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%
\title{Precisely Controlling Level by Level
Behavior
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, measurable cardinal,
level by level equivalence between strong
compactness and supercompactness,
level by level inequivalence between strong
compactness and supercompactness, nonreflecting
stationary set of ordinals.}}
\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{March 2, 2015}
%\date{\today}
\begin{document}
\maketitle
\begin{abstract}
We construct four models containing one supercompact
cardinal in which level by level equivalence between strong
compactness and supercompactness and level by level
inequivalence between strong compactness and supercompactness
are precisely controlled at each non-supercompact measurable cardinal.
%In these models, no cardinal is supercompact up to an inaccessible cardinal.
In these models, no cardinal $\gk$ is ${<} \gk'$ supercompact,
where $\gk'$ is the least inaccessible cardinal greater than $\gk$.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Say that a model containing at least one supercompact cardinal
satisfies {\em level by level equivalence between strong compactness
and supercompactness} if for every pair of regular cardinals
$\gk < \gl$, $\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$
supercompact, except possibly if $\gk$ is a measurable
limit of cardinals $\gd$ which are $\gl$
supercompact.
(The exception is given 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.)
In addition, say that $\gk$ satisfies {\em level by level equivalence between
strong compactness and supercompactness} if for every regular cardinal
$\gl > \gk$, $\gk$ is $\gl$ strongly compact iff $\gk$ is $\gl$ supercompact.
Note that supercompact cardinals automatically satisfy
level by level equivalence between strong compactness and supercompactness.
These notions were first introduced and studied
by the author and Shelah in \cite{AS97a}, where models in which
level by level equivalence between strong compactness and
supercompactness holds nontrivially were constructed.
As a dual to the concepts mentioned in the preceding paragraph,
say that a model containing at least one supercompact cardinal
satisfies {\em level by level inequivalence between
strong compactness and supercompactness} if
for every non-supercompact measurable cardinal
$\gd$, there is some $\gg > \gd$ such that
$\gd$ is $\gg$ strongly compact yet $\gd$
is not $\gg$ supercompact.
The non-supercompact measurable cardinal $\gd$ is then said to satisfy
{\em level by level inequivalence between strong compactness and supercompactness}.
These ideas were first introduced and studied in \cite{A02}.
%This can alternatively be stated by saying that
%{\em level by level inequivalence between strong
%compactness and supercompactness holds at every
%measurable cardinal $\gd$}.
Models containing exactly one supercompact
cardinal in which level by level inequivalence
between strong compactness and supercompactness
holds may be found in %have been constructed in
\cite[Theorem 2]{A02}, \cite[Theorem 2]{A10},
\cite[Theorem 1]{A11}, \cite[Theorems 1--3]{A12},
\cite[Theorem 32(2)]{AGH}, and \cite[Theorem 1.1]{A13}.
%\footnote{Note
%that the dual notion of {\em level by level equivalence
%between strong compactness and supercompactness}
%was first studied by the author and Shelah in
%\cite{AS97a}, to which we refer readers for
%additional details.}
%For the purposes of this paper, this means that for
%any pair of regular cardinals $\gd < \gl$, $\gd$
%is $\gl$ strongly compact iff $\gd$ is $\gl$ supercompact.)
The purpose of this paper is to investigate what sorts of interactions
between these two dual notions are possible in the same model of ZFC.
Specifically, we consider the following
\bigskip\noindent
Question: Is it possible to have a model of ZFC containing
at least one supercompact cardinal in which
level by level
equivalence between strong compactness and supercompactness
and
level by level
inequivalence between strong compactness and supercompactness
are precisely controlled at every non-supercompact measurable cardinal?
\bigskip%\noindent
We provide answers to this question with the
following four theorems.
We will take as notation that for
any cardinal $\gd$, $\gd'$ is the least inaccessible
cardinal greater that $\gd$.
%We will also say that {\em $\gd$ is supercompact up to $\gl$}
%if $\gd$ is $\gg$ supercompact for every $\gg < \gl$.
We will also say that {\em $\gd$ is ${<} \gd'$ supercompact (strongly compact)}
if $\gd$ is $\gg$ supercompact (strongly compact) for every $\gg < \gl$.
In words, this is expressed by saying {\em $\gd$ is supercompact (strongly
compact) up to $\gg$}.
In addition,
we will henceforth, for brevity, write ``level by level equivalence'' and
``level by level inequivalence'' instead of ``level by level equivalence
between strong compactness and supercompactness'' and ``level by level
inequivalence between strong compactness and supercompactness''.
Further, since the statements of our four theorems are quite technical in nature,
we first give intuitive descriptions of what each theorem actually says.
Roughly speaking,
Theorem \ref{t1} provides a model containing exactly one supercompact cardinal
$\gk_0$ and no other strongly compact cardinals
in which no cardinal is supercompact up to an inaccessible cardinal,
level by level {\em inequivalence} holds at every
non-supercompact measurable cardinal which
is a limit of measurable cardinals, and level by level equivalence and
inequivalence at measurable cardinals which are not limits
of measurable cardinals is precisely controlled by a ground model function.
Theorem \ref{t2} provides a model containing exactly one supercompact cardinal
$\gk_0$ and no other strongly compact cardinals
in which no cardinal is supercompact up to an inaccessible cardinal,
level by level {\em equivalence} holds at every measurable cardinal which
is a limit of measurable cardinals, and level by level equivalence and
inequivalence at measurable cardinals which are not limits
of measurable cardinals is precisely controlled by a ground model function.
Theorems \ref{t3} and \ref{t4} provide models containing exactly one
supercompact cardinal $\gk$
and no other strongly compact cardinals
in which no cardinal is supercompact up to an
inaccessible cardinal and an arbitrary ordinal $\ga < \gk$ acts as a ``pivot'',
in the sense that every measurable cardinal $\gd \le \ga$ either satisfies
level by level equivalence or level by level inequivalence, but every
non-supercompact measurable cardinal $\gd > \ga$ satisfies the %opposite
dual property.
Our four theorems are precisely stated as follows:
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH +
$\gk$ is $\gk'$ supercompact''.
%There exist cardinals $\gk < \gl$ such that $\gk$ is $\gl$ supercompact and
%$\gl$ is the least inaccessible cardinal'.
