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\title{Tallness and Level by Level Equivalence
and Inequivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, superstrong cardinal,
strong cardinal, tall cardinal,
level by level equivalence between strong
compactness and supercompactness,
level by level inequivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants
and CUNY
Collaborative Incentive grants.}
\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/apter\\
awapter@alum.mit.edu}
\date{September 7, 2008\\
(revised April 3, 2009)}
\begin{document}
\maketitle
\begin{abstract}
We construct two models
containing exactly one supercompact cardinal in which
%$\gk$ in which
all non-supercompact measurable cardinals are strictly
taller than they are either strongly
compact or supercompact.
In the first of these models,
level by level equivalence between
strong compactness and supercompactness holds.
In the other, level by level inequivalence between
strong compactness and supercompactness holds.
%There are relatively few large cardinals
%above $\gk$ in each universe.
Each universe has only one strongly compact
cardinal and contains relatively few large cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with definitions due to Hamkins which are
found in \cite{H5}.
For $\gk$ a measurable
cardinal and $\ga$ any ordinal, $\gk$ is {\em $\ga$ tall}
if there is an elementary embedding
$j : V \to M$ having critical point $\gk$ such that
$j(\gk) > \ga$ and $M^\gk \subseteq M$. $\gk$ is {\em tall}
if $\gk$ is $\ga$ tall for every ordinal $\ga$.
In \cite{H5}, Hamkins began the study of {\rm tall cardinals}
and showed that they satisfy many
interesting and fundamental properties.
In particular, the existence of any number
of tall cardinals (one tall cardinal,
$\gg$ many tall cardinals for $\gg$
an arbitrary ordinal, or even a proper class of tall cardinals)
is equiconsistent with the same number of strong
cardinals, thereby showing the nontriviality of
tallness. %this concept.
Further, in analogy to strongness, $\ga$ tallness
is witnessed by an extender embedding
(which, without loss of generality, may
be assumed to have rank below the
least strong limit cardinal above $\ga$),
and if $\gd$ is $\ga$ tall, then $\gd$
is $\gb$ tall for every $\gb < \ga$.
We will extend
Hamkins' study of tallness to the context
of level by level
equivalence between strong
compactness and supercompactness and
level by level inequivalence between
strong compactness and supercompactness.
Before doing so, %stating our two main theorems,
we first present a few additional relevant definitions.
For any ordinal $\gd$, let $\gs_\gd$ be the smallest
measurable cardinal greater than $\gd$.
For any non-supercompact measurable cardinal $\gd$, let
$\gth_\gd$ be the smallest cardinal greater than
$\gd$ such that $\gd$ is not $\gth_\gd$ supercompact.
For any non-strongly compact measurable cardinal
$\gd$, let $\gr_\gd$ be the smallest cardinal
greater than $\gd$ such that $\gd$ is not
$\gr_\gd$ strongly compact. Observe that if
$\gd$ is not strongly compact, then both
$\gth_\gd$ and $\gr_\gd$ are defined, and
$\gth_\gd \le \gr_\gd$.
Suppose $V$ is a model of ZFC
in which for all regular cardinals
$\gd < \gl$, $\gd$ is $\gl$ strongly
compact iff $\gd$ 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}.
%We will also say that $\gk$ is a witness to level by
%level equivalence between strong compactness and supercompactness
%iff for every regular cardinal $\gl > \gk$,
%$\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$ supercompact.
Suppose now that $V$ is a model of ZFC in which
if $\gd$ is measurable but not supercompact,
then $\gd$ is $\gth_\gd$ strongly compact.
$V$ is then said to witness
{\em level by level inequivalence between
strong compactness and supercompactness}.
A non-supercompact measurable cardinal
$\gd$ which is $\gth_\gd$ strongly compact
is said to witness
{\em level by level inequivalence between
strong compactness and supercompactness}.
For brevity, in all of these definitions,
we will henceforth eliminate the phrase
``between strong compactness and supercompactness.''
Models containing supercompact cardinals
in which level by level equivalence
%between strong compactness and supercompactness
holds were first constructed in \cite{AS97a}.
A model with exactly one supercompact cardinal
in which level by level inequivalence
%between strong compactness and supercompactness
holds was constructed in \cite{A02a}.
A theorem of Magidor (see \cite[Lemma 7]{AS97a})
shows that if $\gk$ is supercompact, then
there are always cardinals $\gd < \gl < \gk$ such that
$\gl$ is singular of cofinality greater than or
equal to $\gd$, $\gd$ is $\gl$ strongly compact,
but $\gd$ is not $\gl$ supercompact.
As a consequence of Magidor's theorem,
it follows that if
$V \models ``$Level by level equivalence holds and
$\gd$ is measurable but not strongly compact'',
then either
$\gr_\gd = \gth_\gd$ or $\gr_\gd = {(\gth_\gd)}^+$.
The purpose of this paper is to
establish two results which show that
it is possible to have
models for level by level equivalence
and level by level inequivalence containing
exactly one supercompact cardinal and no other
strongly compact cardinals in which
each non-supercompact
measurable cardinal $\gd$ is strictly taller
than it is either strongly compact or supercompact.
%In particular, $\gd$ will be $\gl_\gd$ tall,
%where for any ordinal $\ga$, $\gl_\ga$ is
%the least measurable cardinal above $\ga$.
To avoid trivialities,
our witnesses for tallness for
$\gd$ will be inaccessible cardinals
greater than $\gr_\gd$.\footnote{Hamkins has shown in
\cite[Lemma 2.1]{H5}
that if $\gd$ is $\gl$ tall, then $\gd$ is
$(\gl)^\gd$ tall.
He has further shown in
\cite[Theorem 2.11]{H5}
that if $\gd$ is $\gl$ strongly compact, then
$\gd$ is $\gl^+$ tall.
Thus, any (fully) strongly compact
cardinal must be tall. However, as the referee
has pointed out, it is possible to infer from
\cite[Theorem 2.11]{H5} that if
$\gd$ is measurable and $\gr_\gd = \gl^+$,
then $\gd$ is strictly taller than it is
strongly compact. This is since $\gd$ is
(at least) $\gl^+$ tall, yet $\gd$ is only $\gl$
strongly compact. If, on the other hand, $\gr_\gd$
is a limit cardinal, then the proof given
in \cite[Theorem 2.11]{H5} will not allow us to conclude
that $\gd$ is strictly taller than it is strongly
compact. This is because Hamkins' argument
uses that a $\gl$ strong compactness embedding
actually witnesses $\gl^+$ tallness, so
when $\gr_\gd$ is a limit cardinal, it is unclear
how to find one strong compactness embedding
which gives enough tallness.
