\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{amssymb}
\newcommand{\ga}{\alpha}
\newcommand{\gb}{\beta}
\renewcommand{\gg}{\gamma}
\newcommand{\gd}{\delta}
\newcommand{\gep}{\epsilon}
\newcommand{\gz}{\zeta}
\newcommand{\gee}{\eta}
\newcommand{\gth}{\theta}
\newcommand{\gi}{\iota}
\newcommand{\gk}{\kappa}
\newcommand{\gl}{\lambda}
\newcommand{\gm}{\mu}
\newcommand{\gn}{\nu}
\newcommand{\gx}{\xi}
\newcommand{\gom}{\omicron}
\newcommand{\gp}{\pi}
\newcommand{\gr}{\rho}
\newcommand{\gs}{\sigma}
\newcommand{\gt}{\tau}
\newcommand{\gu}{\upsilon}
\newcommand{\gph}{\phi}
\newcommand{\gch}{\chi}
\newcommand{\gps}{\psi}
\newcommand{\go}{\omega}
\newcommand{\gA}{A}
\newcommand{\gB}{B}
\newcommand{\gG}{\Gamma}
\newcommand{\gD}{\Delta}
\newcommand{\gEp}{E}
\newcommand{\gZ}{Z}
\newcommand{\gEe}{H}
\newcommand{\gTh}{\Theta}
\newcommand{\gI}{I}
\newcommand{\gK}{K}
\newcommand{\gL}{\Lambda}
\newcommand{\gM}{M}
\newcommand{\gN}{N}
\newcommand{\gX}{\Xi}
\newcommand{\gOm}{O}
\newcommand{\gP}{\Pi}
\newcommand{\gR}{P}
\newcommand{\gS}{\Sigma}
\newcommand{\gT}{T}
\newcommand{\gU}{\Upsilon}
\newcommand{\gPh}{\Phi}
\newcommand{\gCh}{X}
\newcommand{\gPs}{\Psi}
\newcommand{\gO}{\Omega}
\newcommand{\bA}{{\bf A}}
\newcommand{\bB}{{\bf B}}
\newcommand{\bG}{\boldGamma}
\newcommand{\bD}{\boldDelta}
\newcommand{\bEp}{{\bf E}}
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bEe}{{\bf H}}
\newcommand{\bTh}{\boldTheta}
\newcommand{\bI}{{\bf I}}
\newcommand{\bK}{{\bf K}}
\newcommand{\bL}{{\bf L}}
\newcommand{\bM}{{\bf M}}
\newcommand{\bN}{{\bf N}}
\newcommand{\bX}{\boldXi}
\newcommand{\bOm}{{\bf O}}
\newcommand{\bP}{\boldPi}
\newcommand{\bR}{{\bf P}}
\newcommand{\bS}{\boldSigma}
\newcommand{\bT}{{\bf T}}
\newcommand{\bU}{\boldUpsilon}
\newcommand{\bPh}{\boldPhi}
\newcommand{\bCh}{{\bf X}}
\newcommand{\bPs}{\boldPsi}
\newcommand{\bO}{\boldOmega}
\newcommand{\rest}{\restriction}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\ov}{\overline}
\newcommand{\add}{{\rm Add}}
\newcommand{\K}{{\mathfrak K}}
\newcommand{\U}{{\cal U}}
%
% Hebrew letters
%
\newcommand{\ha}{\aleph}
\newcommand{\hb}{\beth}
\newcommand{\hg}{\gimel}
\newcommand{\hd}{\daleth}
%
% basic set theory constructions
%
\newcommand{\setof}[2]{{\{\; #1 \; \vert \; #2 \; \} } }
\newcommand{\seq}[1]{{\langle #1 \rangle} }
\newcommand{\card}[1]{{\vert #1 \vert} }
\newcommand{\ot}[1]{\hbox{o.t.($#1$)}}
\newcommand{\forces}{\Vdash}
\newcommand{\decides}{\parallel}
\newcommand{\ndecides}{\nparallel}
\renewcommand{\models}{\vDash}
\newcommand{\powerset}{{\cal P}}
\newcommand{\bool}{{\bf b} }%
%
% stuff for use inside math formulae
%
\newcommand{\dom}{{\rm dom}}
\newcommand{\rge}{{\rm rge}}
\newcommand{\crit}{{\rm crit}}
\renewcommand{\top}{{\rm top}}
\newcommand{\supp}{{\rm supp}}
\newcommand{\support}{{\rm support}}
\newcommand{\cf}{{\rm cf}}
\newcommand{\lh}{{\rm lh}}
\newcommand{\lp}{{\rm lp}}
\newcommand{\up}{{\rm up}}
\newcommand{\FF}{{\mathbb F}}
\newcommand{\FP}{{\mathbb P}}
\newcommand{\FQ}{{\mathbb Q}}
\newcommand{\FR}{{\mathbb R}}
\newcommand{\FS}{{\mathbb S}}
\newcommand{\FT}{{\mathbb T}}
\newcommand{\implies}{\Longrightarrow}
%\newcommand{\commtriangle}[6]
%{
%\medskip
%\[
%\setlength{\dgARROWLENGTH}{6.0em}
%\begin{diagram}
%\node{#1} \arrow[2]{e,t}{#6} \arrow{se,b}{#4} \node[2]{#3} \\
%\node[2]{#2} \arrow{ne,r}{#5}
%\end{diagram}
%\]
%\medskip
%}
%
% This picture tells you what order to put the arguments in
%
%
%
%
% #6
% #1 --------- #3
% \ /
% \#4 / #5
% \ /
% #2/
%
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}[definition]{Remark}
\newtheorem{fact}[definition]{Fact}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{claim}[definition]{Claim}
\newtheorem{conjecture}{Conjecture}
\newenvironment{proof}{\noindent{\bf
Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
\par\bigskip}
\newenvironment{sketch}{\noindent{\bf
Sketch of Proof:}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newenvironment{pf}{\indent{${}$}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newcommand{\lra}{\longrightarrow}
\setlength{\topmargin}{-0.62in}
\setlength{\textheight}{9.10in}
\setlength{\oddsidemargin}{-0.15in}
\setlength{\textwidth}{6.95in}
\setlength{\parindent}{1.5em}
% IndWC.tex
% The following macros are a selection from Joel's general math
% macros used in the document below
%
\def\tlt{\triangleleft}
\def\k{\kappa}
\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\s{\sigma}
\def\t{\tau}
\def\l{\lambda}
\def\lted{{{\leq}\d}}
\def\ltk{{{<}\k}}
%
% Arthur, in the next two macro definitions, use blackboard bold for
% \bm. Latex uses a different name I think.
%
\def\P{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Qdot{\dot\Q}
\def\Pforces{\forces_{\P}}
\def\of{{\subseteq}}
%\def\card#1{\left|#1\right|}
\def\boolval#1{\mathopen{\lbrack\!\lbrack}\,#1\,\mathclose{\rbrack\!
\rbrack}}
\def\restrict{\mathbin{\mathchoice{\hbox{\am\char'26}}{\hbox{\am\char'
26}}{\hbox{\eightam\char'26}}{\hbox{\sixam\char'26}}}}
\def\st{\mid}
\def\set#1{\{\,{#1}\,\}}
\def\th{{\hbox{\fiverm th}}}
\def\muchgt{>>}
\def\cof{\mathop{\rm cof}\nolimits}
\def\iff{\mathrel{\leftrightarrow}}
\def\intersect{\cap}
\def\minus{\setminus}
\def\Union{\bigcup}
\def\union{\bigcup}
\def\and{\mathrel{\kern1pt\&\kern1pt}}
\def\image{\mathbin{\hbox{\tt\char'42}}}
\def\elesub{\prec}
\def\iso{\cong}
\def\<#1>{\langle\,#1\,\rangle}
\def\ot{\mathop{\rm ot}\nolimits}
%
% ------------------------------------------------------------------------------
%
\title{On Level by Level Equivalence and Inequivalence between
Strong Compactness and Supercompactness
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
non-reflecting stationary set of ordinals,
level by level equivalence, level by level
inequivalence}}
\author{Arthur W.~Apter
\thanks{The author's research which led to Theorem \ref{t1}
was partially supported
by PSC-CUNY Grant 61449-00-30 and by the
Logic Institute of the University of Vienna.
In particular, the author wishes to thank
Sy Friedman, Ralf-Dieter Schindler, and
their families for the hospitality shown
him when he visited Vienna during his
sabbatical. The author also wishes to thank
the referee for helpful comments and
suggestions which have been incorporated into
this version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://math.baruch.cuny.edu/$\sim$apter\\
awabb@cunyvm.cuny.edu}
\date{February 21, 2001\\
(revised August 26, 2001)}
\begin{document}
\maketitle
\begin{abstract}
We prove two theorems, one concerning
level by level inequivalence between
strong compactness and supercompactness,
and one concerning level by level
equivalence between strong compactness
and supercompactness.
