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\title{The Enhanced Levinski Property and the Class of Supercompact Cardinals
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
measurable cardinal, normal measure,
level by level equivalence between strong
compactness and supercompactness, lottery sum, enhanced Levinski property.}}
\author{Arthur W.~Apter\\
% \thanks{The
% author's research was partially
% supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{December 27, 2016} \date{January 24, 2017} \date{February 3, 2017}
\date{February 5, 2017}
\begin{document}
\maketitle
\begin{abstract}
We define a generalization of a property
originally due to Levinski \cite{Lev}, show its
relative consistency, and investigate some of its
possible interactions with the class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
%We begin with some key definitions.
%The cardinal $\gk$ is {\em indestructibly supercompact}
%if $\gk$'s supercompactness is preserved after forcing
%with a $\gk$-directed closed partial ordering.
We begin with some terminology and notational conventions.
%Suppose $\gk$ is a regular cardinal.
%For $\gb$ an arbitrary ordinal, the
%partial ordering $\add(\gk, \gb)$ is
%the standard Cohen partial ordering
%for adding $\gb$ many Cohen subsets of $\gk$.
%The partial
%ordering $\FP$ is {\em $\gk$-directed
%closed} if for every directed set $D \subseteq \FP$
%of size less than $\gk$,
%there is a condition in $\FP$
%extending each member of $D$.
As in \cite{A16}, call an ordinal $\ga > 0$ {\em good}
if $\ga$ is definable and is also such that for
any cardinal $\gd$, $\gd^{+ \ga}$ is a regular cardinal below the least
inaccessible cardinal above $\gd$.\footnote{This represents a
slight abuse of notation. Strictly speaking, $\ga$ is a formula
in the language of set theory
with parameter $\gd$, which we will suppress.
This means that there are only countably many good ordinals.
Examples of
good ordinals include $2$, $3$, $75$, $\go + 1$,
the ordinal successor of the least beth fixed point above $\gd$, etc.
This is since $2$, $3$, $75$, $\go +1$, and
the ordinal successor of the least beth fixed point above $\gd$
are all definable with parameter $\gd$. %each of the preceding is definable.
Further, for
$\ga$ any of these values, $\gd^{+ \ga}$ is regular
since it is a successor cardinal.
$\gd^{+ \ga}$ is also below the least inaccessible
cardinal above $\gd$, which must be a (regular) limit cardinal.}
%inaccessible cardinal above $\gd$.\footnote{We will slightly abuse
%notation by also allowing $\ga$
%to be a formula with parameter $\gd$ which always defines an ordinal below the least
%inaccessible cardinal above $\gd$. An
%example is $\ga(\gd) =$ The cardinal successor of the least beth fixed point
%above $\gd$. If this is the case, we will suppress the parameter $\gd$.}
%Suppose throughout $\ga$ is some %suitably
%definable ordinal (including 0)
%such that for any $\gd$, $\gd^{+ \ga}$ is
%below the least inaccessible cardinal above $\gd$.
Say that
a measurable cardinal $\gk$ satisfies the
{\em Levinski property LP($\gk, \ga$)} for a fixed but
arbitrary good ordinal $\ga$
if $2^\gk = \gk^+$, yet there exists a normal measure
$\U_\ga$ over $\gk$ such that $\{\gd < \gk \mid \gd$ is
inaccessible and $2^\gd = \gd^{+ \ga} \} \in \U_\ga$
(so in particular, assuming $\ga > 1$, GCH holds at
$\gk$ in the universe $V$, yet fails at $\gk$ in the
ultrapower $V^\gk / \U_\ga$).
%GCH fails on some final segment of inaccessible cardinals
%below $\gk$ if $\gk$ is a measurable cardinal which is not
%a limit of measurable cardinals, and GCH fails on a
%measure $1$ subset of $\gk$ if $\gk$ is a measurable cardinal
%which is a limit of measurable cardinals).
The normal measure $\U_\ga$ is then said to {\em witness LP($\gk, \ga)$}.
%if for every
%inaccessible cardinal $\gd$ in some final segment
%below $\gk$, $2^\gd = \gd^{+ \ga}$, yet $2^\gk = \gk^+$.
%Refine
Extend the preceding by saying that
a measurable cardinal $\gk$ satisfies the
{\em enhanced Levinski property ELP($\gk$)}
if $2^\gk = \gk^+$, yet for
every good ordinal $\ga$, there exists a normal measure
$\U_\ga$ over $\gk$ such that $\{\gd < \gk \mid \gd$ is
inaccessible and $2^\gd = \gd^{+ \ga} \} \in \U_\ga$.
%(so in particular, assuming $\ga > 1$,
%GCH fails on some final segment of inaccessible cardinals
%below $\gk$ if $\gk$ is a measurable cardinal which is not
%a limit of measurable cardinals, and GCH fails on a
%measure $1$ subset of $\gk$ if $\gk$ is a measurable cardinal
%which is a limit of measurable cardinals).
%Refine the preceding by saying that
%a measurable cardinal $\gk$ satisfies the
%{\em Levinski property LP($\ga$)} for a fixed but
%arbitrary good ordinal $\ga$ if for every
%inaccessible cardinal $\gd$ in some final segment
%below $\gk$, $2^\gd = \gd^{+ \ga}$, yet $2^\gk = \gk^+$.
Both LP($\gk, \ga)$ and
ELP($\gk$) are variants of a property first studied by Levinski in \cite{Lev}.
Suppose $V$ is a model of ZFC
in which for all 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.
Such a universe will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
For brevity, we will henceforth abbreviate this as just
{\em level by level equivalence}.
%Any model witnessing level by
%level equivalence between strong
%compactness and supercompactness
%also witnesses the Kimchi-Magidor
%property \cite{KM} that the classes
%of strongly compact and supercompact
%cardinals coincide precisely,
%except at measurable limit points.
The exception
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.
%Models in which level by level
%equivalence between strong compactness
%and supercompactness holds nontrivially
%were first constructed in \cite{AS97a}.
%We take this opportunity to
%elaborate a bit more on the notion of
%level by level equivalence between strong
%compactness and supercompactness.
Any model of ZFC with this property
also witnesses the Kimchi-Magidor
property \cite{KM} that the classes
of strongly compact and supercompact
cardinals coincide precisely,
except at measurable limit points.
Models in which GCH and level by level
equivalence between strong compactness
and supercompactness hold nontrivially
were first constructed in \cite{AS97a}.
%Beginning now our main narrative,
The purpose of this paper is first to
establish the consistency of ELP($\gk$) relative
to the existence of a measurable cardinal $\gk$,
and then investigate some of its possible interactions
with the class of supercompact cardinals.
Specifically, we will prove the following theorems.
\begin{theorem}\label{t1}
%$\ $
%a) Con(ZFC + $\gk$ is a measurable cardinal) $\implies$
%Con(ZFC + ELP($\gk$) +
%For every good ordinal $\ga$, $\gk$ carries
%$2^{2^\gk}$ many normal measures witnessing LP($\gk, \ga$)).
%b) Con(ZFC + $\gk$ is a measurable cardinal) $\implies$
%Con(ZFC + ELP($\gk$) +
%$\gk$ carries $\gk^+$ many normal measures +
%For every good ordinal $\ga$, $\gk$ carries
%either $2^{2^\gk}$ many normal measures or
%$\gk^+$ many normal measures witnessing LP($\gk, \ga$)).
a) Con(ZFC + GCH + $\gk$ is a measurable cardinal +
$\gl \ge \gk^{++}$ is a regular cardinal) $\implies$
Con(ZFC + %$2^\gk = \gk^+$ +
$2^{\gk^+} = 2^{2^\gk} = \gl$ + ELP($\gk$) +
For every good ordinal $\ga$, $\gk$ carries
$2^{2^\gk}$ many normal measures witnessing LP($\gk, \ga$)).
b) Con(ZFC + GCH + $\gk$ is a measurable cardinal) $\implies$
%+ $\gl \ge \gk^{++}$ is a regular cardinal) $\implies$ \break
Con(ZFC + $2^{\gk^+} = 2^{2^\gk} = \gk^+$ + ELP($\gk$) +
$\gk$ carries $\gk^+$ many normal measures +
For every good ordinal $\ga$, $\gk$ carries
%either $2^{2^\gk}$ many normal measures or
$\gk^+$ many normal measures witnessing LP($\gk, \ga$)).
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$,
no cardinal is supercompact up to an inaccessible cardinal, and
level by level equivalence
%between strong compactness and supercompactness
holds.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + %GCH +
For every measurable cardinal $\gd$,
$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd}= \gd^{++}$ + $\gk$ is supercompact''. In $V^\FP$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence
%between strong compactness and supercompactness
holds.
Further, in $V^\FP$,
every measurable cardinal $\gd$ witnesses ELP($\gd$),
and for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds
with respect to $2^{2^\gd} = \gd^{++}$ many normal measures.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$,
no cardinal is supercompact up to an inaccessible cardinal, and
level by level equivalence
%between strong compactness and supercompactness
holds.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + %GCH + $\gk$ is supercompact''.
For every measurable cardinal $\gd$,
$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{++}$ + $\gk$ is supercompact''. In $V^\FP$,
no cardinal is supercompact up to an inaccessible cardinal,
and level by level equivalence
%between strong compactness and supercompactness
holds.
%Further, in $V^\FP$,
%every measurable cardinal $\gd$ which is a
%limit of measurable cardinals witnesses ELP($\gd$)
%with respect to $2^{2^\gd}$ many normal measures, and
%every measurable cardinal $\gd$ which is not a limit of
%measurable cardinals witnesses ELP($\gd$) with respect to
%$\gd^+$ many normal measures.
Further, in $V^\FP$,
every measurable cardinal $\gd$ witnesses ELP($\gd$).
In addition, for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds with respect to $2^{2^\gd} = \gd^{++}$ many normal measures
if $\gd$ is a limit of measurable cardinals,
but LP($\gd, \ga$) holds with respect to $\gd^+$ many normal measures
if $\gd$ is not a limit of measurable cardinals.
Finally, every measurable cardinal $\gd$ which is not a limit of
measurable cardinals carries only $\gd^+$ many normal measures.
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + GCH + $\K \neq \emptyset$ is
the class of supercompact cardinals''.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\K$ is the class of supercompact cardinals +
For every measurable cardinal $\gd$, $2^\gd = \gd^+$ and
$2^{\gd^+} = 2^{2^\gd} = \gd^{++}$''.
In $V^\FP$, $\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals.
