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\title{Indestructibility, HOD, and the Ground Axiom
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strong cardinal,
%strongly compact cardinal, strong cardinal, strongly unfoldable cardinal,
indestructibility,
%non-reflecting stationary set of ordinals,
HOD, the Ground Axiom.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
the referee, for helpful comments and
suggestions which have been incorporated
into the current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{January 18, 2010\\
(revised July 6, 2010)}
\begin{document}
\maketitle
\begin{abstract}
%Assume $\gl > \gk$ is inaccessible.
Let $\varphi_1$ stand for the statement $V = {\rm HOD}$
and $\varphi_2$ stand for the Ground Axiom.
Suppose $T_i$ for $i = 1, \ldots, 4$ are the theories
``ZFC + $\varphi_1$ + $\varphi_2$'',
``ZFC + $\neg \varphi_1$ + $\varphi_2$'',
``ZFC + $\varphi_1$ + $\neg \varphi_2$'', and
``ZFC + $\neg \varphi_1$ + $\neg \varphi_2$'' respectively.
We show that if $\gk$ is indestructibly supercompact
and $\gl > \gk$ is inaccessible,
then for $i = 1, \ldots, 4$,
$A_i =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
cardinal which is not a limit of inaccessible
cardinals and $V_\gd \models T_i\}$
must be unbounded in $\gk$.
The large cardinal
hypothesis on $\gl$ is necessary,
as we further demonstrate by
constructing via forcing four models in which
$A_i = \emptyset$ for $i = 1, \ldots, 4$.
In each of these models, there is an
indestructibly supercompact cardinal $\gk$,
and no cardinal $\gd > \gk$ is inaccessible.
We show it is also the case that if $\gk$ is
indestructibly supercompact, then $V_\gk \models T_1$,
so by reflection,
$B_1 =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
limit of inaccessible cardinals and $V_\gd
\models T_1\}$ is unbounded in $\gk$. Consequently,
it is not possible to
construct a model in which $\gk$ is indestructibly
supercompact and $B_1 = \emptyset$.
%remove the requirement that
%$A_i$ be composed of inaccessible cardinals which
%are not limits of inaccessible cardinals.
On the other hand, assuming $\gk$ is
supercompact and no cardinal $\gd > \gk$ is inaccessible,
we demonstrate that it is possible
to construct a model in which $\gk$ is indestructibly
supercompact and for every inaccessible cardinal
$\gd < \gk$, $V_\gd \models T_1$.
It is thus not possible to prove in ZFC that
$B_i =_{\rm df} \{\gd < \gk \mid \gd$ is an
inaccessible limit of inaccessible cardinals and
$V_\gd \models T_i\}$ for $i = 2, \ldots, 4$ is
unbounded in $\gk$ if $\gk$ is indestructibly supercompact.
%Analogous results hold if $\gk$ is indestructibly
%strong or indestructibly strongly unfoldable.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s1}
We start 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 $p$}.
If $\gk$ is a cardinal, the partial
ordering $\FP$ is {\em $\gk$-directed
closed} if for every directed set $D$ of
conditions of size less than $\gk$,
there is a condition in $\FP$
extending each member of $D$.
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 $\FP$.
%For $\gd$ any ordinal, $\gd'$ is
%the least cardinal $\gg > \gd$ such that
%$V \models ``\gg$ is measurable''.
%For $\ga < \gb$ ordinals, $[\ga, \gb]$,
%$(\ga, \gb]$, $[\ga, \gb)$, and
%$(\ga, \gb)$ are as in standard interval notation.
We continue with some key definitions.
As in \cite{L}, the cardinal $\gk$
is {\em indestructibly supercompact}
if $\gk$'s supercompactness is preserved after forcing
with a $\gk$-directed closed partial ordering.
The {\em Ground Axiom (GA)} is the assertion
that the universe of sets $V$ is not a generic
extension of any inner model $W \subseteq V$ via
some nontrivial (set) partial ordering $\FP \in W$.
