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\begin{document}
\title[hod-supercompactness]{hod-supercompactness, indestructibility,
and level by level equivalence}
\author{Arthur W. Apter}
\address{A.W. Apter, Department of Mathematics, Baruch College of CUNY,
New York, New York 10010 and the CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, New York 10016 }
\email{awapter@alum.mit.edu}
\thanks{$^*$The research of the first author was partially supported
by various PSC-CUNY grants.}
\urladdr{http://faculty.baruch.cuny.edu/aapter}
\author{Shoshana Friedman}
\address{s. friedman, department of mathematics and computer science, kingsborough community college-cuny, 2001 oriental blvd, brooklyn, new york 11235}
\email{shoshana.friedman4@gmail.com}
%\date{\today}
\date{April 4, 2014}
\thanks{$^{**}$The results obtained in this paper extend a portion of the second author's doctoral dissertation written
at The Graduate Center at CUNY, under the first author, to whom the second author is indebted for his aid and encouragement.}
\thanks{$^{***}$The authors would like to thank
the referee for her/his thorough and conscientious
reading of numerous versions of the manuscript and
many helpful comments and suggestions which have
been incorporated into the current version of the paper.}
\subjclass[2000]{ 03E35, 03E55}
\keywords{Supercompact cardinal, strongly compact cardinal,
measurable cardinal, HOD, level by level equivalence between
strong compactness and supercompactness.}
\maketitle
\begin{abstract}In an attempt to extend the
property of being supercompact but not $\HOD$-supercompact
to a proper class of indestructibly supercompact cardinals,
a theorem is discovered about a
proper class of indestructibly supercompact cardinals
which reveals a surprising incompatibility.
However, it is still possible to force to get a
model in which
the property of being supercompact but not $\HOD$-supercompact
%this property
holds for the
least supercompact cardinal $\kappa_0$,
$\kappa_0$ is indestructibly supercompact, the
strongly compact and supercompact cardinals coincide
except at measurable limit points, and
level by level equivalence between strong compactness
and supercompactness holds above $\gk_0$ but
fails below $\gk_0$.
Additionally, we get
%this property
the property of being supercompact but not $\HOD$-supercompact
at the least supercompact cardinal, in a model where level by level
equivalence between strong compactness and supercompactness holds.
\end{abstract}
\section{Introduction}\label{s1}
In connection to his work in inner model theory, Woodin
introduced the concept of $ N $-supercompactness, where $ N $ is a proper class inner model of $ V $.
As mentioned in \cite{Sargsyan}, at a set theory seminar at Berkeley in 2005, Woodin asked if it were possible to construct a model of set theory in which $ \gk $ is supercompact, but not \HOD -supercompact.
We will extend Sargsyan's result of \cite{Sargsyan}, which answers Woodin's question with the following theorem.
\begin{Thm}\label{T:Sargsyan}{\rm\cite{Sargsyan}} Suppose $ V\models \ZFC+\GCH+ ``\gk$ is a supercompact cardinal.'' Then there is a forcing
extension of $ V $ in which $ \gk $ is supercompact, but not \HOD -supercompact.
\end{Thm}
Note that the cardinal $ \gk $ is $\HOD$\textit{-supercompact} iff $ \gk $ is supercompact and for all strong limit cardinals $ \gl $,
there exists an embedding $ j:V\longrightarrow M $, such that cp$ (j)=\gk,\;j(\gk)>\gl,\; M^\gl\subseteq M, $ and $j(\HOD)\cap V_\gl= \HOD\cap V_\gl $. We follow standard notation in using $ j(N) $, where $ N $ is a proper class, to mean $ j(N)=\bigcup_{\ga <\textsc{ord}}j(V_\ga^N) $. Since $ N=\HOD $ is a definable class, $ j\upharpoonright\HOD:\HOD\rightarrow j(\HOD) $ is fully elementary.
%We will say that a supercompact cardinal $ \gk $, as in Theorem \ref{T:Sargsyan}, satisfies the \textit{Sargsyan property}.