%Further, in $V^\FP$,
%every measurable cardinal $\gd$ witnesses ELP($\gd$)
%with respect to $2^{2^\gd}$ many normal measures.
Further, in $V^\FP$,
every measurable cardinal $\gd$ witnesses ELP($\gd$),
and for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds
with respect to $2^{2^\gd} = \gd^{++}$ many normal measures.
\end{theorem}
\begin{theorem}\label{t5}
Suppose $V \models ``$ZFC + GCH + $\K \neq \emptyset$ is
the class of supercompact cardinals''.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + $\K$ is the class of supercompact cardinals +
For every measurable cardinal $\gd$, $2^\gd = \gd^+$ and
$2^{\gd^+} = 2^{2^\gd} = \gd^{++}$''.
In $V^\FP$, $\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals.
%Further, in $V^\FP$,
%every measurable cardinal $\gd$ which is a
%limit of measurable cardinals witnesses ELP($\gd$)
%with respect to $2^{2^\gd}$ many normal measures, and
%every measurable cardinal $\gd$ which is not a limit of
%measurable cardinals witnesses ELP($\gd$) with respect to
%$\gd^+$ many normal measures.
Further, in $V^\FP$,
every measurable cardinal $\gd$ witnesses ELP($\gd$).
In addition, for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds with respect to $2^{2^\gd} = \gd^{++}$ many normal measures
if $\gd$ is a limit of measurable cardinals,
but LP($\gd, \ga$) holds with respect to $\gd^+$ many normal measures
if $\gd$ is not a limit of measurable cardinals.
Finally, every measurable cardinal $\gd$ which is not a limit of
measurable cardinals carries only $\gd^+$ many normal measures.
\end{theorem}
We take this opportunity to make a few remarks concerning
Theorems \ref{t1} -- \ref{t5}. %our results.
First, we mention that our definition of ``good ordinal'' is arbitrary,
in the sense that it can be changed to allow $\gd^{+\ga}$ to be larger
(e.g., $\ga$ could be taken as a definable ordinal such that $\gd^{+ \ga}$
is a regular cardinal below, say, the least Ramsey cardinal).
This would require certain relatively minor changes in the
definitions of our forcing iterations.
We note in addition that in Theorem \ref{t1}a),
$\gk$ will carry the maximum number of normal measures possible
(namely $2^{2^\gk}$), whereas in Theorem \ref{t1}b), $\gk$ carries
fewer than the maximum number of normal measures (namely $\gk^+$).
Also, in Theorems \ref{t2} and \ref{t3}, $\gk$ is
the only supercompact cardinal.
This is because in each case, in both $V$ and $V^\FP$,
no cardinal is supercompact up to an inaccessible cardinal.
This is in sharp contrast, however, to Theorems \ref{t4} and \ref{t5},
where the class of supercompact cardinals can be arbitrary.
(If the class of supercompact cardinals contains at least two members
$\gk_0 < \gk_1$, then $\gk_0$ is supercompact up to the inaccessible cardinal $\gk_1$
(and much more)).
Finally, Theorems \ref{t2} and \ref{t3} are significant generalizations of
\cite[Theorem 5]{A14}, whose proof is only very briefly sketched
%and not given fully
in \cite{A14}.
In particular, \cite[Theorem 5]{A14} constructs a model for level by level
equivalence with the same limited number of large cardinals as in
Theorems \ref{t2} and \ref{t3}
%containing only one supercompact cardinal in which no
%cardinal is supercompact up to an inaccessible cardinal
in which it is the case that for only one fixed good ordinal $\ga$, LP($\gd, \ga$) holds
for each measurable cardinal $\gd$.
Thus, not only does ELP($\gd$) fail in this model, but no consideration is
given either to the number of normal measures witnessing LP($\gd, \ga$) or to
varying the number of normal measures witnessing LP($\gd, \ga$) depending on whether
$\gd$ is a measurable cardinal which is a limit of measurable cardinals.
We conclude Section \ref{s1}
with a very brief discussion of
some additional preliminary material.
%We presume a basic knowledge
%of large cardinals and forcing.
%A good reference in this regard is \cite{J}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
We will have some slight abuses of notation.
In particular,
when $G$ is $V$-generic over $\FP$,
we %abuse notation slightly and
take both $V[G]$ and
$V^\FP$ as being the generic
extension of $V$ by $\FP$.
We will also, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
%We also %abuse notation slightly by
%occasionally confuse terms with the
%sets they denote, especially for
%ground model sets and variants of the generic object.
For $\ga < \gb$ ordinals, %$[\ga, \gb]$, $(\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, $[\ga, \gb]$, and
$(\ga, \gb)$ are as in standard interval notation.
For any ordinal $\ga$, $\ga'$ is the least inaccessible cardinal
above $\ga$.
For $\gk < \gl$ regular cardinals,
${\rm Coll}(\gk, \gl)$ is the standard L\'evy collapse
of all cardinals in the half-open interval $(\gk, \gl]$ to $\gk$.
For $\gk$ a regular cardinal and $\gl$
an ordinal,
$\add(\gk, \gl)$ is the
standard partial ordering for adding
$\gl$ many Cohen subsets of $\gk$.
The partial ordering
$\FP$ is {\em $\gk$-directed closed} if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
%For $\gk$ a measurable cardinal, the
%normal measure ${\cal U}$ over $\gk$ has
%{\em trivial Mitchell rank} if for
%$j : V \to M$ the elementary embedding
%generated by ${\cal U}$,
%$M \models ``\gk$ is not measurable''.
%$\FP$ is {\em $\gk$-closed} if every increasing
%chain of members of $\FP$ of length $\gk$ has an upper bound.
%$\FP$ is {\em ${<}\gk$-closed} if $\FP$ is
%$\gd$-closed for every $\gd < \gk$.
%$\FP$ is {\em $\gk$-strategically closed}
%if in the two person game in which the
%players construct an increasing sequence
%$\la p_\ga \mid \ga \le \gk \ra$,
%where player I plays odd stages and
%player II plays even stages,
%player II has a strategy ensuring the game
%can always be continued.
%$\FP$ is {\em ${\prec}\gk$-strategically closed}
%if in the two person game in which the
%players construct an increasing sequence
%$\la p_\ga \mid \ga < \gk \ra$,%
%where player I plays odd stages and
%player II plays even stages,
%player II has a strategy ensuring the game
%can always be continued.
%It therefore follows that
%any partial ordering $\FP$ which is
%$\gk$-directed closed is also
%${\prec }\gk$-strategically closed
%and consequently adds no new subsets of
%any cardinal $\gd < \gk$.
%$\FP$ is {\em ${<}\gk$-strategically closed}
%if $\FP$ is $\gd$-strategically closed
%for every $\gd < \gk$.
%$\FP$ is {\em $(\gk, \infty)$-distributive}
%if the intersection of $\gk$ many
%dense open subsets of $\FP$ is dense open.
%It therefore follows that
%any partial ordering $\FP$ which is
%$\gk$-directed closed is also
%${<}\gk$-strategically closed, and any
%partial ordering which is $\gk$-strategically
%closed is $(\gk, \infty)$-distributive.
%It further
We assume familiarity with the
large cardinal notions of
measurability, strong compactness,
and supercompactness.
Readers are urged to consult
\cite{J} for further details.
%We do mention, however, that
%the cardinal $\gk$ is ${<} \gl$
%strongly compact or ${<} \gl$ supercompact if
%$\gk$ is $\gg$ strongly compact or $\gg$
%supercompact for every $\gg < \gl$.
We do note, however, that
we will say {\em $\gk$ is supercompact
up to the inaccessible cardinal $\gl$} if
$\gk$ is $\gd$ supercompact for every
$\gd < \gl$.
We recall for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\it lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorems \ref{t2} -- \ref{t5}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
%$\gd$-strategically closed''.
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ is {\em mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : V \to \ov M$ is
{\em amenable to $V$} when
$j \rest A \in V$ for any
$A \in V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$
that admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an elementary embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
A consequence of Theorem \ref{tgf} is that if
$\FP$ admits a gap at $\ha_1$ and
$V^\FP \models ``\varphi(\gk)$'' where
$\varphi(\gk)$ is either the formula which says
``$\gk$ is supercompact'' or the formula which says
``$\gk$ is measurable'', then
$V \models ``\varphi(\gk)$'' as well.
In addition, it follows from Theorem \ref{tgf} that
if $\FP$ admits a gap at $\ha_1$, $\FP$ is mild with respect to $\gk$, and
$V^\FP \models ``\gk$ is strongly compact'', then
it is also true that
$V \models ``\gk$ is strongly compact''. %as well.
%\noindent A consequence of Theorem \ref{tgf} is that if
%$\FP$ admits a gap at $\ha_1$ and %some regular $\gd < \gk$ and
%$V^\FP \models ``\gd$ is measurable'', then
%$V \models ``\gd$ is measurable'' as well.
%In particular, if $\FP$ admits a gap at $\go$, then
%Finally, at several junctures throughout
To conclude Section \ref{s1},
suppose $\gk \le \gl$ are such that
$V \models ``\gk$ is $\gl$ supercompact''.
We mention that during
the course of this paper, we will be referring to the
``standard lifting arguments'' for lifting a
$\gl$ supercompactness embedding $j : V \to M$
generated by a supercompactness measure over $P_\gk(\gl)$
to a generic extension given by a
suitably defined Easton support iteration.
Although there are numerous references to this
in the literature, we will use the proof found in
\cite[Theorem 4]{A01a} %(see \cite[Section 1]{A14})
as the basis for the sketch we are about to present.
(Readers may also consult the last paragraph of \cite[Section 1]{A14}, which we
are quoting with certain minor modifications.) %nearly verbatim.)
Very briefly, this argument assumes the following.
\begin{enumerate}
\item $V \models %{\rm GCH}$.
``2^\gk = \gk^+$ and there is a cardinal $\gg \in (\gk, \gl]$ such that
$2^\gd = \gd^+$ for every $\gd \in [\gg, \gl]$''.
\item $\gl$ is a regular cardinal.
\item $\FP \ast \dot \FQ =
\la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga \le \gk \ra$
is an Easton support iteration
having length $\gk + 1$ such that $\card{\FP} \le \gk$.
\item For any inaccessible cardinal $\gd < \gk$,
$\forces_{\FP_\gd} ``\card{\dot \FQ_\gd} < \gk$''.
\item For any inaccessible cardinal $\gd < \gk$,
$\forces_{\FP_\gd} ``\dot \FQ_\gd$ is
$\gd$-directed closed''. %${<} \gd$-strategically closed''.