Let
$A = \{\gd < \gk \mid \gd$ is
${<} \gd'$ supercompact$\}$ and
$B = \{\gd < \gk \mid \gd$ is
${<} \gd'$ supercompact but is not a limit of cardinals
$\gl$ which are ${<} \gl'$ supercompact$\}$,
with $f : B \to 2$ a function. 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 + The only measurable cardinals %less than $\gk_0$
are the members of $A \cap (\gk_0 + 1)$''.
%No cardinal $\gd$ is ${<} \gd'$ supercompact''.
In $\ov V$, level by level inequivalence
holds at every non-supercompact measurable
cardinal which is a limit of measurable cardinals,
no measurable cardinal $\gd$ is ${<} \gd'$ supercompact,
the measurable cardinals which are not limits of measurable
cardinals are the members of $B \cap \gk_0$,
and $\gk_0$ is the only strongly compact cardinal. Further,
$\ov V \models ``$If $f(\gd) = 0$, then level by level equivalence
holds at $\gd$, but if $f(\gd) = 1$, then level by level inequivalence
holds at $\gd$''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose $V \models ``$ZFC + GCH +
$\gk$ is $\gk'$ supercompact''.
%There exist cardinals $\gk < \gl$ such that $\gk$ is $\gl$ supercompact and
%$\gl$ is the least inaccessible cardinal'.
Let
$A = \{\gd < \gk \mid \gd$ is
${<} \gd'$ supercompact$\}$ and
$B = \{\gd < \gk \mid \gd$ is
${<} \gd'$ supercompact but is not a limit of cardinals
$\gl$ which are ${<} \gl'$ supercompact$\}$,
with $f : B \to 2$ a function. There is then a partial ordering
$\FP^* \in V$, a submodel $V^* \subseteq V^{\FP^*}$, and a cardinal
$\gk_0 < \gk$ such that $V^* \models ``$ZFC + GCH + $\gk_0$
is supercompact + The only measurable cardinals %less than $\gk_0$
are the members of $A \cap (\gk_0 + 1)$''.
%No cardinal $\gd$ is ${<} \gd'$ supercompact''.
In $V^*$, level by level equivalence
holds at every measurable
cardinal which is a limit of measurable cardinals,
no measurable cardinal $\gd$ is ${<} \gd'$ supercompact,
the measurable cardinals which are not limits of measurable
cardinals are the members of $B \cap \gk_0$,
and $\gk_0$ is the only strongly compact cardinal. Further,
$V^* \models ``$If $f(\gd) = 0$, then level by level equivalence
holds at $\gd$, but if $f(\gd) = 1$, then level by level inequivalence
holds at $\gd$''.
\end{theorem}
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact and
is the only strongly compact cardinal +
No cardinal is supercompact up to an inaccessible cardinal +
Level by level inequivalence holds
at every non-supercompact measurable cardinal $\gd$ (and the
least witness to level by level inequivalence is some
$\gg < \gd'$)''.
Let $\ga < \gk$ be a fixed but arbitrary ordinal.
There is then a partial ordering $\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is supercompact and
is the only strongly compact cardinal +
No cardinal is supercompact up to an inaccessible cardinal +
Level by level inequivalence holds for all measurable cardinals
less than or equal to $\ga$ + Level by equivalence holds for
all measurable cardinals greater than $\ga$''.
In addition, $V$ and $V^\FP$ contain the same measurable cardinals.
\end{theorem}
\begin{theorem}\label{t4}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact and
is the only strongly compact cardinal +
No cardinal is supercompact up to an inaccessible cardinal +
Level by level inequivalence holds
at every non-supercompact measurable cardinal $\gd$ (and the
least witness to level by level inequivalence is some
$\gg < \gd'$)''.
Let $\ga < \gk$ be a fixed but arbitrary ordinal.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is supercompact and
is the only strongly compact cardinal +
No cardinal is supercompact up to an inaccessible cardinal +
Level by level equivalence holds for all measurable cardinals
less than or equal to $\ga$ + Level by inequivalence holds for
all non-supercompact measurable cardinals greater than $\ga$''.
In addition, $V$ and $V^\FP$ contain the same measurable cardinals.
\end{theorem}
We take this opportunity to make a few additional remarks concerning
Theorems \ref{t1} -- \ref{t4}.
The exact hypotheses used to prove Theorems \ref{t3} and \ref{t4}
are GCH together with the existence of cardinals $\gk_1 < \gk_2$ such
that $\gk_1$ is supercompact and $\gk_2$ is inaccessible.
%GCH together with the existence of a cardinal $\gl$ which is $\gl'$ supercompact.
In particular, we first force to construct the ground model $V$
mentioned in the statements of Theorems \ref{t3} and \ref{t4}
(which will be the model witnessing the conclusions of
\cite[Theorem 3]{A12}), and then
force over this model to complete the proofs of these theorems.
To avoid excessive technicalities, we have chosen to
%use the statements
state these theorems as
we did above.
We will discuss in greater detail at the end of this paper
the exact hypotheses we seem to need to construct
models for level by level inequivalence containing exactly
one supercompact cardinal in which no measurable cardinal
$\gd$ is ${<} \gd'$ supercompact.
Also, because a measurable cardinal $\gk$
%with a sufficient degree of supercompactness will reflect
with a normal measure $\mu$ concentrating on measurable cardinals will exhibit
either level by level equivalence
or level by level inequivalence
in the ultrapower via $\mu$,
this will reflect below $\gk$ on a measure 1 set.
Hence, we cannot expect to control
level by level equivalence or level by level inequivalence
%at measurable limits of measurable cardinals
at measurable cardinals of nontrivial Mitchell rank
as arbitrarily as we do in Theorems \ref{t1} and \ref{t2}
at measurable cardinals which are not limits of measurable cardinals.
Further, we note that if $\gk$ is supercompact and $\gl > \gk$ is
inaccessible, then by reflection,
$\{\gd < \gk \mid \gd$ is $\gd'$ supercompact$\}$ is unbounded in $\gk$.
Thus, in the models witnessing the conclusions of \break Theorems \ref{t1} -- \ref{t4},
there cannot be any inaccessible %, weakly compact, Ramsey, measurable, etc$.$
cardinals above the supercompact cardinal in that model.
%the function $f$ of Theorems \ref{t1} and \ref{t2}
%partially supercompact cardinal witnessing enough
Before beginning the proof of Theorems \ref{t1} -- \ref{t4}, we very
briefly mention some preliminary information concerning
notation and terminology.