There are a number of natural questions that arise as a result of this theorem. Three of the questions are as follows:
%\begin{TestQuestion}\mbox{ }
\begin{enumerate}
\item \label{Ques:Sargsyan extend to K?} \textit{Can the property of
being supercompact but not $\HOD$-supercompact
be extended to the class of supercompact cardinals, K,
assuming K has more than one member, and each
$\kappa \in K$ is indestructibly supercompact, i.e.,
is indestructible under $\kappa$-directed closed
forcing as in \cite{Laver:Indest}?}
\item \label{Ques:kimchi-magidor w sarg at least?} \textit{Can
the property of being supercompact but not
$\HOD$-supercompact be extended to the least supercompact cardinal
in a model where the supercompact and strongly compact cardinals coincide
%coincide with the strongly compact cardinals
(except at measurable limit points), and the least
supercompact cardinal is also indestructibly supercompact?}
\item \label{ques:lblequiv}\textit{Can
the property of being supercompact but not
$\HOD$-supercompact hold in a model containing supercompact cardinals
which also satisfies level by level equivalence
between strong compactness and supercompactness?}
%\item \label{Ques: Sarg w/o gch?}\textit{As the proof in \cite{Sargsyan} requires class forcing to achieve \GCH, can this result be obtained by doing set forcing over a model of $\ZFC$ that does not satisfy $ \GCH $?}
\end{enumerate}
%\end{TestQuestion}
These questions will be answered by theorems in the subsequent sections.
The theorem in Section \ref{s2} (Theorem \ref{Apth}) is a
serendipitous result that was discovered
in the course of answering these questions.
We will take this opportunity to mention some preliminary material that will be used throughout this paper.
If $\gk$ is a cardinal and $\FP$ is
a partial ordering, $\FP$ is \textit{$\gk$-closed} if for every $ \gd\leq \gk $, given any sequence
$\langle p_\ga : \ga < \gd \rangle$ of elements of $\FP$ such that
$\beta < \gamma < \gd$ implies $p_\gg \leq p_\gb$ (a decreasing chain of
length less than or equal to
$\gd$), there is some $p \in \FP$ (a lower bound to this chain) such that
$p\leq p_\ga$ for all $\ga < \gd$.
%$P$ is $<\gk$-closed if $P$ is $\delta$-closed for all cardinals $\delta <\gk$.
$\FP$ is \textit{$\gk$-directed closed}
if for every cardinal $\delta < \gk$ and every
directed
set $D = \langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$,
%(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
%for every two distinct elements $p_\rho, p_\nu \in
%\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
%$p_\nu$ have a common lower bound)
there is a
lower bound $p \in \FP$ for the members of $D$.
$\FP$ is \textit{$\gk$-strategically closed} if in the
two person game in which the players construct a decreasing sequence
$\langle p_\ga: \ga \leq\gk\rangle$, where player I plays odd stages and player
II plays even and limit stages, player II has a strategy which ensures the
game
can always be continued.
%***Definition and reference for level by level equivalence****
In \cite{apter-shelah}, Shelah and the first author began the
study of level by level equivalence between
strong compactness and supercompactness
by proving the following theorem.
\begin{Thm}\label{t0}
Let
$V \models ``$ZFC + ${\mathcal K} \neq \emptyset$
is the class of supercompact cardinals.''
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + ${\mathcal K}$ is
the class of supercompact cardinals +
For every pair of regular cardinals $\gk < \gl$,
$\gk$ is $\gl$-strongly compact iff
$\gk$ is $\gl$-supercompact, except
possibly if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$-supercompact.''
\end{Thm}
In any model witnessing the conclusions of
Theorem \ref{t0}, we will say that
{\em level by level equivalence between
strong compactness and supercompactness holds}.
Note that the exception in Theorem \ref{t0}
is provided by a theorem of Menas \cite{menas-stgcpctness},
who showed that if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$-strongly
compact, then $\gk$ is $\gl$-strongly compact
but need not be $\gl$-supercompact.
Observe also that Theorem \ref{t0} is a
strengthening of the result of Kimchi and
Magidor \cite{kimchi-magidor}, who showed it is
consistent for the classes of strongly
compact and supercompact cardinals to
coincide precisely, except at measurable
limit points.
%end definition source level by level equivalence
We also take this opportunity to discuss a generalization of
Hamkins' Gap Forcing Theorem \cite{Hamkins:GenGapForcing},
\cite{Hamkins:GapForcing} (as it is stated in \cite{ACH}),
as its results are used throughout this paper.