%\item $\forces_{\FP_\gk} ``\dot \FQ_\gk$
%is $\gk$-directed closed''.
\item $G_0 \ast G_1$ is $V$-generic over $\FP \ast \dot \FQ$.
\item $\forces_{\FP} ``\card{\dot \FQ} \le \gl$ and
$\dot \FQ$ is $\gk$-directed closed''.
\item $j(\FP \ast \dot \FQ) =
\FP \ast \dot \FQ \ast \dot \FR \ast j(\dot \FQ)$.
\item In $M$, $\forces_{\FP \ast \dot \FQ} ``\dot \FR$ is $\gl^+$-directed closed''.
\end{enumerate}
%Under these circumstances,
%\noindent
Since
%$V \models {\rm GCH}$,
$V \models %{\rm GCH}$.
``2^\gk = \gk^+$ and there is a cardinal $\gg \in (\gk, \gl]$ such that
$2^\gd = \gd^+$ for every $\gd \in [\gg, \gl]$'',
$\card{P_\gk(\gl)} = \gl$. Because
$M[G_0][G_1] \models ``\card{\FR} = j(\gk)$'' and
$V \models ``\card{j(\gk^+)} = \card{j(2^\gk)} = \break
\card{\{f \mid f : P_\gk(\gl) \to \gk^+\}} =
\card{\{f \mid f : \gl \to \gk^+\}} =
\card{\{f \mid f : \gl \to \gl\}} = 2^\gl = \gl^+$'',
$V[G_0][G_1] \models``$There are (at most) $\gl^+ = 2^\gl =
\card{j(\gk^+)} = \card{j(2^\gk)}$ many dense
open subsets of $\FR$ present in $M[G_0][G_1]$''.
Because $M[G_0][G_1]$
remains $\gl$-closed with respect to $V[G_0][G_1]$
and $\FR$ is $\gl^+$-directed closed in both
$M[G_0][G_1]$ and $V[G_0][G_1]$, working in
$V[G_0][G_1]$, it is possible to build an
$M[G_0][G_1]$-generic object $G_2$ over $\FR$ such that
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$.
Still working in $V[G_0][G_1]$, one then lifts $j$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$.
Since $M[G_0][G_1][G_2]$ remains $\gl$-closed
with respect to $V[G_0][G_1]$ and
$V[G_0] \models ``\card{\FQ} \le \gl$'', there is a
master condition $q \in V[G_0][G_1]$
for $\{j(p) \mid p \in G_1\}$.
%Using GCH in $V$,
Because $V \models ``\card{j(\gl^+)} = \card{j(2^\gl)} =
\card{\{f \mid f : P_\gk(\gl) \to \gl^+\}} =
\card{\{f \mid f : \gl \to \gl^+\}} = \card{[\gl^+]^\gl} = \gl^+$'' and
$M[G_0][G_1][G_2] \models ``\card{j(\FQ)} \le j(\gl)$'',
there are (at most) $\gl^+$ many dense open subsets of
$j(\FQ)$ present in $V[G_0][G_1]$.
We may thus build in $V[G_0][G_1]$ an
$M[G_0][G_1][G_2]$-generic object $G_3$ for
$j(\FQ)$ containing $q$.
It is then the case that
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3$,
so we may fully lift $j$ in $V[G_0][G_1]$ to a
$\gl$ supercompactness embedding
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
This argument remains valid (and in fact becomes
even simpler) if no forcing is done at
stage $\gk$ in $V$, i.e., if
$\dot \FQ$ is a term for trivial forcing.
The argument also remains valid if
$j$ is an elementary embedding generated by a normal measure over $\gk$
(which can be canonically identified with a normal measure over $P_\gk(\gk)$) and
$\dot \FQ$ is a term for trivial forcing.
%and GCH is replaced by the weaker assumption $2^\gk = \gk^+$.
\section{The Proofs of Theorems \ref{t1} -- \ref{t5}}\label{s2}
We turn now to the proofs of our theorems,
beginning with the proof of Theorem \ref{t1}. %\ref{t1}a).
\begin{proof}
Suppose $V^* \models ``$ZFC + GCH + $\gk$ is a measurable cardinal +
$\gl \ge \gk^{++}$ is a regular cardinal''.
%Without loss of generality, we assume in addition that $V \models {\rm GCH}$.
We consider two cases.
If $\gl = \gk^{++}$, let $V = V^*$.
If $\gl > \gk^{++}$,
let $V = (V^*)^{\add(\gk^+, \gl)}$.
Regardless of which case holds,
because $\add(\gk^+, \gl)$ is $\gk^+$-directed closed,
we have that $V \models ``\gk$ is a measurable cardinal +
$2^\gd = \gd^+$ for every cardinal $\gd \le \gk$ +
%GCH holds below $\gk$ + $2^\gk = \gk^+$ +
$2^{\gk^+} = 2^{2^\gk} = \gl$''.
The partial ordering used in the proof of Theorem \ref{t1}a) is
defined as
%$\FP_\gk \ast \dot \add(\gk, \gk^+)$, where
$\FP = \FP_{\gk + 1} = \la \la \FP_\gb, \dot \FQ_\gb \ra
\mid \gb \le \gk \ra$, the Easton support iteration of length $\gk + 1$ which
begins by forcing with $\add(\go, 1)$ and then
does nontrivial forcing only at $V$-inaccessible cardinals $\gd \le \gk$.
At such a stage $\gd$, if $\gd$ isn't measurable,
$\dot \FQ_\gd$ is a term for the lottery sum %of
$\oplus \{\add(\gd, \gd^{+ \ga}) \mid \ga \ge 2$ is a good ordinal$\}$.
If, however, $\gd$ is measurable, $\dot \FQ_\gd$ is a term for $\add(\gd, \gd^+)$.
%By the definition of $\FP$,
It now follows that $V^\FP \models ``2^\gk = \gk^+$ and
$2^{\gk^+} = 2^{2^{\gk}} = \gl$''.
This of course is the case if $\gl = \gk^{++}$, since under those
circumstances, $V = V^*$. If, however, $\gl > \gk^{++}$,
then because
$\gk$ remains inaccessible after forcing over $V^*$ with $\add(\gk^+, \gl)$,
by \cite[Lemma 15.4, page 227]{J},
%the usual $\Delta$-system argument continues to be valid
%(see \cite[Theorem 9.19, page 188]{J}). Thus,
$\add(\gk, \gk^+)$
(or indeed, $\add(\gk, \gg)$ for any ordinal $\gg$) is still $\gk^+$-c.c.
Thus, no cardinals are collapsed when forcing with $\add(\gk, \gk^+)$.
This allows us to infer that
$V^\FP \models ``2^\gk = \gk^+$ and $2^{\gk^+} = 2^{2^\gk} = \gl$''.
Also, although the proof of Theorem \ref{t1}
does not require that the definition of $\FP$ begin by forcing with
$\add(\go, 1)$, it is useful to do this so that Theorem \ref{tgf}
may be applied in the proofs of Theorems \ref{t2} -- \ref{t5}.
\begin{lemma}\label{l1}
%It is the case that
$V^\FP \models {\rm ELP}(\gk)$. %$V^\FP \models ``\gk$ is measurable''.
\end{lemma}
\begin{proof}
Let $\ga \ge 2$ be a fixed but arbitrary good ordinal.
Take $j : V \to M$ to be an elementary embedding
witnessing the measurability of $\gk$ in $V$
generated by a normal measure over $\gk$
such that $M \models ``\gk$ isn't measurable''. In
particular, $M^{\gk} \subseteq M$.
We combine several ideas
(including a standard lifting argument, an idea
due to Levinski \cite{Lev}, and an idea due to Magidor \cite{Ma2})
%a form of which is given, e.g., in the proof of \cite[Lemma 2.2]{A10})
to show that $j$ lifts in $V^{\FP_\gk \ast \dot \add(\gk, \gk^+)} = V^\FP$ to
$j : V^{\FP_\gk \ast \dot \add(\gk, \gk^+)}
\to M^{j(\FP_\gk \ast \dot \add(\gk, \gk^+))}$.
We also follow to a certain extent the proof of \cite[Theorem 5]{A16},
quoting verbatim when appropriate.
Specifically, let
$G_0$ be $V$-generic over $\FP_\gk$, and let
$G_1$ be $V[G_0]$-generic over
$\add(\gk, \gk^+)$.
Observe that
$j(\FP_\gk \ast \dot \add(\gk, \gk^+)) =
\FP_\gk \ast \dot \FQ_\gk \ast
\dot \FQ \ast \dot \add(j(\gk), j(\gk^+))$,
where $\dot \FQ_\gk$ is a term for the lottery sum %of
$\oplus \{\add(\gk, \gk^{+ \gb}) \mid \gb \ge 2$ is a good ordinal$\}$.
By forcing above the appropriate condition $p_0$ which opts for
$\add(\gk, \gk^{+ \ga})$ in the stage $\gk$
lottery held in $M^{\FP_\gk}$ in the definition of
$j(\FP_\gk \ast \dot \add(\gk, \gk^+))$, we may in fact assume that
$j(\FP_\gk \ast \dot \add(\gk, \gk^+))$ is forcing equivalent to
$\FP_\gk \ast \dot \add(\gk, \gk^{+ \ga}) \ast
\dot \FQ \ast \dot \add(j(\gk), j(\gk^+))$.
For the remainder of the proof of Lemma \ref{l1},
we assume that we are forcing above $p_0$.
Working in $V[G_0]$,
we first note that since $\FP_\gk$ is $\gk$-c.c.,
$M[G_0]$ remains $\gk$-closed with respect to $V[G_0]$. Next,
we use Levinski's ideas of \cite{Lev} to show that
%working in $V[G_0][G_1]$,
it is possible to rearrange $G_1$ to form an
$M[G_0]$-generic object $H_1$ over
$(\add(\gk, \gk^{+ \ga}))^{M[G_0]}$ in $V[G_0][G_1]$.
Since $V \models ``2^\gk = \gk^+$'' and
$j$ is generated by a normal measure over $\gk$, $(\gk^+)^V = (\gk^+)^M$ and
$(\gk^+)^V < j(\gk) < (\gk^{++})^V$.
In particular,
%because $M[G_0] \models ``(\gk^+)^V \le 2^\gk < j(\gk)$'',
any $\gg \in ((\gk^+)^V, j(\gk))$ which %either $M$, $M[G_0]$, or
$M[G_0]$
believes to be a cardinal actually is an ordinal of cardinality
$\gk^+$ in either $V$, $V[G_0]$, or $V[G_0][G_1]$.
