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\title{A Universal Indestructibility Theorem %for Degrees of Supercompactness
Compatible with
Level by Level Equivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, level by level
equivalence between strong compactness and supercompactness,
indestructibility, lottery sum.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author would like to thank the referee for helpful
comments and suggestions which have been incorporated into
the current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
\date{March 17, 2014\\
(revised January 18, 2015)}
\begin{document}
\maketitle
\begin{abstract}
We prove an indestructibility theorem for
degrees of supercompactness that is compatible with
level by level equivalence between strong
compactness and supercompactness.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
Say that a model $V$ of ZFC 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 which
are $\gl$ supercompact. (A theorem of Menas \cite{Me} shows that if
$\gk$ is a measurable limit of cardinals which are
$\gl$ supercompact, then $\gk$ is $\gl$ strongly compact
but need not be $\gl$ supercompact.)
Models in which level by level equivalence between strong compactness
and supercompactness holds nontrivially were first constructed by the
author and Shelah in \cite{AS97a}.\footnote{In particular, it is shown in
\cite{AS97a} that starting from a model $V$ of ZFC + GCH in which the class of
supercompact cardinals is nonempty but otherwise arbitrary, it is possible to force
and obtain a model $\ov V$ of ZFC + GCH {\em containing exactly the same
measurable and supercompact cardinals} in which level by level equivalence between
strong compactness and supercompactness holds. Note that it is
Hamkins' results of \cite{H2, H3}, a corollary of which is stated here as
Theorem \ref{tgf}, which imply that the classes of measurable and supercompact cardinals
in $V$ and $\ov V$ are the same.}
It is known that in general, indestructibility
for supercompactness in Laver's sense of \cite{L}
is incompatible with level by level equivalence
between strong compactness and supercompactness,
assuming the universe contains enough large cardinals.
Indeed, \cite[Theorem 5]{AH4} shows that if
$\gk < \gl$ are such that $\gk$ is indestructibly supercompact and
$\gl$ is $\gl^+$ supercompact, then level by level equivalence
between strong compactness and supercompactness
fails below $\gk$.
In spite of this, however, as has been done in
\cite{A03a, A03b, A09, A07, A06, AH4},
it is possible to establish theorems which witness
indestructibility in a restricted sense (either
by placing limits on the number of large cardinals
in the universe or constraining the amount of indestructibility witnessed)
but which are compatible with level by level equivalence.
A key feature of all of these theorems, however,
is that there are always measurable cardinals $\gk$
in the witnessing universe
%beyond a certain point, all measurable cardinals $\gk$ in the universe
which have their measurability destroyed
by any ${<} \gk$-closed forcing which adds a subset of $\gk$
(see \cite{H5} for further details on this phenomenon).
This raises the following general
\bigskip\noindent
{\bf Question:} Are there any indestructibility theorems which
are compatible with
level by level equivalence
between strong compactness and supercompactness
in which every measurable cardinal $\gk$
%which is $\gl$ supercompact for $\gl \ge \gk$ regular
witnesses some indestructibility properties?
%there are no limits on the number of measurable cardinals
%witnessing indestructibility properties?
%In particular, are there any indestructibility theorems which
%are compatible with level by level equivalence between
%strong compactness and supercompactness in which
%there is a proper class of measurable cardinals with
%indestructibility properties?
\bigskip\noindent
%Note that the question of indestructibility
%theorems that are compatible with level by level equivalence
%between strong compactness and supercompactness in a
%universe with a restricted large cardinal structure
%has been addressed in \cite{AH4} ....
The purpose of this paper is to provide a positive answer
to the aformentioned Question. Specifically, we prove
the following theorem, where we say for
regular cardinals $\gk \le \gl$ that the $\gl$ supercompactness
of $\gk$ is {\em indestructible under forcing with a partial
ordering $\FP$} if $\gk$ remains $\gl$
supercompact after forcing with $\FP$.
For $\ga \ge 1$ any ordinal, $\add(\gk, \ga)$
is the standard partial ordering for adding
$\ga$ many Cohen subsets of $\gk$, i.e., $\add(\gk, \ga) =
\{f \mid f : \gk \times \ga \to \{0, 1\}$ is a function such that
$\card{\dom(f)} < \gk\}$, ordered by inclusion.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH +
The class of supercompact cardinals is nonempty +
%$\K \neq \emptyset$ is the class of supercompact cardinals +
Level by level
equivalence between strong compactness and
supercompactness holds''.
%Let $\gk \le \gl$ be regular cardinals.