We first show that in a universe
containing a supercompact cardinal but of
restricted size, it is possible to
control precisely the difference
between the degree of strong compactness
and supercompactness that any measurable cardinal exhibits.
We then show that in an unrestricted size
universe containing many supercompact cardinals,
it is possible to have
significant failures of GCH along with
level by level equivalence between
strong compactness and supercompactness,
except possibly at inaccessible levels.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{AS97a}, Shelah and the author began the
study of level by level equivalence between
strong compactness and supercompactness
by proving the following theorem.
\begin{theorem}\label{t0}
Let
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC + GCH + $\K$ is
the class of supercompact cardinals +
For every pair of regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except
possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$
supercompact''.
\end{theorem}
In any model witnessing the conclusions of
Theorem \ref{t0}, we will say that
level by level equivalence between
strong compactness and supercompactness holds.
Note that the exception in Theorem \ref{t0}
is provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
Observe also that Theorem \ref{t0} is a
strengthening of the result of Kimchi and
Magidor \cite{KM}, who showed it is
consistent for the classes of strongly
compact and supercompact cardinals to
coincide precisely, except at measurable
limit points.
The purpose of this paper is to
continue the investigation begun in
\cite{AS97a} by establishing two new
theorems, one concerning level by level
inequivalence between strong compactness and
supercompactness, and one concerning level by
level equivalence between strong compactness and
supercompactness in the context of the
failure of GCH. Specifically, we prove
the following, taking as a notational
convention for the rest of the paper
that for $\gd$ any non-supercompact
measurable cardinal, $\theta_\gd$
is the least cardinal so that $\gd$
isn't $\theta_\gd$ supercompact, and
for $\ga$ an arbitrary ordinal,
$\gl_\ga$ is the least inaccessible cardinal
above $\ga$.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + GCH + $\gk_0$ is
$\rho = \gl_{\gk_0}$ supercompact +
There is some supercompact ultrafilter
$\U$ over $P_{\gk_0}(\rho)$ so that for
$j : V \to M$ for
$j = j_\U$ the associated elementary embedding,
$M \models ``\gk_0$ is $\rho$ supercompact''.
%For any non-supercompact measurable cardinal
%$\gd$, let $\gth_\gd$ be the least cardinal so that
%$\gd$ isn't $\gth_\gd$ supercompact.
There are then cardinals $\gk, \gl < \gk_0$ and
%$\gl \le \gl_0$ and
a partial ordering $\FP \in V$ so that for
$V^\ast$ the model $V^\FP$ truncated at $\gl$,
$V^\ast \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal $\gd > \gk$ is
inaccessible + If $\gd < \gk$ is measurable,
%$\gd$ is ${<}\gth_\gd$ supercompact iff
$\gd$ is $\gth^{+ 5}_\gd$ strongly compact but
%$\gd$
isn't $\gth^{+ 6}_\gd$ strongly compact''.
\end{theorem}
\begin{theorem}\label{t2}
Let
$V \models ``$ZFC + $\K \neq \emptyset$ is the
class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ so that
$V^\FP \models ``$ZFC + $\K$ is the class of
supercompact cardinals + $2^\gd = \gd^{++}$ if
$\gd$ is inaccessible + $2^\gd = \gd^+$ if
$\gd$ isn't inaccessible + For every pair of
regular cardinals $\gk < \gl$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact,
except possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact,
or $\gl$ is inaccessible''.
\end{theorem}
We take this opportunity to make several
comments concerning Theorems \ref{t1} and \ref{t2}.
We note that in Theorem \ref{t1}, we have produced
a model containing a supercompact cardinal $\gk$
with no inaccessibles above it so that the degree
of level by level {\it inequivalence}
between strong compactness and supercompactness
below $\gk$ is
precisely controlled, in the sense we know
that for any measurable cardinal $\gd$,
$\gd$'s degree of supercompactness fails
right at $\theta_\gd$, yet
$\gd$ must be exactly $\gth^{+ 5}_\gd$
strongly compact.
Also, as our proof will indicate, there is nothing
special about $+ 5$ indicating the degree of level by level
inequivalence below $\gk$ between strong compactness and
supercompactness.
It will be possible, e.g., for the degree of
level by level inequivalence between strong
compactness and supercompactness for $\gd$
to be given by $+ \varphi(\gd)$
(meaning that $\gd$ is
$\theta^{+ \varphi(\gd)}_\gd$ strongly compact but
$\gd$ isn't
$\theta^{+ \varphi(\gd) + 1}_\gd$ strongly compact), where
$\varphi(x)$ is a formula in one free variable
that defines a function sending any ordinal
$\ga$ to an ordinal strictly below the
least inaccessible above $\ga$, subject to
one restriction. This is that
$\theta^{+ \varphi(\gd)}_\gd$ can't be
a singular (strong limit)
cardinal of cofinality below $\gd$.
We will comment on this in greater detail immediately
following the proof of Theorem \ref{t1}.
Finally, Theorem \ref{t2} should be contrasted with
Theorem 4 of \cite{A99c}, where a model containing
a supercompact cardinal $\gk$
with no measurables above $\gk$ and in which no cardinal
$\gd$ is $\gl$ supercompact for $\gl > \gd$ measurable
is produced in which there is precise level by level
equivalence below $\gk$ between strong compactness
and supercompactness, yet for every measurable
cardinal $\gd$, $\gd$ is $\gd^+$ supercompact and
$2^\gd = \gd^{++}$.
In Theorem \ref{t2} of this paper, there are no
restrictions on the size of the universe,
yet there remains the ambiguity of whether
$\gk$ is $\gl$ strongly compact iff $\gk$ is
$\gl$ supercompact, whenever $\gl$ is
inaccessible and $\gk$ is not a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
Before giving the proofs of our theorems, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb], [\ga, \gb), (\ga, \gb]$, and $(\ga,
\gb)$ are as
in standard interval notation.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
If we also have that $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
is an Easton support iteration of length $\gk$
so that at stage $\ga$, a non-trivial forcing is done
adding a subset of $\gd_\ga$, then we will say that
$\gd_\ga$ is in the field of $\FP$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
Let $\gk$ be a regular cardinal.
The partial ordering
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$
(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an
upper bound $p \in \FP$. $\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages (choosing the
trivial condition at stage 0),
then player II has a strategy which
ensures the game can always be continued.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$ \FP$ is ${<}\gk$-strategically closed if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
%Note that trivially, if $\FP$ is ${<}\gk$-closed, then $\FP$ is
%${<}\gk$-strategically
%closed and ${\prec}\gk $-strategically closed. The converse of
%both of these facts is false.
Suppose now that $\gk < \gl$ are regular cardinals.
A partial ordering
$\FP(\gk, \gl)$ that will be used in the proof
of Theorem \ref{t1}
is the partial ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\gk$ to $\gl$. Specifically,
$\FP(\gk, \gl) =
\{ p$ : For some
$\ga < \gl$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\gb \in S_p$ implies
$\gb > \gk$ and cof$(\gb) = \gk \}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $\FP(\gk, \gl)$ (see
\cite{Bu}, \cite{A99b}, or \cite{KM}), in $V[G]$,
if we assume GCH holds in $V$,
a non-reflecting stationary
set $S=S[G]=\bigcup\{S_p:p\in G\} \subseteq \gl$
of ordinals of cofinality $\gk$ has been introduced,
the bounded subsets of $\gl$ are the same as those in $V$,
and cardinals, cofinalities, and GCH
have been preserved.
It is also virtually immediate that $\FP(\gk, \gl)$
is $\gk$-directed closed, and it can be shown
(see \cite{Bu}, \cite{A99b}, or \cite{KM}) that
$\FP(\gk, \gl)$
is ${\prec}\gl$-strategically closed.
In the proof of Theorem \ref{t2},
we will employ
the standard Cohen partial ordering
for adding $\gl$ subsets of $\gk$,
$\add(\gk, \gl)$.
As opposed to the most common usage,
however, where $\gk < \gl$ are both
regular cardinals, we will allow
$\gl$ to be an arbitrary ordinal when appropriate.
Assuming GCH holds for cardinals at and above
$\gk$, this will not change the fact that
$\add(\gk, \gl)$ is
$\gk$-directed closed and $\gk^+$-c.c.
We mention that we are assuming familiarity with the
large cardinal notions of measurability,
strong compactness, and supercompactness.
Interested readers may consult \cite{K}
or \cite{SRK}
for further details.
We note only that the cardinal
$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
supercompact for every cardinal $\gd < \gl$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V$ be a model for the hypotheses of
Theorem \ref{t1}, i.e., assume
$V \models ``$ZFC + GCH +
%${\gk_0} < {\gl_{\gk_0}}$ are so that
${\gk_0}$ is $\rho = \gl_{\gk_0}$ supercompact +
%${\gl_{\gk_0}}$ is the least inaccessible above ${\gk_0}$, and
There is some supercompact ultrafilter
$\U$ over $P_{\gk_0}(\rho)$ so that for
$j : V \to M$ for
$j = j_\U$ the associated elementary embedding,
$M \models ``{\gk_0}$ is $\rho$ supercompact''.