%$V \models ``\card{(\gk^{+ \ga})^M} = \gk^+$''.
%In addition, since $\FP$ is an Easton support iteration,
%$\FP_\gk \ast \dot \add(\gk, \gk^+)$ is $\gk$-c.c., which means that
%cardinals at and above $\gk$ are preserved from
%$V$ to $V[G_0][G_1]$ and $M$ to $M[G_0][G_1]$. Hence,
%$(\gk^{+ \ga})^{M[G_0][G_1]} = (\gk^{+ \ga})^M$,
%$(\gk^+)^{V[G_0][G_1]} = (\gk^+)^{V}$, and
Hence, $V[G_0][G_1] \models ``\card{(\gk^{+ \ga})^{M[G_0][H_0]}} = \gk^+$''.
Let $(\gk^{+ \ga})^{M[G_0][H_0]} = \gr$.
%From this, it follows that in $V[G]$, $(\add(\gk, \gk^{+ \ga})^{M[G]}$
%has the form $\add(\gk, \gr)$, where
%$V[G] \models ``\card{\gr} = \gk^+$''.
Working in $V[G_0][G_1]$, %(or $V$),
we may therefore let $f : \gk^+ \to \gr$
be a bijection. For any
$p \in \add(\gk, \gk^+)$, $g(p) = \{\la \la \gs, f(\gb) \ra, \gg \ra
\mid \la \la \gs, \gb \ra, \gg \ra \in p\} \in (\add(\gk, \gr))^{M[G_0]}$.
As can be easily checked (see \cite{Lev}),
$H_1 = \{g(p) \mid p \in G_1\}$ is
%both a $V[G_0]$-generic object and
an $M[G_0]$-generic object over
$(\add(\gk, \gr))^{M[G_0]}$.
Our construction guarantees that
$j '' G_0 \subseteq G_0 \ast H_1$, so $j$ lifts in $V[G_0][G_1]$
to $j : V[G_0] \to M[G_0][H_1]$.
We now use a version of the standard lifting argument
mentioned at the end of Section \ref{s1} to build in $V[G_0][G_1]$
an $M[G_0][H_1]$-generic object $H_2$ over $\FQ$.
At the risk of redundancy, we repeat some of the ideas and details
mentioned earlier, since they will be relevant in the proof of Lemma \ref{l2}.
In $V$, since $M$
is given via an ultrapower by a
normal measure over $\gk$,
$\card{j(\gk)}$ and $\card{j(\gk^+)}$ may be calculated as
$\card{\{f \mid f : \gk \to \gk\}} = 2^\gk = \gk^+$ and
$\card{\{f \mid f : \gk \to \gk^+\}} = \card{[\gk^+]^\gk}= \gk^+$ respectively.
Also, by elementarity, since $V \models ``2^\gk = \gk^+$'',
$M \models ``2^{j(\gk)} = (j(\gk))^+ = j(\gk^+)$''.
Because $(\add(\gk, \gk^+))^{V[G_0]}$ is $\gk^+$-c.c$.$ in $V[G_0]$,
$M[G_0][H_1]$ remains $\gk$-closed with respect to $V[G_0][G_1]$.
In addition, since $M[G_0][H_1] \models ``\FQ$ is an Easton support
iteration of length $j(\gk)$'', $M[G_0][H_1] \models ``\card{\FQ} = j(\gk)$ and
$2^{j(\gk)} = (j(\gk))^+ = j(\gk^+)$''. %$\FQ$ is $j(\gk)$-c.c.''.
This means the number of dense open subsets of $\FQ$
present in $M[G_0][H_1]$ is $j(\gk^+)$. Further, as
$M[G_0][H_1] \models ``\FQ$ is $\gk^+$-directed closed'' and
$M[G_0][H_1]$ is $\gk$-closed with respect to $V[G_0][G_1]$,
$\FQ$ is $\gk^+$-directed closed in $V[G_0][G_1]$ as well.
Since $\gk^+$ is preserved from $V$ to
$V[G_0][G_1]$, we may let
$\la D_\gb \mid \gb < \gk^+ \ra \in V[G_0][G_1]$
enumerate the dense open subsets of $\FQ$ present in $M[G_0][H_1]$.
We may now use the fact that $\FQ$ is
$\gk^+$-directed closed in $V[G_0][G_1]$ to meet each $D_\gb$
and thereby construct in $V[G_0][G_1]$
an $M[G_0][H_1]$-generic
object $H_2$ over $\FQ$. Our construction guarantees that
$j '' G_0 \subseteq G_0 \ast H_1 \ast H_2$,
so $j$ lifts in $V[G_0][G_1]$ to
$j : V[G_0] \to M[G_0][H_1][H_2]$.
By the fact that $\FQ$ is $\gk^+$-directed closed in %both
$M[G_0][H_1]$, %and $V[G_0][G_1]$,
$M[G_0][H_1][H_2]$ remains $\gk$-closed with respect to
$V[G_0][G_1][H_2] = V[G_0][G_1]$.
We now use arguments originally due to
Magidor \cite{Ma2}, which are also given in
\cite[pages 119--120]{AS97a} and are found
other places in the literature as well, to construct
in $V[G_0][G_1]$ an $M[G_0][H_1][H_2]$-generic
object $H_3$ over $(\add(j(\gk), j(\gk^+))^{M[G_0][H_1][H_2]}$
such that $j''(G_0 \ast G_1) \subseteq G_0 \ast H_1 \ast H_2 \ast H_3$.
For the convenience of readers, we present these
arguments below.
For $\gz \in (\gk, \gk^{+})$ and
$p \in \add(\gk, \gk^{+})$, let
$p \rest \gz = \{\la \la \rho, \gs \ra, \eta \ra \in p \mid
\gs < \gz\}$ and
$G_1 \rest \gz = \{p \rest \gz \mid p \in G_1\}$. Clearly,
$V[G_0][G_1] \models ``|G_1 \rest \gz| \le \gk$
for all $\gz \in (\gk, \gk^{+})$''. Thus, since
${\add(j(\gk), j(\gk^{+}))}^{M[G_0][H_1][H_2]}$ is
$j(\gk)$-directed closed and $j(\gk) > \gk^{+}$,
$q_\gz = \bigcup\{j(p) \mid p \in G_1 \rest \gz\}$ is
well-defined and is an element of
${\add(j(\gk), j(\gk^{+}))}^{M[G_0][H_1][H_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\gz) -
\dom(\bigcup_{\gb < \gz} q_\gb)$
($\bigcup_{\gb < \gz} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \gz} j(\gb), j(\gz))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \gz} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \gz$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \gz$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
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 $M[G_0][H_1][H_2] \models ``j(\gk)$ is inaccessible and
$2^{j(\gk)} = j(\gk^+)$'', an application of \cite[Lemma 15.4, page 227]{J} shows that
%\break
$M[G_0][H_1][H_2] \models ``\add(j(\gk),
j(\gk^{+}))$ is
$j(\gk^+)$-c.c$.$ and has
$j(\gk^{+})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][H_1][H_2]$ is a
maximal antichain of \break $\add(j(\gk), j(\gk^{+}))$,
${\cal A} \subseteq \add(j(\gk), \gb)$ for some
$\gb \in (j(\gk), j(\gk^{+}))$. Thus, since
%the fact $V \models ``2^\gk = \gk^+$''
%and the fact $j$ is generated by a normal measure over
%$\gk$ imply that
$V \models ``|j(\gk^{+})| = \gk^{+}$'', we can let
$\la {\cal A}_\gz \mid \gz \in (\gk, \gk^{+}) \ra \in
V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gk), j(\gk^{+}))$
present in
$M[G_0][H_1][H_2]$.
Working in $V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\gz \mid \gz \in (\gk, \gk^{+}) \ra$ of
elements of $\add(j(\gk), j(\gk^{+}))$ such that
$\forall \gz \in (\gk, \gk^{+}) [r_\gz \ge q_\gz$ and
$r_\gz \in \add(j(\gk), j(\gz))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\gz \mid \gz \in (\gk, \gk^{+}) \ra
\exists \gb \in (\gk, \gk^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$H_3 = \{p \in \add(j(\gk), j(\gk^{+})) \mid
\exists r \in \la r_\gz \mid \gz \in (\gk, \gk^{+}) \ra
[r \ge p]\}$ is an
$M[G_0][H_1][H_2]$-generic object over
$\add(j(\gk), j(\gk^{+}))$. To define
$\la r_\gz \mid \gz \in (\gk, \gk^{+}) \ra$, if
$\gz$ is a limit, we let
$r_\gz = \bigcup_{\gb \in (\gk, \gz)} r_\gb$.
By the facts
$\la r_\gb \mid \gb \in (\gk, \gz) \ra$
is (strictly) increasing and
$M[G_0][H_1][H_2]$ is
$\gk$-closed with respect to
$V[G_0][G_1]$, this definition is valid.
Assuming now $r_\gz$ has been defined and
we wish to define $r_{\gz + 1}$, let
$\la {\cal B}_\gb \mid \gb < \eta \le \gk \ra$
be the subsequence of
$\la {\cal A}_\gb \mid \gb \le \gz + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gk), j(\gz + 1))$.
Because
$q_\gz, r_\gz \in \add(j(\gk), j(\gz))$,
$q_{\gz + 1} \in \add(j(\gk), j(\gz + 1))$, and
$j(\gz) < j(\gz + 1)$, the condition
$r_{\gz + 1}' = r_\gz \cup q_{\gz + 1}$ is
well-defined. This is since by our earlier observations,
any new elements of
$\dom(q_{\gz + 1})$ won't be present in either
$\dom(q_\gz)$ or $\dom(r_\gz)$.
We can thus, using the fact
$M[G_0][H_1][H_2]$ is $\gk$-closed
%under $\gk^+$ sequences
with respect to
$V[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb \mid \gb < \eta \ra$ such that
$s_0 \ge r_{\gz + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\gz + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $H_3$ is
$M[G_0][H_1][H_2]$-generic over
$\add(j(\gk), j(\gk^{+}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\gz \mid \gz \in (\gk, \gk^{+}) \ra
\exists \gb \in (\gk, \gk^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\gz) \mid \gz < \gk^{+} \ra$ is
unbounded in $j(\gk^{+})$. To see this, if
$\gb < j(\gk^{+})$ is an ordinal, then for some
$f : \gk \to M$ representing $\gb$,
we can assume that for $\gr < \gk$,
$f(\gr) < \gk^{+}$. Thus, by the regularity of
$\gk^{+}$ in $V$,
$\gb_0 = \bigcup_{\gr < \gk} f(\gr) <
\gk^{+}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\gz \mid \gz <
\gk^{+} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gk, \gk^{+})$ be such that
${\cal A} \subseteq \add(j(\gk), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gk, \gk^{+})$ is such that for some
$\gz \in (\gk, \gk^{+})$, $p = p \rest \gz$,
$H_3$ is such that if
$p \in G_1$, $j(p) \in H_3$.