GA was formulated by Hamkins and Reitz
and studied by Reitz \cite{Rth, R}
and Hamkins, Reitz, and Woodin \cite{HRW}.
Although GA is {\em prima facie} a second order
statement, as Reitz has shown in \cite{Rth, R},
it is actually first-order expressible.
In addition, as was shown in \cite{Fr, ApFr},
if Paul Corazza's {\em Wholeness Axiom (WA)}
(first introduced in \cite{Co}) is consistent,
then it is consistent with GA.
Since Corazza showed in \cite{Co} that WA is consistent
relative to the existence of an ${\rm I}_3$ cardinal
and also showed in \cite{Co} that WA implies the existence of
a cardinal $\gk$ which is super-$n$-huge for every
$n \in \go$, we know that GA is relatively consistent
with some fairly large cardinals.
It is a very interesting fact that the large cardinal
structure of the universe above either a
supercompact or strong cardinal $\gk$
with suitable indestructibility properties
can affect the large cardinal structure below
$\gk$. %, assuming the universe is sufficiently rich.
On the other hand, these effects can be mitigated
if the universe contains relatively few large cardinals.
These sorts of occurrences have been
studied in \cite{AH4, A07, A08, A09, A10}.
%\cite{A07}, \cite{A08}, and \cite{A09}.
The purpose of this paper is to continue
investigating this phenomenon, but in the context of
models of ZFC in which $V = {\rm HOD}$ can be either
true or false and GA can be either true or false.
Specifically, we prove four theorems,
taking as our notation throughout that
$\varphi_1$ stands for the statement $V = {\rm HOD}$,
$\varphi_2$ stands for the Ground Axiom,
$T_i$ for $i = 1, \ldots, 4$ are the theories
``ZFC + $\varphi_1$ + $\varphi_2$'',
``ZFC + $\neg \varphi_1$ + $\varphi_2$'',
``ZFC + $\varphi_1$ + $\neg \varphi_2$'', and
``ZFC + $\neg \varphi_1$ + $\neg \varphi_2$'' respectively,
and for $i = 1, \ldots, 4$,
$A_i =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
cardinal which is not a limit of inaccessible
cardinals and $V_\gd \models T_i\}$.
We begin with the following theorem.
\begin{theorem}\label{t1}
Suppose $\gl > \gk$ is inaccessible
and $\gk$ is indestructibly supercompact.
Then for each $i = 1, \ldots, 4$,
$A_i$ is unbounded in $\gk$.
\end{theorem}
\begin{proof}
Let $V$ be our ground model.
Suppose $\gd$ is any cardinal and $\gr$ is
the least inaccessible cardinal greater than $\gd$.
We describe four $\gd$-directed closed
partial orderings $\FP^*_i$ for $i = 1, \ldots, 4$
such that \break
$V^{\FP^*_i} \models ``V_\gr \models T_i$''.
These partial orderings are as follows:
\begin{enumerate}
\item\label{i1} $\FP^*_1$ is the partial ordering of
\cite[Theorem 11]{R} (see also \cite{Rth, HRW})
as defined in
$V_\gr$ using a coding based on regular cardinals
in the open interval $(\gd, \gr)$
such that $V^{\FP^*_1}_\gr \models T_1$.
For the exact definition of $\FP^*_1$, we refer
readers to \cite{R}.
%We note that $\FP^*_1$ may be intuitively described
%as the partial ordering which first forces GCH
%to hold via a reverse Easton iteration at all cardinals
%in the open interval $(\gd, \gr)$. $\FP^*_1$ then codes
%every $x \subseteq \gg$ for $\gg \in (\gd, \gr)$ into
%the continuum function based on regular cardinals in
%$(\gd, \gr)$ using a reverse Easton iteration
%of an Easton product for controlling the size
%of various continuua.