%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
%{\em $\gd$ 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 $\check x$.
Suppose $\gk > \go$ is a regular cardinal.
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
$\langle p_\ga\mid \ga \le\gk\rangle$, 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
$\langle p_\ga\mid \ga < \gk\rangle$, where player I plays odd
stages and player II plays even
%(which of course includes limit)
stages (again choosing the trivial condition at stage 0),
player II has a strategy which ensures the game can always be continued.
%$\FP$ is {\em ${<} \gk$-strategically closed} if
%$\FP$ is $\gd$-strategically closed for every $\gd < \gk$.
%Note that if $\FP$ is ${\prec} \gk$-strategically closed, then
%$\FP$ is ${<} \gk$-strategically closed as well.
An example of a partial ordering
which is ${\prec} \gk$-strategically closed
and which will be used in the proof of Theorem \ref{t1} is
the partial ordering $\FP(\go, \gk)$ for adding a nonreflecting
stationary set of ordinals of cofinality $\go$ to $\gk$.
%Specifically, $\FP(\go, \gk) = \{p \mid$ For some $\ga < \gk$,
%$p : \ga \to \{0, 1\}$ is a characteristic function of
%$S_p$, a subset of $\ga$ not stationary at its supremum nor
%having any initial segment which is stationary at its
%supremum, such that $\gb \in S_p$ implies $\gb > \go$ and
%${\rm cof}(\gb) = \go\}$, ordered by $q \ge p$ iff
%$q \supseteq p$ and $S_p = S_q \cap \sup(S_p)$, i.e.,
%$S_q$ is an end extension of $S_p$.
For additional details and the exact definition,
readers are urged to consult \cite[second paragraph of
Section 1, page 106]{AS97a}.
%$\FP$ is {\em $(\gk, \infty)$-distributive} if
%given a sequence $\la D_\ga \mid \ga < \gk \ra$
%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 $\delta < \gk$.
We mention that
we are assuming complete familiarity with the notions of
measurability, strong compactness, and supercompactness.
Interested readers may consult \cite{SRK} %or \cite {SRK}
for further details.
We note only that all elementary embeddings witnessing the $\lambda$
supercompactness of $\k$ are presumed to come from some
fine, $\k$-complete, normal
ultrafilter ${\cal U}$ over $P_\k (\l) = \{ x \subseteq \l
\mid | x| < \k \}$, and all elementary embeddings witnessing the $\l$
strong compactness of $\k$ are presumed to come from
some fine, $\k$-complete ultrafilter ${\cal U}$ over
$P_\k(\l)$. An equivalent definition for $\k$
being $\l$ strongly compact is that there is an
elementary embedding $j : V \to M$ having critical
point $\k$ such that for any $x \subseteq M$ with
$|x| \le \l$, there is some $y \in M$ such that
$x \subseteq y$ and $M \models ``|y| < j(\k)$''.
A measurable cardinal $\gk$ has {\em nontrivial Mitchell rank}
if $\gk$ carries a normal measure $\mu$ such that
$\gk$ is measurable in the ultrapower $V^\gk / \mu$.
When discussing the proofs of Theorems \ref{t2} -- \ref{t4},
we will be assuming some familiarity with the work of \cite{AS97a}.
This material, which is rather complicated,
will be taken as a ``black box'', with many definitions
and facts only referenced and not given explicitly.
However, all relevant details can be found in \cite[Sections 1 and 2]{AS97a}.
%In particular, the proofs of all relevant
%facts referenced can be found in \cite[Sections 1 and 2]{AS97a}.
A corollary of Hamkins' work on gap forcing found in
\cite{H2, H3} will be employed in the proofs of our theorems.
We therefore state as a separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\it admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2, H3},
$\FP$ is {\it mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size less than $\gk$ has
a ``nice'' name $\tau$
in $V$ of size less than $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that
$\forces_\FP ``\gt \subseteq \check y$''.
%any ordinal forced by a condition in $\FP$ to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\it amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Hamkins' work from
\cite{H2, H3}
we will be using
is then the following.
\begin{theorem}\label{tgf}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a generic
extension obtained by forcing with $\FP$ that
admits a gap
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
Theorem \ref{tgf} implies that if $\gk$ is
measurable or supercompact in $V^\FP$ and $\FP$ admits
a gap below $\gk$, then $\gk$ was measurable or
supercompact in $V$ as well. In addition, if $\gk$ is
strongly compact in $V^\FP$ and $\FP$ is both mild
with respect to $\gk$ and admits a gap below $\gk$,
then $\gk$ was also strongly compact in $V$.
\section{The Proofs of Theorems \ref{t1} -- \ref{t4}}\label{s2}
We turn now to the proofs of our theorems, beginning
with the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH + $\gk$ is $\gk'$ supercompact''.
The partial ordering $\FP$ used in the proof of Theorem \ref{t1}
is the reverse Easton iteration of length $\gk$ which begins by
adding a Cohen subset of $\go$ and then does nontrivial forcing
only at those $\gd < \gk$ such that either
$\gd$ is a $V$-measurable cardinal but $\gd \not\in A$ ,
or $\gd = \gl^+$ where
$f(\gl) = 0$. At these $\gd$, we force with $\FP(\go, \gd)$.
Note that by its definition, forcing with $\FP$ preserves GCH.
\begin{lemma}\label{l1}
For $\gl = (\gk')^V$, $V^\FP \models ``\gk$ 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 $P_\gk(\gl)$.
Because $M^\gl \subseteq M$, $M \models ``\gl = \gk'$ and
$\gk$ is ${<} \gl$ supercompact''.
By reflection, $\{\gd < \gk \mid \gd$ is ${<} \gd'$ supercompact$\}$
is unbounded in $\gk$ in both $V$ and $M$. From this, we may infer that
%$M \models ``\gk \in j(A \setminus B)$''.
$\gk \in j(A \setminus B)$.
The last three sentences therefore allow us to conclude that
%By the fact that $M \models ``\gk \in j(A \setminus B)$ and $\gl = \gk'$'',
$j(\FP) = \FP \ast \dot \FQ$, where the first ordinal at which
$\dot \FQ$ is forced to do nontrivial forcing is well above $\gl$.
%Laver's original argument from \cite{L}
The argument given by Laver in \cite{L}
now applies and shows that %for any $\gd < \gl$,
$V^\FP \models ``\gk$ is ${<} \gl$ supercompact''.