Thus, working in $V[G_0][G_1]$,
we have shown that
because $j''(G_0 \ast G_1) \subseteq G_0 \ast H_1 \ast H_2 \ast H_3$, $j$ lifts to
$j : V[G_0][G_1] \to M[G_0][H_1][H_2][H_3]$,
i.e.,
$V[G_0][G_1] \models ``\gk$ is measurable''.
Since $M[G_0][H_1] \models ``\FQ \ast \dot \add(j(\gk), j(\gk^+))$ is
$\gk'$-directed closed'' and $(\gk^{+ \ga})^{M[G_0][H_1]} < (\gk')^{M[G_0][H_1]}$,
$M[G_0][H_1][H_2][H_3] \models ``2^\gk = \gk^{+ \ga}$''.
Consequently,
$\gk \in \{\gd < j(\gk) \mid 2^\gd = \gd^{+ \ga}\} =
j(\{\gd < \gk \mid 2^\gd = \gd^{+ \ga}\})$.
Hence, $A_\ga =_{\rm df} \{\gd < \gk \mid 2^\gd = \gd^{+ \ga}\}$
has measure 1 with respect to the normal measure
$\U_\ga \in V[G_0][G_1]$ over $\gk$ generated by $j$.
As $\ga$ was arbitrary and $V^\FP \models ``2^\gk = \gk^+$'',
$V^\FP \models {\rm ELP}(\gk)$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$For every good ordinal $\ga$,
$\gk$ carries $2^{2^\gk} = \gl$ many normal measures
witnessing LP($\gk, \ga$)''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l1} shows that for every good ordinal $\ga$,
$V^\FP \models {\rm LP}(\gk, \ga)$.
However, we can now argue as in \cite[Lemma 1.1]{A01} to
infer that in $V^\FP$, %$\gk$ is a measurable cardinal carrying
there are $2^{2^\gk}$ many normal measures witnessing LP($\gk, \ga$).
%not concentrating on measurable cardinals.
%We first argue as in \cite[Lemma 1.1]{A01} that
%$\gk$ carries $2^{2^\gk}$ many normal measures, and then
%use Theorem \ref{tgf} to show that there are $2^{2^\gk}$ many
%normal measures not concentrating on measurable cardinals.
%To begin,
Specifically, we can show
via a folklore argument
that there are $2^{2^\gk}$ many
different ways of constructing the generic object $H_2$ over $\FQ$.
To do this, as in the proof of Lemma \ref{l1},
let $\la D_\gb \mid \gb < \gk^+ \ra \in V[G_0][G_1]$
enumerate the dense open subsets of $\FQ$ present in $M[G_0][H_1]$.
Since $V[G_0][G_1] \models ``\FQ$ is $\gk^+$-directed closed'',
we can build in $V[G_0][G_1]$ a tree
$\cal T$ of height $\gk^+$ such that:
\begin{enumerate}
\item\label{k1} The root of $\cal T$ is the
empty condition.
\item\label{k2} If $p$ is an element at level
$\gb < \gk^+$ of $\cal T$, then the
successors of $p$ at level $\gb + 1$ are
a maximal incompatible subset of $D_\gb$
extending $p$.
By the definition of %both $\FP_\gk$ and
$\FQ$, there will be
at least two incompatible successors of $p$ at level $\gb + 1$.
\item\label{k3} If $\gl < \gk^+$ is a limit
ordinal, then the elements of $\cal T$ at
height $\gl$ are upper bounds to any
path through $\cal T$ of height $\gl$.
\end{enumerate}
\noindent We observe that any path $I_2$
of height $\gk^+$ through $\cal T$ generates
an $M[G_0][H_1]$-generic object $I^*_2$ over $\FQ$.
Therefore, since there are
$2^{\gk^+}$ many paths of height
$\gk^+$ through $\cal T$, there are
$2^{\gk^+} = 2^{2^\gk} = \gl$ many different
$M[G_0][H_1]$-generic objects over
$\FQ$ present in $V[G_0][G_1]$.
Suppose $H^*_2 \neq H^{**}_2$ are any two distinct
$M[G_0][H_1]$-generic objects over
$\FQ$ generated as above.
If $j^*$ is the lift of $j$ associated with $H^*_2$ and $j^{**}$
is the lift of $j$ associated with $H^{**}_2$,
it will be the case that
$j^*(\la G_0, G_1 \ra) = \la G_0, H_1, H^*_2, H_3 \ra$ and
$j^{**}(\la G_0, G_1 \ra) = \la G_0, H_1, H^{**}_2, H_3 \ra$.
Since \cite[Lemma 1]{C93} tells us that any
$k : V[G_0][G_1] \to N$ witnessing the
measurability of $\gk$ which is a lift of $j$ is generated by the
normal measure over $\gk$ given by
${\cal U} = \{x \subseteq \gk \mid
\gk \in k(x)\}$, there are
$2^{2^\gk} = \gl$ many different normal measures over
$\gk$ in $V[G_0][G_1]$ witnessing LP($\gk, \ga$).
This completes the proof of Lemma \ref{l2}.
\end{proof}
Lemmas \ref{l1} and \ref{l2} complete the proof of Theorem \ref{t1}a).
To prove Theorem \ref{t1}b) and thereby complete the proof of Theorem \ref{t1},
we force over the model $V^\FP$ witnessing the conclusions of Theorem
\ref{t1}a) with $\add(\go, 1) \ast \dot {\rm Coll}(\gk^+, 2^{2^\gk}) =
\add(\go, 1) \ast \dot {\rm Coll}(\gk^+, \gl)$ to obtain the model $\ov V$.
Because $V^\FP \models ``2^\gk = \gk^+$ and $2^{\gk^+} = 2^{2^\gk} = \gl$'',
$\ov V \models ``2^\gk = \gk^+$ and $2^{\gk^+} = 2^{2^\gk} = \gk^{++}$''.
By the proof of \cite[Theorem 1]{ACH},
since $\card{\add(\go, 1)} < \gk$ and $\add(\go, 1)$ is nontrivial,
in $\ov V$, $\gk$ carries exactly
$\card{(2^{2^\gk})^{V^\FP}}
%\card{(2^{2^\gk})^{V^{\add(\go, 1)}}} =
%\card{(2^{2^\gk})^{V^{\add(\go, 1) \ast \dot {\rm Coll}(\gk^+, 2^{2^\gk})}}}
= (\gk^+)^{\ov V}$
many normal measures.
Suppose $\ga \ge 2$ is a fixed but arbitrary good ordinal.
The proof of Theorem \ref{t1}b) will therefore be complete if
we can show that each of the
$(2^{2^\gk})^{V^\FP}$ many normal measures $\U_\ga$ witnessing
LP($\gk, \ga$) in $V^\FP$ has an extension
$\U^*_\ga \supseteq \U_\ga$ witnessing LP($\gk, \ga$) in $\ov V$.
To do this, recall that by the L\'evy-Solovay results \cite{LS},
$\U^*_\ga =_{\rm df} \{x \subseteq \gk \mid \exists y \in \U_\ga
[y \subseteq x]\}$ is a normal measure over $\gk$ in $V^{\FP \ast \dot \add(\go, 1)}$.
Since forcing with $\add(\go, 1)$ preserves cardinals and cofinalities and does not
change the size of power sets, $\U^*_\ga$ witnesses LP($\gk, \ga$) in
$V^{\FP \ast \dot \add(\go, 1)}$.
Because forcing with ${\rm Coll}(\gk^+, 2^{2^\gk})$ adds no subsets of
$\gk$ as ${\rm Coll}(\gk^+, 2^{2^\gk})$ is $\gk^+$-directed closed,
$\U^*_\ga$ witnesses LP($\gk, \ga$) in $V^{\FP \ast \dot \add(\go, 1) \ast \dot
{\rm Coll}(\gk^+, 2^{2^\gk})} = \ov V$ as well.
This completes the proof of both Theorem \ref{t1}b) and \break Theorem \ref{t1}.
\end{proof}
\begin{pf}
Turning to the proof of Theorem \ref{t2},
suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$, no cardinal is supercompact
up to an inaccessible cardinal, and level by level equivalence
holds.
It must be true that $V \models ``$No cardinal above
$\gk$ is inaccessible'' (so in particular,
$V \models ``$No cardinal above $\gk$ is measurable'').
The partial ordering $\FP$ used in the proof of Theorem \ref{t2} may now be taken as
the partial ordering $\FP = \FP_{\gk + 1}$ of Theorem \ref{t1}a)
defined assuming $\gl = \gk^{++}$.
Note that by its definition, forcing with $\FP$ preserves all cardinals, cofinalities, and
the fact %$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{++}$
%for all $V$-measurable cardinals $\gd$.
$2^\gg = \gg^+$ whenever $\gd$ is a $V$-measurable cardinal and
$\gg \in [\gd, \gd')$.
It is of course also the case that
$V^\FP \models ``$No cardinal above $\gk$ is inaccessible''.
\begin{lemma}\label{l3}
Suppose $\gd$ is a measurable cardinal in $V^\FP$. Then
ELP($\gd$) holds,
and for every good ordinal $\ga$, LP($\gd, \ga$) holds with respect to
$2^{2^\gd} = \gd^{++}$ many normal measures.
\end{lemma}
\begin{proof}
As we have just observed, $V \models ``$No cardinal above $\gk$ is measurable'', i.e.,
$V \models ``$If $\gd$ is a measurable cardinal, $\gd \le \gk$''.
Therefore,
by the proof of Theorem \ref{t1}a), if $V \models ``\gd$ is a measurable cardinal'', then
in $V^{\FP_{\gd + 1}}$, ELP($\gd$) holds,
and for every good ordinal $\ga$, LP($\gd, \ga$) holds with respect to
$2^{2^\gd} = \gd^{++}$ many normal measures.
Write $\FP = \FP_{\gd + 1} \ast \dot \FQ$.
By its definition, $\FP$ acts nontrivially only on inaccessible cardinals.