A forcing notion $\FP$ (and the forcing extension to which
it gives rise) \textit{admits a closure point at $\delta$} if
it factors as $\FQ*\dot \FR$, where $\FQ$ is nontrivial,
$|\FQ| \leq\delta$, and $\forces_\FQ ``\dot\FR$ is
$\delta$-strategically closed''. Our arguments, as do Sargsyan's \cite{Sargsyan}, rely on
the following consequence of the main result of \cite{Hamkins2003:ExtensionsWithApproximationAndCoverProperties}.
\begin{Thm}[\cite{Hamkins2003:ExtensionsWithApproximationAndCoverProperties}]\label{Theorem.ClosurePoint}
If $V \subseteq V[G]$ admits a closure point at $\delta$
and $j:V[G]\longrightarrow M[j(G)]$ is an ultrapower embedding in
$V[G]$ with $\delta<{\rm cp}(j)$, then $j\upharpoonright V:V\longrightarrow M$ is
a definable class in $V$.
\end{Thm}
This theorem follows from \cite[Theorem 3, Corollary
14]{Hamkins2003:ExtensionsWithApproximationAndCoverProperties}.
If $j:V[G]\longrightarrow M[j(G)]$ witnesses the $\lambda$-supercompactness of $\kappa$ in $V[G]$, then by
\cite[Corollary 4]{Hamkins2003:ExtensionsWithApproximationAndCoverProperties},
the restriction $j\upharpoonright V: V\longrightarrow
M$ witnesses the $\lambda$-supercompactness of $\kappa$ in
$V$.
This theorem clearly can be applied to measurability embeddings as well,
which gives us the result that
if our forcing exhibits
the closure point property at a sufficiently small cardinal,
we can infer that the measurable and
supercompact cardinals of the forcing extension already
existed in the ground model.
\section{supercompactness, but not $\HOD$ supercompactness and the class of indestructibly supercompact cardinals}\label{s2}
It would be natural to extend
Sargsyan's result of \cite{Sargsyan}
to a proper class of indestructibly supercompact cardinals,
as is postulated by Question \ref{Ques:Sargsyan extend to K?}.
This was attempted in an earlier draft of this paper.
The referee found a gap in the proof for good reason.
In fact we have the following theorem, due to the first author.
\begin{Thm}\label{Apth} If $V\models \ZFC +
``\exists $ a proper class of indestructibly supercompact cardinals'', then $V=\HOD$. \end{Thm}
\begin{proof}
%add what was done on 6/5/12
It suffices to show that every set of ordinals is coded.
Let $A\subseteq\gd$ be a set of ordinals.
We will show that there already is a coding for $A$ in $V$.
Let $\gk$ be the least indestructibly
supercompact cardinal greater than $\gd$.
In a $\gk$-directed closed way,
we can force there to be a block of $\gd$ successor cardinals
where $\GCH$ holds beyond $\gk$ (see the first paragraph
of the proof of Theorem 2.1 of \cite{Apter:MLQ12}
for details).
Then on this block of $\gd$ cardinals where $\GCH$ holds, code $A$,
by forcing $\GCH$ to fail at the $\ga^{th}$ successor
cardinal of the block, according to whether $\ga$ is in $A$,
as in \cite{Sargsyan}. This forcing can be done in a $\gk$-directed closed way. Let us call the partial ordering to force $\GCH$ to hold on a block, followed by the coding, $\FP$. By the indestructibility of $\gk$, $\gk$ remains supercompact after forcing with $\FP$. Let $\gl$ be sufficiently large, and $j:V^{\FP}\longrightarrow M $ be a $\gl$-supercompactness embedding such that $M$ contains this coding. Since $\gd$ is below the critical point of $j$ which is $\gk$, this coding of $A$ reflects unboundedly in $\gk$ in $V^{\FP}$. Since $\FP$ is $\gk$-directed closed,
this coding actually reflects unboundedly in $\gk$ in $V$. Therefore,
$A$ is in $\HOD$ in $V$. Since $A$ was arbitrary, $V=\HOD$.
\end{proof}
Remark 1: Theorem \ref{Apth}
doesn't require a proper class of indestructibly supercompact cardinals.
A proper class of indestructibly strong cardinals, see \cite{Gitik-Shelah},
or even a proper class of indestructibly strongly unfoldable cardinals,
see \cite{Tom-strongunfoldindest}, would suffice.
Remark 2: If $V\models \ZFC + ``\exists $ a proper class of
indestructibly supercompact cardinals'',
then $V \models \GA$.
This is the Ground Axiom of \cite{Reitz:GroundAxiom}.
The above proof essentially shows that $V\models \CCA$,
which is the Continuum Coding Axiom of \cite{Reitz:GroundAxiom}.
Hence, by \cite{Reitz:GroundAxiom},
since $V\models \CCA$, $V\models\GA$.