Whether or not there is a
ZFC proof that every non-strongly compact
measurable cardinal is strictly taller than it is
strongly compact is an open question.}
%as tall as it is strongly compact.}
Specifically, we prove the following two theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact''.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up
to a measurable cardinal''.
In $V^\FP$, level by level equivalence holds, and
$\gk$ is the only strongly compact cardinal.
Further, in $V^\FP$,
every non-supercompact measurable cardinal $\gd$ is
$\gl_\gd$ tall,
for some inaccessible cardinal $\gl_\gd > \gr_\gd$.
In particular, in $V^\FP$, every non-supercompact
measurable cardinal $\gd$ is taller than
it is strongly compact.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact and $\gl$ is the least
inaccessible cardinal above $\gk$ +
Level by level equivalence holds''.
%and there is an elementary embedding $j : V \to M$
%generated by a supercompact ultrafilter over $P_\gk(\gl)$
%such that $M \models ``\gk$ is $\gl$ supercompact''.
There is then a partial ordering $\FP \in V$,
a submodel $\ov V \subseteq V^\FP$, and a
cardinal $\gk_0 < \gk$ such that
$\ov V \models ``$ZFC + GCH + $\gk_0$ is supercompact +
No cardinal is supercompact up to an inaccessible
cardinal''.
In $\ov V$, level by level inequivalence holds, and
$\gk_0$ is the only strongly compact cardinal.
Further, in $\ov V$, every non-supercompact
measurable cardinal $\gd$ is
$\gl_\gd$ tall,
for some inaccessible cardinal $\gl_\gd > \gr_\gd$.
In particular, in $\ov V$, every non-supercompact
measurable cardinal $\gd$ is taller than
it is strongly compact.
\end{theorem}
In fact, for both Theorems \ref{t1} and \ref{t2},
the witnessing models will be such that
every measurable cardinal $\gd$ is $\gl_\gd$
tall, where $\gl_\gd$ is the least weakly
compact cardinal above $\gr_\gd$,
the least Ramsey
cardinal above $\gr_\gd$, the least Ramsey limit
of Ramsey cardinals above $\gr_\gd$, or in general,
where $\gl_\gd$ is any ``reasonable''
large cardinal provably
below the least measurable cardinal above $\gr_\gd$.
We will explain this in greater detail towards
%We will come to this point again towards
the end of the paper.
We now very briefly give some
preliminary information
concerning notation and terminology.
%For anything left unexplained,
%readers are urged to consult \cite{A03},
%\cite{A01a},
%\cite{AS97a}, or \cite{AS97b}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
%and $p \decides \varphi$ means that
%{\em $p$ decides $\varphi$}.
%For $\gk$ a regular cardinal and $\gl$ an ordinal,
%$\add(\gk, \gl)$ is the standard partial ordering for adding
%$\gl$ many Cohen subsets of $\gk$.
%For $\ga < \gb$ ordinals,
%$[\a, \b]$, $[\a, \b)$, $(\a, \b]$, and
%$(\a, \b)$ are as in standard interval notation.
%$[\ga, \gb]$ and $(\ga, \gb]$ 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 $\FP$.
%If $\FP$ is a reverse Easton iteration
%such that at stage $\ga$, a nontrivial
%forcing is done adding a subset
%of $\gd$, then we will say that
%$\gd$ is in the field of $\FP$.
We will, from time to time,
confuse terms with the sets
they denote and write
$x$ when we actually mean
$\dot x$ or $\check x$.
The partial ordering
%$\FP$ is {\em $\gk$-directed closed} 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
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even
%(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures 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
$\langle p_\ga: \ga < \gk\rangle$, where player I plays odd
stages and player
II plays even
%(which of course includes limit)
stages (choosing the
trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
$\FP$ is {\em $(\gk, \infty)$-distributive} if
given a sequence $\la D_\ga : \ga < \gk \ra$
of dense open subsets of $\FP$,
$\bigcap_{\ga < \gk} D_\ga$ is dense open as well.
Note that if $\FP$ is $\gk$-strategically closed,
then $\FP$ is $(\gk, \infty)$-distributive.
Further, if $\FP$ is $(\gk, \infty)$-distributive and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
%$ \FP$ is ${<}\gk$-strategically closed
%if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
A corollary of Hamkins' work on
gap forcing found in
\cite{H2} and \cite{H3}
will be employed in the
proof of Theorem \ref{t2}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2} and \cite{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''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ {\it admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{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 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}, \cite{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} and \cite{H3}
we will be using
is then the following.
\begin{theorem}\label{t3}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing 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}
We assume familiarity with the
large cardinal notions of
measurability, strongness, superstrongness,
strong compactness, and supercompactness.
Readers are urged to consult
\cite{J} for further details.
Note that we will say that
{\em $\gk$ is supercompact (or strongly compact or tall)
up to the cardinal $\gl$} if
$\gk$ is $\gg$ supercompact
(or $\gg$ strongly compact or $\gg$ tall) for every
$\gg < \gl$.
%Finally, for any ordinal $\gd$, $\gd'$
%will denote the least strong cardinal
%in the ground model $V$ greater than $\gd$.
\section{The Proofs of Theorems \ref{t1} and
\ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V \models ``$ZFC + $\gk$ is supercompact''.
By first forcing GCH, then forcing with the
partial ordering of \cite{AS97a}, and then taking
the appropriate submodel if necessary,
we slightly abuse notation and assume in addition that
$V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up
to a measurable cardinal + Level by level
equivalence holds''.
In particular, $V \models ``\gk$ is the
only strongly compact cardinal''.
We are now in a position to define the partial
ordering $\FP$ used in the proof of Theorem \ref{t1}.