It consequently follows that
$\forces_{\FP_{\gd + 1}} ``\dot \FQ$ is
$\gd'$-directed closed'', %for $\gd'$ the least inaccessible cardinal above $\gd$'',
%$(2^{2^\gd})^+$-directed closed''.
%${(2^{[\gl]^{< \gd}})}^{+}$-directed closed'',
so in $V^{\FP_{\gd + 1} \ast \dot \FQ} = V^\FP$,
ELP($\gd$) holds,
and for every good ordinal $\ga$, LP($\gd, \ga$) holds with respect to
$2^{2^\gd} = \gd^{++}$ many normal measures.
The proof of Lemma \ref{l3} will therefore be finished once we have shown that
if $V^\FP \models ``\gd$ is a measurable cardinal'', then
$V \models ``\gd$ is a measurable cardinal'' as well. To do this, write
$\FP = \add(\go, 1) \ast \dot \FR$.
Since $\card{\add(\go, 1)} = \go$, $\add(\go, 1)$ is nontrivial, and
$\forces_{\add(\go, 1)} ``\dot \FR$ is $\ha_2$-directed closed'',
as we observed immediately after the statement of Theorem \ref{tgf},
$\gd$ must be measurable in $V$ as well.
This completes the proof of Lemma \ref{l3}.
%where $\gd_0$ is the least inaccessible cardinal,
%$\FQ$ is the lottery sum of
%$\{\add(\gd_0, \gd_0^{+ \ga}) \mid \ga \ge 2$ is a good ordinal$\}$, and
%$\dot \FR$ is a term for the rest of $\FP$. If $\gd_1$ is the
%second inaccessible cardinal, it is then the case that
\end{proof}
As we have just noted, any cardinal measurable in $V^\FP$
must also have been measurable in $V$.
Therefore, by our remarks in the paragraph immediately
preceding Lemma \ref{l3}, it must be the case that if $\gd$ is
measurable in $V^\FP$, then %$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{++}$.
$2^\gg = \gg^+$ for all $\gg \in [\gd, \gd')$.
\begin{lemma}\label{l4}
If $V \models ``\gd < \gl$ are such that $\gd$ is $\gl$ supercompact
and $\gl$ is regular'', then $V^\FP \models ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
%We significantly modify the proof of \cite[Case 1, Lemma 3.4]{A14}.
Suppose $\gd < \gl$ are as in the hypotheses for Lemma \ref{l4}.
Because $V \models ``$No cardinal is supercompact up to an
inaccessible cardinal'', it must be the case that $\gl \ge \gd^+$ is a successor cardinal
and $\gl < \gd'$.
%We may therefore
As in the proof of Lemma \ref{l3},
write $\FP = \FP_{\gd + 1} \ast \dot \FQ$.
Suppose $j : V \to M$ is an
elementary embedding witnessing the
$\gl$ supercompactness of $\gd$
which is generated by a supercompact ultrafilter over $P_\gd(\gl)$.
Note that since $2^\gd = \gd^+ \le \gl$ and $M^\gl \subseteq M$,
$M \models ``\gd$ is a measurable cardinal''.
In addition, $(\gl^+)^M = (\gl^+)^V$.
Hence, by the definition of $\FP$,
the nine criteria necessary to apply
the standard lifting arguments mentioned in Section \ref{s1}
when forcing with $\FP_{\gd + 1}$ may all be easily verified.
This means that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
Since as we have already observed in the proof of Lemma \ref{l3},
$\forces_{\FP_{\gd + 1}} ``\dot \FQ$ is
$\gd'$-directed closed''. %for $\gd'$ the least inaccessible cardinal above $\gd$'',
%${(2^{[\gl]^{< \gd}})}^{+}$-directed closed'',
$V^{\FP_{\gd + 1} \ast \dot \FQ} = V^\FP \models ``\gd$ is
$\gl$ supercompact''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$Level by level equivalence holds''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd < \gl$ are such that $\gl$ is
regular and $\gd$ is $\gl$ strongly compact''. Because
$V^\FP \models ``$No cardinal above $\gk$ is inaccessible'',
it must be the case that $\gd \le \gk$.
Since it is an immediate corollary of Lemma \ref{l4} that
$V^\FP \models ``\gk$ is supercompact'', we may assume
without loss of generality that $\gd < \gk$.
Note that by its definition, $\FP$ is mild with respect to $\gd$.
Thus, using the factorization of $\FP$ given in the proof of Lemma \ref{l3}
%we may write $\FP = \add(\go, 1) \ast \dot \FQ$, where
%$\forces_{\add(\go, 1)} ``\dot \FQ$ is $\ha_2$-directed closed''. Thus, by
and Theorem \ref{tgf},
$V \models ``\gd$ is $\gl$ strongly compact''.
As $V \models ``$No cardinal is supercompact up to an inaccessible
cardinal'', %as we have noted in Section \ref{s1},
$\gd$ cannot be a measurable limit of cardinals $\gg$ which are $\gl$ supercompact.
Hence, because level by level equivalence holds in $V$,
$V \models ``\gd$ is
$\gl$ supercompact'' as well.
By Lemma \ref{l4}, $V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$V^\FP \models ``$No cardinal is supercompact up to an
inaccessible cardinal''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd < \gl$ are such that $\gl$ is inaccessible
and $\gd$ is $\ga$ supercompact for every $\ga < \gl$''.
By the factorization of $\FP$ given in the proof of Lemma \ref{l3} and Theorem \ref{tgf},
$V \models ``\gd < \gl$ are such that $\gl$ is inaccessible
and $\gd$ is $\ga$ supercompact for every $\ga < \gl$'' also.
This, however, contradicts our hypotheses on $V$ and completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l3} -- \ref{l6} and the intervening remarks complete the proof of Theorem \ref{t2}.
\end{pf}
\begin{pf}
To prove Theorem \ref{t3},
suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Assume in addition that in $V$, no cardinal is supercompact
up to an inaccessible cardinal, and level by level equivalence holds.
We first force with the partial ordering used in the proof of Theorem \ref{t2}
and then abuse notation by calling the resulting generic extension
$V$ as well. In this way,
%Without loss of generality, by first forcing with the partial ordering
%used in the proof of Theorem \ref{t2},
we may also assume that what we will take as our ground model
$V$ satisfies the conclusions of Theorem \ref{t2}
(which of course means that $V \models ``$No cardinal above
$\gk$ is inaccessible'').
The partial ordering used in the proof of Theorem \ref{t3} is now
defined as
%$\FP = \FP_{\gk + 1} = \la \la \FP_\gb, \dot \FQ_\gb \ra
%\mid \gb \le \gk \ra$. This is the Easton support iteration of length $\gk + 1$ which
$\FP = \FP_{\gk} = \la \la \FP_\gb, \dot \FQ_\gb \ra
\mid \gb < \gk \ra$. This is the Easton support iteration of length $\gk$ which
begins by forcing with $\add(\go, 1)$ and then
does nontrivial forcing only at cardinals $\gd < \gk$
which are $V$-measurable cardinals which are not limits of $V$-measurable cardinals.
At such a stage $\gd$,
$\dot \FQ_\gd$ is a term for ${\rm Coll}(\gd^+, \gd^{++}) = {\rm Coll}(\gd^+, 2^{2^\gd})$.
Once again, by its definition, forcing with $\FP$ preserves
$2^\gg = \gg^+$ whenever $\gd$ is a $V$-measurable cardinal and $\gg \in [\gd, \gd')$.
%$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{++}$ for every
%$V$-measurable cardinal $\gd$.
As in the proof of Theorem \ref{t2}, we may now infer that
if $\gd$ is measurable in $V^\FP$, then $2^\gg = \gg^+$ for all $\gg \in [\gd, \gd')$.
%$2^\gd = \gd^+$ and $2^{\gd^+} = \gd^{++}$ for every
%$V^\FP$-measurable cardinal $\gd$.
%Suppose $\gd^*$ is the least $V$-measurable cardinal above $\gd$.
It is also true that by its definition, for every $V$-measurable cardinal
which is a limit of $V$-measurable cardinals, forcing with $\FP$
preserves all cardinals and cofinalities in the half-open interval $[\gd, \gd')$.
\begin{lemma}\label{l7}
Assume $V^\FP \models ``\gd$ is a measurable cardinal
which is not a limit of measurable cardinals''.
Then $V^\FP \models ``$ELP($\gd$) holds + $\gd$ carries
$\gd^+$ many normal measures +
For every good ordinal $\ga$, $\gd$ carries
$\gd^+$ many normal measures witnessing LP($\gd, \ga$)''.
\end{lemma}
\begin{proof}
Suppose $V \models ``\gd$ is a measurable cardinal which is not a
limit of measurable cardinals''.
%$\gd$ is as in the hypotheses for Lemma \ref{l7}.
%$V^\FP \models ``\gd$ is a measurable cardinal''.
We show that
since we have assumed $V$ satisfies the conclusions of Theorem \ref{t2},
forcing with $\FP$ preserves ELP($\gd$).
We also show that after forcing with $\FP$, $\gd$ carries $\gd^+$
many normal measures, and
for every good ordinal $\ga$,
$\gd$ carries
$\gd^+$ many normal measures witnessing LP($\gd, \ga$).
To do this, note that %we consider the following two cases.
%
%\bigskip
%
%\noindent Case 1: $\gd$ is not a limit of measurable cardinals.
by its definition, we may write $\FP = \FQ \ast \dot {\rm Coll}(\gd^+, \gd^{++})
\ast \dot \FR^*$, where %$\card{\FQ} < \gd$ and
$\forces_{\FQ \ast \dot {\rm Coll}(\gd^+, \gd^{++})}
``\dot \FR^*$ is $\gd'$-directed closed''. %$(2^\gd)^+$-directed closed''.
It thus suffices to show that the preceding three facts %ELP($\gd$)
hold after forcing with $\FQ \ast \dot {\rm Coll}(\gd^+, \gd^{++})$.
However, since $\card{\FQ} < \gd$ and
$\FQ$ is nontrivial, this follows as in the proof of Theorem \ref{t1}b)
by using the proof of \cite[Theorem 1]{ACH}.
To finish the proof of Lemma \ref{l7}, it suffices to show that if
$V^\FP \models ``\gg$ is a measurable cardinal which is not a
limit of measurable cardinals'', then
$V \models ``\gg$ is a measurable cardinal which is not a
limit of measurable cardinals'' as well.
As in the proof of Theorem \ref{t2}, we may write
$\FP = \add(\go, 1) \ast \dot \FR$.