We do note, however, that
the work of \cite{Rth, R} shows that this coding
may be done in a way such that
$\FP^*_1$ is $\gd$-directed closed and forcing
with $\FP^*_1$ preserves the inaccessibility of
$\gr$ (which of course means that forcing with
$\FP^*_1$ preserves the fact that $\gr$ is the
least inaccessible cardinal greater than $\gd$).
The work of \cite{Rth, R} additionally shows that
this coding may be done so that
$V^{\FP^*_1}_\gr$ is a model for the
{\em Continuum Coding
Axion (CCA)} of \cite{Rth, R, HRW}, which says that
for every ordinal $\ga$ and every $x \subseteq \ga$,
there is some ordinal $\gth$ such that
$\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gth + \gb + 1}} = \ha_{\gth + \gb + 2}$.\footnote{As
pointed out by the referee, and as mentioned in
\cite{Rth, R}, the CCA implies a strengthening
of itself in which there are unboundedly many
$\gth$ which can be used to code the set of
ordinals $x$. It is this stronger version that
is used to infer GA in \cite{Rth, R}.
Also, CCA clearly implies $V = {\rm HOD}$.}
\item\label{i2} $\FP^*_2$ is the partial ordering of
\cite[Theorem 2]{HRW} as defined in
$V_\gr$ using a coding based on regular cardinals
in the open interval $(\gd, \gr)$
such that $V^{\FP^*_2}_\gr \models T_2$.
For the exact definition of $\FP^*_2$, we refer
readers to \cite{HRW}.
%We note that $\FP^*_2 = \FP^*_1 \ast \dot \FQ$, where
%$\FP^*_1$ is as in (\ref{i1}) above, and
%$\dot \FQ$ is a term for the reverse Easton iteration
%which adds a Cohen subset of each regular cardinal
%$\gg \in (\gd, \gr)$ such that $2^{< \gg} = \gg$.
The work of \cite{HRW} in conjunction with
the work of \cite{Rth, R} once again
show that this may be done in a way such that
$\FP^*_2$ is $\gd$-directed closed and forcing
with $\FP^*_2$ preserves the inaccessibility of
$\gr$.
\item\label{i3} $\FP^*_3$ is the partial ordering of
\cite[Theorem 18]{R} (see also \cite{Rth}) as defined in
$V_\gr$ using a coding based on regular cardinals
in the open interval $(\gd, \gr)$
such that $V^{\FP^*_3}_\gr \models T_3$.
For the exact definition of $\FP^*_3$, we once more refer
readers to \cite{R}.
%We note that $\FP^*_3 = \FP^*_1 \ast \dot \FQ$, where
%$\FP^*_1$ is as in (\ref{i1}) above, and
%$\dot \FQ$ is a term for the partial ordering
%which adds a Cohen subset $x$ of some regular cardinal
%$\gg \in (\gd, \gr)$ and then codes $x$ via the
%continuum function based on regular cardinals
%in $(\gd, \gr)$ greater than $\gg$.
The work of \cite{R} again
shows that this may be done in a way such that
$\FP^*_3$ is $\gd$-directed closed and forcing
with $\FP^*_3$ preserves the inaccessibility of
$\gr$.
\item\label{i4} Fix an arbitrary regular cardinal
$\gg \in (\gd, \gr)$. $\FP^*_4$
is then the partial ordering adding
a Cohen subset of $\gg$.
Clearly, $\FP^*_4$ is $\gd$-directed closed.
By the defintion of GA,
$V^{\FP^*_4}_\gr \models \neg {\rm GA}$, and since
$\FP^*_4$ is {\em almost homogeneous} (i.e., for any
$p, q \in \FP^*_4$, there is an automorphism
$\pi$ of $\FP^*_4$ such that $\pi(p)$ is compatible
with $q$), as in \cite[Theorem 1]{HRW}
(see also \cite[pages 244--245]{Ku}),
$V^{\FP^*_4}_\gr \models V \neq {\rm HOD}$.
Thus, since forcing with $\FP^*_4$ preserves
$\gr$'s inaccessibility, $V^{\FP^*_4}_\gr \models T_4$.
