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%
% ------------------------------------------------------------------------------
%
\title{A New Easton Theorem for Supercompactness
and Level by Level Equivalence
%\title{Another Easton Theorem for Level by Level Equivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
level by level equivalence between strong
compactness and supercompactness, Easton theorem.}}
%\date{May 23, 2016}
%\date{July 25, 2016}
\date{July 30, 2016\\June 22, 2017}
\author{Arthur W.~Apter\\
% \thanks{The
% author's research was partially
% supported by PSC-CUNY grants.}
%\thanks{The author wishes to thank
%Brent Cody for helpful conversations
%on the subject matter of this paper.}
%\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}
\begin{document}
\maketitle
\begin{abstract}
We establish a new Easton theorem
for the least supercompact cardinal $\gk$ that is consistent
with the level by level equivalence between strong compactness and supercompactness.
This theorem is true in any model of ZFC containing
at least one supercompact cardinal, regardless if level by level equivalence holds.
Unlike previous Easton theorems
for supercompactness, %level by level equivalence,
there are no %definability
limits on the Easton functions $F$ used,
other than the usual constraints
given by Easton's theorem and the fact that if
$\gd < \gk$ is regular, then $F(\gd) < \gk$.
%on Easton functions.
%These theorems generalize \cite[Theorem 1]{A05}.
In both our ground model and the
model witnessing the conclusions
of our theorem, there are no
restrictions on the structure
of the class of
supercompact cardinals.
%We also briefly indicate how our
%methods of proof yield an
%Easton theorem that is consistent
%with the level by level equivalence
%between strong compactness and
%supercompactness in a universe
%with a restricted number of large cardinals.
%We conclude by posing some related
%open questions.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
%We also wish to emphasize
%what we mean when we say that
%level by level equivalence between
%strong compactness and supercompactness holds.
%Specifically, we will say
Say that a model of ZFC witnesses
{\em level by level equivalence between
strong compactness and supercompactness}
(which we will henceforth abbreviate as just
{\em level by level equivalence})
iff for every measurable cardinal
$\gk$ and every regular cardinal $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except possibly
if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$
supercompact.
Models containing supercompact
cardinals which also witness the
level by level equivalence between
strong compactness and supercompactness
and satisfy GCH
were first constructed in \cite{AS97a}.
We remark that the exceptions in the previous paragraph
are provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are %either
$\gl$ strongly compact, %or $\gl$ supercompact,
then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
%(Menas' results of \cite{Me} were also a precursor
%to the later work of \cite{Ma1}, \cite{KM}, and \cite{AS97a}.)
When this situation occurs,
we will henceforth say that
{\em $\gk$ is a witness
to the Menas exception at $\gl$}.
If $\gk$ is measurable and
for every regular cardinal
$\gl > \gk$, $\gk$ is $\gl$
strongly compact iff $\gk$ is
$\gl$ supercompact, then we will
say that {\em $\gk$ is a witness to
level by level equivalence between
strong compactness and supercompactness}.
We continue now with the main narrative.
In \cite{A05} and \cite{A12}, the following theorems were proven.
\begin{theorem}\label{t1}{\bf{(\cite{A05})}}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is
either a strong cardinal or the
regular limit of strong cardinals$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i1a} $F(\gd) \in (\gd, \gd^*)$
is a cardinal, where
$\gd^*$ is the least strong
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2a} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3a} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F)(\gd) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''. %+ For every $\gd \not\in A$, $2^\gd = \gd^+$''.
\end{theorem}
\begin{theorem}\label{t2}{\bf{(\cite{A12})}}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is a regular cardinal
which is not the successor of a singular cardinal and
$\neg \exists \gg < \gd[\gg$
is $\ga$ supercompact for every $\ga < \gd]\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i0b} If $\gd_1 < \gd_2$,
%and $\gd_1, \gd_2 \in A$,
then $F(\gd_1) \le F(\gd_2)$.
\item\label{i1b} $F(\gd) \in (\gd, \gd')$
is a cardinal, where
$\gd'$ is the least {Mahlo}
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2b} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3b} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F)(\gd) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$, $2^\gd = F(\gd)$''. %+ For every $\gd \not\in A$, $2^\gd = \gd^+$''.
\end{theorem}
\begin{theorem}\label{t3}{\bf{(\cite{A12})}}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk \mid \gd$ is
either a strong cardinal or the
regular limit of strong cardinals$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i1c} $F(\gd) \in (\gd, \gd^*)$
is a cardinal, where
$\gd^*$ is the least strong
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2c} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3c} If $\gd \in A$ is
$\gl$ supercompact for $\gl \ge \gd^{+ 16}$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that
%either $j(F(\gd)) = F(\gd) = \gd^{+}$ or
$j(F)(\gd) = F(\gd) = \gd^{+ 17}$.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$, $2^\gd = F(\gd)$''. %+ For every $\gd \not\in A$, $2^\gd = \gd^+$''.
\end{theorem}
%\noindent
As in \cite{A12} (from which we quote),
the use of $\gd^{+ 16}$ and $\gd^{+ 17}$ in
the statement of Theorem \ref{t3}
is for ease of presentation and comprehensibility.