%For any $\gd < \gk_0$, let $\gl_\gd$ be
%the least inaccessible cardinal above $\gd$.
This allows us to define the first partial ordering
$\FP^0$ we use in the proof of Theorem \ref{t1}
as the Easton support iteration of length ${\gk_0}$ which
begins by adding a Cohen subset of $\go$ and
then adds, to every measurable cardinal
$\gd < {\gk_0}$ which isn't $\gl_\gd$ supercompact,
%for $\gl_\gd$ the least inaccessible cardinal
%above $\gd$,
a non-reflecting stationary set
of ordinals of cofinality $\go$. Since
$M \models ``{\gk_0}$ is $\rho$ supercompact and
$\rho$ is the least inaccessible cardinal above ${\gk_0}$'',
by reflection,
$A = \{\gd < {\gk_0} : \gd$ is $\gl_\gd$ supercompact$\}$
is unbounded in ${\gk_0}$.
We now sketch the proof of Lemma 3.2 of
\cite{A99b}, which allows us to infer that
$V^{\FP^0} \models ``$For every cardinal
$\gd < {\gk_0}$, $\gd$ is measurable iff
$\gd$ is $\gl_\gd$ strongly compact''.
By Hamkins' work of \cite{H1}, \cite{H2},
and \cite{H3} and the fact that every
element of $\gk_0 - A$ contains a non-reflecting
stationary set of ordinals of
cofinality $\go$,
$V^{\FP^0} \models ``$The only measurable
cardinals $\gd < \gk_0$ are those $\gd$
which were $\gl_\gd$ supercompact in $V$''.
For any $\gd < \gk_0$ which is $\gl_\gd$ supercompact
in $V$, write
$\FP^0 = \FP_\gd \ast \dot \FP^\gd$, where
the field of $\FP_\gd$ is composed of
ordinals below $\gd$.
From this factorization, we see that
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
$2^{\gl_\gd}$-strategically closed''.
Thus, it is the case that
$V^{\FP^0} \models ``\gd$ is $\gl_\gd$
strongly compact'' iff
$V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$
strongly compact''.
The fact that
$V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$
strongly compact'', however, follows
using an argument due to Magidor, unpublished
by him but exposited in \cite{AC1},
\cite{AC2}, \cite{A99b}, and,
in a more complicated form, in Claim 1 of
Case 2 of Lemma \ref{l3} of this paper.
This completes our proof sketch of
Lemma 3.2 of \cite{A99b}.
%Therefore,
%by Lemma 3.2 of \cite{A99b} and the
%succeeding remarks,
In addition, the usual Easton arguments show
%cardinals and cofinalities are preserved
%when forcing with $\FP^0$ and
$V^{\FP^0} \models {\rm GCH}$, and since
$\FP^0$ can be defined so that
$|\FP^0| = \gk_0$,
$V^{\FP^0} \models ``\rho$ is the least
inaccessible cardinal above $\gk_0$''.
\begin{lemma}\label{l1}
$V^{\FP^0} \models ``{\gk_0}$ is ${\rho}$ supercompact''.
\end{lemma}
\begin{proof}
Let $H_0$ be $V$-generic over $\FP^0$, and
let $j : V \to M$ be as in the hypotheses of
Theorem \ref{t1}.
Since
$M \models ``{\rho}$ is the least
inaccessible cardinal above ${\gk_0}$ and
${\gk_0}$ is ${\rho}$ supercompact'',
$j(\FP^0) = \FP^0 \ast \dot \FQ$, where
${\gk_0} \not\in {\rm field}(\dot \FQ)$ and
the least ordinal in the field of
$\dot \FQ$ is above ${\rho}$.
Therefore, as
$M[H_0] \models ``|{\FQ}| = j({\gk_0})$ and
$|{2^{\FQ}}| = {j({\gk_0})}^+ = j(\gk^+_0)$'' and
$V \models ``|({j(\gk^+_0)}| =
|{\{f : f : P_{\gk_0}({\rho}) \to \gk^+_0\}}| =
|{[\gk^+_0]}^{\rho}| = \rho^+$'',
$V[H_0] \models ``$There are $\rho^+$ subsets of
$\FQ$ present in $M[H_0]$''.
Hence, we can let
$\la D_\ga : \ga < \rho^+ \ra$ enumerate in $V[H_0]$
all dense open subsets of $\FQ$ present in $M[H_0]$, where
for the purposes of the argument to be given below, we
assume without loss of generality that
for every dense open set
$D \in \la D_\ga : \ga < \rho^+ \ra$,
$D = D_\gb$ for some odd ordinal $\gb$.
%$D_\ga = D_{\ga + 1}$ for every $\ga < \rho^+$.
By the fact
$M^{\rho} \subseteq M$,
$\forces_{\FP^0} ``\dot \FQ$ is
${\prec}\rho^+$-strategically closed'' in both
$M$ and $V$.
Therefore, since standard arguments show
$M[H_0]$ remains ${\rho}$ closed with respect to $V[H_0]$,
$V[H_0] \models ``\FQ$ is ${\prec}\rho^+$-strategically closed''.
We can now construct an $M[H_0]$-generic object
$H$ over $\FQ$ in $V[H_0]$ as follows.
Players I and II play a game of
length $\rho^+$.
The initial pair of moves is generated by
player II choosing the trivial
condition $q_0$ and player I responding by
choosing $q_1 \in D_1$. Then,
at an even stage $\ga + 2$,
%since $\FQ$ is
%${\prec} \rho^+$-strategically closed,
player II picks $q_{\ga + 2} \ge
q_{\ga + 1}$ by using some fixed
strategy ${\cal S}$, where
$q_{\ga + 1}$ was chosen by player I
to be so that
$q_{\ga + 1} \in D_{\ga + 1}$ and
$q_{\ga + 1} \ge q_\ga$.
%$p_\ga \in D_\ga$ extending
%$\sup(\la q_\gb : \gb < \ga \ra)$
%(initially, $q_{-1}$ is the empty condition)
%and player II responds by picking
%$q_\ga \ge p_\ga$ (so $q_\ga \in D_\ga$).
If $\ga$ is a limit ordinal, player II uses
${\cal S}$ to pick
$q_\ga$ extending each $q_\gb$ for $\gb < \ga$.
%$\sup(\la q_\gb : \gb < \ga \ra)$.
By the ${\prec}\rho^+$-strategic closure of
$\FQ$ in both $M[H_0]$ and $V[H_0]$,
%player II has a winning strategy for this
%game, so
the sequence $\la q_\ga : \ga < \rho^+ \ra$
as just described exists. By construction,
%can be taken
%as an increasing sequence of conditions with
%$q_\ga \in D_\ga$ for $\ga < \gl^+_0$. Clearly,
$H = \{p \in \FQ : \exists \ga <
\rho^+ [q_\ga \ge p]\}$ is our
$M[H_0]$-generic object over $\FQ$.
Thus, $j : V \to M$ extends in $V[H_0]$ to
$j : V[H_0] \to M[H_0][H]$, so
$V[H_0] \models ``{\gk_0}$ is ${\rho}$ supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
We remark that the proof of Lemma \ref{l1}
remains valid under different circumstances.
Specifically, suppose we are forcing over a
ground model $V'$ satisfying GCH.
Suppose further
$\gk' < \gl'$ are so that
$\gk'$ is $\gl'$ supercompact,
$\gl'$ is a cardinal
having cofinality at least $\gk'$,
$B \subseteq \gk'$ is an unbounded set of
regular cardinals with
$\go \not\in B$, and for some
$j' : V' \to M'$ witnessing the $\gl'$
supercompactness of $\gk'$
generated by a supercompact
ultrafilter ${\cal U}'$ over
$P_{\gk'}(\gl')$,
$j'(B) \cap [\gk', \gl'] = \emptyset$. If
$\FQ$ is an Easton support iteration
which begins by adding a Cohen subset of $\go$
and then adds, for every $\gd \in B$,
a non-reflecting stationary set of ordinals
of cofinality $\go$, the proof just
given shows that after forcing with $\FQ$,
$\gk'$ remains $\gl'$ supercompact.
%does not require that
%$\gl_0$ be a regular cardinal.
%The argument is equally valid if $\gl_0$ is
%a singular strong limit cardinal of cofinality
%at least $\gk_0$.
Take now $V_0 = V^{\FP^0}$. Let $\gk$
%\le \gk_0$ and $\gl = \gl_\gk \le \gl_0$
be minimal so that
$V_0 \models ``\gk$ is ${<}\gl_\gk$ supercompact'',
and let $\gl = \gl_\gk$. Since
$V_0 \models ``\gk_0$ is $\rho = \gl_{\gk_0}$
supercompact'', again by reflection,
$\gk, \gl < \gk_0$.