\end{enumerate}
Having completed our description of the $\FP^*_i$,
we now follow the proof of \cite[Theorem 2]{A07}.
Suppose $\gl > \gk$ is inaccessible and
$\gk$ is indestructibly supercompact.
Without loss of generality, assume that $\gl$
is the least inaccessible cardinal above $\gk$.
Let $i$ for $i = 1, \ldots, 4$ be fixed but arbitrary.
Force with one of the partial orderings $\FP^*_i$
as defined over the open interval $(\gk, \gl)$.
After this forcing, which is
$\gk$-directed closed,
$\gl$ remains the least inaccessible
cardinal above $\gk$.
In particular, after the forcing,
$\gl$ is an inaccessible cardinal which
is not a limit of inaccessible cardinals.
Further, by the definition of $\FP^*_i$,
$V^{\FP^*_i} \models ``V_\gl \models T_i$''.
Since $\gk$ is suitably indestructible,
by reflection,
$A_i =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
cardinal which is not a limit of inaccessible cardinals
and $V_\gd \models T_i\}$ is
unbounded in $\gk$ after the
forcing has been performed.
Once more, we infer by the
fact $\FP^*_i$ is $\gk$-directed closed that
$A_i$ is unbounded in $\gk$ in the ground model.
\end{proof}
That the assumption of an inaccessible cardinal
$\gl$ above the supercompact cardinal $\gk$
is necessary is shown by our next theorem.
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
Then for each $i = 1, \ldots, 4$,
there is a partial ordering $\FP \in V$
such that
$V^{\FP} \models ``$ZFC + $\gk$ is indestructibly supercompact +
No cardinal $\gd > \gk$ is inaccessible + $A_i = \emptyset$''.
%In $V^{\FP_i}$, $\gk$ is indestructibly supercompact, and
%$A_i = \emptyset$.
\end{theorem}
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
Without loss of generality, by first
doing a preliminary forcing if necessary,
we assume in addition that $V \models {\rm GCH}$.
Assume $i$ for $i = 1, \ldots, 4$ is given but arbitrary.
Fix $k \neq i$, $k = 1, \ldots, 4$.
Let $\la \gd_j \mid j < \gk \ra$ be the
continuous, increasing enumeration of
$\{\go\} \ \cup \ \{\gd < \gk \mid \gd$ is either
an inaccessible cardinal or a limit of inaccessible cardinals$\}$.
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
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
by four cases as follows, taking as an inductive
hypothesis that $\forces_{\FP_\ga} ``\gd_{\ga + 1}$
is inaccessible'' (so $\forces_{\FP_\ga} ``\gd_{\ga + 1}$
is the least inaccessible cardinal greater than
$\gd_\ga$''):
\begin{enumerate}
\item\label{i0a} $\FP_0 = \{\emptyset\}$.
\item\label{i1a} If $\gd_\ga$ is not an inaccessible limit of
inaccessible cardinals, then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
$\FP^*_k$ of Theorem \ref{t1} defined using
ordinals in the open interval $(\gd_\ga, \gd_{\ga + 1})$,
so that $\forces_{\FP_\ga} ``\dot \FQ_\ga$ is
(at least) $\gd_\ga$-directed closed''.
\item\label{i2a} If $\gd_\ga$ is an inaccessible limit of
inaccessible cardinals and
$f(\gd_\ga) = \la \dot \FQ, \gd \ra$ where
$\gd \in (\gd_\ga, \gd_{\ga + 1})$
and $\forces_{\FP_\ga} ``\dot \FQ$ is $\gd_\ga$-directed
closed and has cardinality less than $\gd_{\ga + 1}$'',
let $\gg'$ be the least (singular) strong limit
cardinal greater than
$\max(\card{{\rm TC}(\dot \FQ)}, \gd)$. Then
%$\card{{\rm TC}(\dot \FQ)}$. Then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ \ast \dot \FQ'
= \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ'$ is a term for the partial ordering
$\FP^*_k$ of Theorem \ref{t1} defined using
ordinals in the open interval $(\gg', \gd_{\ga + 1})$,
so that
$\forces_{\FP_\ga \ast \dot \FQ} ``\dot \FQ'$
is (at least) $\gg'$-directed closed''.