Since $\card{\add(\go, 1)} = \go$, $\add(\go, 1)$ is nontrivial, and as before,
$\forces_{\add(\go, 1)} ``\dot \FR$ is $\ha_2$-directed closed'',
it is once more true that Theorem \ref{tgf} implies
$\gg$ must be measurable in $V$.
If $V \models ``\gg$ is not a limit of measurable cardinals'', then we are done.
If not, then $V \models ``\gg$ is a limit of measurable cardinals'', so
$V \models ``\gg$ is a limit of measurable cardinals $\gr$ which are not themselves
limits of measurable cardinals''.
By our work in the first paragraph, all such $\gr$ remain measurable cardinals in
$V^\FP$, so $V^\FP \models ``\gg$ is a limit of measurable cardinals''.
This contradiction completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
Assume $V^\FP \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals''.
Then $V^\FP \models ``$ELP($\gd$) holds +
For every good ordinal $\ga$, $\gd$ carries
$2^{2^\gd} = \gd^{++}$ many normal measures witnessing LP($\gd, \ga$)''.
\end{lemma}
\begin{proof}
%We argue along the lines of Lemma \ref{l7}.
Suppose $V \models ``\gd$ is a measurable cardinal which is a
limit of measurable cardinals''.
We first show that
since we have assumed $V$ satisfies the conclusions of Theorem \ref{t2},
forcing with $\FP$ preserves ELP($\gd$),
and after forcing with $\FP$, for every good ordinal $\ga$,
$\gd$ carries
$2^{2^\gd} = \gd^{++}$ many normal measures witnessing LP($\gd, \ga$).
To do this, note that by its definition, we may write
$\FP = \FP_\gd \ast \dot \FR$, where
$\forces_{\FP_\gd} ``\dot \FR$ is $\gd'$-directed closed''.\footnote{Note that
if $\gd = \gk$, then $\dot \FR$ is a term for trivial forcing.}
%$(2^{2^\gd})^+$-directed closed''.
Since $V^\FP \models ``2^\gd = \gd^+$'', $V^{\FP_\gd} \models ``2^\gd = \gd^+$''.
It thus suffices to show that the preceding two facts hold after forcing with $\FP_\gd$.
Towards this end, let $\ga$ be a fixed but arbitrary good ordinal.
Take $j : V \to M$ to be an elementary embedding witnessing
the measurability of $\gd$ in $V$ generated by a normal measure $\U$
over $\gd$ witnessing LP($\gd, \ga$).
It is then the case that $A = \{\gg < \gd \mid 2^\gg = \gg^{+ \ga}\} \in \U$,
$\gd \in j(A)$, and $M \models ``2^\gd = \gd^{+ \ga}$''.
%such that $M \models ``\gd$ isn't measurable''.
Note that by the definition of $\FP$, since
$V \models ``\gd$ is a measurable cardinal which is a limit of measurable cardinals'',
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ_\gd \ast \dot \FQ$, where
$\dot \FQ_\gd$ is a term for trivial forcing.
%and $\forces_{\FP_\gd \ast \dot \FQ_\gd} ``\dot \FQ$ is $\gd^+$-directed closed''.
Hence, %by the definition of $\FP$,
the nine criteria necessary to apply
the standard lifting arguments mentioned in Section \ref{s1}
may all once again be easily verified to show that $j$ lifts in $V^{\FP_\gd}$ to
$\bar j : V^{\FP_\gd} \to M^{\bar j(\FP_\gd)}$. Further, in $M$,
$\forces_{\FP_\gd \ast \dot \FQ_\gd} ``\dot \FQ$ is $\gd'$-directed
closed''. %for $\gd'$ the least inaccessible cardinal above $\gd$''.
Therefore, since by its definition, $\gd^{+ \ga} < \gd'$,
the fact
$\ov j (\FP_\gd) = j(\FP_\gd) = \FP_\gd \ast \dot \FQ_\gd \ast \dot \FQ$
allows us to infer that
$M^{\bar j(\FP_\gd)} \models ``2^\gd = \gd^{+ \ga}$''.
This means that $\gd \in \bar j(A)$, from which it is now possible to infer that
the normal measure
$\bar \U = \{x \subseteq \gd \mid \gd \in \bar j(x)\} %\in V^{\FP_\gd}
\supseteq \U$
over $\gd$ witnesses LP($\gd, \ga$) in $V^{\FP_\gd}$.
As $\ga$ was arbitrary, $V^{\FP_\gd} \models {\rm ELP}(\gd)$.
Also, what has really just been shown is that %our work above shows that
for every good ordinal $\ga$,
any normal measure witnessing LP($\gd, \ga$) in $V$ extends to a
normal measure witnessing LP($\gd, \ga$) in $V^{\FP_\gd}$.
Consequently, because
$V$ contains $2^{2^\gd} = \gd^{++}$ many normal measures witnessing LP($\gd, \ga$),
and $\gd^+$ and $\gd^{++}$ are not collapsed in $V^{\FP_\gd}$,
%in either $V^{\FP_\gd}$ or $V^\FP$,
$V^{\FP_\gd}$ contains $2^{2^\gd} = \gd^{++}$ many normal measures witnessing LP($\gd, \ga$)
as well.
As in the proof of Lemma \ref{l7}, we complete the proof of
Lemma \ref{l8} by showing that if
$V^\FP \models ``\gg$ is a measurable cardinal which is a
limit of measurable cardinals'', then
$V \models ``\gg$ is a measurable cardinal which is a
limit of measurable cardinals'' as well.
However, this follows
by the factorization of $\FP$ given in the proof of Lemma \ref{l7}
and another application of Theorem \ref{tgf}.
This completes the proof of Lemma \ref{l8}.
\end{proof}
Suppose $V \models ``\gd < \gl$ are such that $\gd$ is
$\gl$ supercompact and $\gl$ is regular''.
Since $\gl \ge \gd^+$ and $V \models ``2^\gd = \gd^+$'', $\gd$ is
(at least) $2^\gd$ supercompact. Thus, $\gd$ is a measurable
cardinal which is a limit of measurable cardinals, so as
we observed in the proof of Lemma \ref{l8},
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ_\gd \ast \dot \FQ$, where
$\dot \FQ_\gd$ is a term for trivial forcing. In addition, because
$V \models ``$No cardinal is supercompact up to an inaccessible cardinal'',
$\gl < \gd'$.
Also, we know that in $V$, $2^\gg = \gg^+$ for all $\gg \in [\gd, \gd')$.
Hence, the proofs of Lemmas \ref{l4} -- \ref{l6} show that in $V^\FP$,
level by level equivalence holds, $\gk$ is supercompact,
and no cardinal is supercompact up to an inaccessible cardinal.
Since $V^\FP \models ``$No cardinal above $\gk$ is inaccessible'',
this completes the proof of Theorem \ref{t3}.
%Lemmas \ref{l7} and \ref{l8} complete the proof of Theorem \ref{t3}.
\end{pf}
\begin{pf}
To prove Theorem \ref{t4}, suppose $V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals''.
Without loss of generality, by the work of \cite{AS97a},
we may also assume that in $V$, $\gk$ is supercompact iff
$\gk$ is strongly compact, except possibly if $\gk$ is a measurable
limit of supercompact cardinals.
%To define the partial ordering $\FP$ used in the proof of Theorem \ref{t4}, let
%$\Omega = \sup(\{\gd \mid \gd$ is an inaccessible cardinal$\})$ if the inaccessible
%cardinals form a set, or the class of all ordinals if the inaccessible cardinals
%form a proper class.
We now define the partial ordering $\FP$ used in the proof of Theorem \ref{t4}.
$\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd < \gO \ra$
(where $\gO$ is either the ordinal length of the iteration or the class of all ordinals)
is the (possibly proper class) Easton support iteration which
begins by forcing with $\add(\go, 1)$ and then
does nontrivial forcing only at $V$-inaccessible cardinals. %$\gd \in \gO$.
At such a stage $\gd$, if $\gd$ isn't measurable,
$\dot \FQ_\gd$ is a term for the lottery sum %of
$\oplus \{\add(\gd, \gd^{+ \ga}) \mid \ga \ge 2$ is a good ordinal$\}$.
If, however, $\gd$ is measurable, $\dot \FQ_\gd$ is a term for $\add(\gd, \gd^+)$.
By its definition, regardless if $\FP$ is a set or a proper class, $V^\FP \models {\rm ZFC}$.
%$\la \gk_\ga \mid \ga < \Omega \ra$
The proof of Lemma \ref{l3} and our remarks in the
paragraphs immediately preceding and following Lemma \ref{l3} show that in $V^\FP$,
every measurable cardinal $\gd$ is such that $2^\gd = \gd^+$ and
$2^{\gd^+} = 2^{2^\gd} = \gd^{++}$,
every measurable cardinal $\gd$ witnesses ELP($\gd$),
and for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds
with respect to $2^{2^\gd} = \gd^{++}$ many normal measures.
To complete the proof of Theorem \ref{t4}, it therefore remains to show that
$V^\FP \models ``\K$ is the class of supercompact cardinals +
$\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals''.
\begin{lemma}\label{l9}
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
We first show that if $V \models ``\gk$ is supercompact'', then
$V^\FP \models ``\gk$ is supercompact'' as well.
To do this,
let $\gs > \gk$ be a regular cardinal. Define
$\gr = \sup(\{\gd^{+ \ga} \mid \gd \in [\gk, \gs]$ is a cardinal and
$\ga$ is a good ordinal$\})$.
Fix $\gd \in [\gk, \gs]$ a cardinal.
Because $\ga$ is a good ordinal, $\gd^{+ \ga} < \gd'$
%must be below the least inaccessible cardinal above $\gd$ (if it exists).
(if $\gd'$ exists).\footnote{If $\gk$ is the largest inaccessible cardinal in
$V$, then $\gd'$ doesn't exist.}
Hence, as $\gd \le \gs$, $\gd^{+ \ga} < \gs'$ (if $\gs'$ exists).
Since there are only countably many good ordinals
and at most $\gs$ many cardinals in the closed interval
$[\gk, \gs]$, $\gr$ is a singular cardinal below $\gs'$ (if $\gs'$ exists).
%below the least inaccessible cardinal above $\gs$ (if it exists).
Let $\gl = \gr^+$. %(\max(\gr, \gs))^+$.
We prove that $V^\FP \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ may be made arbitrarily large,
this will show that $V^\FP \models ``\gk$ is supercompact''.
