}
%is that $2^\gk = \gk^+$
\begin{pf}
To prove Theorem \ref{t3}, let
$V \models ``\gk'$ is the least supercompact limit of supercompact cardinals''.
Assume without loss of generality that a
reverse Easton iteration
has been done as in \cite[Theorem, Section 18, pages 83--88]{Me76}
so that in addition,
$V \models ``$For every inaccessible cardinal $\gd$,
$2^\gd = \gd^{+ 17}$ and $2^{\gd^{+ 17}} = 2^{2^\gd} = \gd^{+ 95}$''.
Let $k : V \to M'$ be an elementary embedding witnessing
the $2^{\gk'}$-supercompactness of $\gk'$.
By \cite[Lemma 2.1]{AC2} and the succeeding remarks,
$M' \models ``\gk'$ is a strong cardinal''.
Since $\gk'$ is the critical point of $k$, if
$V \models ``\gd < \gk'$ is supercompact'', %then
$M' \models ``k(\gd) = \gd < \gk'$ is supercompact''.
This means that $M' \models ``\gk'$ is a strong cardinal
which is a limit of supercompact cardinals'', so by reflection,
$\{\gd < \gk' \mid \gd$ is a strong cardinal which is a limit
of supercompact cardinals$\}$ is unbounded in $\gk'$ in $V$.
Thus, we can let $\gk < \gk'$ be the least cardinal in $V$
which is both $(\gk + 2)$-strong and is a limit of supercompact cardinals.
%By assuming $j(\gk)$ has been chosen to be minimal,
%in analogy to the proof of \cite[Proposition 2.7]{Me74},
Now, let $j : V \to M$ be an elementary embedding witnessing
the $(\gk + 2)$-strongness of $\gk$.
As in the preceding paragraph, $M \models ``\gk$ is a limit of supercompact cardinals''.
Because $V \models ``\gk$ is the least cardinal which is both
$(\gk + 2)$-strong and a limit of supercompact cardinals'',
$M \models ``j(\gk) > \gk$ is the least cardinal which is both
$(j(\gk) + 2)$-strong and a limit of supercompact cardinals''.
Hence, $M \models ``\gk$ is not $(\gk + 2)$-strong''.
We will show that in $M$, $\gk$ is our desired strongly compact cardinal.
To do this,
%since $\gk$ is the critical point of $j$,
%for any $\gd < \gk$ such that
%$V \models ``\gd$ is supercompact'',
%$M \models ``j(\gd) = \gd$ is supercompact'' as well. In addition,
because $V_{\gk + 2} \subseteq M$ and
a measure over $\gk$
%can be identified with a subset of $2^\gk$,
is a member of $V_{\gk + 2}$,
$M$ contains every (normal or non-normal) measure
over $\gk$. Further,
by the fact that $\gk$ is $(\gk + 2)$-strong in $V$,
%Therefore, since
$V \models ``\gk$ carries $2^{2^\gk}$ many normal measures''.
%as $\gk$ is $(\gk + 2)$-strong'',
It consequently follows that
$M \models ``\gk$ is measurable and carries
$2^{2^\gk}$ many normal measures''. Because $V$ and $M$
are elementarily equivalent,
$M \models
``2^\gk = \gk^{+ 17}$ and $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$''.
Since $M \models ``\gk$ is a measurable limit of
supercompact cardinals'', by Menas' theorem of \cite{Me74}, %we know that
$M \models ``\gk$ is strongly compact''.
Putting the above together, we now have that in $M$,
$\gk$ is a strongly compact cardinal
which is a measurable limit of supercompact cardinals, $\gk$ is
not $(\gk + 2)$-strong,
$2^\gk = \gk^{+ 17}$, $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$, and
$\gk$ carries $2^{2^{\gk}}$ many normal measures.
%We note that an easy reflection argument in conjunction with the proof of
%Theorem \ref{t3} shows that
By reflection, $A = \{\gd < \gk \mid \gd$ is a measurable limit of
supercompact cardinals which is not $(\gd + 2)$-strong,
$2^\gd = \gd^{+ 17}$, $2^{\gd^{+ 17}} = 2^{2^\gd} = \gd^{+ 95}$, and
$\gd$ carries $2^{2^\gd}$ many normal measures$\}$ is unbounded in $\gk$ in $V$.