In essence, $\gd^{+ 17}$
should be seen as representing any suitable regular cardinal $\gl$
for which it will be the case that $2^\gd = \gl$.
The cardinals $\gd^{+ 16}$ and $\gd^{+ 17}$ may be viewed
as a form of ``wild card'' standing in for the
more general possibilities.
Readers are referred to \cite{A12} for additional details.
%, which are somewhat technical in nature and are explicitly stated in \cite[Section 4]{AS97b}.
In Theorems \ref{t1} -- \ref{t3}, there are severe restrictions
placed on the
witnessing Easton functions $F$. %witnessing the conclusions of these theorems.
In particular,
$F(\gd)$ has harsh constraints placed on the values it may take, and specifically
%$F(\gk)$ must be bounded well below $\gk$, and specifically $F$
must be well-behaved with respect to the appropriate supercompactness embeddings.
%These constraints are quite a bit beyond the usual ones associated with Easton
%functions.
This raises the general question of whether it is possible to
prove an Easton theorem for the least supercompact
cardinal consistent with level by level equivalence in which the only
limitation placed on the witnessing Easton function $F$ (beyond the
usual ones for Easton functions) is that for %$\gd < \gk$,
$\gd$ below the least supercompact cardinal $\gk$,
$F(\gd) < \gk$.
The purpose of this paper is to answer this question in the affirmative.
%In particular,
More specifically, we will prove the following theorem.
\begin{theorem}\label{t4}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd < \gk \mid \gd$ is a non-measurable regular cardinal and
$\neg \exists \gg < \gd[\gg$
is $\ga$ supercompact for every $\ga < \gd]\}$.
Then $A$ is a stationary subset of $\gk$.
Further, suppose that $F : A \to \gk$,
$F \in V$
is a function with the following properties.
\begin{enumerate}
\item\label{n1} If $\gd_1 < \gd_2$,
%and $\gd_1, \gd_2 \in A$,
then $F(\gd_1) \le F(\gd_2)$.
\item\label{n2} ${\rm cof}(F(\gd)) > \gd$.
\item\label{n3} $F(\gd) < \gk$ is a cardinal.
\end{enumerate}
\noindent There is then a cardinal and cofinality preserving
partial ordering
$\FP \in V$ which also preserves the
stationarity of $A$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivalence
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''. %+ For every $\gd \not\in A$, $2^\gd = \gd^+$''.
\end{theorem}
Restrictions (\ref{n1}) and (\ref{n2}) are the usual ones for Easton functions.
Restriction (\ref{n3}) is necessary since otherwise, for some $\gd < \gk$,
$V^\FP \models ``2^\gd \ge \gk$'', contradicting the supercompactness
of $\gk$ in $V^\FP$.
Thus, it is possible to find a stationary subset $A$ of the
least supercompact cardinal $\gk$ such that an arbitrary
Easton function $F$ can be realized on $A$ while preserving
all ground model supercompact cardinals, the
stationarity of $A$, and level by level equivalence.
In addition, the generic extension realizing $F$
will contain the same supercompact cardinals as the ground model.
Further, since $F$ is completely arbitrary, it does not have to satisfy
any definability constraints, including those given by Menas in \cite{Me76}.
Consequently, as our proof
(which is quite different from the proofs of Theorems \ref{t1} -- \ref{t3}
and \cite[Theorem, Section 18, pages 83--88]{Me76})
will show, we have in fact established
a completely new Easton theorem for supercompactness, which is
valid in {\em any model of ZFC} containing at least one supercompact cardinal
(including those in which level by level equivalence does not hold).
Before presenting the proof
of our theorem, we briefly
state some preliminary information.
Our notation and terminology
will follow that given in \cite{A05} and \cite{A12}.
We do wish to mention a few
things explicitly, however.
%When forcing, $q \ge p$ means
%that $q$ is stronger than $p$.
For $\gk$ a regular cardinal and
$\ga$ an ordinal,
${\rm Add}(\gk, \ga)$ is the
standard Cohen partial ordering
for adding $\ga$ Cohen subsets
of $\gk$.
A partial ordering $\FP$ is
{\em $\gk$-directed closed} for
$\gk$ a regular cardinal if for
every directed set of conditions $D$
of size less than $\gk$, there is
a single condition extending each member of $D$.
%For $\ga < \gb$ ordinals,
%$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
%$(\a, \b)$ are as in standard interval notation.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
We mention that we are assuming
familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult
\cite{Je} %or \cite{K}
%, \cite{K}, or \cite{SRK} for further details. We do note, however, that
%we will say $\gk$ is {\em supercompact (strongly compact) up to the cardinal $\gl$} if
%$\gk$ is $\gg$ supercompact ($\gg$ strongly compact) for every $\gg < \gl$.
%Also, if $\gk$ is $\gl$ supercompact and $\gl$ is a cardinal, then
%$\gk$ is supercompact up to $\gl^+$, i.e., $\gk$ is $\ga$ supercompact for every $\ga < \gl^+$.
%We note only that the cardinal
%$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
%supercompact for every cardinal $\gd < \gl$.