Working in $V_0$, let
$\la \gd_\ga : \ga < \gk \ra$ be an
increasing enumeration of the measurable
cardinals below $\gk$.
By the choice of
$\gk$ and $\gl$, it must be the case that
$\theta_\gd$ exists for every measurable cardinal
$\gd < \gk$ and
$\theta_\gd < \gl_{\gd}$.
We therefore define the second partial ordering
$\FP^1$ we use in the proof of Theorem \ref{t1}
as the Easton support iteration of length
$\gk$ which begins by adding a Cohen subset of $\go$
and then adds, for every $V_0$-measurable cardinal
$\gd < \gk$, a non-reflecting stationary set of
ordinals of cofinality $\go$ to
${(\theta^{+ 6}_{\gd})}^{V_0}$.
Let $\FP = \FP^0 \ast \dot \FP^1$.
Once again, the usual Easton arguments show
%cardinals and cofinalities are preserved
%when forcing with $\FP^1$ and
$V^{\FP^1}_0 = V^{\FP^0 \ast
\dot \FP^1} = V^\FP \models {\rm GCH}$.
Also, as before, since $\FP^1$ can be
defined in $V_0$ so as to have cardinality
$\gk$, $V^{\FP^1}_0 = V^{\FP^0 \ast \dot \FP^1} =
V^\FP \models ``\gl$ is the least inaccessible cardinal
above $\gk$''.
\begin{lemma}\label{l2}
$V^{\FP^1}_0 = V^\FP \models ``\gk$ is ${<}\gl$ supercompact''.
\end{lemma}
\begin{proof}
Let $\gd < \gl$ be a successor cardinal, and let
$k : V_0 \to M^*$ be an elementary embedding
generated by a supercompact ultrafilter over
$P_\gk(\gd)$ so that
$M^* \models ``\gk$ isn't $\gd$ supercompact''.
By the definition of $\FP^1$ and the choice of $k$,
$k(\FP^1) = \FP^1 \ast \dot \FQ$, where the
least ordinal in the field of $\dot \FQ$ is above
${(\gd^+)}^{M^*} = {(\gd^+)}^{V_0}$. Since
$V_0 \models {\rm GCH}$, this means we can apply
the argument given in Lemma \ref{l1} to show that
$V^{\FP^1}_0 \models ``\gk$ is $\gd$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$For every measurable cardinal
$\gd < \gk$,
$\gd$ is $\theta^{+ 5}_\gd$ strongly compact
but isn't $\theta^{+ 6}_\gd$ strongly compact''.
\end{lemma}
\begin{proof}
Let $\gd < \gk$ be measurable in $V^\FP$.
%For the appropriate value of
%$\theta_\gd$, we will then have
%$V^\FP = V^{\FP_1}_0 \models ``\gd$ is
%${<}\theta_\gd$ supercompact''.
%with $\ga_0$ so that $\gd = \gd_{\ga_0}$.
Write
$\FP^1 = \FP' \ast \dot \FP''$, where
$|\FP'| = \go$ and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
In Hamkins' terminology of \cite{H1}, \cite{H2}, and \cite{H3},
$\FP^1$ ``admits a low gap'', so by the results of
\cite{H1}, \cite{H2}, and \cite{H3},
$V_0 \models ``\gd$ is measurable''.
%${<}\theta_\gd$ supercompact''.
For $n = 0,1$, write
$\FP^n = \FP^n_\gd \ast \dot \FP^{n,\gd}$, where
$\FP^n_\gd$ is the portion of $\FP^n$ whose
field consists of ordinals below $\gd$.
We consider the following two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: $V_0 \models ``|\FP^1_\gd| < \gd$''.
Since by construction,
$V_0 \models ``\gd$ is $\gl_\gd$ strongly compact'',
the L\'evy-Solovay results \cite{LS} imply that
$V^{\FP^1_\gd}_0 \models ``\gd$ is
$\gl_\gd$ strongly compact and
$\theta_\gd = {(\theta_\gd)}^{V_0}$''.
As by its definition,
$\forces_{\FP^1_\gd} ``\FP^{1,\gd}$ is
$\theta^{+ 5}_\gd$-strategically closed and adds a
non-reflecting stationary set of ordinals of
cofinality $\go$ to $\theta^{+ 6}_\gd$'',
the closure properties of $\FP^{1,\gd}$ together
with an application of Theorem 4.8 of
\cite{SRK} and the succeeding remarks yield
$V^{\FP^1_\gd \ast \dot \FP^{1,\gd}}_0 =
V^{\FP^1}_0 =
V^\FP \models
``\gd$ is ${<}\theta_\gd$ supercompact,
$\gd$ is $\theta^{+ 5}_\gd$ strongly compact, and
$\gd$ isn't $\theta^{+ 6}_\gd$ strongly compact''.
\bigskip
Case 2: $V_0 \models ``|\FP^1_\gd| = \gd$''.
Let $\FP^* = \FP^0_\gd \ast \dot \FP^1_\gd$.
Note that by the definition of $\FP$,
with a slight abuse of notation, the
definition of $\FP^*$ just given makes sense.
We prove Case 2 via a series of claims,
with Claim 1 stating that $\gd$ is $\gl_\gd$
strongly compact after forcing over $V$ with
$\FP^*$,
%$\FP^0_\gd \ast \dot \FP^1_\gd$,
Claim 2 stating that $\gth_\gd$ does not
change going from $V_0$ to
$V^{\FP^1}_0 = V^\FP$, and
Claim 3 stating that Claims 1 and 2
allow us to finish as in Case 1.
\bigskip
Claim 1: $V^{\FP^*} = V^{\FP^0_\gd \ast
\dot \FP^1_\gd} \models
``\gd$ is $\gl_\gd$ strongly compact''.
\setlength{\parindent}{1.5em}
\begin{proof}
To prove Claim 1, we use a modification of the
argument due to Magidor
alluded to in the paragraph immediately
preceding Lemma \ref{l1}.
%for the preservation
%of strong compactness, unpublished by him
%but exposited in \cite{AC1}, \cite{AC2},
%\cite{A99}, \cite{A99a}, \cite{A00a}, and \cite{AH4}.
%and \cite{A99b}.
By the definition of $\FP^0$, it must be
the case that
$V \models ``\gd$ is $\gl_\gd$ supercompact''.
Further, the definitions of
$\FP^0$ and $\FP^1$ show that
%coupled with
%the results of \cite{LS} show that
$\gl_\gd$ has the same meaning in
$V$, $V^{\FP^0}$,
$V^{\FP^0 \ast \dot \FP^1} = V^\FP$, and
$V^{\FP^*}$.
Therefore, there is no ambiguity in choosing
$k_1 : V \to M^{**}$ to be an elementary
embedding witnessing the $\gl_\gd$
supercompactness of $\gd$
generated by a supercompact ultrafilter
over $P_\gd(\gl_\gd)$ so that
$M^{**} \models ``\gd$ isn't $\gl_\gd$ supercompact''.
Since $M^{**} \models ``\gd$ is measurable'',
we may choose a normal ultrafilter ${\cal U}''$
of Mitchell order $0$ over $\gd$ so that
$k_2 : M^{**} \to N$ is an elementary embedding witnessing the
measurability of $\gd$ definable in $M^{**}$ with
$N \models ``\gd$ isn't measurable''.
It is the case that if
$k : V \to N$ is an elementary embedding with
critical point $\gd$
and for any $x \subseteq N$ with
$|x| \le \gl_\gd$, there is some $y \in N$
so that $x \subseteq y$ and
$N \models ``|y| < k(\gd)$'',
then $k$ witnesses the $\gl_\gd$
strong compactness of $\gd$.
Using this fact,
it is easily verifiable that
$j^* = k_2 \circ k_1$ is an elementary embedding
witnessing the $\gl_\gd$ strong compactness of $\gd$.
We show that $j^*$ extends to
$j^* : V^{\FP^*} \to N^{j^*(\FP^*)}$.
%for $\FP^* = \FP^0_\gd \ast \dot \FP^1_\gd$.
This extended embedding will witness
the $\gl_\gd$ strong compactness of $\gd$ in
$V^{\FP^*}$.
\setlength{\parindent}{1.5em}
To do this, write
$j^*(\FP^*)$ as
$(\FP^0_\gd \ast \dot \FQ^0 \ast \dot \FR^0) \ast
(\dot \FP^1_\gd \ast \dot \FQ^1 \ast \dot \FR^1)$, where
for $n = 0,1$, the
$\dot \FQ^n$ are terms for the portions of
$j^*(\FP^*)$ between $\gd$ and $k_2(\gd)$ and the
$\dot \FR^n$ are terms for the rest of
$j^*(\FP^*)$, i.e., the parts above $k_2(\gd)$.
Note that by the definition of $\FP^0_\gd$, since
$N \models ``\gd$ is inaccessible but isn't measurable'',
$\gd \not\in {\rm field}(\dot \FQ^0)$.