Let $A = \{\gd < \gk : \gd$ is a non-superstrong
measurable cardinal$\}$. $\FP$ is the reverse
Easton iteration of length $\gk$ which
begins by adding a Cohen subset of $\go$ and then
does nontrivial forcing only at stages
$\gd \in A$, where it adds a
non-reflecting stationary set of ordinals
of cofinality $\go$ to $\gd$.\footnote{The precise
definition of this partial ordering may
be found in either \cite{AS97a} or
\cite{AC2}. We do wish to
note here, however, that it is
${\prec} \gd$-strategically closed.
Also, whenever $\gd$ is inaccessible,
this partial ordering has cardinality $\gd$.}
%each member of $A$ and is trivial everywhere else.
Standard arguments show that $V^\FP \models {\rm GCH}$
and that $V$ and $V^\FP$ have the same
cardinals and cofinalities.
\begin{lemma}\label{l1}
Suppose $\gd < \gl$ are both regular cardinals
and $\gd < \gk$.
If $V \models ``\gd$ is $\gl$ supercompact and
$\gd$ is superstrong'', then
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
%Assume first that $\gd < \gk$.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$
supercompactness of $\gd$ generated
by a supercompact ultrafilter over $P_\gd(\gl)$.
Write $\FP = \FP_\gd \ast \dot \FP^\gd$.
Since $V \models ``$No cardinal is supercompact
up to a measurable cardinal'',
$\gl < \gs_\gd$.
%, where for the rest of this paper, $\gs$ is the least
%$\gl$ is below the least $V$-measurable cardinal above $\gd$.
Therefore, by the definition of $\FP$,
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
${\prec}(\gs_\gd)^V$-strategically closed''.
In addition, because $\card{\FP_\gd} = \gd$,
by the L\'evy-Solovay results \cite{LS},
it is the case that
$\forces_{\FP_\gd} ``(\gs_\gd)^V =
(\gs_\gd)^{V^{\FP_\gd}}$''.
%is the least measurable cardinal above $\gd$''.
%(at least) $2^{[\gl]^{< \gd}}$-strategically closed''.
%where $\eta$ is the least inaccessible cardinal above $\gl$
%(in either $V$ or $V^{\FP_\gd}$).
Thus, to show that $V^\FP \models ``\gd$
is $\gl$ supercompact'', it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl$ supercompact''.
However, by the facts that $\gl < \gs_\gd$ and
$M^\gl \subseteq M$, $M \models ``$No cardinal in the
half-open interval $(\gd, \gl]$ is measurable''.
Further, since by GCH,
$\gl \ge \gd^+ = 2^\gd$, as in \cite[Lemma 2.1]{AC2},
$M \models ``\gd$ is superstrong''.
This means that $j(\FP_\gd) = \FP_\gd \ast \dot \FQ$,
where the first ordinal at which $\dot \FQ$ is
forced to do nontrivial forcing is well above $\gl$.
A standard argument now shows that
$j$ lifts in $V^{\FP_\gd}$ to a
$\gl$ supercompactness embedding
$j: V^{\FP_\gd} \to M^{j(\FP_\gd)}$.\footnote{An
outline of this argument is as follows. Let
$G_0$ be $V$-generic over $\FP_\gd$.
The same method as found, e.g., in the
construction of the generic object $G_1$
in the proof of \cite[Lemma 2.4]{AC2} now
allows us to build in $V[G_0]$ an
$M[G_0]$-generic object $G_1$ over $\FQ$.
This argument uses that by GCH and
the fact that $j$ is given by an
ultrapower embedding, we may let
$\la D_\ga : \ga < \gl^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of $\FQ$ present
in $M[G_0]$. Because $M[G_0]$ remains
$\gl$-closed with respect to $V[G_0]$,
by the ${\prec} \gl^+$-strategic
closure of $\FQ$ in both $M[G_0]$ and $V[G_0]$,
we may work in $V[G_0]$ and
meet each $D_\ga$ in order to construct $G_1$.
Since $j '' G_0 \subseteq G_0 \ast G_1$,
we may lift $j$ in $V[G_0]$ to
$j : V[G_0] \to M[G_0][G_1]$.}
%Hence, if $\gd < \gk$, then $V^\FP \models ``\gd$ is
%$\gl$ supercompact''.
This completes the proof of Lemma \ref{l1}.
\end{proof}
Suppose $\gl > \gk$ is regular.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter over $P_\gk(\gl)$.
Since $V \models ``$No cardinal is supercompact
up to a measurable cardinal'', $V \models ``$There
are no measurable cardinals above $\gk$''. Thus,
$V \models ``\gl$ is not measurable''.
The proof of Lemma \ref{l1} consequently shows that
$j$ lifts in $V^\FP$
as before to $j : V^\FP \to M^{j(\FP)}$.
It therefore follows as an immediate corollary
of the proof of Lemma \ref{l1} that
$V^\FP \models ``\gk$ is supercompact''.
\begin{lemma}\label{l2}
$V^\FP \models ``$No cardinal is supercompact
up to a measurable cardinal''.
\end{lemma}
\begin{proof}
Suppose $\gd \le \gl$ are such that
$V^\FP \models ``\gd$ is $\ga$
supercompact for every $\ga < \gl$
and $\gl$ is measurable''.
By its definition, we may
write $\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
Therefore, by Theorem \ref{t3}, it must be the case that
$V \models ``\gd$ is $\ga$
supercompact for every $\ga < \gl$
and $\gl$ is measurable'' as well.
Since this is contradictory to our hypotheses about $V$,
this completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Level by level equivalence holds''.
\end{lemma}
\begin{proof}
Suppose $\gd \le \gl$ are regular cardinals such that
$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
As has already been observed,
$V \models ``$There are no measurable cardinals above $\gk$''.
Consequently, because $\card{\FP} = \gk$,
the results of \cite{LS} allow us
to infer that $V^\FP \models ``$There are no measurable
cardinals above $\gk$''.
Hence, because $V^\FP \models ``\gk$ is supercompact'',
we may assume without loss of generality that $\gd < \gk$.
Note that from
its definition, $\FP$ is mild with respect to $\gd$.
Thus, by the factorization of $\FP$
(as $\FP' \ast \dot \FP''$) given in
Lemma \ref{l2} and Theorem \ref{t3},
$V \models ``\gd$ is $\gl$ strongly compact''.
Since level by level equivalence holds in $V$,
$V \models ``\gd$ is $\gl$ supercompact'' as well.