(In particular, suppose $\gd < \gl$ is
arbitrary. Let $G_0 \ast G_1$ be $V$-generic over ${\FP} \ast \dot \FQ$.
Lift $j$ in $V[G_0][G_1]$ to $j : V[G_0] \to M[G_0][G_1]$,
%take a master condition $p$ for $j '' G_1$ and a
%$V[G_0][G_1][G_2]$-generic object $G_3$ over $j(\FQ)$ containing $p$, lift $j$
%again in $V[G_0][G_1][G_2][G_3]$ to $j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
and show by the $\gl$-directed closure of $\FQ$ that the supercompactness
measure over ${(P_\gk(\gd))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0]$.)
%Since we have shown that $V^\FP \models ``\gk$ is ${<} \gl$ supercompact'',
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$If $\gd < \gk$ is measurable, then $\gd \in A$''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gd < \gk$ is measurable''.
Write $\FP = \FP_0 \ast \dot \FQ$, where $\card{\FP_0} = \go$,
$\FP_0$ is nontrivial, and $\forces_{\FP_0} ``\dot \FQ$ is $\ha_1$-strategically
closed''. By Theorem \ref{tgf}, because $\FP$ admits a gap at $\ha_1$,
$V \models ``\gd$ is measurable'' as well.
However, by the definition of $\FP$, if $\gd \not\in A$, then
$V^\FP \models ``$There is a non-reflecting stationary subset of $\gd$
composed of ordinals of cofinality $\go$'', which further implies that
$V^\FP \models ``\gd$ is not weakly compact''.
This contradiction completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
If $\gd \in A$, then $V^{\FP_\gd} \models ``\gd$ is
${<} \gd'$ strongly compact''.
\end{lemma}
\begin{proof}
Suppose $\gd \in A$.
Let $\gl < \gd'$, $\gl > \gd$ be a regular cardinal.
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 since $M^\gl \subseteq M$,
$M \models ``\gl < \gd'$''. In addition, because
GCH implies that $\gl \ge \gd^+ = 2^\gd$, $M \models ``\gd$
is a measurable cardinal''.
Therefore, by the definition of $\FP$,
$j(\FP_\gd) = \FP_\gd \ast \dot \FP(\go, \gd) \ast \dot \FR$,
%where $\dot \FP(\go, \gd)$ is a term for the partial ordering
%adding a non-reflecting stationary set of ordinals of
%cofinality $\go$ to $\gd$, and
where the first ordinal at which
$\dot \FR$ 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, due originally to Magidor
but unpublished by him, is as follows. Let
$k : M \to N$ be an elementary embedding
generated by a normal measure ${\cal U} \in M$ over $\gd$
such that $N \models ``\gd$ is not measurable''.
The elementary
embedding $i = k \circ j$ witnesses the
$\gl$ strong compactness of $\gd$ in $V$.
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 $\gd$.
To see this, write $i(\FP_\gd) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$,
where $\dot \FQ^1$ is forced to act nontrivially on
ordinals in the interval $(\gd, k(\gd)]$, and $\dot \FQ^2$
is forced to act nontrivially on ordinals in the interval
$(k(\gd), k(j(\gd))) = (k(\gd), i(\gd))$.
%$j(\FP_\gd) = \FP_\gd \ast \dot \FP(\go, \gd) \ast \dot \FR$,
Next, take $G_0$ to be
$V$-generic over $\FP_\gd$, and build in
$V[G_0]$ generic objects $G_1$ and $G_2$
for $\FQ^1$ and $\FQ^2$ respectively.
The construction of $G_1$ uses that
by GCH and
the fact that $k$ is given by an
ultrapower embedding, we may let
$\la D_\ga \mid \ga < \gd^+ \ra$ enumerate in
$V[G_0]$ the dense open subsets of $\FQ^1$ present
in $N[G_0]$.
Since $N \models ``\gd$ is not measurable'',
the first nontrivial stage of forcing in $\FQ^1$ occurs
well above $\gd$. This implies that
$N[G_0] \models ``\FQ^1$ is ${\prec} \gd^+$-strategically closed''.
Because $N[G_0]$ remains
$\gd$-closed with respect to $V[G_0]$,
by the ${\prec} \gd^+$-strategic
closure of $\FQ^1$ in both $N[G_0]$ 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
%$j : V[G_0] \to M[G_0][G_1]$.
The construction of $G_2$ first requires
building an $M$-generic object $G^{*}_2$
for the term forcing partial ordering
$\FT$ associated with $\dot \FR$ and defined
in $M$ with respect to $\FP_\gd \ast
\dot \FP(\go, \gd)$.
%Unlike the argument given in the
%proof of Lemma \ref{l4} for $G^{**}_2$, however,
$G^{*}_2$ is built using the facts that since
$M^\gl \subseteq M$ and the
first nontrivial stage of forcing in $\FT$
occurs well above $\gl$, $\FT$ 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 $\FQ^2$
This tells us that $i$ lifts in $V[G_0]$ to
$i : V[G_0] \to N[G_0][G_1][G_2]$.}
Also, as $\card{\FP_\gd} = \gd$, $\gd'$ is the same in both
$V$ and $V^{\FP_\gd}$. Consequently, since $\gl > \gd$
was an arbitrary regular cardinal below the least
$V$-inaccessible cardinal above $\gd$, which we know is the same as
the least $V^{\FP_\gd}$-inaccessible cardinal above $\gd$,
the proof of Lemma \ref{l3} has been completed.
\end{proof}
\begin{lemma}\label{l4}
If either $\gd \in A \setminus B$ or $f(\gd) = 1$, then
$V^\FP \models ``\gd$ is ${<} \gd'$ strongly compact''.
\end{lemma}
\begin{proof}
Suppose $\gd$ is as in the hypotheses for Lemma \ref{l4}.
By the definition of $\FP$, in either case,
%only trivial forcing is done at stage $\gd$. Consequently,
we can
write $\FP = \FP_\gd \ast \dot \FP^\gd$, where
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is $\gd'$-strategically closed''.
Since by Lemma \ref{l3}, $V^{\FP_\gd} \models ``\gd$ is ${<} \gd'$
strongly compact'', it follows that
$V^{\FP_\gd \ast \dot \FP^\gd} = V^\FP \models ``\gd$ is ${<} \gd'$
strongly compact'' as well.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
If $f(\gd) = 0$, then $V^\FP \models ``\gd$ is measurable and
level by level equivalence holds at $\gd$''.