There is then a cofinality preserving
partial ordering $\FP \subseteq V$
%and containing the same measurable cardinals as $V$
such that
$V^\FP \models ``$ZFC + GCH + Level by level
equivalence between strong compactness and
supercompactness holds'', and in addition:
%If $\gk \le \gl$ are both regular cardinals, then
%In addition:
\begin{enumerate}
\item\label{i1} If $\gk \le \gl$
are regular cardinals, then
$V \models ``\gk$ is $\gl$ supercompact'' iff
$V^\FP \models ``\gk$ is $\gl$ supercompact''
(so in particular, $V$ and $V^\FP$ contain the
same measurable and supercompact cardinals).
\item\label{i2} If $V^\FP \models ``\gk$
is $\gl$ supercompact and
$\gl \ge \gk$ is inaccessible'', then the $\gl$ supercompactness
of $\gk$ is indestructible
%every measurable cardinal $\gk$ has its measurability indestructible
under forcing with either $\add(\gl, 1)$ or $\add(\gl, \gl^+)$.
\end{enumerate}
\end{theorem}
%\noindent Note that a model witnessing the hypotheses
%of Theorem \ref{t1} was first constructed by the
%author and Shelah in \cite{AS97a}.
We note that in Theorem \ref{t1}, there are no
restrictions whatsoever on the classes of
measurable and supercompact cardinals.
In particular, in both $V$ and $V^\FP$,
there can be a proper class of
measurable cardinals, a proper class of supercompact cardinals, etc.
In addition, if $\gl = \gk$
in clause (\ref{i2}), then Theorem \ref{t1} states that
the $\gk$ supercompactness of $\gk$
%, i.e., the measurability of $\gk$,
is indestructible under forcing with either
$\add(\gk, 1)$ or $\add(\gk, \gk^+)$.
Since $\gk$ is measurable iff $\gk$ is
$\gk$ supercompact iff $\gk$ is $\gk$
strongly compact, this means that
in $V^\FP$, every measurable cardinal $\gk$ has
its measurability indestructible under forcing with either
$\add(\gk, 1)$ or $\add(\gk, \gk^+)$.
Therefore, Theorem \ref{t1} may be regarded as providing a model for
level by level equivalence between strong compactness and supercompactness
with a very weak form of {\em universal indestructibility for
measurability} (where as in \cite{A99}, this means that
for any measurable cardinal $\gk$, $\gk$'s measurability is
indestructible under arbitrary $\gk$-directed closed forcing --- see
\cite{A99} for a discussion of the concept of universal indestructibility
in general).
We conclude Section \ref{s1}
with a very brief discussion of
some preliminary material.
We presume a basic knowledge
of large cardinals and forcing.
A good reference in this
regard is \cite{J}.
When forcing, $q \ge p$ means that
{\em $q$ is stronger than $p$}.
When $G$ is $V$-generic over $\FP$,
we abuse notation slightly and
take both $V[G]$ and
$V^\FP$ as being the generic
extension of $V$ by $G$.
We also abuse notation slightly by
occasionally confusing terms with the
sets they denote, especially for
ground model sets and variants of the generic object.
%For $\ga < \gb$ ordinals, $[\ga, \gb]$,
%$(\ga, \gb]$, $[\ga, \gb)$, and
%$(\ga, \gb)$ are as in standard interval notation.
%For $\gk$ a measurable cardinal, the
%normal measure ${\cal U}$ over $\gk$ has
%{\em trivial Mitchell rank} if for
%$j : V \to M$ the elementary embedding
%generated by ${\cal U}$,
%$M \models ``\gk$ is not measurable''.
Suppose $\gk < \gl$ are regular cardinals.
%For $\ga$ an arbitrary ordinal, the
%partial ordering $\add(\gk, \ga)$ is
%the standard Cohen partial ordering
%for adding $\ga$ many Cohen subsets of $\gk$.
The partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D \subseteq \FP$
of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
$\FP$ is {\em $\gk$-closed} if every increasing
chain of members of $\FP$ of length $\gk$ has an upper bound.
$\FP$ is {\em ${<}\gk$-closed} if $\FP$ is
$\gd$-closed for every $\gd < \gk$.
%$\FP$ is {\em $\gk$-strategically closed}
%if in the two person game in which the
%players construct an increasing sequence
%$\la p_\ga \mid \ga \le \gk \ra$,
%where player I plays odd stages and
%player II plays even stages,
%player II has a strategy ensuring the game
%can always be continued.
%$\FP$ is {\em ${\prec}\gk$-strategically closed}
%if in the two person game in which the
%players construct an increasing sequence
%$\la p_\ga \mid \ga < \gk \ra$,
%where player I plays odd stages and
%player II plays even stages,
%player II has a strategy ensuring the game
%can always be continued.
