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\title{Indestructibility and the Levinski Property
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal,
indestructibility, lottery sum, Levinski property.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
\date{May 29, 2015\\(revised September 12, 2015)}
\begin{document}
\maketitle
\begin{abstract}
We investigate some possible interactions between an indestructibly
supercompact cardinal and a generalization of a property
originally due to Levinski \cite{Lev}.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
%We begin with some key definitions.
%The cardinal $\gk$ is {\em indestructibly supercompact}
%if $\gk$'s supercompactness is preserved after forcing
%with a $\gk$-directed closed partial ordering.
We begin with some terminology and notational conventions.
Suppose $\gk$ is a regular cardinal.
For $\gb$ an arbitrary ordinal, the
partial ordering $\add(\gk, \gb)$ is
the standard Cohen partial ordering
for adding $\gb$ many Cohen subsets of $\gk$.
The partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D \subseteq \FP$
of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
Say that a measurable cardinal $\gk$ satisfies the
{\em Levinski property LP} if $2^\gk = \gk^+$, yet
GCH fails on some final segment of inaccessible cardinals
below $\gk$.
Call an ordinal $\ga > 0$ {\em good}
if $\ga$ is definable and is also such that for
any cardinal $\gd$, $\gd^{+ \ga}$ is a regular cardinal below the least
inaccessible cardinal above $\gd$.\footnote{This represents a
slight abuse of notation. Strictly speaking, $\ga$ is a formula
with parameter $\gd$, which we will suppress. Examples of
good ordinals include $2$, $3$, $75$, $\go + 1$, etc.
This is since $2$, $3$, $75$, and $\go + 1$
are all definable. %each of the preceding is definable.
Further, for
$\ga$ any of these values, $\gd^{+ \ga}$ is regular
since it is a successor cardinal.
$\gd^{+ \ga}$ is also below the least inaccessible
cardinal above $\gd$, which must be a (regular) limit cardinal.}
%inaccessible cardinal above $\gd$.\footnote{We will slightly abuse
%notation by also allowing $\ga$
%to be a formula with parameter $\gd$ which always defines an ordinal below the least
%inaccessible cardinal above $\gd$. An
%example is $\ga(\gd) =$ The cardinal successor of the least beth fixed point
%above $\gd$. If this is the case, we will suppress the parameter $\gd$.}
%Suppose throughout $\ga$ is some %suitably
%definable ordinal (including 0)
%such that for any $\gd$, $\gd^{+ \ga}$ is
%below the least inaccessible cardinal above $\gd$.
Refine the preceding by saying that
a measurable cardinal $\gk$ satisfies the
{\em Levinski property LP($\ga$)} for a fixed but
arbitrary good ordinal $\ga$ if for every
inaccessible cardinal $\gd$ in some final segment
below $\gk$, $2^\gd = \gd^{+ \ga}$, yet $2^\gk = \gk^+$.
Both LP and LP($\ga$) are variants of a property first
studied by Levinski in \cite{Lev}.
Beginning now our main narrative,
it is an interesting and curious fact that the large cardinal
structure of the universe above a
supercompact cardinal $\gk$
with suitable indestructibility properties
can affect the large cardinal structure
below $\gk$ in quite surprising ways. %which are not immediately apparent.
%influence what happens at large cardinals below $\gk$.
On the other hand, these effects may be mitigated
%not be present
%this control can vanish
%it is possible to mitigate these effects
if the universe contains relatively few large cardinals.
These sorts of occurrences have previously
been investigated in \cite{A07, A08, A09, A11, A10, A12, A15, AH4}.
%\cite{A07}, \cite{A08}, and \cite{A09}.
The purpose of this paper is to continue
studying this phenomenon, but in the context of
%looking at
different versions of the Levinski property together
with their interactions
with an indestructibly supercompact cardinal.
We begin with the following theorem,
where as in \cite{L}, $\gk$ is
{\em indestructibly supercompact} if
$\gk$'s supercompactness is preserved by
arbitrary $\gk$-directed closed forcing.
\begin{theorem}\label{t1}
Suppose that $\gk$ is indestructibly supercompact
and there is a measurable cardinal $\gl > \gk$.
%It then follows that
Then for any good ordinal $\ga$,
$A_\ga = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and LP($\ga$) holds for $\gd\}$
is unbounded in $\gk$.
\end{theorem}
The large cardinal
hypothesis on $\gl$ is necessary,
as we further demonstrate by
constructing via forcing models
containing an indestructibly supercompact cardinal $\gk$
with no measurable cardinal above it in which for fixed
but arbitrary good $\ga$, every measurable cardinal
$\gd < \gk$ which is not a limit of measurable cardinals
satisfies LP($\ga$).
%With a limited large cardinal structure above $\gk$,
%Theorem \ref{t1} need not be true.
Specifically we have:
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable''.
Let $\ga$ be a good ordinal.
There is then a partial ordering $\FP \in V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is measurable +
$\gk$ is indestructibly supercompact +
If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then
LP($\ga$) holds''.
\end{theorem}
We also show the necessity of the large cardinal
hypothesis on $\gl$ by constructing via forcing
models containing an indestructibly supercompact
cardinal $\gk$ with no measurable cardinals above it such that
for every measurable cardinal $\gd < \gk$ which is not a
limit of measurable cardinals, $2^\gd > \gd^+$. In particular we have:
\begin{theorem}\label{t3}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable''.
There is then a partial ordering $\FP \subseteq V$
such that
$V^\FP \models ``$ZFC +
No cardinal $\gz > \gk$ is measurable +
$\gk$ is indestructibly supercompact + If $\gd < \gk$
is a measurable cardinal which is not a limit of
measurable cardinals, then $2^\gd = \gd^{++}$''.
\end{theorem}
We take this opportunity to make a few remarks concerning
Theorems \ref{t2} and \ref{t3}. In the conclusions of Theorem \ref{t3}, there
is nothing special about having $2^\gd = \gd^{++}$ for
every measurable cardinal $\gd < \gk$ which is not a
limit of measurable cardinals.
As our methods of proof will show, it is also possible to have
$2^\gd = \gd^{+3}$, $2^\gd = \gd^{+4}$, etc.
In addition, for both Theorems \ref{t2} and \ref{t3},
the measurable cardinal $\gd < \gk$ cannot in general be a
limit of measurable cardinals. This is since for any $n < \go$,
$(\add(\gk^+, 1) \ast \dot \add(\gk, \gk^{+ n}))^{V^\FP}$
is $\gk$-directed closed in $V^\FP$.
Standard arguments (see \cite[Exercise 15.16]{J}) show that
after forcing with $\add(\gk^+, 1)$, $2^\gk = \gk^+$.