Further, it must be the case that
$V \models ``\gd$ is superstrong''.
(If not, then by the definition of $\FP$,
$V^\FP \models ``\gd$ contains a non-reflecting
stationary set of ordinals of cofinality
$\go$ and thus is not weakly compact''.)
Therefore, by Lemma \ref{l1},
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$Every measurable cardinal
$\gd$ is tall up to $(\gs_\gd)^V$''.
\end{lemma}
\begin{proof}
Let $\gd \neq \gk$ be such that
$V^\FP \models ``\gd$ is measurable''.
As in the proof of Lemma \ref{l3},
we may assume without loss of generality that
$\gd < \gk$ and that
$V \models ``\gd$ is superstrong''. Let
$\gl < (\gs_\gd)^V$ be an arbitrary inaccessible
cardinal which is not a limit of
inaccessible cardinals. By the factorization of
$\FP$ (as $\FP_\gd \ast \dot \FP^\gd$) given in
Lemma \ref{l1}, to show that
$V^\FP \models ``\gd$ is $\gl$ tall'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl$ tall''.
To do this, we first observe that because
$V \models ``\gd$ is superstrong'',
$V \models ``\gd$ is $\gl$ strong''.
(This follows since an easy reflection argument
shows that any superstrong cardinal is strong
well beyond the least measurable cardinal above it.)
Using this fact, we will now argue in analogy
to the proof of \cite[Theorem 4.1]{H5}
(which itself uses ideas originally due to
Magidor that are found in the proofs of
\cite[Lemma 2.4]{AC2} and \cite[Lemma 2.3]{AH03},
as well as elsewhere in the literature),
quoting freely (or perhaps verbatim)
from the relevant arguments when appropriate.
Let $j : V \to M$ be an elementary
embedding witnessing the $\gl$ strongness of
$\gd$ generated by a $(\gd, \gl)$-extender.
We may assume without loss of generality that
$M^\gd \subseteq M$ and that there is no
$(\gd, \gl)$-extender ${\cal F} \in M$
(so that in particular, $M \models ``\gd$
is measurable but not superstrong'').
The cardinal $\gl$ is large enough so that
we may choose a normal measure $\U \in M$ over $\gd$
having trivial Mitchell rank and let
$k : M \to N$ be the elementary embedding
generated by $\U$.
As in \cite[Theorem 4.1]{H5}, $i = k \circ j$
is an elementary embedding witnessing the
$\gl$ tallness of $\gd$. We show that
$i : V \to N$ lifts in $V^{\FP_\gd}$ to
$i : V^{\FP_\gd} \to N^{i(\FP_\gd)}$.
Because this lifted embedding will witness
the $\gl$ tallness of $\gd$ in $V^{\FP_\gd}$
and $\gl$ was arbitrary,
this will prove Lemma \ref{l4}.
Let $G_0$ be $V$-generic over $\FP_\gd$. Since
$N \models ``\gd$ is not measurable'', only
trivial forcing is done at stage $\gd$ in
$N^{\FP_\gd}$ in the definition of $i(\FP_\gd)$.
Thus, we may write $i(\FP_\gd) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$, where
$\dot \FQ^1$ is forced to do nontrivial forcing
at ordinals in the half-open interval $(\gd, k(\gd)]$, and
$\dot \FQ^2$ is forced to do nontrivial forcing
at ordinals in the open interval $(k(\gd), k(j(\gd)))$,
i.e., at ordinals in the open interval $(k(\gd), i(\gd))$.
We will build in $V[G_0]$ generic objects $G_1$
and $G_2$ for $\FQ^1$ and $\FQ^2$ respectively.
As it will be the case that
$i '' G_0 \subseteq G_0 \ast G_1 \ast G_2$,
$i$ will lift in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$, and the
proof of Lemma \ref{l4} will be complete.
To construct $G_1$, note that since $k$ is
generated by the ultrafilter $\U$ over $\gd$,
by GCH in $V$, $M$, and $N$,
$V \models ``\card{k(\gd^+)} = \card{k(2^\gd)} =
\card{\{f : f : \gd \to \gd^+\}} =
\card{[\gd^+]^{\gd}} = \gd^+$''. Thus, as
$N[G_0] \models ``\card{\wp(\FQ^1)} = \card{k(2^\gd)}$'',
we can let $\la D_\ga : \ga < \gd^+ \ra \in V[G_0]$
enumerate the dense open subsets of $\FQ^1$
present in $N[G_0]$. Using the fact that
$\FQ^1$ is ${\prec}\gd^+$-strategically closed
in $N[G_0]$, $M[G_0]$, and $V[G_0]$,
the same argument as given
in the construction of the generic object $G_1$
in the proof of \cite[Lemma 2.4]{AC2} is once
again applicable and allows us to build in
$V[G_0]$ an $N[G_0]$-generic object
$G_1$ over $\FQ^1$.
To construct $G_2$, we first write
$j(\FP_\gd) = \FP_\gd \ast \dot \FR^1
\ast \dot \FR^2$. Since
$M \models ``\gd$ is measurable but not
superstrong and there are no measurable
cardinals in the half-open interval
$(\gd, \gl]$''\footnote{It is because
$M \models ``\gl$ is an inaccessible
cardinal which is not a limit of
inaccessible cardinals'' (which of course
follows since $V_\gl \subseteq M$) that
we may infer that $M \models ``$There are
no measurable cardinals in the half-open
interval $(\gd, \gl]$''.},
$\dot \FR^1$ is a term
for the partial ordering adding a non-reflecting
stationary set of ordinals of cofinality $\go$ to
$\gd$, and the first ordinal at which
$\dot \FR^2$ is forced to do nontrivial forcing is
well beyond $\gl$. Next, working in
$M$, we consider the
``term forcing'' partial ordering $\FT^{}$
(see \cite{F} for the first published
account of term forcing or \cite[Section 1.2.5, page 8]{C};
the notion is originally due to Laver)
associated with $\dot \FR^2$. This is given by
$\gt \in \FT^{}$ iff $\gt$ is a term in
the forcing language with respect to
$\FP_\gd \ast \dot \FR^1$ and
$\forces_{\FP_\gd \ast \dot \FR^1} ``\gt \in \dot \FR^2$'',
with ordering $\gt_1 \ge \gt_0$ iff
$\forces_{\FP_\gd \ast \dot \FR^1} ``\gt_1 \ge \gt_0$''.