%It therefore follows that
%any partial ordering $\FP$ which is
%$\gk$-directed closed is also
%${\prec }\gk$-strategically closed
%and consequently adds no new subsets of
%any cardinal $\gd < \gk$.
%$\FP$ is {\em ${<}\gk$-strategically closed}
%if $\FP$ is $\gd$-strategically closed
%for every $\gd < \gk$.
%$\FP$ is {\em $(\gk, \infty)$-distributive}
%if the intersection of $\gk$ many
%dense open subsets of $\FP$ is dense open.
%It therefore follows that
%any partial ordering $\FP$ which is
%$\gk$-directed closed is also
%${<}\gk$-strategically closed, and any
%partial ordering which is $\gk$-strategically
%closed is $(\gk, \infty)$-distributive.
%It further
We recall for the benefit of readers the
definition given by Hamkins in
\cite[Section 3]{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the {\it lottery sum} is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra \mid \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and then
forces with it.\footnote{The terminology
``lottery sum'' is due to Hamkins, although
the concept of the lottery sum of partial
orderings has been around for quite some
time and has been referred to at different
junctures via the names ``disjoint sum of partial
orderings,'' ``side-by-side forcing,'' and
``choosing which partial ordering to force
with generically.''}
A corollary of Hamkins' work on
gap forcing found in
\cite{H2, H3}
will be employed in the
proof of Theorem \ref{t1}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
%$\gd$-strategically closed''.
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ is {\em mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\dot \tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that
$\forces_{\FP} ``\dot \tau \subseteq \check y$''.
%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
{\em 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 elementary embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH + The class of supercompact
%$\K \neq \emptyset$ is the class of supercompact cardinals
cardinals in nonempty + Level by level
equivalence between strong compactness and
supercompactness holds''.
Let $\Omega$ be the class of all ordinals
if there is a proper class of measurable cardinals, or
the supremum of the set of all measurable cardinals otherwise.
The partial ordering
$\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd \in \Omega \ra$ used
in the proof of Theorem \ref{t1}
will be the (possibly proper class) reverse Easton iteration
which begins by forcing with
$\add(\go, 1)$ and then (potentially) performs nontrivial forcing only
at those stages $\gd$ which are inaccessible cardinals.
At such a $\gd$, %we force with
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for
the lottery sum of $\{\emptyset\}$,
$\add(\gd, 1)$, and $\add(\gd, \gd^+)$.
Standard arguments show that $V^\FP \models {\rm ZFC}$
and forcing with $\FP$ preserves all cofinalities and GCH.
\begin{lemma}\label{l1}
Suppose $\gk \le \gl$ are regular cardinals such that
$V \models ``\gk$ is $\gl$ supercompact''. Then
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
Write $\FP = \FP_{\gl + 1} \ast \dot \FS =
\FP_\gk \ast \dot \FR \ast \dot \FS$, where
$\dot \FR$ is a term for the portion of
$\FP$ acting on inaccessible cardinals in the
closed interval $[\gk, \gl]$, and
$\dot \FS$ is a term for the portion of
$\FP$ acting on inaccessible cardinals above $\gl^+$. Since
$\forces_{\FP_\gk \ast \dot \FR} ``\dot \FS$ is
$\eta$-directed closed for $\eta$
the least strong limit cardinal above $\gl$'', to show that
$V^\FP \models ``\gk$ is $\gl$ supercompact'',
it suffices to show that
$V^{\FP_{\gl + 1}} =
V^{\FP_\gk \ast \dot \FR} \models ``\gk$ is $\gl$ supercompact''.
To do this, let $j : V \to M$ be an elementary
embedding witnessing the $\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter over $P_\gk(\gl)$.
Let $G_0$ be $V$-generic over $\FP_\gk$ and $G_1$ be
$V[G_0]$-generic over $\FR$. We consider now the
following two cases.
\bigskip\noindent Case 1: $\card{G_1} \le \gl$, i.e.,
%by forcing above the appropriate condition, $\FP_\gk \ast \dot \FR$ is forcing equivalent to
the part of $\FP_\gk \ast \dot \FR$ above a suitable condition
$p_0$ is forcing equivalent to
a partial ordering having cardinality at most $\gl$.
%at stage $\gl$ in the definition of $\FP$,
%either $\gl$ is not inaccessible (so trivial forcing is done),
%or $\gl$ is inaccessible, and
%the partial ordering selected in the lottery by $G_1$
%is either $\{\emptyset\}$ or $\add(\gl, 1)$.
Without loss of generality, but with an abuse of terminology and notation,
we will take $\FP_\gk \ast \dot \FR$ as $\FP_\gk \ast \dot \FR$ above $p_0$.