\end{lemma}
\begin{proof}
Suppose that $\gd$ is as in the hypotheses for Lemma \ref{l5}.
By the definition of $\FP$, we can write
$\FP = \FP_\gd \ast \dot \FP(\go, \gd^+) \ast \dot \FP^\gd$.
Lemma \ref{l3} implies that $V^{\FP_\gd} \models ``\gd$ is measurable''. Since
$\forces_{\FP_\gd} ``\dot \FP(\go, \gd^+)$ is $\gd$-strategically closed'',
$V^{\FP_\gd \ast \dot \FP(\go, \gd^+)} \models ``\gd$ is measurable'' as well.
As $V^{\FP_\gd \ast \dot \FP(\go, \gd^+)} \models ``$There is a nonreflecting
stationary subset of $\gd^+$ composed of ordinals of cofinality $\go$'',
by \cite[Theorem 4.8]{SRK}, $V^{\FP_\gd \ast \dot \FP(\go, \gd^+)} \models ``\gd$ is not
$\gd^+$ strongly compact''.
It therefore follows that
$V^{\FP_\gd \ast \dot \FP(\go, \gd^+)} \models ``$Level by level equivalence holds
at $\gd$''. Because $\forces_{\FP_\gd \ast \dot \FP(\go, \gd^+)} ``\dot \FP^\gd$ is
$\gd'$-strategically closed'',
$V^{\FP_\gd \ast \dot \FP(\go, \gd^+) \ast \dot \FP^\gd} = V^\FP \models ``\gd$
is measurable and $\gd$ is not $\gd^+$ strongly compact'', i.e.,
%$V^{\FP_\gd \ast \dot \FP(\go, \gd^+) \ast \dot \FP^\gd} =
$V^\FP \models ``$Level
by level equivalence holds at $\gd$''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
Let now $\gk_0$ be the least cardinal such that
$V^\FP \models ``\gk_0$ is ${<} \gk_0'$ supercompact''.
Lemma \ref{l1} implies that $\gk_0$ exists and $\gk_0 \le \gk$.
To see that in fact $\gk_0 < \gk$, let $\gl = \gk'$ and
$j : V \to M$ 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}, $\gk \in j(A \setminus B)$, and
$j(\FP) = \FP \ast \dot \FQ = \FP_\gk \ast \dot \FQ$, where the first ordinal at which
$\dot \FQ$ is forced to do nontrivial forcing is well above $\gk'$.
Because $M^{\gk'} \subseteq M$ and
$V^\FP \models ``\gk$ is ${<} \gk'$ supercompact'', it follows that in $M$, both
$\forces_{\FP} ``\gk$ is ${<} \gk'$ supercompact'' and
$\forces_{\FP_\gk \ast \dot \FQ} ``\gk$ is ${<} \gk'$ supercompact''.
In other words, in $M$,
$\forces_{j(\FP)} ``\gk$ is ${<} \gk'$ supercompact''.
By reflection,
$C = \{\gd < \gk \mid \ \forces_{\FP} ``\gd$ is ${<} \gd'$ supercompact''$\}$
is unbounded in $\gk$.
The smallest member $\gk_0$ of $C$ witnesses that $\gk_0 < \gk$.
%$\{\gd < \gk \mid \ \forces_{\FP_\gd} ``\gd$ is ${<} \gd'$ supercompact$\}$
%is unbounded in $\gk$.
%Let $\gk_0 < \gk$ be least such that $\forces_{\FP_{\gk_0}} ``\gk_0$ is
%${<} \gk_0'$ supercompact''. Write $\FP = \FP_{\gk_0} \ast \dot \FP^{\gk_0}$.
%Because as in Lemma \ref{l2},
%we can write $\FP_{\gk_0} = \FP_0 \ast \dot \FQ$, where
%$\card{\FP_0} = \go$, $\FP_0$ is nontrivial, and $\forces_{\FP_0} ``\dot \FQ$ is
%$\ha_1$-strategically closed'', by Theorem \ref{tgf}, $V \models ``\gk_0$ is
%${<} \gk_0'$ supercompact''.
%Consequently, by the definition of $\FP$, $\forces_{\FP_{\gk_0}} ``\dot \FP^{\gk_0}$ is
Define $\ov V = (V_{\gk_0'})^{V^\FP}$. It follows that $\ov V$ is a model of
ZFC + GCH containing no measurable cardinals greater than $\gk_0$ and
$\ov V \models ``\gk_0$ is supercompact and no cardinal
$\gd < \gk_0$ is ${<} \gd'$ supercompact''.
By Lemmas \ref{l2}, \ref{l4}, and \ref{l5}, the only measurable cardinals in
$\ov V$ less than $\gk_0$ are the members of
$A \cap \gk_0 = \{\gd < \gk_0 \mid \gd$ is ${<} \gd'$ supercompact in $V\}$.
Since $B \cap \gk_0 = \{\gd < \gk_0 \mid \gd$ is
${<} \gd'$ supercompact in $V$ but is not a limit of cardinals
$\gl$ which are ${<} \gl'$ supercompact in $V\}$, this means that in $\ov V$,
the measurable cardinals which are not limits of measurable cardinals
are the members of $B \cap \gk_0$, and the measurable limits of measurable
cardinals below $\gk_0$ are the members of $(A \minus B) \cap \gk_0$. Therefore,
the fact $\gk_0$ is the smallest cardinal such that
$V^\FP \models ``\gk_0$ is ${<} \gk_0'$ supercompact''
%the leastness of $\gk_0$
and Lemma \ref{l4} together imply that if either
$\gd < \gk_0$ is a measurable limit of measurable cardinals or $f(\gd) = 1$, then
$\ov V \models ``$Level by level inequivalence holds at $\gd$
(and the least witness to level by level inequivalence is some $\gg < \gd')$''.
Lemma \ref{l5} implies that $\ov V \models ``$If $f(\gd) = 0$, then
$\gd$ is measurable and
level by level equivalence holds at $\gd$''.
The proof of Theorem \ref{t1} is consequently completed by the
following lemma.
\begin{lemma}\label{l6}
$\ov V \models ``\gk_0$ is the only strongly compact cardinal''.