A corollary of Hamkins' work on gap forcing found in
\cite{H2, H3} will be employed in the proofs of our theorems.
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
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\it admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2, H3},
$\FP$ is {\it mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size less than $\gk$ has
a ``nice'' name $\tau$
in $V$ of size less than $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that
$\forces_\FP ``\gt \subseteq \check y$''.
%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
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
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 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}
Theorem \ref{tgf} implies that if $\gk$ is
%measurable or
supercompact in $V^\FP$ and $\FP$ admits
a gap below $\gk$, then $\gk$ was %measurable or
supercompact in $V$ as well. In addition, if $\gk$ is
strongly compact in $V^\FP$ and $\FP$ is both mild
with respect to $\gk$ and admits a gap below $\gk$,
then $\gk$ was also strongly compact in $V$.
\section{The Proof of Theorem \ref{t4}}\label{s2}
We turn now to the proof of Theorem \ref{t4}.
\begin{proof}
%Turning now to the proof of Theorem \ref{t4}, suppose
%$V$ and $A$ are as in the hypotheses for this theorem.
Suppose $V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A$ and $F : A \to \gk$
%be an Easton function satisfying properties (\ref{n1}) -- (\ref{n3}) of
be as in the hypotheses of Theorem \ref{t4}.
We begin by showing that
%the set $A$ of the hypotheses of Theorem \ref{t4}
$A$ is a stationary subset of $\gk$.
\begin{lemma}\label{l1}
In $V$, $A = \{\gd < \gk \mid \gd$ is a non-measurable regular cardinal and
$\neg \exists \gg < \gd[\gg$
is $\ga$ supercompact for every $\ga < \gd]\}$ is a stationary subset of $\gk$.
\end{lemma}
\begin{proof}
Let $j : V \to M$ be an elementary embedding
generated by a normal measure $\mu$ over $\gk$ such that
$M \models ``\gk$ is not measurable''.
By \cite[Lemma 2.1]{AC2} and the succeeding remarks,
in both $V$ and $M$, $\gk$ is a limit of strong cardinals.
Further, for no $\gg < \gk$ is it the case that
$M \models ``\gg$ is $\ga$ supercompact for every $\ga < \gk$''.
This is since otherwise, for $\eta < \gk$ such that
$M \models ``\eta$ is a strong cardinal'',
$M \models ``\gg$ is $\ga$ supercompact for every $\ga < \eta$''.
Thus, by \cite[Lemma 1.1]{A02a}, $M \models ``\gg$ is supercompact''.
However, since $M \models ``j(\gg) = \gg$ is supercompact'',
$V \models ``\gg$ is supercompact'' as well.
This contradicts that $\gg < \gk$ and $V \models ``\gk$ is the least
supercompact cardinal''.
As $M \models ``\gk$ is regular'', it is therefore the case that
$\gk \in j(A)$. Hence, $A \in \mu$, from which it immediately
follows that $V \models ``A$ is a stationary subset of $\gk$''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
We next turn to the definition of the partial ordering $\FP$
used in the proof of Theorem \ref{t4}.
We take as our convention for the duration of this paper that
all product partial orderings have Easton support.
$\FP$ will be defined as
$\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)$, where both $\FP^1$
and $\FP^0$ are products. To define $\FP^1$ and $\FP^0$, let
$\gd_0$ be the least inaccessible cardinal (which is the same in either $V$ or
$V_0 = V^{\add(\go, 1)}$). Write $A = A_0 \cup A_1$, where
$A_0 = \{\gg \in A \mid \gg < \gd_0\}$ and
$A_1 = \{\gg \in A \mid \gg \ge \gd_0\}$.
Note that by the definition of $A$, $A_0$ consists of
all of the regular cardinals less than $\gd_0$.
We can now working in $V_0$ let
$\FP^1 = \prod_{\gg \in A_1} \add(\gg, F(\gg))$ and
$\FP^0 = \prod_{\gg \in A_0} \add(\gg, F(\gg))$.
Since by the definitions of $\FP^1$ and $\FP^0$, $\FP^1 \times \FP^0$ is
the Easton support product $\prod_{\gg \in A} \add(\gg, F(\gg))$,
the standard arguments concerning
Cohen, iterated, and Easton forcing (see \cite{Je}) show that
forcing with $\FP$ preserves cardinals and cofinalities and
$V^\FP \models ``$For every $\gg \in A$, $2^\gg = F(\gg)$''.
%+ For every $\gg \not\in A$, $2^\gg = \gg^+$''.
A few words are perhaps now in order concerning the intuition
behind the above definition of $\FP$.
$\FP$ begins by forcing with $\add(\go, 1)$ to introduce a
sufficiently low gap, so that Theorem \ref{tgf} can be applied.
%The components $\FP^0$ and $\FP^1$ of
%$\prod_{\gg \in A} \add(\gg, F(\gg))$
%are used in the proofs of Lemmas \ref{l2} -- \ref{l4}.
The partial orderings $\FP^0$ and $\FP^1$
are used in the proofs of Lemmas \ref{l2} -- \ref{l4}, which is why
$\prod_{\gg \in A} \add(\gg, F(\gg))$ is factored as $\FP^1 \times \FP^0$.
\begin{lemma}\label{l2}
Suppose $V \models ``\gd < \gl$ are such that $\gd$ is $\gl$ supercompact
and $\gl$ is regular''.