As in the proof of \cite[Theorem 4.1]{H5}
(see also the proof of \cite[Lemma 2.3]{AH03}),
we may assume without loss of generality that
$M \models ``\card{\FT^{}} = j(\gd)$''. Since
$\forces_{\FP_\gd \ast \dot \FR^1} ``\dot \FR^2$ is
${\prec}\gd^+$-strategically closed'', it can
easily be verified that $\FT^{}$ is also
${\prec}\gd^+$-strategically closed in $M$, and,
since $M^\gd \subseteq M$, in $V$ as well.
In addition, because
$\forces_{\FP_\gd \ast \dot \FR^1} ``\dot \FR^2$ is
$\gl$-strategically closed'',
$M \models ``\FT^{}$ is $\gl$-strategically
closed and hence is also $(\gl, \infty)$-distributive''.
Let $X = \{j(f)(\gd, \gl) : f \in V\}$.
%It is easy to verify that $X \prec M$.
%that $X$ is closed under $\gd$ sequences in $V$,
%and that $\gd, \gl, \FT^{} \in X$.
As in the proof of \cite[Theorem 4.1]{H5},
$X \prec M$ and
$X$ is isomorphic to the ultrapower of $V$
via the measure $\mu$ on $\gd \times \gd$ given by
$A \in \mu$ iff $\la \gd, \gl \ra \in j(A)$.
If we let $j_0 : V \to M_0$ be the ultrapower embedding
by $\mu$,
then $j = j_1 \circ j_0$,
where $j_0 = \pi \circ j$ for $\pi$ the Mostowski
collapse of $X$ to $M_0$,
$j_1 : M_0 \to M$ has critical point greater than $\gd$,
and $j_1 = \pi^{-1}$.
The situation is given by the commutative diagram
\commtriangle{V}{M_0}{M}{j_0}{j_1}{j}
If $\FT_0 = \pi(\FT^{})$ and $\gl_0 = \pi(\gl)$, then
$\FT_0$ has size $j_0(\gd)$ in $M_0$. Further, $\FT_0$ is
${\prec}\gd^+$-strategically closed in %both
$M_0$ and, since $(M_0)^\gd \subseteq V$, in $V$ as well.
The argument used in the construction of $G_1$ may
therefore be employed to construct in $V$ an
$M_0$-generic object $G^*_2$ for $\FT_0$.
%Let $G^{**}_2 = j_1 '' G^*_2$.
We claim that the filter $G^{**}_2 \subseteq \FT^{}$
generated by $j_1 '' G^*_2$ is
$M$-generic over $\FT^{}$. To see this, let
$D \in M$ be a dense open subset of $\FT^{}$.
Hence, $D = j(f)(s)$ for some
$f : V_\gd \to V$, where $f \in V$ and $s \in V_\gl$.
It therefore follows that $D = j_1(r)(s)$, where
$r = j_0(f) \rest {(V_{\gl_0})}^{M_0}$. We may assume that
$r(t)$ is a dense open subset of $\FT_0$ for
every $t \in {(V_{\gl_0})}^{M_0}$.
%In addition, since
%$\gl_0 \le \gl$, $\card{{(V_{\gl_0})}^{M_0}} \le
%\card{{(V_{\gl})}^{M}} = \gl$.
Consequently, since
%$\card{{(V_{\gl_0})}^{M_0}} = \gl_0$
$M_0 \models ``\card{V_{\gl_0}} = \gl_0$
and $\FT_0$ is $(\gl_0, \infty)$-distributive'',
$\ov D = \bigcap_{t \in {(V_{\gl_0})}^{M_0}} r(t)$
is a dense open subset of $\FT_0$ in $M_0$. It then
follows that $\ov D \cap G^*_2 \neq \emptyset$, which
further implies that $j_1(\ov D) \cap G^{**}_2 \neq
\emptyset$. As $j_1(\ov D) \subseteq D$,
$D \cap G^{**}_2 \neq \emptyset$, i.e.,
$G^{**}_2$ is $M$-generic over $\FT^{}$.
Note now that since $N$ is the ultrapower of $M$
via the normal measure ${\cal U} \in M$
over $\gd$, \cite[Fact 2 of Section 1.2.2]{C}
tells us that $k '' G^{**}_2$ generates an
$N$-generic object $G^{***}_2$ over $k(\FT)$.
By elementarity, $k(\FT)$ is the term
forcing partial ordering defined in $N$
with respect to $k(j(\FP_\gd)_{\gd + 1}) =
\FP_\gd \ast \dot \FQ^1$.
Therefore, since $i(\FP_\gd) = k(j(\FP_\gd)) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$,
$G^{***}_2$ is $N$-generic over $k(\FT)$, and
$G_0 \ast G_1$ is $N$-generic over
$k(\FP_\gd \ast \dot \FR^1)$,
\cite[Fact 1 of Section 1.2.5]{C} (see also \cite{F})
tells us that $G_2 = \{i_{G_0 \ast G_1}(\gt) :
\gt \in G^{***}_2\}$ is $N[G_0][G_1]$-generic over $\FQ^2$.
It now follows that $i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$, i.e.,
$V[G_0] \models ``\gd$ is $\gl$ tall''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
$V^\FP \models ``$Every measurable cardinal
$\gd \neq \gk$ is $\gl_\gd$ tall, for some
inaccessible cardinal $\gl_\gd > \gr_\gd$''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l4},
we may assume without loss of generality that $\gd < \gk$.
In addition, as in the proof of Lemma \ref{l3}, $\FP$
is mild with respect to $\gd$. Therefore,
by the factorization of $\FP$ (as
$\FP' \ast \dot \FP''$) given in Lemma \ref{l2}
and Theorem \ref{t3}, if $V^\FP \models ``\gd$ is
$\ga$ strongly compact'', then $V \models ``\gd$
is $\ga$ strongly compact'' as well. Consequently,
since $\gd < \gk$, ${(\gr_\gd)}^V$ is defined, and
${(\gr_\gd)}^{V^\FP} \le {(\gr_\gd)}^V < (\gs_\gd)^V$.