Since for any $\gd \in A$,
$M \models ``j(\gd) = \gd$ is a measurable limit of
supercompact cardinals'', $\gk$ is not the least measurable
limit of supercompact cardinals in either $V$ or $M$.
This completes the proof of Theorem \ref{t3}.
\end{pf}
\begin{pf}
Turning now to the proof of Theorem \ref{t4},
we will use the cardinal $\gk$ and the model $M$ witnessing the conclusions of
Theorem \ref{t3} in our proof. %of Theorem \ref{t4}.
First, let us observe that in $M$, since $\gk < j(\gk)$ and
$M \models ``j(\gk)$ is the least cardinal which is both
$(j(\gk) + 2)$-strong and a limit of supercompact cardinals'',
%supercompact limit of supercompact cardinals'',
$\gk$ is below the least supercompact
limit of supercompact cardinals. Keeping this in mind,
%to prove Theorem \ref{t4},
we take $M$ as our ground model.
Let $\FP$ be the Magidor iteration of Prikry forcing \cite{Ma76} which
adds a cofinal $\go$ sequence to each supercompact cardinal
below $\gk$.
By \cite[Theorem 3.4]{Ma76},
$M \models ``\gk$ is strongly compact''.
Because $V$ and $M$ are elementarily equivalent,
$M \models ``$For every inaccessible cardinal $\gd$, $2^\gd = \gd^{+ 17}$''.
Consequently, since the supercompact cardinals are unbounded in $\gk$ in $M$,
as in the proof of \cite[Theorem 4.5]{Ma76},
$M^\FP \models ``$There are unboundedly in $\gk$ many singular strong
limit cardinals violating GCH''.
By Solovay's theorem of \cite{So}, this means we may now infer that
$M^\FP \models ``$No cardinal $\gd < \gk$ is strongly compact'', i.e.,
$M^\FP \models ``\gk$ is the least strongly compact cardinal''.
%By Magidor's work, since
%$M \models ``\gk$ is strongly compact and is a limit of
%supercompact cardinals'', $M^\FP \models ``\gk$ is the least strongly compact cardinal''.
For any $M$-measurable cardinal $\gd$,
%< \gk$ which is not supercompact, as we mentioned in
by the definition of the Magidor iteration of Prikry
forcing given in Section \ref{s1}, it is possible to write
$\FP = \FP_\gd \ast \dot \FR$. If $\gd$ is not supercompact,
%By the definition of $\FP$,
%By the choice of $\gd$
then because forcing with $\FP$ does not add a cofinal $\go$ sequence
to $\gd$, the definition of $\FP$ and \cite[Lemma 2.4]{Ma76} yield that
%we have that
$\forces_{\FP_\gd} ``$Forcing with $\dot \FR$ adds no new subsets of
the least inaccessible cardinal above $\gd$ (and so in particular adds
no new subsets of $2^\gd$)''.
If $\card{\FP_\gd} < \gd$, then by the L\'evy-Solovay results \cite{LS},
$M^{\FP_\gd} \models ``\gd$ is measurable''.
If $\card{\FP_\gd} = \gd$, then by \cite[Theorem 2.5]{Ma76}, it again follows that
$M^{\FP_\gd} \models ``\gd$ is measurable''.
It is thus the case that
$M^{\FP_\gd \ast \dot \FR} = M^\FP \models ``\gd$ is measurable'' as well.
In addition, because $\gk$ is in $M$ a limit of supercompact cardinals,
there are in $M$ unboundedly many in $\gk$ measurable cardinals
which are not supercompact.
Consequently, we may now infer that $M^\FP \models ``\gk$ is
a limit of measurable cardinals''.
Because $\gk$ is below the least supercompact limit of
supercompact cardinals, no normal measure over $\gk$
concentrates on $A = \{\gd < \gk \mid \gd$ is supercompact$\}$.\footnote{This
%supercompact cardinals.\footnote{This
is since otherwise, if $\mu$ were a normal measure over $\gk$ concentrating on
%the set of
supercompact cardinals, with $j_\mu : M \to N$ the associated
elementary embedding, then $N \models ``\gk$ is supercompact''.
Further, as $j_\mu$ has critical point $\gk$, for any $\gd < \gk$ such that
$M \models ``\gd$ is supercompact'', $N \models ``j_\mu(\gd) = \gd$ is supercompact''.