If $F(\gg) < \gd$ for every $\gg < \gd$, then $V^\FP \models ``\gd$ is $\gl$ supercompact''.
%Let $\eta = \{\gg < \gd \mid F(\gg) < \gd\}$.
\end{lemma}
\begin{proof}
Since $\card{\add(\go, 1)} = \go < \gd$, by the L\'evy-Solovay results
\cite{LS}, $V_0 \models ``\gd$ is $\gl$ supercompact''.
We continue to work in $V_0$ for the remainder of the proof of this lemma.
By hypothesis, $F(\gg) < \gd$ for every $\gg < \gd$.
Since $\gd_0$ is the least inaccessible cardinal in either $V$ or $V_0$,
$\gd_0 < \gd$. Hence, $F(\gg) < \gd$ for every $\gg \in A_0$.
Therefore, by the definition of $\FP$,
$\card{\FP^0} < \gd$.
This means by the results of \cite{LS} that $V^\FP = V_0^{\FP^1 \times \FP^0}
%= V_0^{\FP^0 \times \FP^1}
\models ``\gd$ is $\gl$
supercompact'' iff $V_0^{\FP^1} \models ``\gd$ is $\gl$ supercompact''.
We thus show that $V_0^{\FP^1} \models ``\gd$ is $\gl$ supercompact''.
To do this, let $A_1 = A_2 \cup A_3$, where
$A_2 = \{\gg \in A_1 \mid \gg < \gd\}$ and
$A_3 = \{\gg \in A _1\mid \gg \ge \gd\}$.
Write
$\FP^1 = \FQ^0 \times \FQ^1$, where
$\FQ^1 = \prod_{\gg \in A_3} \add(\gg, F(\gg))$ and
$\FQ^0 = \prod_{\gg \in A_2} \add(\gg, F(\gg))$.
By the definitions of $A$ and $\FP$ and our hypothesis that
$F(\gg) < \gd$ for $\gg < \gd$, $\card{\FQ^0} \le \gd$,
$\FQ^0$ is $\gd$-c.c., and
the first regular cardinal on which $\FQ^1$ acts is at least $\gl^{++}$.
Thus, by its definition, $\FQ^1$ is $\gl^{++}$-directed closed,
so forcing with $\FQ^1$ over $V_0$ adds no new subsets of $\gl^+$.
Also, since $V_0 \models {\rm GCH}$, $V^{\FQ^0}_0 \models
``2^\gg = \gg^+$ for all cardinals $\gg \ge \gd$'', from which it immediately follows that
$V^{\FQ^0}_0 \models ``2^{[\gl]^{< \gd}} = 2^\gl = \gl^+$''.
By \cite[Lemma 15.19]{Je}, we hence infer that forcing with
$\FQ^1$ over $V^{\FQ^0}_0$ adds no new subsets of $2^{[\gl]^{< \gd}}$.
This means that $V_0^{\FP^1} = V_0^{\FQ^1 \times \FQ^0} =
V_0^{\FQ^0 \times \FQ^1} \models ``\gd$ is $\gl$
supercompact'' iff $V_0^{\FQ^0} \models ``\gd$ is $\gl$ supercompact''.
We consequently now
show that $V_0^{\FQ^0} \models ``\gd$ is $\gl$ supercompact'' by
considering the following two cases.
\bigskip\noindent Case 1: $\card{\FQ^0} < \gd$. Once again, by the
results of \cite{LS}, $V_0^{\FQ^0} \models ``\gd$ is $\gl$ supercompact''.
\bigskip\noindent Case 2: $\card{\FQ^0} = \gd$. Let $j : V_0 \to M$
be an elementary embedding witnessing the $\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over $\gd$.
%such that $M \models ``\gd$ is not $\gl$ supercompact''.
By GCH in both $V_0$ and $M$, $M \models ``\gd$ is $\ga$ supercompact
for every $\ga < \gl$''.
%This means that by the definition of $\FP$,
The definition of $\FP$ therefore implies that
$j(\FQ^0) = \FQ^0 \times \FR$, where the first cardinal in $M$ on which
$\FR$ acts is at least $\gl^+$.
Further, the definition of $\FP$ implies that
$M \models ``\card{\FR} \le j(\gd)$ and $\FR$ is $\gl^+$-directed closed''.
Consequently, because $M^\gl \subseteq M$, $V_0 \models ``\FR$ is $\gl^+$-directed closed''
as well. In addition, the number of dense open subsets of $\FR$ present in $M$ is at most
$2^{j(\gd)}$, which by GCH in both $V_0$ and $M$ is $(j(\gd))^+ = j(\gd^+)$. However, since
$\gl \ge \gd^+$, by GCH in $V_0$ it follows that
$\card{j(\gd^+)} = |\{f \mid f : P_\gd(\gl) \to \gd^+$ is a function$\}| =
|\{f \mid f : \gl \to \gd^+$ is a function$\}| = |\{f \mid f : \gl \to \gl$ is a function$\}| =
2^\gl = \gl^+$. We can therefore use
the usual diagonalization arguments (as given, e.g.,
in the construction of the generic object $G_1$ in \cite[Lemma 2.4]{AC2}) to build in
$V_0$ an $M$-generic object $H$ over $\FR$.