\item\label{i3a} If $\gd_\ga$ is an inaccessible limit of
inaccessible cardinals and Case \ref{i2a} does not hold, then
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
$\FP^*_k$ of Theorem \ref{t1} defined using
ordinals in the open interval $(\gd_\ga, \gd_{\ga + 1})$,
so that $\forces_{\FP_\ga} ``\dot \FQ_\ga$ is
(at least) $\gd_\ga$-directed closed''.
\end{enumerate}
\noindent An easy induction shows that for any
$\ga < \gk$, $\card{\FP_\ga} < \gd_{\ga + 1}$.
From this, it follows that the inductive hypothesis
holds and $\FP$ is well-defined, i.e., that
$V^{\FP_\ga \ast \dot \FQ_\ga} = V^{\FP_{\ga + 1}} \models
``\gd_{\ga + 1}$ is inaccessible and $V_{\gd_{\ga + 1}}
\models T_k$''.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is indestructibly supercompact''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 2.1]{A07}.
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
$j(f)(\gk) = \la \dot \FQ, \gg \ra$. Since
$V \models ``$No cardinal $\gd > \gk$ is inaccessible''
%, $\gg \ge 2^{{[\gk^+]}^{<\gk}}$,
and $M^\gg \subseteq M$,
the definition of ${\FP}$ implies that
$j({\FP} \ast \dot \FQ) = {\FP} \ast
\dot \FQ \ast \dot \FR \ast
j(\dot \FQ)$, 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}
In $V^\FP$, $A_k = \{\gd < \gk \mid \gd$ is an
inaccessible cardinal which is not a limit of
inaccessible cardinals$\}$.
\end{lemma}
\begin{proof}
For any $\gd < \gk$ such that
$V \models ``\gd$ is an inaccessible cardinal
which is not a limit of inaccesssible cardinals'',
let $\ga < \gk$ be such that $\gd = \gd_{\ga + 1}$.
Write $\FP = \FP_\ga \ast \dot \FQ_\ga \ast \dot \FR =
\FP_{\ga + 1} \ast \dot \FR$.
As we have already observed,
$V^{\FP_{\ga + 1}} \models
``\gd_{\ga + 1}$ is inaccessible and $V_{\gd_{\ga + 1}}
\models T_k$''. Since $\forces_{\FP_{\ga + 1}} ``\dot \FR$
is $\gd_{\ga + 1}$-directed closed'',
$V^{\FP_{\ga + 1} \ast \dot \FR} = V^\FP \models
``\gd_{\ga + 1}$ is inaccessible and $V_{\gd_{\ga + 1}}
\models T_k$''. In addition, because
$V \models ``\gd_{\ga + 1}$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals'',
$V^\FP \models ``\gd_{\ga + 1}$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals'' as well.
Consequently, the proof of Lemma \ref{l2} will be complete
once we have shown that if $V^\FP \models ``\gd$ is
an inaccessible cardinal which is not a limit of
inaccessible cardinals'', then
$V \models ``\gd$ is an inaccessible cardinal which
is not a limit of inaccessible cardinals''. If not,
$V \models ``\gd$ is an inaccessible limit of
inaccessible cardinals'', so
$V \models ``\gd$ is an inaccessible limit of inaccessible
cardinals which are not themselves limits of
inaccessible cardinals''. As we have just shown, such
cardinals are preserved to $V^\FP$, so
$V^\FP \models ``\gd$ is an inaccessible limit of
inaccessible cardinals''.
This contradiction completes the proof of Lemma \ref{l2}.
\end{proof}
Since by Lemma \ref{l2}, $A_i = \emptyset$, and
since forcing cannot create a new inaccessible cardinal,
Lemmas \ref{l1} and \ref{l2} complete the proof of
Theorem \ref{t2}.