Thus, if $V^\FP \models ``\gk$ is indestructibly supercompact'',
$V^{\FP \ast \dot \add (\gk^+, 1) \ast \dot \add(\gk, \gk^{+ n})} \models ``\gk$
is supercompact + $2^\gk = \gk^{+ n}$ + $\gk$ is a measurable
limit of measurable cardinals''. Hence, by reflection, %for any $n < \go$, in
in $V^{\FP \ast \dot \add (\gk^+, 1) \ast \dot \add(\gk, \gk^{+ n})}$,
$B_n = \{\gd < \gk \mid \gd$ is a measurable limit of measurable cardinals and
$2^\gd = \gd^{+ n}\}$ is unbounded in $\gk$. Because
$(\add(\gk^+, 1) \ast \dot \add(\gk, \gk^{+ n}))^{V^\FP}$
is $\gk$-directed closed in $V^\FP$, $B_n$ is unbounded in $V^\FP$ as well.
This precludes the conclusions of both Theorems \ref{t2} and \ref{t3}
holding for $\gd$ when $\gd < \gk$ is a measurable limit of measurable cardinals.
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 $p$}.
We will have two slight abuses of notation.
In particular,
when $G$ is $V$-generic over $\FP$,
we %abuse notation slightly and
take both $V[G]$ and
$V^\FP$ as being the generic
extension of $V$ by $\FP$.
We also %abuse notation slightly by
occasionally confuse terms with the
sets they denote, especially for
ground model sets and variants of the generic object.
For $\ga < \gb$ ordinals, %$[\ga, \gb]$, $(\ga, \gb]$,
$[\ga, \gb)$ 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''.
%$\FP$ 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 Theorems \ref{t2} and \ref{t3}.
We therefore state as a
separate theorem
what is relevant for this paper, along
with some associated terminology,
quoting from \cite{H2, H3}
when appropriate.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} < \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
%$\gd$-strategically closed''.
$\gd^+$-directed closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\em admits a gap at $\gd$}.
%In Hamkins' terminology of
%\cite{H2, H3},
%$\FP$ is {\em mild
%with respect to a cardinal $\gk$}
%iff every set of ordinals $x$ in
%$V^\FP$ of size below $\gk$ has
%a ``nice'' name $\tau$
%in $V$ of size below $\gk$,
%i.e., there is a set $y$ in $V$,
%$|y| <\gk$, such that any ordinal
%forced by a condition in $\FP$
%to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2, H3} and elsewhere,
an embedding
$j : \ov V \to \ov M$ 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}
\noindent A consequence of Theorem \ref{tgf} is that if
$\FP$ admits a gap at some regular $\gd < \gk$ and
$V^\FP \models ``\gd$ is measurable'', then
$V \models ``\gd$ is measurable'' as well.
\section{The Proofs of Theorems \ref{t1} -- \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t1},
whose proof will depend on the existence of
a certain partial ordering $\FP(\gd, \gl, \ga)$.
We isolate the existence of this key forcing
notion in the following theorem.
%which will be established via a sequence of lemmas.
\begin{theorem}\label{t4}
Suppose $V \models ``\gd < \gl$ are such that $\gd$
is a regular cardinal and $\gl$ is the least measurable
cardinal greater than $\gl$''. Let $\ga$ be a good ordinal.
There is then a $\gd$-directed closed
partial ordering $\FP(\gd, \gl, \ga)$ such that
$V^{\FP(\gd, \gl, \ga)} \models ``\gl$ is the least measurable
cardinal greater than $\gd$ + LP($\ga$) holds for $\gl$''.
\end{theorem}
\begin{proof}
Assume $\gd$, $\gl$, and $\ga$ are as in the hypotheses of Theorem \ref{t4}.
We define $\FP(\gd, \gl, \ga)$ as $\FP^1 \ast \dot \FP^2$, where $\FP^1 =
\add(\gl^+, 1)$. Because $\add(\gl^+, 1)$ is $\gl^+$-directed closed,
$V^{\FP^1} \models ``\gl$ is the least measurable cardinal greater
than $\gd$''. As we have already observed, standard arguments show
%Standard arguments (see \cite[Exercise 15.16]{J}) show
that $V^{\FP^1} \models ``2^\gl = \gl^+$''.
Work now in $\ov V = V^{\FP^1}$. $\FP^2$ is defined as $\FP_\gl
\ast \dot \add(\gl, \gl^+)$, where $\FP_\gl = \la \la \FP_\gb, \dot \FQ_\gb \ra
\mid \gb < \gl \ra$ is the reverse Easton iteration of length $\gl$ which
begins by forcing with $\add(\gd, 1)$ and then
does nontrivial forcing only at those $\gg \in (\gd, \gl)$ which are inaccessible
cardinals in $\ov V$. At such a stage $\gg$, $\dot \FQ_\gg$ is a term for
$\add(\gg^+, 1) \ast \dot \add(\gg, \gg^{+ \ga})$. Standard arguments once again
show that $\ov V^{\FP^1} \models ``2^\gg = \gg^{+ \ga}$ if $\gg \in (\gd, \gl)$
is inaccessible''.
In addition, by its definition, $\FP(\gd, \gl, \ga)$ is $\gd$-directed closed.
It is also the case that $\ov V^{\FP^2} \models ``\gl$ is measurable''.
To see this,
let $j : \ov V \to M$ be an elementary embedding
witnessing the measurability of $\gl$ in $\ov V$
generated by a normal measure over $\gl$. In
particular, $M^{\gl} \subseteq M$.
We combine several ideas
(including a standard lifting argument, an idea
due to Levinski \cite{Lev}, and an idea due to Magidor \cite{Ma2})
%a form of which is given, e.g., in the proof of \cite[Lemma 2.2]{A10})
to show that $j$ lifts in $\ov V^{\FP_\gl \ast \dot \add(\gl, \gl^+)}$ to
$j : \ov V^{\FP_\gl \ast \dot \add(\gl, \gl^+)}
\to M^{j(\FP_\gl \ast \dot \add(\gl, \gl^+))}$. Specifically, let
$G_0$ be $\ov V$-generic over $\FP_\gl$, and let
$G_1$ be $\ov V[G_0]$-generic over
$\add(\gl, \gl^+)$.
Observe that $j(\FP_\gl \ast \dot \add(\gl, \gl^+)) =
\FP_\gl \ast \dot \add(\gl^+, 1) \ast \dot \add(\gl, \gl^{+ \ga}) \ast
\dot \FQ \ast \dot \add(j(\gl), j(\gl^+))$.