Since the supercompact cardinals are unbounded in $\gk$ in $M$,
this means that $N \models ``\gk$ is a supercompact limit of supercompact cardinals''.
By reflection, the set of supercompact limits of supercompact cardinals is
unbounded below $\gk$ in $M$.
%Since any supercompact cardinal is also a strong cardinal,
This contradicts that in $M$, $\gk$ is below the least supercompact limit of
supercompact cardinals.}
%then $M^\gk / \mu \models ``\gk$ is supercompact''.
%This fact, combined with Magidor's work, shows that
%any normal measure over $\gk$ in $M$ extends to a
%normal measure over $\gk$ in $M^\FP$. Since the Magidor
%iteration does not change the size of power sets, we may now infer that
Hence, as in the last paragraph of the proof of Theorem \ref{t2}, we may now infer that
in $M^\FP$,
%``\gk$ is the least strongly compact cardinal +
%which is not $\gk + 2$ strong +
$\gk$ carries $2^{2^{\gk}}$ many normal measures, %as well as
%$M^\FP \models
$2^\gk = \gk^{+ 17}$, and $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$.
We have therefore completed the proof of Theorem \ref{t4}, unless it also happens to be true that
$M^\FP \models ``\gk$ is $(\gk + 2)$-strong''.
If this is the case, then %there is a cardinal $\gz \le \gk$ such that
$M \models \varphi(\gk)$,
where $\varphi(x)$ is the formula in one free variable in the language of ZFC which says
``$x$ is a limit of supercompact cardinals, $x$ is not $(x + 2)$-strong,
$x$ is measurable and carries $2^{2^x}$ many normal measures,
$2^x = x^{+ 17}$, $2^{x^{+ 17}} = 2^{2^x} = x^{+ 95}$,
and forcing with the Magidor iteration of Prikry forcing which changes the
cofinality of each supercompact cardinal below $x$ to $\go$
makes $x$ into an $(x + 2)$-strong cardinal''.
%such that $N^{j(\FP)} \models ``\gk$ is not $(\gk + 2)$-strong''.
Without loss of generality, %but with a slight abuse of notation,
assume $\gk$ is least such that $M \models \varphi(\gk)$.
Let $j : M^\FP \to N^{j(\FP)}$ be an
elementary embedding witnessing the $(\gk + 2)$-strongness of
$\gk$.
We will show that in $N^\FP$, $\gk$ is our desired strongly
compact cardinal.
We first show that
$N \models ``\gk$ is a measurable limit of supercompact cardinals''
(and hence is strongly compact in $N$, by Menas' theorem from \cite{Me74}).
To do this,
consider $j \rest M : M \to N$, which is still an elementary
embedding having critical point $\gk$.
As before, for any $\gd < \gk$ such that $M \models ``\gd$ is
supercompact'', $N \models ``j(\gd) = \gd$ is supercompact''.
Thus, since $M \models ``\gk$ is a limit of supercompact cardinals'',
$N \models ``\gk$ is a limit of supercompact cardinals''.
Further, because $M \models ``\gk$ is below the least
supercompact limit of supercompact cardinals'',
$N \models ``j(\gk) > \gk$ is below the least supercompact limit of supercompact cardinals''.
Hence, $N \models ``\gk$ is below the
least supercompact limit of supercompact cardinals'' as well.
It therefore immediately follows that $N \models ``\gk$ is not supercompact''.
This means that by the definition of $\FP$ given in Section \ref{s1},
$j(\FP)$ factors as $\FP_\gk \ast \dot \FQ = \FP \ast \dot \FQ$,
where the first ordinal to which $\dot \FQ$ is forced to add
a cofinal $\go$ sequence is well above $\gk$.
Consequently, as in the second paragraph of the proof of this theorem,
$\forces_{\FP} ``$Forcing with $\dot \FQ$ adds no new subsets of
the least inaccessible cardinal above $\gk$ (and so in particular adds
no new subsets of $2^\gk$)''.
Also, exactly as in the proof of Theorem \ref{t3},
because $M^\FP \models ``\gk$ is a measurable cardinal
carrying $2^{2^\gk}$ many normal measures'' and $j$ is
an elementary embedding witnessing that $\gk$ is
$(\gk + 2)$-strong in $M^\FP$,
%is strongly compact and hence measurable'', by elementarity,
$N^{j(\FP)} \models ``\gk$ is a measurable
cardinal carrying $2^{2^\gk}$ many normal measures'' as well.