By Lemma \ref{l4} and its proof,
$V^\FP \models ``$There is an
inaccessible cardinal $\gl_\gd > \gr_\gd$ such that
$\gd$ is $\gl_\gd$ tall''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
Since $V^\FP \models ``$Level by level equivalence
holds and there are no measurable cardinals
above $\gk$'', $V^\FP \models ``\gk$ is the
only strongly compact cardinal''. This fact, together with
Lemmas \ref{l1} -- \ref{l5} and the intervening
remarks, complete the proof of Theorem \ref{t1}.
\end{proof}
Having completed the proof of Theorem \ref{t1},
we turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH + $\gk < \gl$ are
such that $\gk$ is $\gl$ supercompact and $\gl$
is the least inaccessible cardinal above $\gk$ +
Level by level equivalence holds''.
Without loss of generality (but with a
slight abuse of notation), we assume that $\gk$
is the least cardinal which is $\gg$ supercompact
for some inaccessible cardinal $\gg > \gk$.
This immediately implies that no cardinal
$\gd < \gk$ is either supercompact
or strongly compact up to a measurable cardinal
(and in fact, that no cardinal $\gd < \gk$ is either
supercompact or strongly compact
up to the second inaccessible cardinal above it).
Let $B = \{\gd < \gk : $ Either $\gd$ is a non-superstrong
measurable cardinal or $\gd$ is not supercompact
up to the least inaccessible cardinal above it$\}$.
The partial ordering $\FP$ used in the proof
of Theorem \ref{t2} is then defined as the reverse
Easton iteration of length $\gk$ which begins by
adding a Cohen subset of $\go$ and then does nontrivial
forcing only at stages $\gd \in B$, where it adds a
non-reflecting stationary set of ordinals of
cofinality $\go$ to $\gd$. Standard arguments once
again show that $V^\FP \models {\rm GCH}$ and that
$V$ and $V^\FP$ have the same cardinals and cofinalities.
%In addition, the exact same arguments as given in the
%proof of Lemma \ref{l4} show that
%$V^\FP \models ``$Every measurable cardinal $\gd < \gk$
%is $\gl_\gd$ tall''.
\begin{lemma}\label{l1a}
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter over $P_\gk(\gl)$.
As in the proof of Lemma \ref{l1}, note that
$M \models ``\gk$ is superstrong''. Therefore,
by the definition of $\FP$ and the fact that
$M \models ``\gl$ is the least inaccessible cardinal
greater than $\gk$ and $\gk$ is
supercompact up to $\gl$'', $j(\FP) = \FP \ast \dot \FQ$,
where the first ordinal at which $\dot \FQ$ is
forced to do nontrivial forcing is well above $\gl$.
The same arguments as used in the proof of Lemma \ref{l1}
now show that $j$ lifts in $V^\FP$ to a $\gl$
supercompactness embedding $j : V^\FP \to M^{j(\FP)}$.
This completes the proof of Lemma \ref{l1a}.
\end{proof}
Using Lemma \ref{l1a} and reflection, we may
now let $\gk_0 < \gk$ be the smallest cardinal
such that $V^\FP \models ``\gk_0$ is supercompact
up to some inaccessible cardinal $\gl_0$''.
Without loss of generality, we assume in addition that
$V^\FP \models ``\gl_0$ is the least inaccessible
cardinal above $\gk_0$''.
\begin{lemma}\label{l2a}
$V^\FP \models ``$If $\gd < \gk_0$
is measurable, then $\gd$ is strongly
compact up to the least inaccessible
cardinal above it''.
\end{lemma}
\begin{proof}
We may assume that
$V \models ``\gd$ is both superstrong and
supercompact up to the least inaccessible
cardinal above it''.
(If this is not the case, then by the definition of $\FP$,
$V^\FP \models ``\gd$ contains a non-reflecting
stationary set of ordinals of cofinality $\go$''.)
Since $\card{\FP_\gd} = \gd$,
this means that we may write
$\FP = \FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$
adds no bounded subsets to the least measurable
cardinal above $\gd$''.
In particular,
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$
adds no subsets to the least inaccessible
cardinal above $\gd$''.
Thus, in order to
prove Lemma \ref{l2a}, it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is strongly compact
up to the least inaccessible cardinal above it''.
To do this, let $\gl > \gd$ be a
regular cardinal below
the least $V$-inaccessible cardinal above $\gd$
(which, by the factorization of
$\FP$ given in the preceding paragraph,
is the same as the least $V^\FP$-inaccessible or
least $V^{\FP_\gd}$-inaccessible
cardinal above $\gd$). Take $j : V \to M$
as an elementary embedding witnessing the
$\gl$ supercompactness of $\gd$ generated
by a supercompact ultrafilter over $P_\gd(\gl)$ such that
$M \models ``\gd$ is not $\gl$ supercompact''.
Note that
$M \models ``\gl$ is below the least inaccessible
cardinal above it''.
Therefore, by the definition of $\FP$,
$j(\FP_\gd) = \FP_\gd \ast \dot \FR^1 \ast \dot \FR^2$,
where $\dot \FR^1$ is a term for the partial ordering
adding a non-reflecting stationary set of ordinals of
cofinality $\go$ to $\gd$, and the first ordinal at which
$\dot \FR^2$ is forced to do nontrivial forcing is
well above $\gl$.
The same argument as found in the proofs of
\cite[Lemma 2.4]{AC2} and \cite[Lemma 2.3]{AH03}
now shows that $V^{\FP_\gd} \models ``\gd$ is
$\gl$ strongly compact''.\footnote{An outline
of this argument is as follows. Let
$k : M \to N$ be an elementary embedding
generated by a normal measure ${\cal U} \in M$
having trivial Mitchell rank. The elementary
embedding $i = k \circ j$ witnesses the
$\gl$ strong compactness of $\gd$ in $V$.
As in the proof of Lemma \ref{l4}, this
embedding lifts in $V^{\FP_\gd}$
to an elementary embedding
$i : V^{\FP_\gd} \to N^{i(\FP_\gd)}$
witnessing the $\gl$ strong compactness of $\gd$.
This is shown by writing $i(\FP_\gd) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$,
%$j(\FP_\gd) = \FP_\gd \ast \dot \FR^1 \ast \dot \FR^2$,
taking $G_0$ to be
$V$-generic over $\FP_\gd$, and building in
$V[G_0]$ generic objects $G_1$ and $G_2$
for $\FQ^1$ and $\FQ^2$ respectively.