Working in $\ov V[G_0]$,
we first note that since $\FP_\gl$ is $\gl$-c.c.,
$M[G_0]$ remains $\gl$-closed with respect to $\ov V[G_0]$.
This means that $(\add(\gl^+, 1))^{M[G_0]}$ (which by the $\gl$-closure
of $M[G_0]$ with respect to $\ov V[G_0]$ has the
same definition in both $\ov V[G_0]$ and $M[G_0]$)
is $\gl^+$-directed closed in both $M[G_0]$ and $\ov V[G_0]$.
Consequently, without fear of ambiguity, we write
$\add(\gl^+, 1)$ for $(\add(\gl^+, 1))^{M[G_0]}$.
Since $M[G_0]{} \models ``\card{\add(\gl^+, 1)}
= \card{[\gl^+]^\gl} = 2^ \gl$'' and $M$ and $M[G_0]$
have the same cardinals at and above $\gl$,
the number of dense open subsets of $\add(\gl^+, 1)$ present
in $M[G_0]{}$ is $(2^{2^\gl})^M < j(\gl)$.
In $\ov V$, since $M$
is given via an ultrapower by a
normal measure over $\gl$,
$\card{j(\gl)}$ may be calculated as
$\card{\{f \mid f : \gl \to \gl\}} = 2^\gl = \gl^+$.
Since $\gl^+$ is preserved from $\ov V$ to
$\ov V[G_0]{}$, we may let
$\la D_\gb \mid \gb < \gl^+ \ra \in \ov V[G_0]{}$
enumerate the dense open subsets of $\add(\gl^+, 1)$ present in $M[G_0]{}$.
We may now use the fact that $\add(\gl^+, 1)$ is
$\gl^+$-directed closed in $\ov V[G_0]{}$ to meet each $D_\gb$
and thereby construct in $\ov V[G_0]{}$
an $M[G_0]{}$-generic
object $H_0$ over $\add(\gl^+, 1)$. Our construction guarantees that
$j '' G_0 \subseteq G_0 \ast H_0$,
so $j$ lifts in $\ov V[G_0]{}$ to
$j : \ov V[G_0] \to M[G_0]{}[H_0]$.
Note that
because $\add(\gl^+, 1)$ is $\gl^+$-directed closed in both
$M[G_0]$ and $\ov V[G_0]$,
$M[G_0][H_0]$ remains $\gl$-closed
with respect to $\ov V[G_0][H_0] = \ov V[G_0]$.
We use now Levinski's ideas of \cite{Lev} to show that
it is possible to rearrange $G_1$ to form an
$M[G_0][H_0]$-generic object $H_1$ over
$(\add(\gl, \gl^{+ \ga}))^{M[G_0][H_0]}$ in $\ov V[G_0][G_1]$.
Since $\ov V \models ``2^\gl = \gl^+$'' and
$j$ is generated by an ultrafilter over $\gl$,
$\gl^+ < j(\gl) < \gl^{++}$.
In particular, because $M[G_0] \models
``\gl^+ \le 2^\gl = \card{(\add(\gl^+, 1))} < j(\gl)$'', any
$\gg \in (\gl^+, j(\gl))$ which %either $M$, $M[G_0]$, or
$M[G_0][H_0]$
believes to be a cardinal actually is an ordinal of cardinality
$\gl^+$ in either $\ov V$, $\ov V[G_0]$, or $\ov V[G_0][G_1]$.
%$\ov V \models ``\card{(\gl^{+ \ga})^M} = \gl^+$''.
%In addition, since $\FP$ is an Easton support iteration,
%$\FP_\gl \ast \dot \add(\gl, \gl^+)$ is $\gl$-c.c., which means that
%cardinals at and above $\gl$ are preserved from
%$\ov V$ to $\ov V[G_0][G_1]$ and $M$ to $M[G_0][G_1]$. Hence,
%$(\gl^{+ \ga})^{M[G_0][G_1]} = (\gl^{+ \ga})^M$,
%$(\gl^+)^{\ov V[G_0][G_1]} = (\gl^+)^{\ov V}$, and
Hence, $\ov V[G_0][G_1] \models ``\card{(\gl^{+ \ga})^{M[G_0][H_0]}} = \gl^+$''.
Let $(\gl^{+ \ga})^{M[G_0][H_0]} = \gr$.
%From this, it follows that in $V[G]$, $(\add(\gl, \gl^{+ \ga})^{M[G]}$
%has the form $\add(\gl, \gr)$, where
%$V[G] \models ``\card{\gr} = \gl^+$''.
Working in $\ov V[G_0][G_1]$, %(or $V$),
we may therefore let $f : \gl^+ \to \gr$
be a bijection. For any
$p \in \add(\gl, \gl^+)$, $g(p) = \{\la \la \gs, f(\gb) \ra, \gg \ra
\mid \la \la \gs, \gb \ra, \gg \ra \in p\} \in (\add(\gl, \gr))^{M[G_0][H_0]}$.
As can be easily checked (see \cite{Lev}),
$G_1 = \{g(p) \mid p \in H\}$ is an $M[G_0][H_0]$-generic object over
$(\add(\gl, \gr))^{M[G_0][H_0]}$.
As before, our construction guarantees that
$j '' G_0 \subseteq G_0 \ast H_0 \ast H_1$, so $j$ lifts in $\ov V[G_0][G_1]$
to $j : \ov V[G_0] \to M[G_0][H_0][H_1]$.
Because $(\add(\gl, \gl^+))^{\ov V[G_0]}$ is $\gl^+$-c.c$.$ in $\ov V[G_0]$,
$M[G_0][H_0][H_1]$ remains $\gl$-closed with respect to $\ov V[G_0][G_1]$.
In addition, since $M[G_0][H_0][H_1] \models ``\FQ$ is a reverse Easton
iteration of length $j(\gl)$'', $M[G_0][H_0][H_1] \models ``\card{\FQ} = j(\gl)$ and
$\FQ$ is $j(\gl)$-c.c.''. This means the number of antichains of $\FQ$
present in $M[G_0][H_0][H_1]$ is $j(\gl)$. Further, as
$M[G_0][H_0][H_1] \models ``\FQ$ is $\gl^+$-directed closed'' and
$M[G_0][H_0][H_1]$ is $\gl$-closed with respect to $\ov V[G_0][G_1]$,
$\FQ$ is $\gl^+$-directed closed in $\ov V[G_0][G_1]$ as well.
This means that the argument used in the construction of $H_0$
may be used to construct in $\ov V[G_0][G_1]$ an
$M[G_0][H_0][H_1]$-generic object $H_2$ over $\FQ$.