Write $j(\FP_\gk \ast \dot \FR) =
\FP_\gk \ast \dot \FR \ast \dot \FR' \ast j(\dot \FR)$, where
$\dot \FR'$ is a term for the portion of the forcing acting
on $M$-inaccessible cardinals in the open interval $(\gl, j(\gk))$.
Because %by forcing above the appropriate condition,
%the part of $\FP_\gk \ast \dot \FR$ above $p_0$ is forcing equivalent to
$\FP_\gk \ast \dot \FR$ is forcing equivalent to
a partial ordering having cardinality at most $\gl$, we may assume that
$\FP_\gk \ast \dot \FR$ is $\gl^+$-c.c.
Consequently, as
$M^\gl \subseteq M$, $M[G_0][G_1]^\gl \subseteq M[G_0][G_1]$.
Further, since $j$ is generated by a
supercompact ultrafilter over $P_\gk(\gl)$,
$2^\gl = \gl^+$ in $V$, and
$M[G_0][G_1] \models ``\card{\FR'} = j(\gl)$'', the number of dense open
subsets of $\FR'$ present in $M[G_0][G_1]$ is
$(2^{j(\gl)})^M = (2^{j(\gl)})^{M[G_0][G_1]} =
j(\gl^+)$.
This is calculated in either $V$ or $V[G_0][G_1]$ as
$\card{\{f \mid f : P_\gk(\gl) \to 2^\gl\}} =
\card{\{f \mid f : \gl \to 2^\gl\}} =
\card{\{f \mid f : \gl \to \gl^+\}} = [\gl^+]^\gl = \gl^+$.
We may therefore let
$\la D_\ga \mid \ga < \gl^+ \ra \in V[G_0][G_1]$ enumerate the
dense open subsets of $\FR'$ present in $M[G_0][G_1]$.
As $M[G_0][G_1]^\gl \subseteq M[G_0][G_1]$,
by the definition of $\FP$,
$\FR'$ is $\gl^+$-directed closed in both
$M[G_0][G_1]$ and $V[G_0][G_1]$.
%We may hence in $V[G_0][G_1]$ meet each $D_\ga$ and thereby
%construct in $V[G_0][G_1]$ an $M[G_0][G_1]$-generic object $G_2$ over $\FR'$.
We may hence in $V[G_0][G_1]$ construct an increasing sequence
$\la q_\ga \mid \ga < \gl^+ \ra$ by letting $q_0 \in D_0$, and for
$\ga > 0$, letting $q_\ga \in D_\ga$ extend
$\sup(\la q_\gb \mid \gb < \ga \ra)$. Clearly,
$G_2 = \{p \in \FR' \mid \exists \ga < \gl^+ [q_\ga \ge p] \}$
is $M[G_0][G_1]$-generic over $\FR'$.
Since by construction,
$j '' G_0 \subseteq G_0 \ast G_1 \ast G_2$,
$j$ lifts in $V[G_0][G_1]$ to $j : V[G_0] \to M[G_0][G_1][G_2]$.
Now, since
$M[G_0][G_1][G_2]^\gl \subseteq M[G_0][G_1][G_2]$ in $V[G_0][G_1]$,
%$V[G_0] \models ``$The part of $\FR$
%above a suitable condition is forcing equivalent to a
%partial ordering having cardinality at most $\gl$'',
$V[G_0] \models ``\FR$ is forcing equivalent to a
partial ordering having cardinality at most $\gl$'',
$M[G_0][G_1][G_2] \models ``j(\FR)$ is
$j(\gl)$-directed closed'', and
$j(\gl) > \gl^+$, there is a master condition
$q \in j(\FR)$ for $j '' G_1$.
Further, the number of dense open subsets of
$j(\FR)$ present in $M[G_0][G_1][G_2]$ is at most
$(2^{j(\gl)})^M$. As in the preceding paragraph,
since $(2^{j(\gl)})^M = (2^{j(\gl)})^{M[G_0][G_1][G_2]}$,
this is calculated in either $V$ or $V[G_0][G_1]$ as $\gl^+$.
Consequently, we can once again use the same argument as
given in the previous paragraph and build in
$V[G_0][G_1]$ an $M[G_0][G_1][G_2]$-generic object $G_3$
over $j(\FR)$ containing $q$. Since by construction,
$j '' (G_0 \ast G_1) \subseteq G_0 \ast G_1 \ast G_2 \ast G_3$,
$j$ now fully lifts to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$.
Hence, $V[G_0][G_1] \models ``\gk$ is $\gl$ supercompact'',
i.e.,
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
\bigskip\noindent Case 2: $\card{G_1} = \gl^+$, i.e.,
$\gl$ is inaccessible, and at stage $\gl$ in the definition of $\FP$,
the partial ordering selected in the lottery by $G_1$ is $\add(\gl, \gl^+)$.