\end{lemma}
\begin{proof}
By the definition of $\FP$, unboundedly many cardinals
$\gd < \gk_0$ (e.g., those $\gd$ which were measurable in $V$
but not ${<} \gd'$ supercompact in $V$) have nonreflecting
stationary subsets composed of ordinals of cofinality $\go$.
By \cite[Theorem 4.8]{SRK} and the succeeding remarks,
this means that $\ov V \models ``$No cardinal less than $\gk_0$
is strongly compact''.
This completes the proof of both Lemma \ref{l6} and Theorem \ref{t1}.
\end{proof}
\end{proof}
\begin{pf}
Turning now to the proof of Theorem \ref{t2}, let $V$ be such that
$V \models ``$ZFC + GCH + $\gk$ is $\gk'$ supercompact''.
Because $V$ satisfies the same hypotheses as the ground model
of Theorem \ref{t1}, we may let $\FP \in V$ and $\ov V \subseteq V^\FP$ be such that
$\ov V$ witnesses the conclusions of Theorem \ref{t1}.
We will take $\ov V$ as our ground model and force over $\ov V$ with
a class partial ordering $\FQ$ defined in $\ov V$ (which will be a set partial ordering
defined in $V^\FP$). To define $\FQ$, let
$\mathfrak D = \la \gd_\ga \mid \ga \le \gk_0 \ra$ enumerate
the members of
$((A \setminus B) \cap \gk_0) \cup \{\gk_0\}$. $\FQ$ is then taken as the reverse Easton
proper class iteration which begins by adding a Cohen subset of $\go$ and then
is the partial ordering $P$ of \cite[page 114]{AS97a} defined using the
members of $\mathfrak D$.
By the L\'evy-Solovay results \cite{LS} and
\cite[Lemmas 8--11, pages 114--120]{AS97a}, $\ov V^\FQ \models ``$ZFC + GCH +
Every $\gd \in {\mathfrak D}$ is measurable + Level by
level equivalence holds for each $\gd \in {\mathfrak D}$ + $\gk_0$ is supercompact''.
In addition, as in the proof of Lemma \ref{l2}, we can write
$\FQ = \FQ_0 \ast \dot \FR$, where $\card{\FQ_0} = \go$, $\FQ_0$ is nontrivial, and
$\forces_{\FQ_0} ``\dot \FR$ is $\ha_1$-strategically closed''.
Further, by its definition, $\FQ$ is mild with respect to any $\ov V$-measurable cardinal.
Hence, by Theorem \ref{tgf},
any cardinal measurable in $\ov V^\FQ$ had to have been measurable in $\ov V$,
and no cardinal $\gd < \gk_0$ is either strongly compact or ${<} \gd'$ supercompact.
The work of the preceding paragraph indicates
that the members of $(A \setminus B) \cap \gk_0$ remain measurable
in $\ov V^\FQ$. To see that the members of $B \cap \gk_0$ remain measurable in
$\ov V^\FQ$ as well, suppose $\gd \in B \cap \gk_0$. By the definition of $\FQ$, we may write
$\FQ = \FQ_\gd \ast \dot \FQ^\gd$, where
%$\FQ_\gd$ is forcing equivalent to a partial ordering having size less than $\gd$ and
$\card{\FQ_\gd} < \gd$ and
$\forces_{\FQ_\gd} ``\dot \FQ^\gd$ is $\gd'$ strategically closed''.
By the results of %the L\'evy-Solovay results
\cite{LS}, $\ov V^{\FQ_\gd} \models ``\gd$ is measurable'',
from which it immediately follows that
$\ov V^{\FQ_\gd \ast \dot \FQ^\gd} = \ov V^\FQ \models ``\gd$ is measurable'' as well.
Further, if $f(\gd) = 0$, then since
%$\FQ_\gd$ is forcing equivalent to a partial ordering having size less than $\gd$ and
$\card{\FQ_\gd} < \gd$ and
$\forces_{\FQ_\gd} ``\dot \FQ^\gd$ is $\gd'$ strategically closed'',
the fact that $\ov V \models ``$Level
by level equivalence holds at $\gd$ and $\gd$ is not $\gd^+$ strongly compact''
is preserved to both $\ov V^{\FQ_\gd}$ and $\ov V^{\FQ_\gd \ast \dot \FQ^\gd} = \ov V^\FQ$.
If $f(\gd) = 1$, then reasoning similar to that used in the preceding sentence shows that
$\ov V \models ``$Level by level inequivalence holds at $\gd$
(and the least witness to level by level inequivalence is some $\gg < \gd')$''
is also preserved to both $\ov V^{\FQ_\gd}$ and $\ov V^{\FQ_\gd \ast \dot \FQ^\gd} = \ov V^\FQ$.
%It therefore follows that many of the properties of the measurable cardinals in $\ov V$
%are preserved to $\ov V^\FQ$, i.e., that
%the only measurable cardinals less than $\gk_0$ are the members of $A$,
%the measurable cardinals which are not limits of measurable cardinals are the members of $B$,
%the measurable limits of measurable cardinals are the members of $A \setminus B$,
Let now $\FP^* = \FP \ast \dot \FQ$ and
$V^* = \ov V^\FQ = (V^{\FP \ast \dot \FQ})_{\gk_0'}$.
The preceding two paragraphs tell us the facts that
the only measurable cardinals less than $\gk_0$ are the members of $A \cap \gk_0$,
the measurable cardinals which are not limits of measurable cardinals
are the members of $B \cap \gk_0$,
and the measurable limits of measurable cardinals are the members of
$(A \setminus B) \cap \gk_0$
are preserved from $\ov V$ to $\ov V^\FQ$.
Thus, $V^*$ is our desired model.
This completes the proof of Theorem \ref{t2}.
\end{pf}
\begin{pf}
To prove Theorem \ref{t3}, suppose $V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up to an inaccessible cardinal +
Level by level inequivalence holds at every non-supercompact
measurable cardinal $\gd$ (and the least witness to
level by level inequivalence is some $\gg < \gd'$)''. Let $\ga < \gk$ be fixed but arbitrary.
The partial ordering $\FP$ used in the proof of Theorem \ref{t3} is once again
the reverse Easton proper class iteration which begins by adding a Cohen
subset of $\go$ and then is the partial ordering $P$ of \cite[page 114]{AS97a}
defined using ${\mathfrak D} = \{\gd \mid \gd$
is a measurable cardinal greater than $\ga\}$.