%$(N^{j(\FP)})^{2^\gk} \subseteq N^{j(\FP)}$,
%and a measure over $\gk$ can be identified with
%a subset of $2^\gk$,
%$N^{j(\FP)} \models ``\gk$ is measurable'', and
%$M^\FP \models ``$The measures $\gk$ carries are
%the same as in $N^{j(\FP)}$''.
%In $N$, as $\forces_{\FP} ``$Forcing with $\dot \FQ$ adds no
%new subsets of $2^\gk$, preserves all cardinals, and
%does not change the size of power sets'',
The preceding two sentences now allow us to infer that
$N^\FP \models ``\gk$ is a measurable cardinal carrying
$2^{2^\gk}$ many normal measures''.
%$N^\FP \models ``\gk$ is measurable'', and
%$M^\FP \models ``$The measures $\gk$ carries are
%the same as in $N^\FP$''.
Therefore, since %the Magidor iteration of Prikry forcing
by \cite[Theorem 3.1]{Ma76}, forcing with $\FP$
creates no new measurable cardinals,
$N \models ``\gk$ is a measurable cardinal'', i.e.,
$N \models ``\gk$ is a measurable limit of supercompact cardinals''.
We now know that
$N \models ``\gk$ is strongly compact and is a
limit of supercompact cardinals''. In addition, by elementarity, it is again the case that
$N \models ``$For every inaccessible cardinal $\gd$, $2^\gd = \gd^{+ 17}$''.
This means that
as in the first two paragraphs of the proof of this theorem, we may infer that
$N^\FP \models ``\gk$ is the least strongly compact cardinal
and is a limit of measurable cardinals''.
Because $M$ and $N$ are elementarily equivalent
and as we noted in Section \ref{s1},
forcing with $\FP$ neither collapses any cardinals nor
changes the value of $\card{2^\gd}$ for cardinals $\gd \ge \gk$,
%power set function for cardinals greater than or equal to $\gk$,
%and forcing with $\FP$ preserves both cardinals and the size of power sets,
$N^\FP \models ``2^\gk = \gk^{+ 17}$ and $2^{\gk^{+ 17}} =
2^{2^\gk} = \gk^{+ 95}$''.
Also, because $M \models ``\gk$ is the least cardinal such that
$\varphi(\gk)$ is true'', by elementarity,
$N \models ``j(\gk) > \gk$ is the least cardinal such that
$\varphi(j(\gk))$ is true''.
By our work above, it therefore follows that
$N^\FP \models ``\gk$ is not $(\gk + 2)$-strong''.
%In addition,
%in $N$, since $\forces_{\FP} ``$Forcing with $\dot \FQ$ adds no
%new subsets of the least inaccessible cardinal above $\gk$''
%of $2^{2^\gk}$'', a measure over
%$P_\gk(2^\gk)$ can be identified with
%a subset of $2^{2^\gk}$,
%and $N^{j(\FP)} \models ``\gk$ is not $(\gk + 2)$-strong'',
%$N^\FP \models ``\gk$ is not $(\gk + 2)$-strong'' as well.
%Finally, because $M^\FP$ and $N^\FP$ have the same normal
%measures over $\gk$ and
%$M^\FP \models ``\gk$ carries
%$2^{2^\gk}$ many normal measures'',
%$N^\FP \models ``\gk$ carries
%$2^{2^\gk}$ many normal measures'' as well.
Since we have already seen that
$N^\FP \models ``\gk$ carries $2^{2^\gk}$ many normal measures'', this
completes the proof of
Theorem \ref{t4}.
\end{pf}
\section{Concluding Remarks}\label{s3}
We conclude with a few observations.
As we remarked in Section \ref{s1}, the non-$(\gk + 2)$-strong strongly compact cardinals
$\gk$ witnessing the conclusions of Theorems \ref{t3} and \ref{t4} are neither the least
measurable limit of supercompact cardinals nor the least measurable cardinal.
This is since the methods used in the proofs of Theorems \ref{t1} and \ref{t2}
do not seem to be adaptable to the situation where $2^\gk > \gk^+$.
The reason is that in the proofs of Theorems \ref{t1} and \ref{t2}, we need to know the partial
ordering $\FP_\gk$ of Theorem \ref{t1} increases the number of normal measures
over the strongly compact cardinal $\gk$ in question
not concentrating on measurable cardinals to $2^{2^\gk}$.