Since once again it is the case that $j '' G_0 \subseteq G_0 \ast
H_0 \ast H_1 \ast H_2$,
we may then in $\ov V[G_0][G_1]$ lift $j$ to
$j : \ov V[G_0] \to M[G_0][H_0][H_1][H_2]$.
By the fact that $\FQ$ is $\gl^+$-directed closed in both
$M[G_0][H_0][H_1]$ and $\ov V[G_0][G_1]$,
$M[G_0][H_0][H_1][H_2]$ remains $\gl$-closed with respect to
$\ov V[G_0][G_1][H_2] = \ov V[G_0][H_0][G_1][H_2] = \ov V[G_0][G_1]$.
We now use arguments originally due to
Magidor \cite{Ma2}, which are also given in
\cite[pages 119--120]{AS97a} and are found
other places in the literature as well, to construct
in $\ov V[G_0][G_1]$ an $M[G_0][H_0][H_1][H_2]$-generic
object $H_3$ over $(\add(j(\gl), j(\gl^+))^{M[G_0][H_0][H_1][H_2]}$.
For the convenience of readers, we present these
arguments below.
For $\gz \in (\gl, \gl^{+})$ and
$p \in \add(\gl, \gl^{+})$, let
$p \rest \gz = \{\la \la \rho, \gs \ra, \eta \ra \in p \mid
\gs < \gz\}$ and
$G_1 \rest \gz = \{p \rest \gz \mid p \in G_1\}$. Clearly,
$\ov V[G_0][G_1] \models ``|G_1 \rest \gz| \le \gl$
for all $\gz \in (\gl, \gl^{+})$''. Thus, since
${\add(j(\gl), j(\gl^{+}))}^{M[G_0][H_0][H_1][H_2]}$ is
$j(\gl)$-directed closed and $j(\gl) > \gl^{+}$,
$q_\gz = \bigcup\{j(p) \mid p \in G_1 \rest \gz\}$ is
well-defined and is an element of
${\add(j(\gl), j(\gl^{+}))}^{M[G_0][H_0][H_1][H_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\gz) -
\dom(\bigcup_{\gb < \gz} q_\gb)$
($\bigcup_{\gb < \gz} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \gz} j(\gb), j(\gz))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \gz} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \gz$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \gz$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][H_0][H_1][H_2] \models ``j(\gl)$ is inaccessible and
$2^{j(\gl)} = j(\gl^+)$'', an application of \cite[Lemma 15.4]{J} shows that
%\break
$M[G_0][H_0][H_1][H_2] \models ``\add(j(\gl),
j(\gl^{+}))$ is
$j(\gl^+)$-c.c$.$ and has
$j(\gl^{+})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][H_0][H_1][H_2]$ 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 the fact
$\ov V \models ``2^\gl = \gl^+$''
and the fact $j$ is generated by a normal measure over
$\gl$ imply that
$V \models ``|j(\gl^{+})| = \gl^{+}$'', we can let
$\la {\cal A}_\gz \mid \gz \in (\gl, \gl^{+}) \ra \in
\ov V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gl), j(\gl^{+}))$
present in
$M[G_0][H_0][H_1][H_2]$.
Working in $\ov V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\gz \mid \gz \in (\gl, \gl^{+}) \ra$ of
elements of $\add(j(\gl), j(\gl^{+}))$ such that
$\forall \gz \in (\gl, \gl^{+}) [r_\gz \ge q_\gz$ and
$r_\gz \in \add(j(\gl), j(\gz))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\gz \mid \gz \in (\gl, \gl^{+}) \ra
\exists \gb \in (\gl, \gl^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$H_3 = \{p \in \add(j(\gl), j(\gl^{+})) \mid
\exists r \in \la r_\gz \mid \gz \in (\gl, \gl^{+}) \ra
[r \ge p]$ is an
$M[G_0][H_0][H_1][H_2]$-generic object over
$\add(j(\gl), j(\gl^{+}))$. To define
$\la r_\gz \mid \gz \in (\gl, \gl^{+}) \ra$, if
$\gz$ is a limit, we let
$r_\gz = \bigcup_{\gb \in (\gl, \gz)} r_\gb$.
By the facts
$\la r_\gb \mid \gb \in (\gl, \gz) \ra$
is (strictly) increasing and
$M[G_0][H_0][H_1][H_2]$ is
$\gl$-closed with respect to
$\ov V[G_0][G_1]$, this definition is valid.
Assuming now $r_\gz$ has been defined and
we wish to define $r_{\gz + 1}$, let
$\la {\cal B}_\gb \mid \gb < \eta \le \gl \ra$
be the subsequence of
$\la {\cal A}_\gb \mid \gb \le \gz + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gl), j(\gz + 1))$.
Since
$q_\gz, r_\gz \in \add(j(\gl), j(\gz))$,
$q_{\gz + 1} \in \add(j(\gl), j(\gz + 1))$, and
$j(\gz) < j(\gz + 1)$, the condition
$r_{\gz + 1}' = r_\gz \cup q_{\gz + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\gz + 1})$ won't be present in either
$\dom(q_\gz)$ or $\dom(r_\gz)$.
We can thus, using the fact
$M[G_0][H_0][H_1][H_2]$ is $\gl$-closed
%under $\gl^+$ sequences
with respect to
$\ov V[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb \mid \gb < \eta \ra$ such that
$s_0 \ge r_{\gz + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\gz + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $H_3$ is
$M[G_0][H_0][H_1][H_2]$-generic over
$\add(j(\gl), j(\gl^{+}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\gz \mid \gz \in (\gl, \gl^{+}) \ra
\exists \gb \in (\gl, \gl^{+})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\gz) \mid \gz < \gl^{+} \ra$ is
unbounded in $j(\gl^{+})$. To see this, if
$\gb < j(\gl^{+})$ is an ordinal, then for some
$f : \gl \to M$ representing $\gb$,
we can assume that for $\gr < \gl$,
$f(\gr) < \gl^{+}$. Thus, by the regularity of
$\gl^{+}$ in $\ov V$,
$\gb_0 = \bigcup_{\gr < \gl} f(\gr) <
\gl^{+}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\gz \mid \gz <
\gl^{+} \ra$, ${\cal A} = {\cal A}_\rho$,
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
$\gz \in (\gl, \gl^{+})$, $p = p \rest \gz$,
$H_3$ is such that if
$p \in G_1$, $j(p) \in H_3$.
Thus, working in $\ov V[G_0][G_1]$,
we have shown that $j$ lifts to
$j : \ov V[G_0][G_1] \to M[G_0][H_0][H_1][H_2][H_3]$,
i.e.,
$\ov V[G_0][G_1] \models ``\gl$ is measurable''.