The construction of $G_1$ is the same as
that found for the generic object $G_1$
in the proof of Lemma \ref{l4}.
The construction of $G_2$ first requires
building an $M$-generic object $G^{**}_2$
for the term forcing partial ordering
$\FT$ associated with $\dot \FR^2$ and defined
in $M$ with respect to $\FP_\gd \ast
\dot \FR^1$.
%Unlike the argument given in the
%proof of Lemma \ref{l4} for $G^{**}_2$, however,
The current $G^{**}_2$ is built using the fact that since
$M^\gl \subseteq M$, $\FT$ is ${\prec}\gl$-strategically
closed in both $M$ and $V$, which means
that the diagonalization argument employed
in the construction of $G_1$ may be applied
in this situation as well.
$k '' G^{**}_2$ now generates an $N$-generic
object $G^{***}_2$ for $k(\FT)$ and an
$N[G_0][G_1]$-generic object $G_2$ for $\FQ^2$
%in the same way that $k '' G^{**}_2$ does
as in the proof of Lemma \ref{l4}. This means that
$i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$.}
Since $\gl$ was an arbitrary regular cardinal below
the least $V^{\FP_\gd}$-inaccessible cardinal above $\gd$,
the proof of Lemma \ref{l2a} has consequently been completed.
\end{proof}
\begin{lemma}\label{l4a}
$V^\FP \models ``$Every measurable cardinal
$\gd < \gk_0$ witnesses level by level inequivalence''.
\end{lemma}
\begin{proof}
By the choice of $\gk_0$, if
$V^\FP \models ``\gd < \gk_0$ is measurable'', then
$V^\FP \models ``\gth_\gd$ is below the least
inaccessible cardinal above it''. However, by
Lemma \ref{l2a}, $V^\FP \models ``\gd$ is strongly
compact up to the least inaccessible cardinal
above it''.
This completes the proof of Lemma \ref{l4a}.
\end{proof}
Let $\ov V = {(V_{\gl_0})}^{V^\FP}$.
By the definition of $\FP$,
$\ov V \models ``$There are unboundedly in
$\gk_0$ many cardinals $\gd < \gk_0$ containing
non-reflecting stationary sets of ordinals
of cofinality $\go$''. Therefore, by
\cite[Theorem 4.8]{SRK} and the succeeding remarks,
$\ov V \models ``$No cardinal $\gd < \gk_0$ is
strongly compact''.
In addition, using the factorization of $\FP$
(as $\FP_\gd \ast \dot \FP^\gd$) given in Lemma \ref{l2a},
the same arguments as found in the proofs of
Lemmas \ref{l4} and \ref{l5} show that
$V^\FP \models ``$Every measurable cardinal
$\gd < \gk_0$ is $\gl_\gd$ tall,
for some inaccessible cardinal $\gl_\gd > \gr_\gd$''.
These facts, together with
Lemmas \ref{l1a} -- \ref{l4a}, the intervening remarks,
and the fact that $\ov V \models ``$No cardinal
above $\gk_0$ is inaccessible'',
complete the proof of Theorem \ref{t2}.
\end{proof}
Since $\card{\FP_\gd} = \gd$ in the above proofs,
the results of
\cite{LS} imply that for any ordinal $\gg$ in
the open interval $(\gd, \gs_\gd)$,
the least weakly compact cardinal above $\gg$,
the least Ramsey cardinal above $\gg$, the least
Ramsey limit of Ramsey cardinals above $\gg$,
or in general, any large cardinal
provably in the open interval $(\gg, \gs_\gd)$
for which the results of \cite{LS} hold
are the same in
$V$ and $V^{\FP_\gd}$. It is for this reason,
together with the fact that
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
${\prec}(\gs_\gd)^V$-strategically closed and
$(\gs_\gd)^V = (\gs_\gd)^{V^{\FP_\gd}}$'', that
the witnessing models for
Theorems \ref{t1} and \ref{t2}
have each non-supercompact measurable cardinal
$\gd$ exhibit $\gl_\gd$ tallness for $\gl_\gd$
the least weakly compact cardinal above $\gr_\gd$,
the least Ramsey cardinal above $\gr_\gd$,
the least Ramsey limit of Ramsey cardinals
above $\gr_\gd$, etc.
On the other hand, our methods of proof seem to
require ground models with a severely restricted
large cardinal structure
(although the same proof techniques will allow,
e.g., the definition of $\gl$ in Theorem \ref{t2}
to be changed to the least weakly compact cardinal
above $\gk$, thereby giving a model witnessing the
conclusions of Theorem \ref{t2} with a slightly
richer large cardinal structure).
We conclude by asking if it is possible
to remove these restrictions, and obtain results
analogous to those of this paper in which
the large cardinal structure of the universe
can be arbitrary.
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As in \cite{H4},
if ${\cal A}$ is a collection of partial orderings, then
the {\em lottery sum} is the partial ordering
$\oplus {\cal A} =
\{\la \FP, p \ra : \FP \in {\cal A}$
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
$\oplus {\cal A}$, then $G$
first selects an element of
${\cal A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\cal A}$'') and then
forces with it. 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 expressions ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''
$V^\FP \models ``\gk_0$ is the smallest
cardinal supercompact up to an inaccessible
cardinal'', it must be the case that
$V \models ``\gd$ is not supercompact up to
the least inaccessible cardinal $\gl' > \gd$''.
Let $B = \{\gd < \gk : \gd$ is a non-superstrong
measurable cardinal which is not supercompact
up to the least inaccessible cardinal above $\gd\}$.
As in the proofs of Lemmas \ref{l3} and
\ref{l4}, we may assume without loss of generality that
$V \models ``\gd$ is superstrong''. In addition, as in
the proof of Lemma \ref{l3}, we may further assume that
$V \models ``\gd$ is supercompact up to the least
inaccessible cardinal above it''.
\begin{theorem}\label{t1}
Suppose
%$V \models ``$ZFC + GCH + $\gk$ is
%supercompact + No cardinal is supercompact up
%to an inaccessible cardinal + Level by level
%equivalence holds''.
$V \models ``$ZFC + $\gk$ is supercompact''.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up
to a measurable cardinal + Level by level
equivalence holds''.