We then have that
%above the appropriate condition,
the part of $\FR$ above a suitable condition $q_0$ is forcing equivalent
to a partial ordering of the form $\FR^* \ast \dot \add(\gl, \gl^+)$.
In analogy to Case 1 above,
%without loss of generality, but with an abuse of terminology and notation,
we will take $\FR$ as $\FR$ above $q_0$.
This allows us to write
$G_1 = H_0 \ast H_1$, where $H_0$ is $V[G_0]$-generic over
$\FR^*$, and $H_1$ is $V[G_0][H_0]$-generic over $\add(\gl, \gl^+)$.
The arguments given in Case 1 above then show that $j$ lifts in
$V[G_0][G_1]$ to
$j : V[G_0][H_0] \to M[G_0][G_1][G_2][G_3]$, where
$G_2$ and $G_3$ are constructed as in Case 1.
Note it is once again the case that
$M[G_0][G_1][G_2][G_3]^\gl \subseteq M[G_0][G_1][G_2][G_3]$ in
$V[G_0][G_1]$.
To lift $j$ fully to $V[G_0][G_1]$,
we now use an idea originally due to
Magidor \cite{Ma2} but also found in \cite[Lemma 2.2]{A12b}
(and elsewhere in the literature as well ---
readers may consult \cite[Lemma 2.2]{A12b} for
additional references).
We again feel free to quote verbatim as needed.
We will construct in $V[G_0][G_1]$ an
$M[G_0][G_1][G_2][G_3]$-generic object over
$\add(j(\gl), j(\gl^+))$.
For $\ga \in (\gl, \gl^+)$ and
$p \in \add(\gl, \gl^+)$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p \mid
\gs < \ga\}$ and
$H_1 \rest \ga = \{p \rest \ga \mid p \in H_1\}$. Clearly,
$V[G_0][G_1] \models ``|H_1 \rest \ga| \le \gl$
for all $\ga \in (\gl, \gl^+)$''. Thus, since
${(\add(j(\gl), j(\gl^+))}^{M[G_0][G_1][G_2][G_3]}$ is
$j(\gl)$-directed closed and $j(\gl) > \gl^+$,
$q_\ga = \bigcup\{j(p) \mid p \in H_1 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gl), j(\gl^+))}^{M[G_0][G_1][G_2][G_3]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in H_1 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$H_1 \rest \gb$ and $H_1 \rest \ga$, for
$p \rest \gb = q \in H_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2][G_3] \models
``2^{j(\gl)} = j(\gl^+)$'',
%``$GCH holds for all cardinals at or above $j(\gl)$'',
$M[G_0][G_1][G_2][G_3] \models ``\add(j(\gl),
j(\gl^+))$ is
$j(\gl^+)$-c.c$.$ and has
$j(\gl^+)$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2][G_3]$ is a
maximal antichain of $\add(j(\gl), j(\gl^+))$,
${\cal A} \subseteq \add(j(\gl), \gb)$ for some
$\gb \in (j(\gl), j(\gl^+))$.
Thus, since $V \models ``2^\gl = \gl^+$''
and the fact $j$ is generated by a
supercompact ultrafilter over $P_\gk(\gl)$ imply that
$V \models ``|j(\gl^+)| = \gl^+$'', we can let
$\la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra \in
V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gl), j(\gl^+))$
present in $M[G_0][G_1][G_2][G_3]$.
Working in $V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga \mid \ga \in (\gl, \gl^+) \ra$ of
elements of $\add(j(\gl), j(\gl^+))$ such that
$\forall \ga \in (\gl, \gl^+) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gl), j(\ga))]$ and such that
%$\forall {\cal A} \in \la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra
%\exists \gb \in (\gl, \gl^+) \exists r \in {\cal A} [r_\gb \ge r]$.
$\forall \ga \in (\gl, \gl^+) \exists \gb \in (\gl, \gl^+)
\exists r \in {\cal A}_\ga [r_\gb \ge r]$.
Assuming we have such a sequence,
%$G_4 = \{p \in \add(j(\gl), j(\gl^+)) \mid
%\exists r \in \la r_\ga \mid \ga \in (\gl, \gl^+) \ra [r \ge p]$ is an
$G_4 = \{p \in \add(j(\gl), j(\gl^+)) \mid
\exists \ga \in (\gl, \gl^+) [r_\ga \ge p]$ is an
$M[G_0][G_1][G_2][G_3]$-generic object over
$\add(j(\gl), j(\gl^+))$. To define
$\la r_\ga \mid \ga \in (\gl, \gl^+) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gl, \ga)} r_\gb$.