Thus, the field of $\dot \FQ^0$
is composed of an unbounded subset of
$N$-measurable cardinals in the interval
$(\gd, k_2(\gd)]$. As
$M^{**} \models ``\gd$ is measurable but
isn't $\gl_\gd$ supercompact'',
by the definition of $\FP^0_\gd$,
$\gd \in {\rm field}(k_1(\FP^0_\gd))$.
This means that by elementarity,
$k_2(\gd) \in {\rm field}(\dot \FQ^0)$. Further,
the field of $\dot \FR^0$ is also composed of an
unbounded subset of the
$N$-measurable cardinals in the interval
$(k_2(\gd), k_2(k_1(\gd)))$.
Let $H = H_0 \ast H_1$ be $V$-generic over
$\FP^* = \FP^0_\gd \ast \dot \FP^1_\gd$.
We construct in $V[H_0]$ an
$N[H_0]$-generic object $H^0_1$ over
$\FQ^0$ and an
$N[H_0][H^0_1]$-generic object $H^0_2$ over
$\FR^0$.
We then construct in
$V[H_0][H_1] = V[H]$ an
$N[H_0][H^0_1][H^0_2][H_1]$-generic object
$H^1_1$ over $\FQ^1$ and an
$N[H_0][H^0_1][H^0_2][H_1][H^1_1]$-generic object
$H^1_2$ over $\FR^1$.
Our construction will guarantee that
${j^*} '' (H_0 \ast H_1)
\subseteq (H_0 \ast H^0_1 \ast H^0_2) \ast
(H_1 \ast H^1_1 \ast H^1_2)$.
This means that
$j^* : V \to N$ extends to
$j^* : V[H_0 \ast H_1] \to N[H_0][H^0_1][H^0_2]
[H_1][H^1_1][H^1_2]$ in $V[H] = V[H_0][H_1]$,
meaning that
$V^{\FP^*} \models ``\gd$ is $\gl_\gd$ strongly
compact''.
To obtain $H^0_1$, note that since $k_2$
is generated by an
ultrafilter over $\gd$ and
since GCH holds in both $V$ and $N$,
$|k_2(\gd^+)| = |k_2(2^\gd)| =
|\{ f : f : \gd \to \gd^+$ is a function$\}| =
|{[\gd^+]}^\gd| = \gd^+$. Thus, as
$N[H_0] \models ``|\wp(\FQ^0)| = k_2(2^\gd)$'',
we can let
$\la D_\ga : \ga < \gd^+ \ra$ enumerate in
either $V[H_0]$ or
$M^{**}[H_0]$ the dense open subsets of
$\FQ^0$ found in $N[H_0]$ so that for
every dense open set
$D \in \la D_\ga : \ga < \gd^+ \ra$,
$D = D_\gb$ for some odd ordinal $\gb$.
Since
$N[H_0] \models ``\FQ^0$ is
${\prec}\gd^+$-strategically closed'',
the argument given in the proof of Lemma \ref{l1}
for the construction of the generic object $H$
is applicable here as well and allows us to build
$H^0_1$ in either $V[H_0]$ or $M^{**}[H_0]$ in the same manner.
We next construct in $V[H_0]$ the
desired $N[H_0][H^0_1]$-generic object
$H^0_2$ over $\FR^0$.
To do this, we first note that as
$\gd \in {\rm field}(k_1(\FP^0_\gd))$,
we can write
$k_1(\FP^0_\gd)$ as
$\FP^0_\gd \ast \dot \FS^0 \ast \dot \FT^0$, where
$\forces_{\FP^0_\gd} ``\dot \FS^0 = \dot \FP(\go, \gd)$'', and
$\dot \FT^0$ is a term for the rest of
$k_1(\FP^0_\gd)$.
Note now that
$M^{**} \models ``$There are no measurable cardinals
in the interval $(\gd, \gl_\gd]$''.
Thus, the field of
$\dot \FT^0$ is composed of an unbounded subset of
the $M^{**}$-measurable cardinals
in the interval $(\gl_\gd, k_1(\gd))$
which implies that in $M^{**}$,
$\forces_{\FP^0_\gd \ast \dot \FS^0}
``\dot \FT^0$ is ${\prec}\gl_\gd^+$-strategically
closed''. Further, since
$V \models {\rm GCH}$ and $\gl_\gd$
has cofinality at least $\gd$,
$|{[\gl_\gd]}^{< \gd}| = \gl_\gd$ and $2^{\gl_\gd} = \gl_\gd^+$.
Therefore, as $k_1$ is
generated by a supercompact ultrafilter over
$P_\gd(\gl_\gd)$,
$|{(k_1(\gd))}^+| =
|k_1(\gd^+)| = |k_1(2^{\gd)}| =
|2^{k_1(\gd)}| =
|\{ f : f : P_\gd(\gl_\gd) \to \gd^+$ is a function$\}| =
|{[\gd^+]}^{\gl_\gd}| = \gl^+_\gd$.
%|{[\gl_\gd]}^{\gl_\gd}| = \gg^{{+5}^+}$.
Work until otherwise specified in $M^{**}$. Consider the
``term forcing'' partial ordering $\FT^*$
(see \cite{C}, Section 1.2.5, page 8) associated with
$\dot \FT^0$, i.e., $\tau \in \FT^*$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^0_\gd \ast \dot \FS^0$ and
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau \in
\dot \FT^0$'', ordered by $\tau \ge \sigma$ iff
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau \ge \sigma$''.
Although $\FT^*$ as defined is technically a proper
class,
%by restricting the terms forced to appear in
%$\dot \FT^0$ to be a set,
it is possible to restrict the terms
appearing in it to a sufficiently large
set-sized collection, with the additional
crucial property that any term $\tau$
forced to be in $\dot \FT^0$ is also forced
to be equal to an element of $\FT^*$.
As we will show below,
this can be done in such a way that
$M^{**} \models ``|\FT^*| = k_1(\gd)$''.
Clearly, $\FT^* \in M^{**}$. Also, since
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\dot \FT^0$ is
${\prec}\gl_\gd^+$-strategically closed'',
it can easily be verified that $\FT^*$ itself is
${\prec}\gl_\gd^+$-strategically closed in $M^{**}$ and, since
${(M^{**})}^{\gl_\gd} \subseteq M^{**}$, in $V$ as well.
Observe that
$M^{**} \models ``k_1(\gd)$ is measurable and
$|\FP^0_\gd \ast \dot \FS^0| < k_1(\gd)$'' and
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\dot \FT^0$ is
an Easton support iteration of length $k_1(\gd)$ and
$|\dot \FT^0| = k_1(\gd)$''.
We can thus let $\dot f$ be a term so that
$\forces_{\FP^0_\gd \ast \dot \FS^0}
``\dot f : k_1(\gd) \to \dot \FT^0$ is
a bijection''.
Since
$M^{**} \models ``|\FP^0_\gd \ast \dot \FS^0| < k_1(\gd)$'',
for each $\ga < k_1(\gd)$, let
$S_\ga = \{ r^\ga_\gb : \gb < \eta^\ga < k_1(\gd) \}$
be a maximal incompatible set of elements of
$\FP^0_\gd \ast \dot \FS^0$ so that for some term
$\tau^\ga_\gb$,
$r^\ga_\gb \forces ``\tau^\ga_\gb = \dot f(\ga)$''.
Define $T_\ga = \{\tau^\ga_\gb : \gb < \eta^\ga \}$ and
$T = \bigcup_{\ga < k_1(\gd)} T_\ga$. Clearly,
$|T| = k_1(\gd)$, so we can let
$\la \tau_\ga : \ga < k_1(\gd) \ra$ enumerate the
members of $T$.
%Each sequence
%$\la \tau^\ga_\gb : \gb < \eta^\ga < k_1(\gd) \ra$
%can be used to define a sequence of terms
$\la \tau_\ga : \ga < k_1(\gd) \ra$ is so that if
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau \in \dot \FT^0$'',
then for some $\ga < k_1(\gd)$,
$\forces_{\FP^0_\gd \ast \dot \FS^0} ``\tau = \tau_\ga$''.
Therefore, we can restrict the set of terms we choose so that
we can assume that in $M^{**}$,
$|\FT^*| = k_1(\gd)$. Since
$M^{**} \models ``2^{k_1(\gd)} = {(k_1(\gd))}^+ =
k_1(\gd^+)$'',
this means we can let
$\la D_\ga : \ga < \gl_\gd^+ \ra$
enumerate in $V$ the dense open subsets of $\FT^*$
found in $M^{**}$,
so that as before, for every
dense open subset
$D \subseteq \FT^*$ present in $M^{**}$,
for some odd ordinal $\gb$,
$D = D_\gb$,
and argue as we did in Lemma \ref{l1} to construct in
$V$ an $M^{**}$-generic object $H^*$ over $\FT^*$.