By the definition of $\FP$,
we may write $\FP = \FP_0 \ast \dot \FQ$, where $\card{\FP_0} = \go$,
$\FP_0$ is nontrivial, and $\forces_{\FP_0} ``\dot \FQ$ is $\ga'$-strategically closed''.
Consequently, as in the proof of Theorem \ref{t2}, Theorem \ref{tgf} implies that
every cardinal measurable in $V^\FP$ had to have been measurable in $V$.
In addition, exactly as in the proof of Theorem \ref{t2},
$V^\FP \models ``$ZFC + GCH + Every $V$-measurable
cardinal $\gd > \ga$ remains measurable and satisfies level by level equivalence +
No cardinal %$\gd$ is ${<} \gd'$ supercompact +
is supercompact up to an inaccessible cardinal +
$\gk$ is supercompact and is the only strongly compact cardinal''.
The above factorization of $\FP$, the results of
\cite{LS}, and the fact that every non-supercompact
$V$-measurable cardinal $\gd$ witnesses
level by level inequivalence in $V$ via some $\gg < \gd'$ then allow us to infer that
in both $V^{\FP_0}$ and $V^{\FP_0 \ast \dot \FQ} = V^\FP$,
the measurable cardinals less than or equal to $\ga$ are the same as in $V$
and satisfy level by level inequivalence.
%every $V$-measurable cardinal $\gd < \ga$ remains measurable and satisfies
%level by level inequivalence.
In particular, we now know that $V$ and $V^\FP$
contain the same measurable cardinals.
This completes the proof of Theorem \ref{t3}.
\end{pf}
\begin{pf}
Finally, to prove Theorem \ref{t4}, suppose $V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up to an inaccessible cardinal +
Level by level inequivalence holds at every
non-supercompact measurable cardinal $\gd$ (and the least witness to
level by level inequivalence is some $\gg < \gd'$)''. Let $\ga < \gk$ be fixed but arbitrary.
In analogy to the proof of Theorem \ref{t3},
the partial ordering $\FP$ used in the proof of Theorem \ref{t4} is
the reverse Easton iteration which begins by adding a Cohen
subset of $\go$ and then is the partial ordering $P$ of \cite[page 114]{AS97a}
defined using ${\mathfrak D} = \{\gd \mid \gd$
is a measurable cardinal less than or equal to $\ga\}$.
Note that unlike the proof of Theorem \ref{t3},
the current partial ordering is a set, not a proper class.
Once again, by the definition of $\FP$,
we may write $\FP = \FP_0 \ast \dot \FQ$, where $\card{\FP_0} = \go$,
$\FP_0$ is nontrivial, and $\forces_{\FP_0} ``\dot \FQ$ is $\ha_1$-strategically closed''.
Consequently, as before, Theorem \ref{tgf} implies that
every cardinal measurable in $V^\FP$ had to have been measurable in $V$.
In addition, in analogy to the proofs of Theorems \ref{t2} and \ref{t3},
$V^\FP \models ``$ZFC + GCH + Every $V$-measurable
cardinal $\gd \le \ga$ remains measurable and satisfies level by level equivalence +
No cardinal %$\gd$ is ${<} \gd'$ supercompact +
is supercompact up to an inaccessible cardinal +
$\gk$ is supercompact and is the only strongly compact cardinal''.
The above factorization of $\FP$ consequently tells us that the measurable
cardinals less than or equal to $\ga$ in $V$, $V^{\FP_0}$,
and $V^{\FP_0 \ast \dot \FQ} = V^\FP$ are exactly the same.
The fact that every non-supercompact $V$-measurable cardinal $\gd$ witnesses
level by level inequivalence in $V$ via some $\gg < \gd'$ and
the definition of $\FP$ then allow us to infer that for $\ga^*$ the
least $V$-measurable cardinal greater than $\ga$, $\card{\FP} < \ga^*$.
Thus, in both $V^{\FP_0}$ and $V^{\FP_0 \ast \dot \FQ} = V^\FP$,
the non-supercompact
measurable cardinals greater than or equal to $\ga^*$ are the same as in $V$
and satisfy level by level inequivalence.
%every $V$-measurable cardinal $\gd < \ga$ remains measurable and satisfies
%level by level inequivalence.
In particular, we now know that $V$ and $V^\FP$
contain the same measurable cardinals.
This completes the proof of Theorem \ref{t4}.
\end{pf}
In conclusion to this paper,
we note that the exact consistency
strength of level by level inequivalence is discussed in \cite{A15}.
%For the purposes of this paper,
The current evidence seems to suggest that
hypotheses slightly stronger than the existence of a supercompact cardinal
are needed to establish Theorems \ref{t1} -- \ref{t4}.
%(This is unlike the situation with level by level equivalence --- see \cite{AS97a} for further details.)
We therefore ask if Theorems \ref{t1} -- \ref{t4} can be established
using only one supercompact cardinal.
Finally, the techniques available at the present time do not
seem to allow results analogous to Theorems \ref{t1} -- \ref{t4}
in which the models constructed contain more than one supercompact cardinal.
We end by asking if this is indeed possible.
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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''. 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 $i(\FP_\gd) =
\FP_\gd \ast \dot \FQ^1 \ast \dot \FQ^2$,
%$j(\FP_\gd) = \FP_\gd \ast \dot \FR^1 \ast \dot \FR$,
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 $\dot \FR$ 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
$i : V[G_0] \to N[G_0][G_1][G_2]$.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
In all of the earlier models constructed, when
a measurable cardinal $\gd$ witnesses level by level
inequivalence, there is always some maximal {\em regular}
cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact. In our situation, this is
never going to be true.
%not necessarily going to be true
%(e.g., when in the ground model $V$,
%To see this,
This follows since by the proof of Lemma \ref{l3},
in the model $V^\FP$ witnessing the conclusions of Theorem \ref{t1},
for any measurable cardinal $\gd < \gk$,
there is always a singular limit cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
(So in particular, for any regular $\gg < \gl$ such that
$V^\FP \models ``\gd$ is $\gg$ strongly compact yet
$\gd$ is not $\gg$ supercompact'', $\gg^+ < \gl$ is also such that
$V^\FP \models ``\gd$ is $\gg^+$ strongly compact yet
$\gd$ is not $\gg^+$ supercompact.)