By the facts
$\la r_\gb \mid \gb \in (\gl, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2][G_3]^\gl \subseteq M[G_0][G_1][G_2][G_3]$
in $V[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb \mid \gb < \eta < \gl^+ \ra$
be the subsequence of
$\la {\cal A}_\gb \mid \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gl), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gl), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gl), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2][G_3]^\gl \subseteq M[G_0][G_1][G_2][G_3]$
in $V[G_0][G_1]$, define by induction an increasing sequence
$\la s_\gb \mid \gb < \eta \ra$ such that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_4$ is
$M[G_0][G_1][G_2][G_3]$-generic over
$\add(j(\gl), j(\gl^+))$, we must show that
$\forall \ga \in (\gl, \gl^+) \exists \gb \in (\gl, \gl^+)
\exists r \in {\cal A}_\ga [r_\gb \ge r]$.
%$\forall {\cal A} \in
%\la {\cal A}_\ga \mid \ga \in (\gl, \gl^+) \ra
%\exists \gb \in (\gl, \gl^+)
%\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) \mid \ga < \gl^+ \ra$ is
unbounded in $j(\gl^+)$. To see this, if
$\gb < j(\gl^+)$ is an ordinal, then for some
$f : P_\gk(\gl) \to M$ representing $\gb$,
we can assume that for $\ga < \gl$,
$f(\ga) < \gl^+$. Thus, by the regularity of
$\gl^+$ in $V$,
$\gb_0 = \bigcup_{\ga < \gl} f(\ga) <
\gl^+$, and $j(\gb_0) \ge \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga \mid \ga <
\gl^+ \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gl, \gl^+)$ be such that
${\cal A} \subseteq \add(j(\gl), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gl, \gl^+)$ is such that for some
$\ga \in (\gl, \gl^+)$, $p = p \rest \ga$,
$G_4$ is such that if
$p \in H_1$, $j(p) \in G_4$.
Thus, working in $V[G_0][G_1]$,
we have shown that $j$ lifts to
$j : V[G_0][H_0][H_1] \to M[G_0][G_1][G_2][G_3][G_4]$, i.e.,
$j$ lifts to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3][G_4]$.
This means that $V[G_0][G_1] \models ``\gk$ is $\gl$
supercompact'', i.e., $V^\FP \models ``\gk$ is $\gl$
supercompact''.
\bigskip
Cases 1 and 2 complete the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2}
follows closely
the proof of \cite[Lemma 1.3]{A05}.
We again feel free to quote verbatim as needed.
Suppose
$V^\FP \models ``\gk < \gl$ are
regular cardinals such that
$\gk$ is $\gl$ strongly compact and
$\gk$ is not a measurable limit of
cardinals which are $\gl$ supercompact''.
%As we observed in the first
%paragraph of the proof of
%Theorem \ref{t1}, forcing with
%$\FP$ preserves all cardinals
%and cofinalities.
By Lemma \ref{l1}, any cardinal
which is $\gl$ supercompact in $V$ remains $\gl$ supercompact in $V^\FP$.
This means that
$V \models ``\gk < \gl$ are
regular cardinals such that
%$\gk$ is $\gl$ strongly compact and
$\gk$ is not a measurable limit of
cardinals which are
$\gl$ supercompact''.
Note that it is possible to
write $\FP =
\FP^0 \ast \dot \FP^1$, where
$|\FP^0| = \go$,
$\FP^0$ is nontrivial, and
$\forces_{\FP^0} ``\dot \FP^1$ is
$\ha_2$-directed closed''.
Further, by the definition of
$\FP$, it is easily seen that
$\FP$ is mild with respect to $\gk$.
Therefore, by Theorem \ref{tgf},
$V \models ``\gk$ is $\gl$ strongly compact''.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gk$ is $\gl$ supercompact'',
so another application of Lemma \ref{l1}
yields that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
%Assume $\gl \ge \gk$ is a regular cardinal and
Assume $V^\FP \models ``\gl \ge \gk$ is inaccessible and
$\gk$ is $\gl$ supercompact''. Then
$V^\FP \models ``$The $\gl$ supercompactness of $\gk$
is indestructible under forcing with either
%In $V^\FP$, every measurable cardinal $\gk$ has its
%measurability indestructible under forcing with either
$\add(\gl, 1)$ or $\add(\gl, \gl^+)$''.
\end{lemma}
\begin{proof}
Suppose $V^\FP \models ``\gl \ge \gk$ is inaccessible and
$\gk$ is $\gl$ supercompact''.
As in the proof of Lemma \ref{l1},
write $\FP = \FP_{\gl + 1} \ast \dot \FS$. It is then true that
$\forces_{\FP_{\gl + 1}} ``\dot \FS$ is $\eta$-directed closed
for $\eta$ the least strong limit cardinal above $\gl$''.