More specifically,
let $\la D_\ga \mid \ga < \gl^+ \ra$ enumerate in $V_0$ the dense open subsets of $\FR$
present in $M$. As $M^\gl \subseteq M$, by the $\gl^+$-directed closure of $\FR$ in
%both $M$ and
$V_0$, we may work in $V_0$ and form an increasing sequence
$\la q_\ga \mid \ga < \gl^+ \ra$ of elements of $\FR$ such that
$q_\ga \in D_\ga$ for every $\ga < \gl^+$.
$H = \{p \in \FR \mid \exists \ga < \gl^+ [q_\ga \ge p]\}$ is our desired generic object.
Because $H \in V_0$ and $M \subseteq V_0$, $M[H] \subseteq V_0$.
%(An outline of this argument is as follows.
%Let $\la D_\ga \mid \ga < \gl^+ \ra$ enumerate in $V_0$ the dense open subsets of $\FR$
%present in $M$. As $M^\gl \subseteq M$, by the $\gl^+$-directed closure of $\FR$ in
%both $M$ and
%$V_0$, we may work in $V_0$ and meet each $D_\ga$ in order to create $H$.)
%By the $\gl^+$-directed closure of $\FR$ in $M$ and the fact that
%$(\gl^+)^M = \gl^+ \ge \gd^+ > \gd$, $M[H]$ and $M$ contain the same subsets of $\gd$.
%As $M$ and $V_0$ also contain the same subsets of $\gd$, this means that
%$M[H]$ and $V_0$ contain the same subsets of $\gd$ as well.
Hence, if $G$ is
$V_0$-generic over $\FQ^0$, $G$ is also $M[H]$-generic over $\FQ^0$.
Since $j '' G \subseteq G \times H$ and
by the product lemma, $M[H][G] = M[G][H]$, $j$ lifts in $V_0[G]$ to
$j : V_0[G] \to M[G][H]$. Thus, $V_0^{\FQ^0} \models ``\gd$ is $\gl$ supercompact''.
\bigskip
Cases 1 and 2 complete the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Every measurable cardinal $\gd$ is
either a witness to level by level equivalence or is a witness to
the Menas exception at $\gl$ for some $\gl > \gd$'', i.e.,
$V^\FP \models ``$Level by level equivalence holds''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ strongly compact and $\gl$ is regular''.
We prove Lemma \ref{l3} by considering the following three cases,
where we adopt throughout the notation and terminology of Lemma \ref{l2}.
\bigskip\noindent Case 1: $\gd > \gk$. When this occurs,
then by its definition, $\card{\FP} = \gk < \gd$.
Consequently, since $V \models ``$Level by level equivalence holds'',
by the results of \cite{LS}, $V^\FP \models ``$Either level by level
equivalence holds at $\gd$, or $\gd$ is a witness to the Menas exception
at $\gl$''.
\bigskip\noindent Case 2: $\gd = \gk$. We then have that the analysis
given in Lemma \ref{l2} (specifically, Case 2) shows that for any regular
$\gl > \gk$, $V^\FP \models ``\gk$ is $\gl$ supercompact''. Thus,
$V^\FP \models ``\gk$ is supercompact''. In this situation,
$A_3 = \emptyset$, and $\FQ^1$ is trivial forcing.
\bigskip\noindent Case 3: $\gd < \gk$. It is then true that
$V^\FP = V^{\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)} \models
``\gd$ is $\gl$ strongly compact''.
Note it must be true that for all $\gg \in A_0$, $F(\gg) < \gd$.
This is since if $\gg \in A_0$, $\gg < \gd_0 < \gd$, so if $F(\gg) \ge \gd$,
then by the definition of $\FP$, $V^\FP \models ``2^\gg \ge \gd$''.
This contradicts the fact that $V^\FP \models ``\gd$ is $\gl$ strongly compact''.
Because $V^\FP \models ``\gd$ is Mahlo''
and forcing can't create a new Mahlo cardinal,
$V \models ``\gd$ is Mahlo and hence inaccessible''.
Since $\gd_0 < \gd$ and $\gd$ is inaccessible,
it therefore follows by the definition of $\FP$ that
$\forces_{\add(\go, 1)} ``\card{\dot \FP^0} < \gd$''.
Thus, by the results of \cite{LS},
$V^\FP = V^{\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)} \models
``\gd$ is $\gl$ strongly compact'' iff
$V^{\add(\go, 1) \ast \dot \FP^1} \models
``\gd$ is $\gl$ strongly compact''.
It is hence true that
$V^{\add(\go, 1) \ast \dot \FP^1} \models
``\gd$ is $\gl$ strongly compact''.
By its definition, $\add(\go, 1) \ast \dot \FP^1$
is such that $\card{\add(\go, 1)} = \go$, $\add(\go, 1)$ is nontrivial, and
$\forces_{\add(\go, 1)} ``\dot \FP^1$ is $\ha_2$-directed closed''.