The proof of Theorem \ref{t4} will be finished once we have shown
that $\ov V[G_0][G_1] \models ``$No cardinal in the interval $(\gd, \gl)$
is measurable''. To see that this is the case, write
$\FP^2 = \add(\gd, 1) \ast \dot \FR$. Since this definition shows that
$\FP^2$ admits a gap at $\gd$, by Theorem \ref{tgf}, any cardinal
in the interval $(\gd, \gl)$ which is measurable in $\ov V[G_0][G_1]$
had to have been measurable in $\ov V[G_0]$. However, since
$\ov V[G_0] \models ``\gl$ is the least measurable cardinal greater
than $\gd$'', $\ov V[G_0][G_1] \models ``\gl$ is the least measurable
cardinal greater than $\gd$'' as well.
This completes the proof of Theorem \ref{t4}.
\end{proof}
\begin{pf}
With the proof of Theorem \ref{t4} having been established,
we can now prove Theorem \ref{t1}.
We follow the proofs of \cite[Theorem 2]{A07}
and \cite[Theorem 1]{A10}.
Suppose that
$\gk$ is indestructibly supercompact and there
is a measurable cardinal $\gl > \gk$.
We show that for any good ordinal $\ga$,
$A_\ga = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and LP($\ga$) holds for $\gd\}$
is unbounded in $\gk$.
Let $\eta > \gk$ be the least measurable cardinal.
Force with $\FP(\gk, \eta, \ga)$.
After this forcing,
which is $\gk$-directed closed,
LP($\ga$) holds for $\eta$, and
$\eta$ remains the least measurable
cardinal above $\gk$.
In particular, after the forcing,
$\eta$ is a measurable cardinal which
is not a limit of measurable cardinals at which
LP($\ga$) holds.
Since $\gk$ is indestructibly supercompact,
by reflection,
$A_\ga = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and LP($\ga$) holds for $\gd\}$
is unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact $\FP(\gk, \eta, \ga)$
is $\gk$-directed closed that
$A_\ga$ is unbounded in $\gk$ in the ground model.
This completes the proof of Theorem \ref{t1}.
\end{pf}
Having finished the proof of Theorem \ref{t1}, we
turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gz > \gk$ is measurable''.
Let $\ga$ be a fixed but arbitrary good ordinal.
%Without loss of generality, by first
%doing a preliminary forcing if necessary,
%we assume in addition that $V \models {\rm GCH}$.
Take $\la \gd_j \mid j < \gk \ra$ to be the
continuous, increasing enumeration of
$\{\go\} \ \cup \ \{\gd < \gk \mid \gd$ is either
a measurable cardinal or a limit of measurable cardinals$\}$.
For any measurable cardinal $\gd$, $\gd = \gd_j$, let $\gth_\gd$
be the least cardinal $\gth \in (\gd_j, \gd_{j + 1})$ such that $\gd$
is not $\gth$ supercompact in $V$, or $\gd$ if $\gd$ is $\gd_{j + 1}$
supercompact in $V$.
%Let $f$ be a Laver function \cite{L} for $\gk$, i.e.,
%$f : \gk \to V_\gk$ is such that
%for every $x \in V$ and every
%$\gl \ge \card{{\rm TC}(x)}$, there is
%an elementary embedding
%$j : V \to M$ generated by a
%supercompact ultrafilter over
%$P_\gk(\gl)$ such that
%$j(f)(\gk) = x$.
%Suppressing the index $i$,
We define now
a length $\gk$ reverse Easton iteration
$\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd < \gk \ra$
%which does trivial forcing except in the following four cases, taking as an inductive
by four cases as follows, taking as an inductive
hypothesis that if $\gd = \gd_j$ is a measurable cardinal,
then $\forces_{\FP_\gd} ``\gd_{j + 1}$
is the least measurable cardinal greater
than $\gd$'':
%(so by the definition we are about to give
%and the proof of Theorem \ref{t4},
%$\forces_{\FP_{\gb + 1}} ``\gd_{\gb + 1}$
%is the least measurable cardinal greater than
%$\gd_\gb$''):
\begin{enumerate}
\item\label{i0a} $\FP_0 = \add(\go, 1) = \add(\gd_0, 1)$.
\item\label{i1a} If $\gd = \go$
or $\gd$ is in $V$ either a non-measurable limit of measurable cardinals
or a measurable cardinal which is not a limit of
measurable cardinals, let $j$ be such that $\gd = \gd_j$. Then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for the partial ordering
$\FP(\gd^{++}_j, \gd_{j + 1}, \ga)$ of Theorem \ref{t4}.
%defined using ordinals in the open interval $(\gd_\gb, \gd_{\gb + 1})$.
%so that $\forces_{\FP_\gb} ``\dot \FQ_\gb$ is (at least) $\gd_\gb$-directed closed''.
\item\label{i2a} If $\gd$ is in $V$ a measurable limit of measurable cardinals with
$\gd = \gd_j$, then $\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ \ast \dot
\FP(\dot \eta, \gd_{j + 1}, \ga) = \FP_\gd \ast \dot \FQ_\gd$. Here,
$\forces_{\FP_\gd} ``\dot \FQ$ is the lottery sum of all $\gd$-directed
closed partial orderings having rank less than $\gd_{j + 1}$'', and
$\forces_{\FP_\gd \ast \dot \FQ} ``\dot \eta$ is the least inaccessible
cardinal greater than $\max(\gth_\gd, \card{{\rm TC}(\dot \FR)})$, where
$\dot \FR$ is the partial ordering selected in the stage $\gd$ lottery''.
\item\label{i3a}
%If $\gd_\gb$ is a measurable limit of measurable cardinals and Case \ref{i2a} does not hold,
If neither Cases \ref{i0a} -- \ref{i2a} holds, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for trivial forcing $\{\emptyset\}$.
\end{enumerate}
By induction, it follows that
%using the L\'evy-Solovay results \cite{LS}
for any $j < \gk$, $\FP_{\gd_j}$ is forcing equivalent to a
partial ordering having size at most $2^{\gd_j} < \gd_{j + 1}$.
From this, the L\'evy-Solovay results \cite{LS} show
that the inductive hypothesis holds and $\FP$ is well-defined.
%, i.e., that
%$V^{\FP_\gb \ast \dot \FQ_\gb} = V^{\FP_{\gb + 1}} \models
%``\gd_{\gb + 1}$ is measurable''.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is indestructibly supercompact''.
\end{lemma}
\begin{proof}
We slightly modify the proofs of \cite[Lemma 2.1]{A07},
\cite[Lemma 2.1]{A10}, \cite[Lemma 3.1]{A12},
and \cite[Lemma 2.6]{A15},
quoting verbatim when appropriate.