Note that since $N$ is given
by an ultrapower of $M^{**}$ via a normal ultrafilter
${\cal U}'' \in M^{**}$ over $\gd$,
Fact 2 of Section 1.2.2 of \cite{C} tells us that
$k_2 '' H^*$ generates an $N$-generic object
$H^*_2$ over $k_2(\FT^*)$. By elementarity,
$k_2(\FT^*)$ is the term forcing in $N$
defined with respect to
$k_2(k_1(\FP^0_\gd)_{\gd + 1}) =
\FP^0_\gd \ast \dot \FQ^0$.
Therefore, since
$j^*(\FP^0_\gd) = k_2(k_1(\FP^0_\gd)) =
\FP^0_\gd \ast \dot \FQ^0 \ast
\dot \FR^0$,
$H^*_2$ is $N$-generic over
$k_2(\FT^*)$, and $H_0 \ast H^0_1$ is
$k_2(\FP^0_\gd \ast \dot \FS^0)$-generic over
$N$, Fact 1 of Section 1.2.5 of \cite{C}
tells us that for
$H^0_2 = \{i_{H_0 \ast H^0_1}(\tau) : \tau \in
H^*_2\}$, $H^0_2$ is $N[H_0][H^0_1]$-generic over
$\FR^0$.
%Thus, in $V[H_0]$,
%$j : V \to N$ extends to
%$j : V[H_0] \to N[H_0][H^0_1][H^0_2]$.
%This means
%$V^{\FP^0_\gd} \models ``\gd$ is $\gl_\gd$
%strongly compact''.
Working in $V[H_0][H_1]$, we build the
generic objects $H^1_1$ and $H^1_2$.
To construct $H^1_1$, we note that
by the strategic closure properties
of the partial orderings over which
$H^0_1$ and $H^0_2$ are generic,
$N[H_0][H^0_1][H^0_2]$ remains
$\gd$ closed with respect to
$V[H_0][H^0_1][H^0_2]$ = $V[H_0]$.
Therefore, since $\FP^1_\gd$ is
$\gd$-c.c.,
$N[H_0][H^0_1][H^0_2][H_1]$ remains
$\gd$ closed with respect to
$V[H_0][H^0_1][H^0_2][H_1] = V[H_0][H_1]$.
This means we can construct $H^1_1$ in
$V[H_0][H_1]$ in the same way $H^0_1$
was constructed in $V[H_0]$.
To build $H^1_2$,
we once again work in $M^{**}$.
Write $k_1(\FP^0_\gd \ast \dot \FP^1_\gd)$ as
$\FP^0_\gd \ast \dot \FS^0 \ast \dot \FT^0
\ast \dot \FP^1_\gd \ast \dot \FT^1$, where
$\dot \FS^0$ and $\dot \FT^0$ are as before, and
$\dot \FT^1$ is a term for the portion of
$k_1(\FP^0_\gd \ast \dot \FP^1_\gd)$ defined in
$M^{**}$ between stages $\gd$ and $k_1(\gd)$.
Since there are no measurable cardinals in $M^{**}$
in the interval $(\gd, \gl_\gd)$,
the field of $\dot \FT^1$ is actually composed
of ordinals in the interval $(\gl_\gd, k_1(\gd))$.
Let $\FT^{**}$ be
the term forcing partial ordering
associated with $\FT^1$, i.e.,
$\tau \in \FT^{**}$ iff $\tau$ is a
term in the forcing language with respect to
$\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd$ and
$\forces_{\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd} ``\tau \in
\dot \FT^1$'', ordered by
$\tau \ge \sigma$ iff
$\forces_{\FP^0_\gd \ast \dot \FS^0
\ast \dot \FT^0 \ast \dot \FP^1_\gd}
``\tau \ge \gs$''.
A similar analysis to that given for
the term forcing $\FT^*$, using the
observations made in the preceding paragraph,
now allows us to construct in $V$ an
$M^{**}$-generic object $H^{**}$ for $\FT^{**}$,
infer that $k_2 '' H^{**}$ generates an $N$-generic
object $H^{**}_2$
for the relevant term forcing partial
ordering in $N$, and working in
$V[H_0][H_1]$, use
$H^{**}_2$ to create an
$N[H_0][H^0_1][H^0_2][H_1][H^1_1]$-generic object
$H^1_2$ over $\FT^1$.
Thus, in $V[H_0][H_1]$,
$j^* : V \to N$ extends to
$j^* : V[H_0][H_1] \to N[H_0][H^0_1][H^0_2][H_1]
[H^1_1][H^1_2]$.
This means
$V^{\FP^0_\gd \ast \dot \FP^1_\gd} =
V^{\FP^*} \models ``\gd$ is $\gl_\gd$
strongly compact''.
\end{proof}
\setlength{\parindent}{0pt}
\bigskip
Claim 2:
$V^{\FP^1}_0 = V^\FP \models ``\theta_\gd =
{(\theta_\gd)}^{V_0}$''.
\setlength{\parindent}{1.5em}
\begin{proof}
To prove Claim 2, we assume inductively that
$\gd$ is the least cardinal
in $V_0$ for which this is not the case.
As before, working in $V_0$,
we can factor $\FP^1$ as
$\FP' \ast \dot \FP''$, where
$|\FP'| = \go$ and
$\forces_{\FP'} ``\dot \FP''$ is
$\ha_1$-strategically closed''.
Thus, by the results of \cite{H1},
\cite{H2}, and \cite{H3}, if
$V^{\FP^1}_0 \models ``\gd$ is $\gg$
supercompact'',
$V_0 \models ``\gd$ is $\gg$ supercompact''.
This means that
$V^{\FP^1}_0 = V^\FP \models ``\theta_\gd \le
{(\theta_\gd)}^{V_0}$'',
so by our assumptions on $\gd$,
$V^{\FP^1}_0 = V^\FP \models ``\theta_\gd <
{(\theta_\gd)}^{V_0}$''.
Let
$\gg < {(\theta_\gd)}^{V_0}$
be either a regular cardinal
or a singular (strong limit)
cardinal of cofinality at least
$\gd$. Choose
$i : V_0 \to M^{***}$ an elementary embedding
witnessing the $\gg$ supercompactness of
$\gd$ so that
$V_0 \models ``\gd$ isn't $\gg$ supercompact''.
By elementarity, since
$V_0 \models ``\gd$ is the least cardinal for which
%$\theta_\gd$
the first cardinal at which
$\gd$'s supercompactness fails
decreases in size when forcing with
$\FP^1$'',
$M^{***} \models ``i(\gd) > \gd$ is the least cardinal
for which the the first cardinal at which
$i(\gd)$'s supercompactness fails decreases
in size when forcing with $i(\FP^1)$''.
By GCH in $V_0$ and $M^{***}$,
$M^{***} \models ``\theta_\gd = \gg$''. Therefore,
by the definition of $\FP^1$ and the choice of $i$,
$i(\FP^1) = \FP^1 \ast \dot \FQ'$, where
the least ordinal in the field of
$\dot \FQ'$ is above
${(\gg^+)}^{M^{***}} = {(\gg^+)}^{V_0}$.
Another application of GCH in $V_0$ now allows us
once again to apply the argument given in
Lemma \ref{l1} to show that for
$G$ $V_0$-generic over $\FP^1$ and
$G_\gd = G \rest \FP^1_\gd$,
%where $\FP^1_\gd$ is the portion of $\FP^1$
%defined through stage $\gd$,
in $V_0[G_\gd]$, $i$ extends to
$i : V_0[G_\gd] \to M^{***}[G_\gd][G^*]$, where
$G^*$ is an $M^{***}[G_\gd]$-generic object over
$i(\FP^1_\gd)/G_\gd$ constructed in $V_0[G_\gd]$.
If we now write
$G = G_\gd \ast G^\gd$,
then since $G^\gd$ is
$V_0[G_\gd]$-generic over a partial ordering
which is
${(\theta^{+5}_\gd)}^{V_0}$-strategically closed and
$\gg < {(\theta_\gd)}^{V_0} < {(\theta^{+5}_\gd)}^{V_0}$,
$V^{\FP^1}_0 = V^\FP \models ``\gd$ is $\gg$ supercompact''.
Since $\gg$ can be any regular cardinal or
singular strong limit cardinal of cofinality
at least $\gd$ below
${(\theta_\gd)}^{V_0}$,
and since $\gd$'s supercompactness can't first fail
at the successor of a singular (strong limit) cardinal
of cofinality below $\gd$
(this is since if $\eta > \gd$ is a singular
strong limit cardinal of cofinality below $\gd$,
then $\gd$ is $\eta$ supercompact iff
$\gd$ is $\eta^+$ supercompact),
we have shown in contradiction to what we deduced above that
$V^{\FP^1}_0 = V^\FP \models ``\theta_\gd =
{(\theta_\gd)}^{V_0}$''.
\end{proof}
\setlength{\parindent}{0pt}
\bigskip
Claim 3: Claims 1 and 2 allow us to
finish as in Case 1.