In addition, by the same analysis as given in the preceding paragraph,
if $\gg \in A_2$, $F(\gg) < \gd$ (so in particular, since
$A_0 \cup A_2 = \{\gg \in A \mid \gg < \gd\}$,
$F(\gg) < \gd$ for every $\gg < \gd$). From this, the factorization in $V_0$ of $\FP^1$ as
$\FQ^0 \times \FQ^1$, and the fact that both $\FQ^0$ and $\FQ^1$ are
Easton products, it further follows that $\add(\go, 1) \ast \dot \FP^1$ is
mild with respect to $\gd$. Consequently, by Theorem \ref{tgf},
$V \models ``\gd$ is $\gl$ strongly compact''.
Because level by level equivalence holds in $V$,
$V \models ``$Either $\gd$ is $\gl$ supercompact, or
$\gd$ is a measurable limit of cardinals $\eta$ which are $\gl$ supercompact''.
If $V \models ``\gd$ is $\gl$ supercompact'', then by Lemma \ref{l2},
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
If $V \models
``\gd$ is a measurable limit of cardinals $\eta$ which are $\gl$ supercompact'',
then let $\eta_0 < \gd$ be least such that $V \models ``\eta_0$ is $\gl$ supercompact''.
By its definition, there is no $\gg \in A$ with $\eta_0 \le \gg \le \gl^+$.
Thus, we can write $A = A_4 \cup A_5$, where
$A_4 = \{\gg \in A \mid \gg < \eta_0\}$ and
$A_5 = \{\gg \in A \mid \gg \ge \gl^{++}\}$.
If $\gg \in A_4$, then
$\gg < \eta_0 < \gd < \gl$. Hence, by the same analysis as
in the preceding two paragraphs, $F(\gg) < \gd$.
%This is since otherwise,
%$V^\FP \models ``2^\gg \ge \gd$'', which contradicts that
%$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
Also, by its definition, $\card{A_4} \le \eta_0$.
This means that if we define in $V_0$
$\FQ^2 = \prod_{\gg \in A_4} \add(\gg, F(\gg))$
and
$\FQ^3 = \prod_{\gg \in A_5} \add(\gg, F(\gg))$, then
$V_0 \models ``\card{\FQ^2} < \gd$ and $\FQ^3$ is $\gl^{++}$-directed closed''.
It consequently follows from the results of \cite{LS} that
$V^{\FQ^2}_0 \models ``\gd$ is a measurable limit of cardinals
$\eta$ which are $\gl$ supercompact''.
Because GCH in $V_0$ yields that
$V^{\FQ^2}_0 \models ``2^\gg = \gg^+$ for all cardinals $\gg > \card{\FQ^2}$'',
$V^{\FQ^2}_0 \models ``2^{[\gl]^{< \gd}} = 2^\gl = \gl^+$''.
Again by \cite[Lemma 15.19]{Je}, as in the proof of Lemma \ref{l2},
we may now infer that forcing with $\FQ^3$ over $V^{\FQ^2}_0$
adds no new subsets of $2^{[\gl]^{< \gd}}$. This means that
$V^{\FQ^2 \times \FQ^3}_0 = %V^{\FQ^3 \times \FQ^2}_0 =
V^\FP \models ``\gd$ is a measurable limit of cardinals $\eta$ which are $\gl$ supercompact''.
Consequently, we have that
$V^\FP \models ``$Either $\gd$ is $\gl$ supercompact, or $\gd$ is
a measurable limit of cardinals $\eta$ which are $\gl$ supercompact'', i.e.,
$V^\FP \models ``$Either level by level equivalence holds at $\gd$, or
$\gd$ is a witness to the Menas exception at $\gl$''.
%has the form
%$\FR^0 \ast \dot \FR^1$, where $\card{\FR^0} = \go$, $\FR^0$ is
%nontrivial, and $\forces_{\FR^0} ``\dot \FR^1$ is $\ha_1$-directed closed''.
\bigskip
Cases 1 -- 3 complete the proof of Lemma \ref{l3}.
\end{proof}
We note that the same analysis given in the first paragraph of
Case 3 of Lemma \ref{l3} allows us to infer that if
$V^\FP = V^{\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)} \models
``\gd$ is $\gl$ supercompact'', then
$V^{\add(\go, 1) \ast \dot \FP^1} \models ``\gd$ is $\gl$ supercompact'' as well.
%is valid for {\em any} measurable cardinal $\gd$,
%not just for measurable cardinals $\gd < \gk$. It is also valid
%is valid if ``$\gl$ strongly compact'' is replaced by ``$\gl$ supercompact''.
We will make use of
%these facts
this fact in the proof of Lemma \ref{l4}.
\begin{lemma}\label{l4}
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
We adopt the notation of the preceding lemmas.
Since $\card{\FP} = \gk$, by the results of \cite{LS},
$V^\FP \models ``$The class of supercompact cardinals above
$\gk$ is the same as in $V$'', i.e., $V^\FP \models ``\K \setminus \{\gk\}$
is the class of supercompact cardinals above $\gk$''.
We also know that by Case 2 of Lemma \ref{l3},
$V^\FP \models ``\gk$ is supercompact''.