In order to show that this is indeed the case, as the proof of %\cite[Lemma 1.1]{A01}
Theorem \ref{t1} indicates, we have to be able to
construct $2^{2^\gk}$ many generic objects for a certain $\gk^+$-directed closed
partial ordering $\FR$ by meeting all of
the dense open subsets of $\FR$ %of a certain partial ordering
present in a generic extension $M[G]$
of a $\gk$-closed inner model $M$ of the ground model $V_1$
(where we adopt the same notation as in the proof of Theorem \ref{t1}).
%Here, $G$ is $V$-generic over $\FP_\gk$, $\FR \in M[G] \subseteq V[G]$,
%the construction takes place in $V[G]$,
%and $j : V \to M$ is an ultrapower embedding generated by a normal
%measure over $\gk$ not concentrating on measurable cardinals.
If $2^\gk = \gk^+$, then this is not a problem, as we have already seen. However,
%as the proof of Theorem \ref{t1} indicates. However,
if $2^\gk > \gk^+$, then the calculation given in the
proof of Theorem \ref{t1} for the
number of dense open subsets of $\FR$ present in $M[G]$ yields some $\gl \ge \gk^{++}$.
%It will be necessary to handle
Building the generic object for $\FR$ via the induction given previously does not work,
as there are $\gl$ many dense open subsets which must be met.
The construction will break down at stage $\gk^+$, because $\FR$ is only
$\gk^+$-directed closed.
%The preceding induction will break down at stage $\gk^+$.
It is not at all clear at the moment how to overcome this obstacle. %predicament.
%proceed from $\gk^+$ and beyond.
Theorems \ref{t1} -- \ref{t4}
%and some related generalizations
only barely scratch the surface of what
we feel is possible for non-$(\gk + 2)$-strong strongly compact
cardinals $\gk$. We finish by making this precise via the following \bigskip
\noindent Conjecture: For any non-$(\gk + 2)$-strong strongly compact cardinal
$\gk$ (such as the ones considered earlier), it is relatively consistent for $\gk$ to carry
%It is relatively consistent to have a strongly compact cardinal $\gk$ which carries
exactly $\gd$ many normal
measures. Here, $1 \le \gd \le 2^{2^{\gk}}$ is any cardinal,
and the values of both $2^\gk$ and $2^{2^\gk}$ can be freely manipulated
in a way compatible with the value of $\gd$. In particular,
it is relatively consistent to have a non-($\gk + 2$)-strong strongly
compact cardinal $\gk$ which carries exactly 1, 2, 3, 98, $\ha_{64}$,
$\gd$ for $\gd$ the least inaccessible cardinal, $\gk^{+ 99}$, etc$.$ many normal
measures, with %either $2^\gk = \gk^+$ or $2^\gk > \gk^+$.
arbitrary values for either $2^\gk$ or $2^{2^\gk}$ which are compatible
with $\gd$ many normal measures over $\gk$.
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%{\it Annals of Pure and Applied Logic 154}, 2008, 191--208.
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\end{document}
In Theorems \ref{t3} and \ref{t4}, there is nothing special about
``$\gk^{+ 17}$'' and ``$\gk^{+ 95}$''. They stand for any
``reasonably definable cardinals appropriate for reverse Easton
iterations'' (i.e., values of an Easton function
%defined only at inaccessible cardinals
defined as in \cite[Theorem, Section 18, pages 83--88]{Me76}).
To prove Theorem \ref{t3}, let
$V \models ``\gk$ is the least supercompact limit of supercompact cardinals''.
Assume without loss of generality that a
reverse Easton iteration
has been done as in \cite[Theorem, Section 18, pages 83--88]{Me76}
%and \cite[Theorem 3.17]{FH} so that in addition,
so that in addition,
$V \models ``$For every inaccessible cardinal $\gd$,
$2^\gd = \gd^{+ 17}$ and $2^{\gd^{+ 17}} = 2^{2^\gd} = \gd^{+ 95}$''.
%\footnote{The
%arguments of \cite[Theorem, Section 18, pages 83--88]{Me76} yield that the
%reverse Easton iteration preserves each supercompact cardinal.