\setlength{\parindent}{1.5em}
\begin{proof}
Writing now
$\FP^1 = \FP_\gd^1 \ast \dot \FQ^*$, it will
be the case that in $V_0$,
$\forces_{\FP^1_\gd} ``\dot \FQ^*$ is
${(\theta^{+5}_\gd)}^{V_0}$-strategically closed and
adds a non-reflecting stationary set of
ordinals of cofinality $\go$ to
${(\theta^{+6}_\gd)}^{V_0}$''.
By Claims 1 and 2,
we can therefore conclude as we did in Case 1
above that
$V^{\FP^1_\gd \ast \dot \FQ^*}_0 =
V^{\FP^1}_0 = V^\FP \models
``\gd$ is ${<}\theta_\gd$ supercompact,
$\gd$ is $\theta^{+ 5}_\gd$ strongly compact, and
$\gd$ isn't $\theta^{+ 6}_\gd$ strongly compact''.
This completes the proof of Claim 3.
\end{proof}
The proof of Claim 3 completes the proof of
Lemma \ref{l3}.
\end{proof}
Let now $V^*$ be $V^\FP$ truncated at $\gl$.
Lemmas \ref{l1} - \ref{l3} show that $V^*$
is our desired model.
This completes the proof of Theorem \ref{t1}.
\end{proof}
We take this opportunity to make some
concluding observations. First, we note
that the proof of Theorem \ref{t1}
remains valid if the cofinality of the
ordinals in the non-reflecting stationary
sets added is changed to any regular cardinal
below the least $V$-measurable cardinal.
Also, as we remarked in Section \ref{s1},
and as the proof just given indicates,
there is nothing special about $+ 5$
indicating the degree of level by level
inequivalence between strong compactness
and supercompactness below $\gk$.
There {\it are} two things critical to our proof,
however, concerning this notion.
In Section \ref{s1}, we observed that
it is possible for the degree of
level by level inequivalence between
strong compactness and supercompactness
for a measurable cardinal $\gd < \gk$ to be given by
$\varphi(\gd)$, where $\varphi(\gd)$ defines an
ordinal below $\gl_\gd$, subject to the restriction that
$\theta^{+ \varphi(\gd)}_\gd$ can't be a
singular strong limit cardinal of cofinality below $\gd$.
This is since if $\gd$ is $\gg$ strongly
compact where $\gg$ is a singular strong limit
cardinal of cofinality $\eta < \gd$, then
$\gd$ in fact must be $\gg^+$ strongly
compact.
Also, because we do not yet know how
to obtain a model in which every
measurable cardinal manifests as
much strong compactness as desired
(see the discussion given after the
proofs of Theorems 1 and 2 of \cite{A99b}),
we of necessity have to restrict the size of
our universe in the proof just given.
Thus, a question we pose to conclude
Section \ref{s2} is whether it is possible
to prove a version of Theorem \ref{t1}
in a universe containing as many large
cardinals as desired.
\section{The Proof of Theorem \ref{t2}}\label{s3}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Let
$V \models ``$ZFC + $\K$ is the class of
supercompact cardinals''.
Without loss of generality, by first doing
a preliminary forcing if necessary, we may
also assume
that $V$ is as in Theorem \ref{t0}, i.e.,
that GCH and level by level equivalence between
strong compactness and supercompactness
hold in $V$.
This allows us to define in $V$ our partial
ordering $\FP$ as the Easton support iteration
which first adds a Cohen real and then,
at any inaccessible cardinal $\gd$, forces with
$\add(\gd, \gd^{++})$.
Standard arguments in tandem with the results of
\cite{H1}, \cite{H2}, and \cite{H3} then show
that cardinals and cofinalities are preserved
when forcing with $\FP$ and
$V^\FP \models ``$ZFC + $\K$ is the class of
supercompact cardinals + $2^\gd = \gd^{++}$ if
$\gd$ is inaccessible + $2^\gd = \gd^+$ if
$\gd$ isn't inaccessible''.
Thus, the proof of Theorem \ref{t2} will be
complete once we have established the following
two lemmas.
\begin{lemma}\label{m1}
If
$V \models ``\gk < \gl$ are so that
$\gk$ is $\gl$ supercompact and $\gl$
is regular but not inaccessible'', then
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
If $\gk$ and $\gl$ are as in the hypotheses of Lemma \ref{m1},
then we consider the following two cases.
\setlength{\parindent}{0pt}
\bigskip
Case 1: $\gl$ is not the successor of an
inaccessible cardinal. Write
$\FP = \FP_\gl \ast \dot \FP^\gl$, where
the field of $\FP_\gl$ consists of
ordinals below $\gl$, and $\FP^\gl$
consists of the rest of $\FP$.
By the choice of $\gl$, $|\FP_\gl| \le \gl$.
Hence, by the definition of $\FP_\gl$,
if $j : V \to M$ is an embedding witnessing the
$\gl$ supercompactness of $\gk$,
$\FP_\gl$ is an initial segment of
$j(\FP_\gl)$.
Therefore, the standard reverse
Easton arguments show
$V^{\FP_\gl} \models ``\gk$ is $\gl$ supercompact''.
Since
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
${(2^{[\gl]^{< \gk}})}^{+}$-directed closed'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models ``\gk$ is
$\gl$ supercompact''.
\bigskip
Case 2: $\gl$ is the successor of an inaccessible cardinal.
Once again, write
$\FP = \FP_\gl \ast \dot \FP^\gl$, where
the field of $\FP_\gl$ consists of
ordinals below $\gl$, and $\FP^\gl$ is the rest of
$\FP$, i.e., the field of
$\FP^\gl$ consists of ordinals above $\gl$.
In this instance, it is not the case that
$|\FP_\gl| \le \gl$, since for the $\gd$ so that
$\gl = \gd^+$,
$|\FP_\gl| = \gd^{++} = \gl^+ > \gl$.
However, the arguments given on pages
119--120 of \cite{AS97a} or pages
555-556 of \cite{A99flm} (which are
originally due to Magidor and are
also found earlier in
\cite{JMMP}, \cite{JW}, and
\cite{Ma2}) will yield that
$V^{\FP_\gl} \models ``\gk$ is $\gl = \gd^+$
supercompact''.
For the convenience of readers, we give these
arguments below.
\setlength{\parindent}{1.5em}
Write
$\FP_\gl = \FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++})$,
where the field of $\FQ_0$ consists of
ordinals below $\gk$, and the field of
$\dot \FQ^1$ is composed of all remaining ordinals
in the interval $[\gk, \gl)$. Let $G$ be $V$-generic
over $\FP_\gl$, with
$G_0 \ast G_1 \ast G_2$ the corresponding
factorization of $G$. Fix
$j : V \to M$ an elementary embedding witnessing
the $\gl = \gd^+$ supercompactness of
$\gk$ which is generated by a supercompact
ultrafilter $\U$ over $P_\gk(\gl)$. We then have
$j(\FP_\gl) = \FQ_0 \ast \dot \FQ_1
\ast \dot \add(\gd, \gd^{++}) \ast \dot \FR_0
\ast \dot \FR_1$, where $\dot \FR_1$ is a term for
$\add(j(\gd), j(\gd^{++}))$ as computed in
$M^{\FQ_0 \ast \dot \FQ_1 \ast \dot
\add(\gd, \gd^{++}) \ast \dot \FR_0}$.
Therefore, by using the argument given in Lemma \ref{l1},
since $M[G_0][G_1][G_2]$ remains $\gl$ closed with respect to
$V[G_0][G_1][G_2]$ and $V \models {\rm GCH}$,
it is possible working in $V[G_0][G_1][G_2]$
to construct an $M[G_0][G_1][G_2]$-generic object
$G_3$ over $\FR_0$ and extend $j$ to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
It is then the case that
$M[G_0][G_1][G_2][G_3]$ remains $\gl$ closed
with respect to $V[G_0][G_1][G_2]$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_2 \rest \ga = \{p \rest \ga : p \in G_2\}$. Clearly,
$V[G_0][G_1][G_2] \models ``|G_2 \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2][G_3]}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) : p \in G_2 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2][G_3]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal so that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_2 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_2 \rest \gb$ and $G_2 \rest \ga$, for
$p \rest \gb = q \in G_2 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2][G_3] \models ``$GCH holds
for all cardinals $\ge j(\gd)$'',
$M[G_0][G_1][G_2][G_3] \models \break ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2][G_3]$ is a
maximal antichain of $\add(j(\gd), j(\gd^{++}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{++}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a supercompact ultrafilter over
$P_\gk(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra \in
V[G_0][G_1][G_2]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G_0][G_1][G_2][G_3]$.