Let $\FQ \in V^{\FP}$ be such that
$V^{\FP} \models ``\FQ$ is
$\gk$-directed closed''.
Take $\dot \FQ$ as a term for
$\FQ$ such that
$\forces_{{\FP}} ``\dot \FQ$ is
$\gk$-directed closed''.
Suppose $\gl \ge \card{{\rm TC}(\dot \FQ)}$
is an arbitrary cardinal, and let
$\gg = 2^{\card{[\gl]^{< \gk}}}$. Take
$j : V \to M$ as an elementary
embedding witnessing the $\gg$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gg)$ such that $M \models ``\gk$ is not $\gg$ supercompact''. Since
%$j(f)(\gk) = \dot \FQ$. Since
$V \models ``$No cardinal $\gz > \gk$ is measurable''
%, $\gg \ge 2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
the definition of ${\FP}$ implies that in $M$, above the appropriate condition,
$j({\FP} \ast \dot \FQ)$ is forcing equivalent to ${\FP} \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, where
the first stage at which
$\dot \FR$ is forced to do nontrivial forcing
is well above $\gg$.
%$2^{[\gg]^{< \gk}}$.
Laver's original argument from \cite{L} now applies
and shows that
$V^{{\FP} \ast \dot \FQ} \models
``\gk$ is $\gl$ supercompact''.
(Simply let $G_0 \ast G_1
\ast G_2$ be $V$-generic over
${\FP} \ast \dot \FQ \ast \dot \FR$,
lift $j$ in $V[G_0][G_1][G_2]$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$,
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 $\gg^+$-directed closure of
$\FR \ast j(\dot \FQ)$ that the supercompactness
measure over ${(P_\gk(\gl))}^{V[G_0][G_1]}$
generated by $j$ is actually a member of
$V[G_0][G_1]$.)
As $\gl$ and $\FQ$ were arbitrary,
this completes the proof of
Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
%Suppose $\ga$ is a good ordinal.
$V^\FP \models ``$If $\gd < \gk$ is a measurable cardinal which
is not a limit of measurable cardinals, then LP($\ga$) holds''.
\end{lemma}
\begin{proof}
Let $\gg < \gk$ be such that $V \models ``\gd = \gd_\gg$ is
a measurable cardinal which is not a limit of measurable cardinals''.
By the definition of the $\gd_j$'s, it must be the case that
$\gg$ is a successor ordinal.
Let $\gg = \gb + 1$, $\gs = \gd_\gb$, and
write $\FP = \FP_{\gs + 1} \ast \dot \FP^{\gs + 1}$. By the definition
of $\FP$ and Theorem \ref{tgf}, it must also be true that
$V^{\FP_{\gs + 1}} \models ``\gd_{\gb + 1}$ is the least measurable
cardinal greater than $\gd_\gb$ + LP($\ga$) holds for $\gd_{\gb + 1}$''.\footnote{If
we are in Case \ref{i1a} of the definition of $\FP$ at stage $\gs + 1$,
then this follows by the results of \cite{LS}, since $\FP_{\gd_\gb}$ is
forcing equivalent to a partial ordering having size at most
$2^{\gd_\gb} < \gd_{\gb + 1}$.
%this is straightforward.
If we are in Case \ref{i2a} of the definition of $\FP$ at stage $\gs + 1$, with
$\FP_{\gs + 1} = \FP_\gs \ast \dot \FQ \ast \dot \FP(\dot \eta, \gd_{\gb + 1}, \ga)$,
then because forcing with $\FQ$ is forcing equivalent to forcing with a partial
ordering having size less than $\gd_{\gb + 1}$, an application of the results of
\cite{LS} shows that
$\forces_{\FP_\gs \ast \dot \FQ} ``\gd_{\gb + 1}$ is measurable''.
Because $\FP_\gs \ast \dot \FQ = \FP_0 \ast \dot \FR$, where
$\FP_0 = \add(\go, 1)$ and $\forces_{\FP_0} ``\dot \FR$ is
$\ha_2$-directed closed'', $\FP_\gs \ast \dot \FQ$
admits a gap at $\go$. Therefore, by Theorem \ref{tgf}, any cardinal measurable in
$V^{\FP_\gs \ast \dot \FQ}$ had to have been measurable in $V$.
This means that $\forces_{\FP_\gs \ast \dot \FQ} ``\gd_{\gb + 1}$ is the
least measurable cardinal greater than $\gd_\gb$''.
This fact is then preserved
after forcing with $\FP(\eta, \gd_{\gb + 1}, \ga$).}
Since $\forces_{\FP_{\gs + 1}} ``\dot \FP^{\gs + 1}$ is
$\gd^{++}_{\gb + 1}$-directed closed'', $V^{\FP_{\gs + 1} \ast \dot \FP^{\gs + 1}}
= V^{\FP} \models ``\gd_{\gb + 1}$ is the least measurable
cardinal greater than $\gd_\gb$ + LP($\ga$) holds for $\gd_{\gb + 1}$'' as well.
The proof of Lemma \ref{l2} will therefore be complete once we have shown that
in $V^\FP$, any measurable cardinal $\gd < \gk$ which is not a limit of
measurable cardinals is such that $\gd = \gd_{\gb + 1}$ for some $\gb < \gk$.
To see this, assume to the contrary that $\gd \neq \gd_{\gb + 1}$ for any $\gb < \gk$.
Write $\FP = \FP_0 \ast \dot \FQ$, where $\FP_0 = \add(\go, 1)$ and
%is nontrivial, $\card{\FP_0} < \ha_1$, and
$\forces_{\FP_0} ``\dot \FQ$ is $\ha_2$-directed closed''.
Since $\FP$ admits a gap at $\go$,
by Theorem \ref{tgf}, any cardinal measurable in $V^\FP$ had to have
been measurable in $V$.
This means that $\gd = \gd_\gl$ for some limit ordinal $\gl < \gk$, i.e., in $V$,
$\gd$ is a measurable limit of measurable cardinals. In particular, in $V$,
$\gd$ is a limit of measurable cardinals which are not themselves limits
of measurable cardinals. It consequently follows that
in $V$, $\gd$ is a limit of measurable cardinals
which have the form $\gd_{\gb + 1}$ for some $\gb < \gk$.
However, the arguments of the preceding paragraph show that any such
measurable cardinal remains measurable in $V^\FP$. From this, we
immediately infer that in $V^\FP$, $\gd$ is a measurable limit of
measurable cardinals. This contradiction completes the proof of Lemma \ref{l2}.
\end{proof}
Since $\FP$ may be defined so that $\card{\FP} = \gk$,
by the results of \cite{LS}, $V^\FP \models ``$No cardinal $\gz > \gk$ is measurable''.