In $V^\FP$, every measurable cardinal $\gd \neq \gk$ is
$\gl_\gd$ tall, where $\gl_\gd$ is the least
weakly compact cardinal above $\gth_\gd$.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + GCH + $\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact and $\gl$ is the least
inaccessible cardinal above $\gk$''.
%and there is an elementary embedding $j : V \to M$
%generated by a supercompact ultrafilter over $P_\gk(\gl)$
%such that $M \models ``\gk$ is $\gl$ supercompact''.
There is then a partial ordering $\FP \in V$,
a submodel $\ov V \subseteq V^\FP$, and a
cardinal $\gk_0 < \gk$ such that
$\ov V \models ``$ZFC + GCH + $\gk_0$ is supercompact +
No cardinal is supercompact up to an inaccessible
cardinal + Level by level inequivalence holds''.
In $\ov V$, $\gk_0$ is the only strongly compact cardinal.
Further, in $\ov V$, every measurable cardinal $\gd \neq \gk_0$ is
$\gl_\gd$ tall.%, where $\gl_\gd$ is the least
%weakly compact cardinal above $\gth_\gd$.
\end{theorem}
\begin{lemma}\label{l3a}
$V^\FP \models ``$Every measurable cardinal
$\gd < \gk_0$ is $\gl_\gd$ tall''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l4}, we may assume
without loss of generality that
$V \models ``\gd$ is superstrong''. Also, since
$V \models ``\gk$ is the least cardinal which is
$\gg$ supercompact for some inaccessible $\gg > \gk$''
and $\gd < \gk_0 < \gk$, ${(\gth_\gd)}^V$ is no larger than
the least inaccessible cardinal in $V$ above it.
Note that as in the proof of Lemma \ref{l2}, we may write
$\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
Therefore, by Theorem \ref{t3},
${(\gth_\gd)}^{V^\FP} \le {(\gth_\gd)}^V$.
Further, by writing $\FP = \FP_\gd \ast \dot \FP^\gd$,
since the first cardinal on which $\dot \FP^\gd$ is
forced to act nontrivially is the least $V$-measurable
cardinal above $\gd$, ${(\gth_\gd)}^{V^\FP} =
{(\gth_\gd)}^{V^{\FP_\gd}}$. Hence, as
$\card{\FP_\gd} = \gd$, the preceding facts,
together with the results of
\cite{LS} and an analogous argument to
the one given in the
first paragraph of the proof of Lemma \ref{l4},
allow us to infer that the least
inaccessible cardinal above $\gd$
is the same in $V$, $V^{\FP_\gd}$,
and $V^\FP$ and that $\gl_\gd$
(the least weakly compact cardinal above $\gth_\gd$)
is the same in both $V$ and $V^{\FP_\gd}$.
In addition, this line of reasoning allows us to infer that
to show that $V^\FP \models ``\gd$ is $\gl_\gd$ tall'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$ tall''.
We may now apply the exact same proof as given in
Lemma \ref{l4} in order to do this and
thereby complete the proof of Lemma \ref{l3a}.
%demonstrate that $V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$ tall''.
%This completes the proof of Lemma \ref{l3a}.
\end{proof}
We remark that the proof of Lemma \ref{l2}
actually shows that for any $\ga$,
if $V^\FP \models ``\gd$ is $\ga$ supercompact'', then
$V \models ``\gd$ is $\ga$ supercompact'' as well.
In combination with Lemma \ref{l1}, we may consequently
infer that if $\gd < \gk$ and
$V \models ``\gd$ is superstrong'', then
${(\gth_\gd)}^V = {(\gth_\gd)}^{V^\FP}$. However, since
we may write $\FP_\gd = \FQ' \ast \dot \FQ''$, where
$\card{\FQ'} = \go$, $\FQ'$ is nontrivial, and
$\forces_{\FQ'} ``\dot \FQ''$ is $\ha_1$-strategically closed'',
the proofs of Lemmas \ref{l1} and \ref{l2}
actually allow us to infer that in addition,
${(\gth_\gd)}^V = {(\gth_\gd)}^{V^{\FP_\gd}}$ .
\begin{lemma}\label{l3a}
$V^\FP \models ``$Every measurable cardinal
$\gd < \gk_0$ is $\gl_\gd$ tall''.
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l4}, we may assume
without loss of generality that
$V \models ``\gd$ is superstrong''. Also, since
$V \models ``\gk$ is the least cardinal which is
$\gg$ supercompact for some inaccessible $\gg > \gk$''
and $\gd < \gk_0 < \gk$, ${(\gth_\gd)}^V$ is no larger than
the least inaccessible cardinal in $V$ above it.
Note that as in the proof of Lemma \ref{l2}, we may write
$\FP = \FP' \ast \dot \FP''$, where
$\card{\FP'} = \go$, $\FP'$ is nontrivial, and
$\forces_{\FP'} ``\dot \FP''$ is $\ha_1$-strategically closed''.
Therefore, by Theorem \ref{t3},
${(\gth_\gd)}^{V^\FP} \le {(\gth_\gd)}^V$.
Further, by writing $\FP = \FP_\gd \ast \dot \FP^\gd$,
as in Lemma \ref{l2a},
$\card{\FP_\gd} = \gd$ and
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$
adds no bounded subsets of the least measurable
cardinal above $\gd$''.
Hence, ${(\gth_\gd)}^{V^\FP} =
{(\gth_\gd)}^{V^{\FP_\gd}}$, and
as in the proof of Lemma \ref{l2a},
the least
inaccessible cardinal above $\gd$
is the same in $V$, $V^{\FP_\gd}$,
and $V^\FP$.
In adition, $\gl_\gd$
(the least weakly compact cardinal above $\gth_\gd$)
is the same in $V$, $V^{\FP_\gd}$, and $V^\FP$, and
to show that $V^\FP \models ``\gd$ is $\gl_\gd$ tall'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$ tall''.
We may now apply the exact same proof as given in
Lemma \ref{l4} in order to do this and
thereby complete the proof of Lemma \ref{l3a}.
%demonstrate that $V^{\FP_\gd} \models ``\gd$ is $\gl_\gd$ tall''.
%This completes the proof of Lemma \ref{l3a}.
\end{proof}