Working in $V[G_0][G_1][G_2]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$ of
elements of $\add(j(\gd), j(\gd^{++}))$ so that
$\forall \ga \in (\gd, \gd^{++}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and so that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_4 = \{p \in \add(j(\gd), j(\gd^{++})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2][G_3]$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2][G_3]$ is
$\gd^+$ closed with respect to
$V[G_0][G_1][G_2]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ so that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2][G_3]$ is closed under
$\gd^+$ sequences with respect to
$V[G_0][G_1][G_2]$, define by induction
an increasing sequence
$\la s_\gb : \gb < \eta \ra$ so that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is so that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_4$ is
$M[G_0][G_1][G_2][G_3]$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gk(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gk(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gk(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{++})$ be so that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ so that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{++})$ is so that for some
$\ga \in (\gd, \gd^{++})$, $p = p \rest \ga$,
$G_4$ is so that if
$p \in G_2$, $j(p) \in G_4$.
Thus, working in $V[G_0][G_1][G_2]$,
we have shown that $j$ extends to
$j : V[G_0][G_1][G_2] \to M[G_0][G_1][G_2][G_3][G_4]$,
i.e.,
$V[G_0][G_1][G_2] \models ``\gk$ is $\gl = \gd^+$
supercompact''.
Since as in Case 1,
$\forces_{\FP_\gl} ``\dot \FP^\gl$ is
${(2^{[\gl]^{< \gk}})}^{+}$-directed closed'',
$V^{\FP_\gl \ast \dot \FP^\gl} = V^\FP \models
``\gk$ is $\gl$ supercompact''.
This completes the proof of Case 2 and Lemma
\ref{m1}.
\end{proof}
We observe that the proof of Lemma \ref{m1}
breaks down if $\gl$ is inaccessible.
This is since the inner model $M$ will not
have enough closure to allow either of the
proofs given in Cases 1 and 2 above to remain
valid.
\begin{lemma}\label{m2}
$V^\FP \models ``$For every pair of regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except possibly if
$\gk$ is a measurable limit of cardinals $\gd$
which are $\gl$ supercompact, or $\gl$ is
inaccessible''.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gk < \gl$ are regular,
$\gl$ isn't inaccessible,
$\gk$ is $\gl$ strongly compact, and
$\gk$ isn't a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact''.
Since forcing with $\FP$ preserves all
cardinals and cofinalities,
$V \models ``\gl$ is regular but isn't inaccessible''.
Therefore,
by Lemma \ref{m1}, any cardinal $\gd$ so that
$V \models ``\gd$ is $\gl$ supercompact''
remains $\gl$ supercompact in $V^\FP$.
This means
$V \models ``\gk < \gl$ are regular,
$\gl$ isn't inaccessible,
%$\gk$ is $\gl$ strongly compact,
and $\gk$ isn't a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact'',
and by the results of \cite{H1},
\cite{H2}, and \cite{H3},
$V \models ``\gk$ is $\gl$ strongly compact''.
Hence, by level by level equivalence in
$V$,
$V \models ``\gk$ is $\gl$ supercompact'',
so another application of Lemma \ref{m1}
implies that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{m2}.
\end{proof}
Lemmas \ref{m1} and \ref{m2} complete the
proof of Theorem \ref{t2}.
\end{proof}
%In conclusion to Section \ref{s3},
We note that any cardinal $\gk$ in $V^\FP$
which is a measurable limit of cardinals
$\gd$ which are $\gl$ strongly compact
where $\gl > \gk$ is regular but not
inaccessible must be in $V^\FP$ a measurable limit
of cardinals $\gd$ which are $\gl$ supercompact.
This is since the results of \cite{H1}, \cite{H2},
and \cite{H3}, which tell us that there are no
new instances of measurability, strong compactness,
or supercompactness in $V^\FP$, imply
that $\gk$ must be in $V$ a measurable limit
of cardinals $\gd$ which are $\gl$ strongly compact.
$\gk$ can then be written in $V$ as a
measurable limit of cardinals $\gd$ which are
$\gl$ strongly compact where each such $\gd$
is not itself a measurable limit of cardinals
$\gg$ which are $\gl$ strongly compact.
By level by level equivalence between strong
compactness and supercompactness in $V$,
each such cardinal $\gd$ must be
$\gl$ supercompact in $V$.
Lemma \ref{m1} then implies that each of these
cardinals remains $\gl$ supercompact in
$V^\FP$.
%\section{Concluding Remarks}\label{s4}
In conclusion to this paper,
we mention that Theorem \ref{t2} raises
a number of open questions.
Some of these are as follows.
\begin{enumerate}
\item In the model for Theorem \ref{t2},
is there precise level by level equivalence
between strong compactness and supercompactness?
If not,
is there any model for precise level by
level equivalence between strong compactness and
supercompactness in which $2^\gd = \gd^{++}$ for
every inaccessible cardinal $\gd$?
\item Is it possible to get a model witnessing
the conclusions of Theorem \ref{t2} in which
there are different kinds of failures of GCH
at the inaccessible cardinals,
e.g., in which $2^\gd = \gd^{+++}$ for every
inaccessible cardinal $\gd$, or
$2^\gd = \gd^{++}$ at every regular cardinal?
\item In general,
what sorts of failures of GCH are possible
in a model for precise level by level equivalence
between strong compactness and supercompactness?
\end{enumerate}
\begin{thebibliography}{99}
%\bibitem{A99} A.~Apter, ``Aspects of Strong
%Compactness, Measurability, and Indestructibility'',
%to appear in the
%{\it Archive for Mathematical Logic}.
\bibitem{A99flm} A.~Apter, ``Forcing the Least
Measurable to Violate GCH'',
{\it Mathematical Logic Quarterly 45}, 1999, 551--560.
%\bibitem{A99a} A.~Apter, ``On the Level by Level Equivalence
%between Strong Compactness and Strongness'', to appear in
%the {\it Journal of the Mathematical Society
%of Japan}.
%\bibitem{A00a} A.~Apter, ``Some Remarks on Indestructibility
%and Hamkins' Lottery Preparation'', submitted for publication
%to the {\it Archive for Mathematical Logic}.
\bibitem{A99c} A.~Apter, ``Some Structural Results
Concerning Supercompact Cardinals'', to appear in the
{\it Journal of Symbolic Logic}.
\bibitem{A99b} A.~Apter, ``Strong Compactness,
Measurability, and the
Class of Supercompact Cardinals'',
{\it Fundamenta Mathematicae 167}, 2001, 65--78.
\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness'',
{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness II: Strong Cardinals'',
{\it Archive for Mathematical Logic 40}, 2001, 25--38.
%\bibitem{AH4} A.~Apter, J.~D.~Hamkins,
%``Indestructibility and the Level-by-Level Agreement
%between Strong Compactness and Supercompactness'',
%submitted for publication to the {\it Journal of
%Symbolic Logic}.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong
Equality between Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349},
1997, 103--128.
%\bibitem{AS97b} A.~Apter, S.~Shelah, ``Menas' Result is
%Best Possible'',
%{\it Transactions of the American Mathematical Society 349},
%1997, 2007--2034.
\bibitem{Bu} J.~Burgess, ``Forcing'', in:
J.~Barwise, editor,
{\it Handbook of Mathematical Logic},
North-Holland, Amsterdam, 1977, 403--452.
\bibitem{C} J.~Cummings, ``A Model in which GCH Holds at
Successors but Fails at Limits'',
{\it Transactions of the American Mathematical
Society 329}, 1992, 1--39.
\bibitem{H1} J.~D.~Hamkins,
``Destruction or Preservation As You
Like It'',
{\it Annals of Pure and Applied Logic 91},
1998, 191--229.
\bibitem{H2} J.~D.~Hamkins, ``Gap Forcing'',
to appear in the
{\it Israel Journal of Mathematics}.
\bibitem{H3} J.~D.~Hamkins, ``Gap Forcing:
Generalizing the L\'evy-Solovay Theorem'',
{\it Bulletin of Symbolic Logic 5}, 1999, 264--272.
%\bibitem{J} T.~Jech, {\it Set Theory},
%Academic Press, New York and San
%Francisco, 1978.
\bibitem{JMMP} T.~Jech, M.~Magidor,
W.~Mitchell, K.~Prikry, ``Precipitous Ideals'',
{\it Journal of Symbolic Logic 45}, 1980, 1--8.
\bibitem{JW} T.~Jech, W.H.~Woodin,
``Saturation of the Closed Unbounded Filter on the
Set of Regular Cardinals'',
{\it Transactions of the American Mathematical
Society 292}, 1985, 345--356.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite}, Springer-Verlag,
Berlin and New York, 1994.
\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
between the Concepts of Compactness and Supercompactness'',
circulated manuscript.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
%\bibitem{Ma} M.~Magidor, ``How Large is the First
%Strongly Compact Cardinal?'', {\it Annals of
%Mathematical Logic 10}, 1976, 33--57.
\bibitem{Ma2} M.~Magidor, ``On the Existence of
Nonregular Ultrafilters and the Cardinality of
Ultrapowers'', {\it Transactions of the American
Mathematical Society 249}, 1979, 97--111.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
``Strong Axioms of Infinity and Elementary Embeddings'',
{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}