This fact, together with
Lemmas \ref{l1} and \ref{l2}, complete the proof of Theorem \ref{t2}.
\end{proof}
Having finished the proof of Theorem \ref{t2}, we turn now to
the proof of Theorem \ref{t3}.
\begin{proof}
Suppose $V \models ``$ZFC + $\gk$ is indestructibly supercompact +
No cardinal $\gz > \gk$ is measurable''.
Without loss of generality, by first forcing GCH if necessary and then
forcing with the (possibly proper class) reverse Easton iteration which
is trivial except at inaccessible stages $\gd$, where the partial ordering
used is $\add(\gd, \gd^{++})$, we may assume in addition that
$V \models ``2^\gd = \gd^{++}$ for every inaccessible cardinal $\gd$''.
As in the proof of Theorem \ref{t2},
let $\la \gd_j \mid j < \gk \ra$ be the
continuous, increasing enumeration of
$\{\go\} \ \cup \ \{\gd < \gk \mid \gd$ is either
a measurable cardinal or a limit of measurable cardinals$\}$.
%Let $f$ as before be a Laver function for $\gk$.
%, i.e., $f : \gk \to V_\gk$ is such that
%for every $x \in V$ and every
%$\gl \ge \card{{\rm TC}(x)}$, there is
%an elementary embedding
%$j : V \to M$ generated by a
%supercompact ultrafilter over
%$P_\gk(\gl)$ such that $j(f)(\gk) = x$.
We define now
a length $\gk$ reverse Easton iteration
$\FP = \la \la \FP_\gd, \dot \FQ_\gd \ra \mid \gd < \gk \ra$
by three cases as follows:
%, again taking as an inductive
%hypothesis that if $\gd = \gd_j$ is a measurable cardinal, then
%$\forces_{\FP_\gd} ``\gd_{j + 1}$
%is the least measurable cardinal greater than $\gd$'':
%(so by the definition we are about to give
%and the proof of Theorem \ref{t4},
%$\forces_{\FP_{\gb + 1}} ``\gd_{\gb + 1}$
%is the least measurable cardinal greater than
%$\gd_\gb$''):
\begin{enumerate}
\item\label{j0a} $\FP_0 = \add(\go, 1) = \add(\gd_0, 1)$.
%\item\label{j1a} If $\gd$
%is either a non-measurable limit of measurable cardinals
%or is a measurable cardinal which is not a limit of
%measurable cardinals, let $j$ be such that $\gd = \gd_j$. Then
%$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
%$\dot \FQ_\gd$ is a term for the partial ordering
%$\FP(\gd^{++}_j, \gd_{j + 1}, \ga)$ of Theorem \ref{t4}.
%defined using ordinals in the open interval $(\gd_\gb, \gd_{\gb + 1})$.
%so that $\forces_{\FP_\gb} ``\dot \FQ_\gb$ is (at least) $\gd_\gb$-directed closed''.
\item\label{j1a} If $\gd$ is in $V$ a measurable limit of
measurable cardinals with $\gd = \gd_j$, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\forces_{\FP_\gd} ``\dot \FQ_\gd$ is
the lottery sum of all $\gd$-directed closed partial orderings
having rank less than $\gd_{j + 1}$''.
\item\label{j2a}
%If $\gd_\gb$ is a measurable limit of measurable cardinals and Case \ref{i2a} does not hold,
If neither Cases \ref{j0a} nor \ref{j1a} holds, then
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$, where
$\dot \FQ_\gd$ is a term for trivial forcing $\{\emptyset\}$.
\end{enumerate}
%As in the proof of Theorem \ref{t2}, an easy induction shows that
%for any $j < \gk$, $\card{\FP_{\gd_j}} < \gd_{j + 1}$. Then, again
%as in the proof of Theorem \ref{t2}, this fact in tandem
%with the arguments of \cite{LS} yield that $\FP$ is well-defined.
The same reasoning as given for Theorem \ref{t2}
allows us to infer that
$V^\FP \models ``\gk$ is indestructibly supercompact +
No cardinal $\gz > \gk$ is measurable''.
The proof of Theorem \ref{t3} will therefore be completed by
the following lemma.
\begin{lemma}\label{l3}
$V^\FP \models ``$If $\gd < \gk$ is a measurable cardinal
which is not a limit of measurable cardinals, then $2^\gd = \gd^{++}$''.
\end{lemma}
\begin{proof}
We argue in analogy to the proof of Lemma \ref{l2}.
Let $\gg < \gk$ be such that $V \models ``\gd = \gd_\gg$ is
a measurable cardinal which is not a limit of measurable cardinals''.
As before, by the definition of the $\gd_j$'s, it must be the case that
$\gg$ is a successor ordinal.
Let $\gg = \gb + 1$, $\gs = \gd_\gb$, and
write $\FP = \FP_{\gs + 1} \ast \dot \FP^{\gs + 1}$.
By the definition of $\FP$, it inductively follows that
$\FP_{\gs + 1}$ is
forcing equivalent to a partial ordering having size less than $\gd_{\gb + 1}$. Since
$\forces_{\FP_{\gs + 1}} ``\dot \FP^{\gs + 1}$ is (at least) $\gd^{+ 3}$-directed closed'',
in both $V^{\FP_{\gs + 1}}$ and $V^{\FP_{\gs + 1} \ast \dot \FP^{\gs + 1}} = V^\FP$,
$\gd$ is a measurable cardinal which is not a limit of measurable cardinals and
$2^\gd = \gd^{++}$.
The same proof as given in Lemma \ref{l2} now shows that if
$V^\FP \models ``\gd$ is a measurable cardinal which is not a limit of measurable
cardinals'', then $\gd = \gd_{\gr + 1}$ for some $\gr < \gk$.
This completes the proof of both Lemma \ref{l3} and Theorem \ref{t3}.
\end{proof}
\end{proof}
In conclusion to this paper, we note that Theorems \ref{t1} -- \ref{t3}
remain valid if the definition of good ordinal is changed to allow
$\gd^{+ \ga}$ to be a regular cardinal above the least inacessible
cardinal greater than $\gd$.
The definition used in this paper was chosen as
a matter of convenience and ease of presentation.
In addition, we observe that
results analogous to Theorems \ref{t1} -- \ref{t3}
hold if $\gk$ is either an indestructible
strong cardinal in Gitik and Shelah's sense of \cite{GS}
or an indestructible strongly unfoldable cardinal
in Johnstone's sense of \cite{Joth, Jo}.
(See \cite{Joth, Jo} for the definition
of strongly unfoldable cardinal.)