Towards this end, take
$j : V \to M$ to be an elementary embedding witnessing the $\gl$ supercompactness
of $\gk$ generated by a supercompactness measure over $P_\gk(\gl)$.
Write
$\FP = \FP_\gk \ast \dot \FQ \ast \dot \FR$, where $\dot \FQ$ is a term for the
portion of $\FP$ acting on cardinals in the closed interval $[\gk, \gl]$, and
$\dot \FR$ is a term for the rest of $\FP$. Since
$\forces_{\FP_\gk \ast \dot \FQ} ``\dot \FR$ is $(2^{[\gl]^{< \gk}})^+$-directed closed'',
%$\gl'$-directed closed'',
%for $\gl'$ the least inaccessible cardinal above $\gl$'',
it suffices to show that
$V^{\FP_\gk \ast \dot \FQ} \models ``\gk$ is $\gl$ supercompact''.
However, by the definitions of $\gl$ and $\FP_\gk \ast \dot \FQ$,
the nine criteria necessary to apply
the standard lifting arguments mentioned in Section \ref{s1}
may all be easily verified.
This means that
$V^{\FP_\gk \ast \dot \FQ} \models ``\gk$ is $\gl$ supercompact'', i.e.,
$V^\FP \models ``\gk$ is supercompact''.
The proof of Lemma \ref{l9} will be finished if we can show that if
$V^\FP \models ``\gk$ is supercompact'', then $V \models ``\gk$ is
supercompact'' as well.
Note that we can write $\FP = \add(\go, 1) \ast \dot \FQ'$.
Since $\card{\add(\go, 1)} = \go$, $\add(\go, 1)$ is nontrivial, and as before,
$\forces_{\add(\go, 1)} ``\dot \FQ'$ is $\ha_2$-directed closed'',
by our remarks immediately following the statement of Theorem \ref{tgf},
$\gk$ must be supercompact in $V$.
This completes the proof of Lemma \ref{l9}.
\end{proof}
\begin{lemma}\label{l10}
$V^\FP \models ``\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gk$ is strongly compact''.
Note that by its definition,
since it must be the case that $V \models ``\gk$ is inaccessible'',
$\FP$ is mild with respect to $\gk$.
Therefore, by the factorization of $\FP$ given in Lemma \ref{l9}
and our remarks immediately following the statement of
Theorem \ref{tgf}, $V \models ``\gk$ is strongly compact'' as well.
By our assumptions on $V$, it must then be the case that
$V \models ``$Either $\gk$ is supercompact, or $\gk$ is a measurable
limit of supercompact cardinals''.
Since by the proof of Lemma \ref{l9}, forcing with $\FP$ preserves all
$V$-supercompact cardinals, we may now infer that
$V^\FP \models ``$Either $\gk$ is supercompact, or $\gk$ is a measurable
limit of supercompact cardinals''.
This completes the proof of Lemma \ref{l10}.
\end{proof}
Lemmas \ref{l9} and \ref{l10} complete the proof of Theorem \ref{t4}.
\end{pf}
\begin{pf}
Finally, to prove Theorem \ref{t5},
suppose $V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals''.
As in the proof of Theorem \ref{t4},
%without loss of generality,
by the work of \cite{AS97a},
we may first assume that in $V$, $\gk$ is supercompact iff
$\gk$ is strongly compact, except possibly if $\gk$ is a measurable
limit of supercompact cardinals.
We then assume, by forcing over $V$ with the partial ordering
used in the proof of Theorem \ref{t4}, that $V$ satisfies the
conclusions of Theorem \ref{t4}. We again abuse notation somewhat
by relabelling the resulting generic extension as $V$ and taking
this as our ground model.
We are now in a position to define the partial ordering $\FP$ used
in the proof of Theorem \ref{t5}.
$\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd < \gO \ra$
(where $\gO$ is as before either the ordinal length of the iteration or the class of all ordinals)
is the (possibly proper class) Easton support
iteration which begins by forcing with $\add(\go, 1)$ and then
does nontrivial forcing only at cardinals $\gd$
which are in $V$ measurable cardinals which are not limits of measurable cardinals.
At such a stage $\gd$,
$\dot \FQ_\gd$ is a term for ${\rm Coll}(\gd^+, \gd^{++}) = {\rm Coll}(\gd^+, 2^{2^\gd})$.
As before, by its definition, regardless if $\FP$ is a set or a proper class,
$V^\FP \models {\rm ZFC}$.
The proof of Theorem \ref{t3} shows that
$V^\FP \models ``$For every measurable cardinal $\gd$,
$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{++}$''.
Assume now that $V^\FP \models ``\gd$ is a measurable cardinal
which is not a limit of measurable cardinals''.
The proof of Lemma \ref{l7} shows that
$\gd$ witnesses ELP($\gd$), $\gd$ carries $\gd^+$ many normal measures,
and for every good ordinal $\ga$, LP($\gd, \ga$) holds
with respect to $\gd^{+}$ many normal measures.
Assume next that $V^\FP \models ``\gd$ is a measurable cardinal
which is a limit of measurable cardinals''.
The proof of Lemma \ref{l8} shows that $\gd$ witnesses ELP($\gd$),
and for every good ordinal $\ga$ and every measurable cardinal $\gd$,
LP($\gd, \ga$) holds
with respect to $2^{2^\gd} = \gd^{++}$ many normal measures.
To complete the proof of Theorem \ref{t5}, it therefore once again remains to show that
$V^\FP \models ``\K$ is the class of supercompact cardinals +
$\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals''.
\begin{lemma}\label{l11}
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l9},
we first show that if $V \models ``\gk$ is supercompact'', then
$V^\FP \models ``\gk$ is supercompact'' as well.
To do this, we let $A = \{\gd > \gk \mid \gd$ is an inaccessible cardinal$\}$.
Suppose $A$ is a set.
Because $V$ is obtained by forcing over a model $\ov V$ of
GCH with a partial ordering $\FQ$ which preserves all inaccessible cardinals,
the collection of inaccessible cardinals in $\ov V$ is a set and not a proper class.
Thus, $\FQ$ is also a set and not a proper class, and so forcing with $\FQ$
preserves GCH on a final segment of cardinals.
This means we can
let $\gg > \sup(A)$ be a regular cardinal such that for every cardinal
$\gd \ge \gg$, $2^\gd = \gd^+$ and take $\gl > \gg$ to be an
arbitrary regular cardinal.
If $A$ is a proper class, then let $\gd > \gk$ be a non-measurable inaccessible cardinal.
As in the preceding paragraph, let $\FQ$ be the forcing used to obtain
$V$ from $\ov V$.
Although $\FQ$ is now a proper class and not a set, by its definition,
since $\ov V \models {\rm GCH}$, there must be some cardinal
$\gg \in (\gd, \gd')$ such that $2^\gs = \gs^+$ for all cardinals
$\gs \in [\gg, \gd')$. Fix $\gl \in (\gg, \gd')$ as such a regular cardinal.
Regardless of whether $A$ is a set or a proper class,
the definition of $\gl$, combined with the fact that $\gl$ may be made
arbitrarily large, now allow us to argue as in the proof of Lemma \ref{l9}
to show that $V^\FP \models ``\gk$ is supercompact'' and then infer that
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
This completes the proof of Lemma \ref{l11}.
\end{proof}
Since the same proof as given in Lemma \ref{l10} shows that
$V^\FP \models ``\gk$ is supercompact iff $\gk$ is strongly compact,
except possibly if $\gk$ is a measurable limit of supercompact cardinals'',
the proof of Theorem \ref{t5} is now complete.
\end{pf}
%We conclude with two questions.
In conclusion, we ask whether it is possible to establish analogues of
Theorems \ref{t2} and \ref{t3} in which the large cardinal structure
of both our ground models and generic extensions is richer.
In particular, can there be models witnessing
the conclusions of Theorems \ref{t2} and \ref{t3}
in which the class of supercompact cardinals is
arbitrary, or must there be some restrictions of necessity? Finally, we ask
whether it is possible to establish analogues of each of our theorems in which
for measurable cardinals $\gd$ and good ordinals $\ga$, the cardinality
of the number of normal measures witnessing LP($\gd, \ga$) is
different from either $\gd^+$ or $2^{2^\gd}$.
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\end{thebibliography}
\end{document}
We take this opportunity to observe that in Theorems \ref{t4} and \ref{t5},
it is possible to have $2^{2^\gd} > \gd^{++}$ for each measurable cardinal $\gd$.
As an example, suppose we wish to have $2^{2^\gd} = \gd^{+ 3}$ for
every measurable cardinal $\gd$.
We first force over our initial model $V$ satisfying GCH and level by level equivalence with
the Easton support iteration which begins %by forcing
with $\add(\go, 1)$ and
then does nontrivial forcing only at %those stages $\gd$ which are
inaccessible cardinals in $V$.
At such a stage $\gd$, our partial ordering is $\add(\gd^+, \gd^{+ 3})$.
The arguments of this paper show that after this initial %forcing has taken place,
portion of our new construction has taken place,
the class of supercompact cardinals is the same as in $V$,
the supercompact and strongly compact cardinals coincide, except at
measurable limits of supercompact cardinals, and for all inaccessible cardinals $\gd$,
$2^\gd = \gd^+$ and $2^{\gd^+} = 2^{2^\gd} = \gd^{+ 3}$.
If we then continue as in our original proofs of Theorems \ref{t4} and \ref{t5}
(except that for the new version of Theorem \ref{t5}, we use %force with
${\rm Coll}(\gd^+, \gd^{+ 3})$ since $2^{\gd^+} = 2^{2^\gd} = \gd^{+ 3}$),
the arguments of this paper show that the models constructed are as desired.
We leave it to any interested readers to fill in the missing details for themselves.
\begin{lemma}\label{l11}
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l9},
we first show that if $V \models ``\gk$ is supercompact'', then
$V^\FP \models ``\gk$ is supercompact'' as well.
To do this, we let $A = \{\gd > \gk \mid \gd$ is a measurable cardinal$\}$.
If $A$ is a set, then let $\gl > \sup(A)$ be the successor of a strong limit cardinal.
If $A$ is a proper class, then let $\gd > \gk$ be a measurable cardinal, and take
$\gl = \gd^{+ 4}$.
The definition of $\gl$, combined with the fact that $\gl$ may be made
arbitrarily large, now allow us to argue as in the proof of Lemma \ref{l9}
to show that $V^\FP \models ``\gk$ is supercompact'' and then infer that
$V^\FP \models ``\K$ is the class of supercompact cardinals''.
This completes the proof of Lemma \ref{l11}.
\end{proof}