Thus, to complete the proof of Lemma \ref{l4}, it suffices to show that
$V^\FP \models ``$No cardinal $\gd < \gk$ is supercompact''
%Thus, it suffices to show that $V^\FP \models ``\gk$ is the least supercompact cardinal''.
To see that this is true, suppose to the contrary that
$V^\FP \models ``\gd < \gk$ is supercompact''.
Since $V \models ``\gk$ is the least supercompact cardinal'',
let $\gl > \gd$ be such that $V \models ``\gd$ is not $\gl$ supercompact''.
We must have that $V^\FP \models ``\gd$ is $\gl$ supercompact''.
As we have just observed, the analysis given
in Case 3 of Lemma \ref{l3} shows that %for any $\gd < \gk$ and any $\gl \ge \gd$,
$V^\FP = V^{\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)} \models
``\gd$ is $\gl$ supercompact'' iff
$V^{\add(\go, 1) \ast \dot \FP^1} \models
``\gd$ is $\gl$ supercompact''.
The factorization of $\add(\go, 1) \ast \dot \FP^1$ given in the first
sentence of the second paragraph of Case 3 in Lemma \ref{l3}
together with Theorem \ref{tgf} then show that
$V \models ``\gd$ is $\gl$ supercompact'', a contradiction.
%Further, since $V \models ``\gk$ is the least supercompact cardinal'' and $\gd < \gk$,
%we know that it is possible to find some $\gl > \gk$ such that
%$V \models ``\gk$ is not $\gl$ supercompact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Since by its definition, $\FP$ is $\gk$-c.c., by \cite[Exercise H2, page 247]{Ku},
$A$ remains stationary in $V^\FP$.
Lemmas \ref{l1} -- \ref{l4} and the intervening remarks therefore complete
the proof of Theorem \ref{t4}.
\end{proof}
As in \cite{A05} and \cite{A12}, %we ask
we conclude by asking
what other types of Easton theorems are
consistent with the level by level equivalence
between strong compactness and supercompactness.
In particular, is it possible to include
regular cardinals in the domain of the Easton
function $F$ which are above the least supercompact
cardinal $\gk$?
This is precluded by the restrictions
on $A$ given in Theorem \ref{t4}.
Is it possible for $F$'s domain
%the domain of the Easton function $F$
to be all regular cardinals,
as in Easton's original result?
Which cardinals may be included in $F$'s range?
Finally, %as the referee has asked,
are there any GCH patterns which are incompatible
with the level by level equivalence between strong
compactness and supercompactness?
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%Nonregular Ultrafilters and the Cardinality of
%Ultrapowers'', {\it Transactions of the American
%Mathematical Society 249}, 1979, 97--111.
\bibitem{Me76} T.~Menas, ``Consistency Results Concerning
Supercompactness'', {\it Transactions of the American Mathematical
Society 223}, 1976, 61--91.
\bibitem{Me} T.~Menas, ``On Strong Compactness and
Supercompactness'', {\it Annals of Mathematical Logic 7},
1974, 327--359.
%\bibitem{So} R.~Solovay, ``Strongly Compact Cardinals
%and the GCH'', in: {\it Proceedings of the Tarski
%Symposium}, {\bf Proceedings of Symposia in Pure
%Mathematics 25}, American Mathematical Society,
%Providence, 1974, 365--372.
%\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}
Note that the results of \cite{AS97a} generalize
the fundamental work of Magidor \cite{Ma1} where
it is shown, relative to a supercompact cardinal,
that it is consistent for the least strongly compact
and supercompact cardinals to coincide precisely.\footnote{Magidor
also showed in \cite{Ma1} that it is consistent,
relative to the existence of a strongly compact
cardinal, for the least strongly compact and
measurable cardinals to coincide precisely.
In this situation, the least strongly compact cardinal
is not the least supercompact cardinal.}
These results also generalize the later work of
Kimchi and Magidor \cite{KM} who showed, relative
to the existence of a class of supercompact cardinals,
that it is consistent for the classes of strongly
compact and supercompact cardinals to coincide precisely,
except at measurable limit points.
We remark that the exceptions in the previous paragraph
are provided by a theorem of Menas \cite{Me},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are either $\gl$ strongly
compact or $\gl$ supercompact,
then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.
(Menas' results of \cite{Me} were also a precursor
to the later work of \cite{Ma1}, \cite{KM}, and \cite{AS97a}.)
When this situation occurs,
we will henceforth say that
{\em $\gk$ is a witness
to the Menas exception at $\gl$}.
If $\gk$ is measurable and
for every regular cardinal
$\gl > \gk$, $\gk$ is $\gl$
strongly compact iff $\gk$ is
$\gl$ supercompact, then we will
say that {\em $\gk$ is a witness to
level by level equivalence between
strong compactness and supercompactness}.
It is unfortunately the case that
the Easton function $F$ is defined
only on a restricted set of
inaccessible cardinals at and below the least
supercompact cardinal $\gk$.
Furthermore, because of the restrictions
placed on $F$'s range by clause (\ref{i3a}),
if $\gd \in A$, then $2^\gd \le \gd^{++}$.