Readers may
%We leave it to readers to
work out the details for themselves.
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\end{document}
\begin{abstract}
Say that a measurable cardinal $\gk$ satisfies the
{\em Levinski property LP} if $2^\gk = \gk^+$, yet
GCH fails on some final segment of measurable cardinals
below $\gk$. Suppose throughout $\ga$ is some %suitably
definable ordinal (including 0)
such that for any $\gd$, $\gd^{+ \ga}$ is
below the least measurable cardinal above $\gd$.
Refine the preceding by saying that
a measurable cardinal $\gk$ satisfies the
{\em Levinski property LP($\ga$)} if for every
measurable cardinal $\gd$ in some final segment
below $\gk$, $2^\gd = \gd^{+ \ga}$, yet $2^\gk = \gk^+$.
Suppose that $\gk$ is indestructibly supercompact
and there is a measurable cardinal $\gl > \gk$.
It then follows that for any $\ga$, %as just described,
$A_\ga = \{\gd < \gk \mid \gd$ is
measurable, $\gd$ is not a limit of
measurable cardinals, and LP($\ga$) holds for $\gd\}$
is unbounded in $\gk$.
The large cardinal
hypothesis on $\gl$ is necessary,
as we further demonstrate by
constructing via forcing models
containing an indestructibly supercompact cardinal $\gk$
with no measurable cardinal above it in which for fixed
but arbitrary $\ga$, every measurable cardinal
$\gd < \gk$ which is not a limit of measurable cardinals
satisfies LP($\ga$).
We also show the necessity of the large cardinal
hypothesis on $\gl$ by constructing via forcing
models containing an indestructibly supercompact
cardinal $\gk$ with no measurable cardinals above it such that
for every measurable cardinal $\gd < \gk$ which is not a
limit of measurable cardinals, $2^\gd > \gd^+$.
\end{abstract}
In the proof of Theorem \ref{t1},
we will use the partial ordering
$\FP(\gk, \gl)$, the standard notion of
forcing for adding a nonreflecting stationary
set of ordinals of cofinality $\gk$ to $\gl$.
For further details on the definition of this
partial ordering, we refer readers to either
\cite{AC1} or \cite{A01}.
We note only that $\FP(\gk, \gl)$ is
$\gk$-directed closed and ${\prec} \gl$-strategically closed.
In the proof of Theorem \ref{t3},
we will refer to our
partial ordering $\FP$ as being a
{\it Gitik style iteration of
Prikry-like forcings.}
By this we will mean an Easton
support iteration
as first given by Gitik in \cite{G}.
%(and elaborated upon further in
%\cite{G2}),
The ordering,
roughly speaking, is the
usual one associated with
reverse Easton iterations, except that
when extending Prikry conditions,
we take larger stems only finitely often.
For a more precise definition, we urge
readers to consult either \cite{G}
%\cite{G2},
or \cite{AG}.
%By Lemmas 1.2 and 1.3 of
%\cite{G} and \cite{G2}
%respectively and Lemma 1.4 of \cite{G},
%if $\gd_0$ is the first stage in the
%definition of $\FP$ at which a non-trivial
%forcing is done, then forcing with
%$\FP$ adds no bounded subsets to $\gd_0$,
%assuming the forcing done at
%any non-trivial stage $\gd$
%is $\gd$-directed closed forcing
%followed by either Prikry forcing over $\gd$
%or trivial forcing.
As in the proof of Theorem \ref{t2},
let $\la \gd_j \mid j < \gk \ra$ be the
continuous, increasing enumeration of
$\{\go\} \ \cup \ \{\gd < \gk \mid \gd$ is either
a measurable cardinal or a limit of measurable cardinals$\}$.
Let $f$ once again be a Laver function for $\gk$.
%, i.e., $f : \gk \to V_\gk$ is such that
%for every $x \in V$ and every
%$\gl \ge \card{{\rm TC}(x)}$, there is
%an elementary embedding
%$j : V \to M$ generated by a
%supercompact ultrafilter over
%$P_\gk(\gl)$ such that $j(f)(\gk) = x$.
We define now
a length $\gk$ reverse Easton iteration
$\FP = \la \la \FP_\gb, \dot \FQ_\gb \ra \mid \gb < \gk \ra$
by four cases as follows, taking as an inductive
hypothesis that $\forces_{\FP_\gb} ``\gd_{\gb + 1}$
is the least measurable cardinal greater
than $\gd_\gb$'':
%(so by the definition we are about to give
%and the proof of Theorem \ref{t4},
%$\forces_{\FP_{\gb + 1}} ``\gd_{\gb + 1}$
%is the least measurable cardinal greater than
%$\gd_\gb$''):
\begin{enumerate}
\item\label{j0a} $\FP_0 = \add(\go, 1) = \add(\gd_0, 1)$.
\item\label{j1a} If $\gd_\gb$
is either a non-measurable limit of measurable cardinals
or is not a limit of
measurable cardinals, then
$\FP_{\gb + 1} = \FP_\gb \ast \dot \FQ_\gb$, where
$\dot \FQ_\gb$ is a term for the partial ordering
$\FP(\gd^{++}_\gb, \gd_{\gb + 1}, \ga)$ of Theorem \ref{t4}.
%defined using ordinals in the open interval $(\gd_\gb, \gd_{\gb + 1})$.
%so that $\forces_{\FP_\gb} ``\dot \FQ_\gb$ is (at least) $\gd_\gb$-directed closed''.
\item\label{j2a} If $\gd_\gb$ is a measurable limit of
measurable cardinals and
$f(\gd_\gb) = \dot \FQ$ where
$\forces_{\FP_\gb} ``\dot \FQ$ is $\gd_\gb$-directed
closed and has cardinality less than $\gd_{\gb + 1}$'',
then $\dot \FQ_\gb = \dot \FQ$ and
$\FP_{\gb + 1} = \FP_\gb \ast \dot \FQ_\gb$.
\item\label{j3a} If $\gd_\gb$ is a measurable limit of
measurable cardinals and Case \ref{i2a} does not hold, then
$\FP_{\gb + 1} = \FP_\gb \ast \dot \FQ_\gb$, where
$\dot \FQ_\gb$ is a term for trivial forcing $\{\emptyset\}$.
\end{enumerate}
\end{proof}
\item\label{i2a} If $\gd$ is a measurable limit of
measurable cardinals with $\gd = \gd_j$ and
$f(\gd) = \dot \FQ$ where
$\forces_{\FP_\gd} ``\dot \FQ$ is $\gd$-directed
closed and has cardinality less than $\gd_{j + 1}$'',
then $\dot \FQ_\gd = \dot \FQ$ and
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$.