Hence, to prove Lemma \ref{l3}, it suffices to show that
in $V^{\FP_{\gl + 1}}$, $\gk$ has its $\gl$ supercompactness
indestructible under forcing with either
$\add(\gl, 1)$ or $\add(\gl, \gl^+)$.
To do this, we observe that
at each inaccessible cardinal $\ga$,
if we first force
with either $\{\emptyset\}$, $\add(\ga, 1)$, or $\add(\ga, \ga^+)$,
and then follow this by forcing
with $\add(\ga, 1)$ or $\add(\ga, \ga^+)$,
then this is equivalent to forcing with
either $\add(\ga, 1)$ or $\add(\ga, \ga^+)$.
Therefore,
%assuming we have forced above the appropriate condition,
$\FP_\gl \ast \dot \FQ_\gl \ast \dot \add(\gl, 1)$ and
$\FP_\gl \ast \dot \FQ_\gl \ast \dot \add(\gl, \gl^+)$
are forcing equivalent to
$\FP_\gl \ast \dot \add(\gl, 1)$ and $\FP_\gl \ast \dot \add(\gl, \gl^+)$ respectively.
%as well as to $\FP_{\gl + 1}$.
%to $\FP_\gl \ast \dot \FQ_\gl = \FP_{\gl + 1}$.
By the proof of Lemma \ref{l1},
$V^{\FP_\gl \ast \dot \add(\gl, 1)} \models ``\gk$ is $\gl$ supercompact'' and
$V^{\FP_\gl \ast \dot \add(\gl, \gl^+)} \models ``\gk$ is $\gl$ supercompact''.
%$V^{\FP_{\gl + 1}} \models ``\gk$ is $\gl$ supercompact''.
Since there are forcing conditions $r_0$ and $s_0$ such that the
part of $\FP_\gl \ast \dot \FQ_\gl = \FP_{\gl + 1}$ above $r_0$
is forcing equivalent to $\FP_\gl \ast \dot \add(\gl, 1)$ and the
part of $\FP_\gl \ast \dot \FQ_\gl = \FP_{\gl + 1}$ above $s_0$
is forcing equivalent to $\FP_\gl \ast \dot \add(\gl, \gl^+)$,
this completes the proof of Lemma \ref{l3}.
\end{proof}
Note that by Theorem \ref{tgf} and the factorization of
$\FP$ given in Lemma \ref{l2}, if $V^\FP \models ``\gk$ is
$\gl$ supercompact'', then $V \models ``\gk$ is $\gl$
supercompact'' as well.
Lemmas \ref{l1} -- \ref{l3}
therefore complete the proof of Theorem \ref{t1}.
\end{proof}
Observe that if $\gk \in V^\FP$ is supercompact, then
$\gk$ is not indestructibly supercompact.
This follows from the fact that if it were, then
$V^{\FP \ast \dot \add(\gk, \gk^{++})} \models ``\gk$
is measurable and $2^\gk = \gk^{++}$''.
Thus, in $V^{\FP \ast \dot \add(\gk, \gk^{++})}$,
$A = \{\gd < \gk \mid 2^\gd = \gd^{++}\}$ is unbounded
in $\gk$. However, since $\add(\gk, \gk^{++})$ is
$\gk$-directed closed, $A$ is unbounded in $V^\FP$ as well.
This contradicts the fact that $V^\FP \models {\rm GCH}$.
We feel that Theorem \ref{t1} should be viewed as a
first step in establishing indestructibility theorems
which are compatible with level by level equivalence
between strong compactness and supercompactness in which
there are no limits on the number of measurable cardinals
with indestructibility properties.
We consequently conclude by posing the general question of
what other theorems along these lines are possible.
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\end{document}
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH + $\K \neq \emptyset$ is
the class of supercompact cardinals + Level by level
equivalence between strong compactness and
supercompactness holds''.
There is then a partial ordering $\FP \subseteq V$
preserving all cofinalities and containing the
same measurable cardinals as $V$ such that
$V^\FP \models ``$ZFC + GCH + $\K \neq \emptyset$ is
the class of supercompact cardinals + Level by level
equivalence between strong compactness and
supercompactness holds''.
In $V^\FP$, every measurable cardinal $\gk$
has its measurability indestructible under
forcing with either $\add(\gk, 1)$ or $\add(\gk, \gk^+)$.
\end{theorem}
For the proofs of Lemmas \ref{l1} -- \ref{l2}, we use
well known ideas.
%ideas that have appeared extensively in the literature.
Our reference is \cite[Theorem 2.1]{A12}
for Lemmas \ref{l1} and \ref{l2} and \cite[Lemma 2.2]{A12b}
for Lemma \ref{l2},
from which we feel free to quote verbatim if necessary.