It therefore becomes desirable to see if
it is possible to remove these constraints.
The purpose of this paper is to show that
this is indeed the case, and to address further
the general question of what GCH patterns are
consistent with the level by level equivalence between
strong compactness and supercompactness.
Specifically, we prove the following two theorems.
Before presenting the proof
of our theorem, we briefly
state some preliminary information.
Our notation and terminology
will follow that given in \cite{A05} and \cite{A06}.
We do wish to mention a few
things explicitly, however.
When forcing, $q \ge p$ means
that $q$ is stronger than $p$.
For $\gk$ a regular cardinal and
$\ga$ an ordinal,
${\rm Add}(\gk, \ga)$ is the
standard Cohen partial ordering
for adding $\ga$ Cohen subsets
of $\gk$.
For $\ga < \gb$ ordinals,
$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
$(\a, \b)$ are as in standard interval notation.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly
and use both $V[G]$ and $V^\FP$
to indicate the universe obtained
by forcing with $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
A partial ordering $\FP$ is
{\em $\gk$-directed closed} for
$\gk$ a cardinal if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga \mid \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$ \FP$ is {\em ${<}\gk$-strategically closed}
if $\FP$ is $\delta$-strategically
closed for all cardinals $\delta < \gk$.
Note that if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
In addition, if $\FP$ is $\gk$-directed closed, then
$\FP$ is ${<}\gk$-strategically closed.
%If $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga,
%\dot \FQ_\ga \ra : \ga < \gk + 1 \ra$
%is a reverse Easton iteration of length
%$\gk + 1$
%such that at stage $\ga$, a non-trivial forcing is done
%adding a subset of some ordinal
%$\gd_\ga$, then we will say that
%$\gd_\ga$ is in the field of $\FP$.
Finally, we mention that we are assuming
familiarity with the
large cardinal notions of measurability, strongness,
strong compactness, and supercompactness.
Interested readers may consult
\cite{J} or \cite{K}
%, \cite{K}, or \cite{SRK}
for further details.
We do note, however, that
we will say $\gk$ is {\em supercompact (strongly compact)
up to the cardinal $\gl$} if
$\gk$ is $\gg$ supercompact ($\gg$ strongly compact) for every
$\gg < \gl$.
Also, if $\gk$ is $\gl$ supercompact and
$\gl$ is a cardinal, then
$\gk$ is supercompact up to $\gl^+$, i.e.,
$\gk$ is $\ga$
supercompact for every $\ga < \gl^+$.
%We note only that the cardinal
%$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
%supercompact for every cardinal $\gd < \gl$.
We next turn to the definition of the partial ordering $\FP$
used in the proof of Theorem \ref{t4}. $\FP$ will be taken as
$\add(\go, 1) \ast (\dot \FP^1 \times \dot \FP^0)$, where both $\FP^1$
and $\FP^0$ are Easton products. To define $\FP^1$ and $\FP^0$, let
$\gd_0$ be the least inaccessible cardinal (in either $V$ or
$V_0 = V^{\add(\go, 1)}$). Write $A = A_0 \cup A_1$, where
$A_0 = \{\gd \in A \mid \gd < \gd_0\}$ and
$A_1 = \{\gd \in A \mid \gd \ge \gd_0\}$. We can now let
$\FP^1 = \prod_{\gd \in A_1} \add(\gd, F(\gd))$ and
$\FP^0 = \prod_{\gd \in A_0} \add(\gd, F(\gd))$.
\begin{lemma}\label{l2}
Suppose $V \models ``\gd < \gl$ are such that $\gd$ is $\gl$ supercompact
and $\gl$ is regular''.
If $F(\gg) < \gd$ for every $\gg < \gd$, then $V^\FP \models ``\gd$ is $\gl$ supercompact''.
%Let $\eta = \{\gg < \gd \mid F(\gg) < \gd\}$.
\end{lemma}
\begin{proof}
Since $\card{\add(\go, 1)} = \go < \gd$, by the L\'evy-Solovay results
\cite{LS}, $V_0 \models ``\gd$ is $\gl$ supercompact''.
We continue to work in $V_0$ for the remainder of the proof of this lemma.
Write $\FP^1 = \FQ^0 \times \FQ^1$, where
Since by its definition, the first regular cardinal on which $\FP^1$ acts is
at least $\gl^{++}$, $V^\FP = V_0^{\FP^1 \times \FP^0} =
V_0^{\FP^0 \times \FP^1} \models ``\gd$ is $\gl$
supercompact'' iff $V_0^{\FP^0} \models ``\gd$ is $\gl$ supercompact''.
We thus show that $V_0^{\FP^0} \models ``\gd$ is $\gl$ supercompact''.
To do this, we consider the following two cases.
\bigskip\noindent Case 1: $\card{\FP^0} < \gd$. Once again, by the
results of \cite{LS}, $V_0^{\FP^0} \models ``\gd$ is $\gl$ supercompact''.
\bigskip\noindent Case 2: $\card{\FP^0} = \gd$. Let $j : V \to M$
be an elementary embedding witnessing the $\gl$ supercompactness of $\gd$.
\end{proof}