%It follows from \cite[Theorem 3.17]{FH} that this iteration
%preserves the strongness of $\gk$.}
By assuming $j(\gk)$ has been chosen to be minimal,
in analogy to the proof of \cite[Proposition 2.7]{Me74},
let $j : V \to M$ be an elementary embedding witnessing
the $(\gk + 2)$-strongness of $\gk$ such that
$M \models ``\gk$ is not $(\gk + 2)$-strong''.
We will show that in $M$, $\gk$ is our desired strongly compact cardinal.
Turning now to the proof of Theorem \ref{t4},
we will use the cardinal $\gk$ and the model $M$ witnessing the conclusions of
Theorem \ref{t3} in our proof. %of Theorem \ref{t4}.
First, let us observe that in $M$, since $\gk < j(\gk)$ and
$M \models ``j(\gk)$ is the least cardinal which is both
$(\gk + 2)$-strong and a limit of supercompact cardinals'',
%supercompact limit of supercompact cardinals'',
$\gk$ is below the least supercompact
limit of supercompact cardinals. Keeping this in mind,
%to prove Theorem \ref{t4},
we take $M$ as our ground model.
Let $\FP$ be the Magidor iteration of Prikry forcing \cite{Ma76} which
adds a cofinal $\go$ sequence to each supercompact cardinal
below $\gk$.
By \cite[Theorem 3.4]{Ma76},
$M \models ``\gk$ is strongly compact''.
Because $V$ and $M$ are elementarily equivalent,
$M \models ``$For every inaccessible cardinal $\gd$, $2^\gd = \gd^{+ 17}$''.
Consequently, since the supercompact cardinals are unbounded in $\gk$ in $M$,
as in the proof of \cite[Theorem 4.5]{Ma76},
$M^\FP \models ``$There are unboundedly in $\gk$ many singular strong
limit cardinals violating GCH''.
By Solovay's theorem of \cite{So}, this means we may now infer that
$M^\FP \models ``$No cardinal $\gd < \gk$ is strongly compact'', i.e.,
$M^\FP \models ``\gk$ is the least strongly compact cardinal''.
%By Magidor's work, since
%$M \models ``\gk$ is strongly compact and is a limit of
%supercompact cardinals'', $M^\FP \models ``\gk$ is the least strongly compact cardinal''.
For any $M$-measurable cardinal $\gd < \gk$ which is not supercompact,
as we mentioned in Section \ref{s1}, it is possible to write
$\FP = \FP_\gd \ast \dot \FR$.
%By the definition of $\FP$,
%By the choice of $\gd$
Because forcing with $\FP$ does not add a cofinal $\go$ sequence
to $\gd$, the definition of $\FP$ and \cite[Lemma 2.4]{Ma76} yield that
%we have that
$\forces_{\FP_\gd} ``$Forcing with $\dot \FR$ adds no new subsets of $2^\gd$''.
If $\card{\FP_\gd} < \gd$, then by the L\'evy-Solovay results \cite{LS},
$M^{\FP_\gd} \models ``\gd$ is measurable''.
If $\card{\FP_\gd} = \gd$, then by \cite[Theorem 2.5]{Ma76}, it again follows that
$M^{\FP_\gd} \models ``\gd$ is measurable''.
It is thus the case that
$M^{\FP_\gd \ast \dot \FR} = M^\FP \models ``\gd$ is measurable'' as well.
In addition, because $\gk$ is in $M$ a limit of supercompact cardinals,
there are in $M$ unboundedly many in $\gk$ measurable cardinals
which are not supercompact.
Consequently, we may now infer that $M^\FP \models ``\gk$ is
a limit of measurable cardinals''.
Because $\gk$ is below the least supercompact limit of
supercompact cardinals, no normal measure over $\gk$
concentrates on %the set of
supercompact cardinals.\footnote{This
is since otherwise, if $\mu$ were a normal measure over $\gk$ concentrating on
%the set of
supercompact cardinals, with $j_\mu : M \to N$ the associated
elementary embedding, then $N \models ``\gk$ is supercompact''.
Further, as $j$ has critical point $\gk$, for any $\gd < \gk$ such that
$M \models ``\gd$ is supercompact'', $N \models ``j_\mu(\gd) = \gd$ is supercompact''.
Since the supercompact cardinals are unbounded in $\gk$ in $M$,
this means that $N \models ``\gk$ is a supercompact limit of supercompact cardinals''.