We therefore end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every measurable cardinal $\gd$, there is
some maximal regular cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
In all of the earlier models constructed, when
a measurable cardinal $\gd$ witnesses level by level
inequivalence, there is always some {\em regular}
cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact. In our situation, this may
not be true.
To see this, suppose that our ground model $V$,
in addition to satisfying the properties stipulated
earlier, also is as in \cite{AS97a}, i.e., satisfies
level by level equivalence between strong compactness
and supercompactness. Let $\FP$ be as in the proof of Theorem \ref{t1}.
In $V^\FP$, let $\gd$ be the least member of $A$,
i.e., $\gd$ is the least cardinal below
our supercompact cardinal $\gk$ such that for some
$\gl > \gd$ having cofinality $\gd$, $\gd$ is $\gg$ supercompact for
every $\gg < \gl$, $\gd$ is not $\gl$ supercompact, yet $\gd$ is $\gl$
strongly compact. We claim that
$V^\FP \models ``$There is no regular cardinal $\xi$ such that
$\gd$ is $\xi$ strongly compact yet $\gd$ is not $\xi$ supercompact''.
This follows since if $\xi$ were such a cardinal, then
by the choice of $\gd$, $\xi \ge \gl^+$.
%Without loss of generality, we may assume that $\xi = \gl^+$.
By Theorem \ref{t2} and the fact that
$\xi$ must be regular in $V$,
%and the fact that forcing with $\FP$
%preserves all cardinals and cofinalities,
it would have to be the case that
$V \models ``\gd$ is $\xi$ strongly compact''.
%, and $\gd$ is not $\xi$ supercompact''.
Since level by level equivalence between strong compactness and
supercompactness holds in $V$,
$V \models ``\gd$ is $\xi$ supercompact''.
We therefore end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every measurable cardinal $\gd$, there is
some maximal regular cardinal $\gl > \gd$ such that
$\gd$ is $\gl$ strongly compact yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier in which level by level inequivalence between
strong compactness and supercompactness holds.
In fact, there is a key difference.
In all of the earlier models,
for any measurable cardinal $\gd$
at which level by level inequivalence holds,
there is always some regular cardinal $\gl > \gd$
such that $\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
The construction given here does not necessarily do this,
and relies in many cases on singular instances of
level by level inequivalence between strong
compactness and supercompactness.
To see that there may be measurable cardinals
for which there is no regular witness to
level by level inequivalence, we may suppose by the results of
\cite{AS97a} and by truncating the universe if necessary
that our ground model $V$ is such that
$V \models ``$ZFC + GCH + $\gk$ is supercompact + There are
no cardinals $\gd < \gl$ such that $\gd$ is $\gl$ supercompact
and $\gl$ is inaccessible + For every pair of regular cardinals
$\gd < \gl$, $\gd$ is $\gl$ strongly compact iff $\gd$
is $\gl$ supercompact''.
Let $\FP$ be as before.
It will still be true that $V^\FP$ is a model for
level by level inequivalence between strong compactness and supercompactness.
However, in $V^\FP$, there will be unboundedly in
$\gk$ many measurable cardinals $\gd$ such that for no regular cardinal
$\gl > \gd$ is it the case that $\gd$ is $\gl$ strongly compact yet
$\gd$ is not $\gl$ supercompact.
In fact, any $\gd \in A$ for $A$ as earlier
(defined in $V^\FP$) is such a measurable cardinal.
This follows since if $\gd \in A$, then there is some singular
$\eta > \gd$ of cofinality $\gd$
such that $\gd$ is $\gg$ supercompact for every $\gg < \eta$,
$\gd$ is not $\eta$ supercompact, yet $\gd$ is $\eta$ strongly compact.
If there were some regular $\gl > \eta$ such that
$\gd$ is $\gl$ strongly compact in $V^\FP$ (and of necessity, since
$\gl > \eta$, $\gd$ is not $\gl$ supercompact in $V^\FP$), then
by the factorization of $\FP$ given in the proof of Lemma \ref{l2}
and the fact $\FP$ is mild with respect to $\gd$,
it must also be the case that
$V \models ``\gd$ is $\gl$ strongly compact''.
By our assumptions on $V$, we have in addition that
$V \models ``\gd$ is $\gl$ supercompact''. However,
We therefore %ask
end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every non-supercompact
measurable cardinal $\gd$,
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gl$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
In conclusion to this paper, we return to the question raised earlier
of whether there is something different about the model witnessing
the conclusions of Theorem \ref{t1} from the models constructed
earlier witnessing level by level inequivalence between
strong compactness and supercompactness.
In fact, there is a key difference.
When all of the earlier models were constructed,
essential use was made of the property that when
a measurable cardinal $\gd$ witnesses
level by level inequivalence in $V^\FP$,
there is always some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gg$ supercompact.
The construction given here does not do this,
and relies on singular witnesses to
level by level inequivalence between strong
compactness and supercompactness.
We therefore %ask
end by asking whether the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds +
For every non-supercompact
measurable cardinal $\gd$,
there is some inaccessible
cardinal $\gl > \gd$ and some $\gr \in (\gd, \gl)$
such that $\gd$ is $\gg$ strongly compact
for every $\gg \in [\gr, \gl)$ yet $\gd$ is not
$\gg$ supercompact''
is stronger in consistency strength than the theory
``ZFC + There exists a supercompact cardinal +
Level by level inequivalence between strong
compactness and supercompactness holds''.
We conjecture that this is indeed the case.
compact$\}$.\footnote{A sketch of %this proof
the proof that $A$ is unbounded in $\gk$
is as follows. Let $\gl > \gk$ be a singular
strong limit cardinal of cofinality $\gk$.
Let $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of
$\gk$ such that $M \models ``\gk$ is not
$\gl$ supercompact''. Because $M^\gl \subseteq M$,
$M \models ``\gk$ is $\gg$ supercompact for every
$\gg < \gl$''. By the argument from \cite[Lemma 7]{AS97a})
(which will be given in some detail in
the last paragraph of the proof of Lemma \ref{l2}),
since $M \models ``\gk$ is $\gg$ supercompact for
every $\gg < \gl$ and $\gl$ has cofinality greater
than or equal to $\gk$'',
$M \models ``\gk$ is $\gl$ strongly compact''.
By reflection, because
$M \models ``\gk$ is not $\gl$ supercompact'',
the set $A$ is unbounded in $\gk$.}