\end{proof}
We observe that in the proof we have just given for
Theorem \ref{t2}, $A_j = \emptyset$ for $j \neq k$.
Our method of proof allows for other possible values
for the $A_j$,
which we leave for readers to work out for themselves.
Further, our method of proof shows that if $k = 1$, then
$A_1 = \{\gd < \gk \mid \gd$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals and
$V_\gd \models {\rm CCA}\}$.
%Thus, if $\gk$ is indestructibly supercompact and
%$\gl > \gk$ is inaccessible, then for $i = 1, \ldots, 4$,
%the sets $A_i$ are all unbounded in $\gk$.
Note that by definition, the $A_i$ are mutually disjoint, and
regardless if $\gk$ is also indestructible,
$\bigcup_{i = 1, \ldots, 4} A_i =
\{\gd < \gk \mid \gd$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals$\}$.
%On the other hand,
%for any fixed $i$, a particular $A_i$
%need not be unbounded in $\gk$
%may consistently be empty
%if $\gk$ is indestructibly supercompact
%and there is no inaccessible cardinal above $\gk$ (although
Observe also that in spite of Theorem \ref{t2},
by the last sentence, at least one of
the $A_i$ must be unbounded in $\gk$ if
$\gk$ is supercompact.
One may wonder if Theorem \ref{t1}
can be improved. To make this more precise,
one may ask if it is possible to infer anything
along the lines of Theorem \ref{t1} without the
additional assumption of an inaccessible cardinal
above the supercompact cardinal $\gk$.
By Theorem \ref{t2}, one would then of necessity
have to work with inaccessible limits of inaccessible cardinals.
In fact, this question has a positive answer, as
the following theorem shows.
\begin{theorem}\label{t3}
Suppose $\gk$ is indestructibly supercompact. Then
$V_\gk \models T_1$, so by reflection,
$B_1 =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
limit of inaccessible cardinals and $V_\gd \models T_1\}$
is unbounded in $\gk$.
\end{theorem}
\begin{proof}
As has already been mentioned,
%The work of \cite{Rth, R} shows that
${\rm ZFC} \vdash ``{\rm CCA} \implies$ GA +
$V = {\rm HOD}$''.
%if $M$ is any model and
%$M \models ``$ZFC + CCA'', then $M \models T_1$.
Thus, if $\gk$ is indestructibly supercompact,
to prove Theorem \ref{t3}, it suffices to
show that $V_\gk \models {\rm CCA}$.
To do this, fix $\ga < \gk$ and $x \subseteq \ga$.
Let $\FP$ be the $\gk$-directed closed partial ordering
which first forces GCH for all cardinals $\gd$ in the
closed interval $[\gk, \gk^{+ \ga + \ga}]$ and then
forces failures of GCH in this interval so that
$\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gk + \gb + 1}} = \ha_{\gk + \gb + 2}$.
By indestructibility, let $\gl > \gk$ be
sufficiently large with $j : V^\FP \to M$ an
elementary embedding witnessing the $\gl$
supercompactness of $\gk$ generated by
a supercompact ultrafilter ${\cal U} \in V^\FP$ over
$P_\gk(\gl)$ such that
$M \models ``\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gk + \gb + 1}} = \ha_{\gk + \gb + 2}$''.
Since $\ga < \gk$ and $\gk$ is the critical point
of $j$, by reflection, there are unboundedly many %(in $\gk$)
$\gth \in (\ga, \gk)$ such that in both $V^\FP$
and ${(V_\gk)}^{V^\FP}$,
$\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gth + \gb + 1}} = \ha_{\gth + \gb + 2}$.
Since $\FP$ is $\gk$-directed closed, this last
fact must be true in $V$ and $V_\gk$ as well.
Thus, $V_\gk \models {\rm CCA}$.
This completes the proof of Theorem \ref{t3}.