By reflection, the set of supercompact limits of supercompact cardinals is
unbounded below $\gk$ in $M$.
%Since any supercompact cardinal is also a strong cardinal,
This contradicts that in $M$, $\gk$ is below the least supercompact limit of
supercompact cardinals.}
%then $M^\gk / \mu \models ``\gk$ is supercompact''.
%This fact, combined with Magidor's work, shows that
%any normal measure over $\gk$ in $M$ extends to a
%normal measure over $\gk$ in $M^\FP$. Since the Magidor
%iteration does not change the size of power sets, we may now infer that
Consequently, as in the last paragraph of the proof of Theorem \ref{t2}, we may now infer that
in $M^\FP$,
%``\gk$ is the least strongly compact cardinal +
%which is not $\gk + 2$ strong +
$\gk$ carries $2^{2^{\gk}}$ many normal measures, %as well as
%$M^\FP \models
$2^\gk = \gk^{+ 17}$, and $2^{\gk^{+ 17}} = 2^{2^\gk} = \gk^{+ 95}$.
We have therefore completed the proof of Theorem \ref{t4}, unless it also happens to be true that
$M^\FP \models ``\gk$ is $(\gk + 2)$-strong''.
If this is the case, then let $j : M^\FP \to N^{j(\FP)}$ be an
elementary embedding witnessing the $(\gk + 2)$-strongness of
$\gk$ such that
$N^{j(\FP)} \models ``\gk$ is not $(\gk + 2)$-strong''.
Note that by the definition of $\FP$ given in Section \ref{s1},
$j(\FP)$ factors as $\FP_\gk \ast \dot \FQ = \FP \ast \dot \FQ$.
We will show that in $N^\FP$, $\gk$ is our desired strongly
compact cardinal.
We first show that
$N \models ``\gk$ is a measurable limit of supercompact cardinals''
(and hence is strongly compact in $N$, by Menas' theorem from \cite{Me74}).
To do this,
consider $j \rest M : M \to N$, which is still an elementary
embedding having critical point $\gk$.
As before, for any $\gd < \gk$ such that $M \models ``\gd$ is
supercompact'', $N \models ``j(\gd) = \gd$ is supercompact''.
Thus, since $M \models ``\gk$ is a limit of supercompact cardinals'',
$N \models ``\gk$ is a limit of supercompact cardinals''.
Further, exactly as in the proof of Theorem \ref{t3},
because $M^\FP \models ``\gk$ is a measurable cardinal
carrying $2^{2^\gk}$ many normal measures'' and $j$ is
an elementary embedding witnessing that $\gk$ is
$(\gk + 2)$-strong in $M^\FP$,
%is strongly compact and hence measurable'', by elementarity,
$N^{j(\FP)} \models ``\gk$ is a measurable
cardinal carrying $2^{2^\gk}$ many normal measures'' as well.
%$(N^{j(\FP)})^{2^\gk} \subseteq N^{j(\FP)}$,
%and a measure over $\gk$ can be identified with
%a subset of $2^\gk$,
%$N^{j(\FP)} \models ``\gk$ is measurable'', and
%$M^\FP \models ``$The measures $\gk$ carries are
%the same as in $N^{j(\FP)}$''.
In $N$, as $\forces_{\FP} ``$Forcing with $\dot \FQ$ adds no
new subsets of $2^\gk$, preserves all cardinals, and
does not change the size of power sets'',
$N^\FP \models ``\gk$ is a measurable cardinal carrying
$2^{2^\gk}$ many normal measures''.
%$N^\FP \models ``\gk$ is measurable'', and
%$M^\FP \models ``$The measures $\gk$ carries are
%the same as in $N^\FP$''.
Therefore, since %the Magidor iteration of Prikry forcing
by \cite[Theorem 3.1]{Ma76}, forcing with $\FP$
creates no new measurable cardinals,
$N \models ``\gk$ is a measurable cardinal''.
We now know that
$N \models ``\gk$ is strongly compact and is a
limit of supercompact cardinals''. In addition, by elementarity, it is again the case that
$N \models ``$For every inaccessible cardinal $\gd$, $2^\gd = \gd^{+ 17}$''.
This means that
as in the first two paragraphs of the proof of this theorem, we may infer that
$N^\FP \models ``\gk$ is the least strongly compact cardinal
and is a limit of measurable cardinals''.