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%\bibitem{A08} A.~Apter, ``Indestructibility and
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%\bibitem{A09} A.~Apter, ``Indestructibility and
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%{\it Mathematical Logic Quarterly 55}, 2009, 228--236.
%\bibitem{A11} A.~Apter, ``Indestructibility, HOD,
%and the Ground Axiom'',
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%\bibitem{A10} A.~Apter, ``Indestructibility,
%Instances of Strong Compactness, and
%Level by Level Inequivalence'',
%{\it Archive for Mathematical Logic 49}, 2010, 725--741.
%\bibitem{A12} A.~Apter, ``Indestructibilty, Measurability,
%and Degrees of Supercompactness'', {\it Mathematical Logic
%Quarterly 58}, 2012, 75--82.
To do this, we show that
at each inaccessible cardinal $\ga$,
%$\FQ_\ga \cong \FQ_\ga \ast \dot \FQ^*_\ga$, where
%$\dot \FQ^*_\ga$ is a term
%for the lottery sum of $\{\emptyset\}$, $\add(\ga, 1)$, and
%$\add(\ga, \ga^+)$.
%This in essence is the same thing as saying that
if we first force
with either $\{\emptyset\}$, $\add(\ga, 1)$, or $\add(\ga, \ga^+)$,
and then follow this by forcing
with $\add(\ga, 1)$ or $\add(\ga, \ga^+)$,
%then we have forced with a partial ordering which is forcing equivalent to
%the lottery sum
%of $\{\emptyset\}$, $\add(\ga, 1)$, or $\add(\ga, \ga^+)$.
then this is forcing equivalent to forcing
with either $\add(\ga, 1)$ or $\add(\ga, \ga^+)$.
If we first force with $\{\emptyset\}$ and then follow this
by forcing with either
$\add(\ga, 1)$ or $\add(\ga, \ga^+)$,
then this is clearly forcing equivalent to forcing with either
$\add(\ga, 1)$ or $\add(\ga, \ga^+)$.
If we first force with $\add(\ga, 1)$ and then follow this
by forcing with either
$\add(\ga, 1)$ or $\add(\ga, \ga^+)$,
then this is forcing equivalent to forcing with either
$\add(\ga, 1)$ or $\add(\ga, \ga^+)$.
This follows since by their definitions
and the fact that each partial ordering is
$\ga$-directed closed,
$\add(\ga, 1) \ast \dot \add(\ga, 1) \cong \add(\ga, 1)$,
and $\add(\ga, 1) \ast \dot \add(\ga, \ga^+) \cong \add(\ga, \ga^+)$.
Finally, if we first force with $\add(\ga, \ga^+)$ and then follow this
by forcing with either
$\add(\ga, 1)$ or $\add(\ga, \ga^+)$,
then this is forcing equivalent to forcing with
$\add(\ga, \ga^+)$.
This follows since by their definitions
and the fact that each partial ordering is
$\ga$-directed closed,
$\add(\ga, \ga^+) \ast \dot \add(\ga, 1) \cong
\add(\ga, \ga^+)$,
and $\add(\ga, \ga^+) \ast \dot \add(\ga, \ga^+) \cong \add(\ga, \ga^+)$.
%We now know that for each inaccessible cardinal
%$\ga$, $\FQ_\ga \ast \dot \FQ^*_\ga \cong \FQ_\ga$.
%In $V^{\FP_{\gk + 1}}$, let $\dot \FQ^*_\gk$
We now know by the work of the preceding paragraph that
%by forcing above the appropriate condition,
first forcing with the lottery sum of
$\{\emptyset\}$, $\add(\gl, 1)$, and $\add(\gl, \gl^+)$
followed by forcing with either
$\add(\gl, 1)$ or $\add(\gl, \gl^+)$ is forcing
equivalent to forcing with %the lottery sum of
either $\add(\gl, 1)$ or $\add(\gl, \gl^+)$.
In particular, assuming we have forced above
the appropriate condition,
$\FP_\gl \ast \dot \FQ_\gl \ast \dot \add(\gl, 1)$ and
$\FP_\gl \ast \dot \FQ_\gl \ast \dot \add(\gl, \gl^+)$
are both forcing equivalent to
$\FP_\gl \ast \dot \FQ_\gl = \FP_{\gl + 1}$.
By the proof of Lemma \ref{l1},
$V^{\FP_{\gl + 1}} \models ``\gk$ is $\gl$ supercompact''.
%Since $\gk$ being $\gk$ supercompact is equivalent to
%$\gk$ being measurable,
This completes the proof of Lemma \ref{l3}.