\end{proof}
In light of Theorem \ref{t3}, one may also wonder
if Theorem \ref{t2} can be improved.
To be more explicit, one may ask if it is possible,
assuming $\gk$ is supercompact and no cardinal
$\gd > \gk$ is inaccessible, to obtain a model
of ZFC in which $\gk$ is indestructibly supercompact and
for {\em every} inaccessible cardinal
$\gd < \gk$, $V_\gd \models T_1$.
Once again, this question has a positive answer,
as the following theorem shows.
\begin{theorem}\label{t4}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
Then there is a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is indestructibly supercompact +
No cardinal $\gd > \gk$ is inaccessible + For
every inaccessible cardinal $\gd < \gk$,
$V_\gd \models T_1$''.
\end{theorem}
%\section{The Proofs of Theorems \ref{t1} -- \ref{t4}}\label{s2}
%We begin with the proof of Theorem \ref{t1}.
%Having completed the proof of Theorem \ref{t1},
%we turn now to the proof of Theorem \ref{t2}.
%Having completed our proof and discussion of
%Theorem \ref{t2}, we turn now to the proof
%of Theorem \ref{t3}.
%Finally, we turn to the proof of Theorem \ref{t4}.
\begin{proof}
Suppose
$V \models ``$ZFC + $\gk$ is supercompact +
No cardinal $\gd > \gk$ is inaccessible''.
As in the proof of Theorem \ref{t2},
we assume in addition that $V \models {\rm GCH}$.
We then let $\FP$ be the partial ordering used
in the proof of Theorem \ref{t2} as defined when $k = 1$.
From this, the proof of Theorem \ref{t2} and the succeeding
remarks allow us to infer immediately that
$V^\FP \models ``$ZFC + $\gk$ is indestructibly supercompact +
No cardinal $\gd > \gk$ is inaccessible + $A_1 =
\{\gd < \gk \mid \gd$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals$\} =
\{\gd < \gk \mid \gd$ is an inaccessible cardinal
which is not a limit of inaccessible cardinals
and $V_\gd \models {\rm CCA}\}$''.
%In addition, as was already mentioned,
Therefore, as in the proof of Theorem \ref{t3},
to prove Theorem \ref{t4}, it suffices to
show that if $V^\FP \models ``\gd$
is an inaccessible cardinal which is a limit
of inaccessible cardinals'', then
${(V_\gd)}^{V^\FP} \models {\rm CCA}$.
To see this, work for the rest of the proof in $V^\FP$.
Let $\ga < \gd$ with $x \subseteq \ga$, and
let $\gr$ be the least inaccessible cardinal greater
than $\ga$. Clearly, $\gr < \gd$,
$\gr \in A_1$, and $x \in V_\gr$.
Since $V_\gr \models {\rm CCA}$, there is some
$\gth \in (\ga, \gr)$ such that $V_\gr \models
``\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gth + \gb + 1}} = \ha_{\gth + \gb + 2}$''.
But then,
$\gth \in (\ga, \gd)$ and $V_\gd \models
``\gb \in x$ iff for every $\gb < \ga$,
$2^{\ha_{\gth + \gb + 1}} = \ha_{\gth + \gb + 2}$''.
Thus, $V_\gd \models {\rm CCA}$.
This completes the proof of Theorem \ref{t4}.
\end{proof}
Observe that by Theorem \ref{t4}, it is impossible to
improve Theorem \ref{t3}. In other words, in ZFC
alone, only $B_1$ need be unbounded in $\gk$ if
$\gk$ is indestructibly supercompact, and not
$B_i =_{\rm df} \{\gd < \gk \mid \gd$ is an inaccessible
limit of inaccessible cardinals and $V_\gd \models T_i\}$
for $i = 2, \ldots, 4$.
In conclusion to this paper, we note that
results analogous to Theorems \ref{t1} -- \ref{t4}
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.)
We leave it to readers to
work out the details for themselves.
%\section{Concluding Remarks}\label{s4}
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\end{document}