Because $M$ and $N$ are elementarily equivalent and forcing
with $\FP$ preserves both cardinals and the size of power sets,
$N^\FP \models ``2^\gk = \gk^{+ 17}$ and $2^{\gk^{+ 17}} =
2^{2^\gk} = \gk^{+ 95}$''.
In $N$, since $\forces_{\FP} ``$Forcing with $\dot \FQ$ adds no
new subsets of the least inaccessible cardinal above $\gk$''
%of $2^{2^\gk}$'', a measure over
%$P_\gk(2^\gk)$ can be identified with
%a subset of $2^{2^\gk}$,
and $N^{j(\FP)} \models ``\gk$ is not $(\gk + 2)$-strong'',
$N^\FP \models ``\gk$ is not $(\gk + 2)$-strong'' as well.
%Finally, because $M^\FP$ and $N^\FP$ have the same normal
%measures over $\gk$ and
%$M^\FP \models ``\gk$ carries
%$2^{2^\gk}$ many normal measures'',
%$N^\FP \models ``\gk$ carries
%$2^{2^\gk}$ many normal measures'' as well.
Since we have already seen that
$N^\FP \models ``\gk$ carries $2^{2^\gk}$ many normal measures'', this
completes the proof of
Theorem \ref{t4}.
We conclude with a few observations.
As we remarked in Section \ref{s1}, the non-$(\gk + 2)$-strong strongly compact cardinals
$\gk$ witnessing the conclusions of Theorems \ref{t3} and \ref{t4} are neither the least
measurable limit of supercompact cardinals nor the least measurable cardinal.
This is since the methods used in the proofs of Theorems \ref{t1} and \ref{t2}
do not seem to be adaptable to the situation where $2^\gk > \gk^+$.
The reason is that in the proofs of Theorems \ref{t1} and \ref{t2}, we need to know the partial
ordering $\FP_\gk$ of Theorem \ref{t1} increases the number of normal measures
over the strongly compact cardinal $\gk$ in question
not concentrating on measurable cardinals to $2^{2^\gk}$.
In order to show that this is indeed the case, as the proof of %\cite[Lemma 1.1]{A01}
Theorem \ref{t1} indicates, we have to be able to
construct a generic object for a certain $\gk^+$-directed closed
partial ordering $\FR$ by meeting all of
the dense open subsets (or maximal antichains) of $\FR$ %of a certain partial ordering
present in a generic extension $M[G]$
of a $\gk$-closed inner model $M$ of the ground model $V$. Here,
$G$ is $V$-generic over $\FP_\gk$,
$\FR \in M[G] \subseteq V[G]$,
the construction takes place in $V[G]$,
and $j : V \to M$ is an ultrapower embedding generated by a normal
measure over $\gk$ not concentrating on measurable cardinals.
If $2^\gk = \gk^+$, then this is not a problem, since
$M[G] \models ``\card{\FR} = j(\gk)$'',
$M[G]$ remains $\gk$-closed with respect to $V[G]$,
and we must only %need to
meet $\card{j(\gk^+)} = \card{2^{j(\gk)}} = \card{\{f \mid f : \gk \to \gk^+\}} =
\card{[\gk^+]^\gk} = 2^\gk = \gk^+$ many dense open subsets.
We can do this by letting $\la D_\ga \mid \ga < \gk^+ \ra$ enumerate in $V[G]$
all of the dense open subsets of $\FR$ present in $M[G]$ and defining
via an induction of length $\gk^+$
an increasing sequence $\la p_\ga \mid \ga < \gk^+ \ra$ of members of $\FR$
such that $p_\ga \in D_\ga$. Because $\FR$ is $\gk^+$-directed closed,
there is no problem whatsoever in achieving this goal. However,
if $2^\gk > \gk^+$, then the preceding calculation of $\card{2^{j(\gk)}}$ and hence the
number of dense open subsets of $\FR$ present in $M[G]$ yields some $\gl \ge \gk^{++}$.
%It will be necessary to handle
Building the generic object for $\FR$ via the preceding induction does not work,
as there are $\gl$ many dense open subsets which must be met.
The construction will break down at stage $\gk^+$, because $\FR$ is only
$\gk^+$-directed closed.
%The preceding induction will break down at stage $\gk^+$.
It is not at all clear at the moment how to overcome this obstacle. %predicament.
%proceed from $\gk^+$ and beyond.