Remark 3: Even though Theorem \ref{Apth} shows
that Sargsyan's result of \cite{Sargsyan} can't
be extended to a proper class of indestructibly
supercompact cardinals, we can ask if this result
can be extended to a proper class of supercompact
cardinals which aren't indestructibly supercompact.
This is a question we are currently unable to answer
(and in fact, we don't even know how to extend
Sargsyan's result to two supercompact cardinals).
%end 6/5/12
\section{Indestructibility and hod-supercompactness}\label{s3}
We now come to Question \ref{Ques:kimchi-magidor w sarg at least?},
namely
\textit{Can
the property of being supercompact but not
$\HOD$-supercompact be extended to the least supercompact cardinal
in a model where the supercompact and strongly compact cardinals coincide
(except at measurable limit points), and the least
supercompact cardinal is also indestructibly supercompact}?
This question is answered in the affirmative by the following theorem.
\begin{Thm}\label{levelbylevelindest}
Let $V \models \ZFC +``K$ is the class of supercompact cardinals" +
``$\gk_0 $ is the least supercompact cardinal.'' Then there is an
Easton support iteration of length $\gk_0+1$, $\FP \in V$, such that:
%and a model $ V \subseteq \FV $, with $K, \;\FP \subseteq V and\; V \models \ZFC$
\begin{description}
%\item[(1)]$ V^{\FP} \models \ZFC + $``If $ \gk \in K $, then $ \gk $ is a supercompact cardinal whose supercompactness is indestructible under $ \gk $-directed closed forcing."
\item[(1)] $ V^{\FP} \models$ ``K is the class of supercompact cardinals."
\item[(2)] $ V^{\FP}\models $ `` $ \gk_0 $ is indestructibly supercompact, but not \HOD -supercompact."
In addition,
$V^\FP\models$ ``Every supercompact cardinal greater than $\gk_0$ is superdestructible.''
\item[(3)] Suppose $V\models ``\gd>\gk_0 $ is
$\gd^+$-supercompact." Then $ V^{\FP}\models $ ``Level by level equivalence between strong compactness and supercompactness holds above $\gk_0$ but fails below $\gk_0$."
\item[(4)]$V^{\FP} \models$ ``The only strongly compact cardinals are the elements of K or their measurable limit points."
\end{description}
\end{Thm}
Prior to beginning the proof of Theorem \ref{levelbylevelindest}, we
give some definitions.
Note that as in the Main Theorem of \cite{Hamkins98:SmallForcing},
a cardinal $\gk$ is said to be {\em superdestructible}
if any partial ordering adding a subset of $\gk$
which is $\gd$-closed for every $\gd < \gk$
destroys $\gk$'s weak compactness.
For $\gk$ a regular cardinal and $\gg$ an ordinal,
${\rm Add}(\gk, \gg)$ is the standard partial ordering
for adding $\gg$ many Cohen subsets of $\gk$.
The following partial ordering, a version of which is used by Sargsyan in \cite{Sargsyan}, is designed to code sets of
ordinals into the continuum function and hence, into $\HOD$.
\begin{Def}\label{D:codeA wo gch1} Suppose $\gk<\gl $ are
cardinals and A is a subset of $ \gk$.
Let $ \gl_{\ga} $ be the
$ (\ga+1)^{{\rm st}}$ strong limit cardinal strictly greater than $ \gl$.
Let
$$ \FS_{\gk,\gl}(A)=\prod_{\ga\in A}\add(\gl_\ga,\gl_\ga^{++}).$$
\end{Def}
If \textsc{gch} holds above $\gl$ in $V$, then
in $V^{\FS_{\gk,\gl}(A)},\; \ga\in A $ iff $ 2^{\gl_\ga}=\gl_\ga^{++}$. This implies that in $V^{\FS_{\gk,\gl}(A)}, \;A \in \HOD $.
We define a building block of our forcing,
which we call the {\em lottery sum} after
Hamkins \cite{Hamkins2000:LotteryPreparation}.
\begin{Def}
The {\em lottery sum} of a collection $ A $ of partial
orderings is defined as $\oplus A=\{\langle \FQ,p\rangle :\FQ\in A $ and $ p\in \FQ\}\cup\{{\textbf{1}}\}$, ordered with $ \textbf{1} $ above everything and $ \langle\FQ,p\rangle \leq\langle\FQ^\prime,q\rangle$ when $ \FQ=\FQ^\prime $ and $ p\leq_\FQ q $.
\end{Def}
Since all compatible conditions
in the generic object over $\oplus A$
must be in the same partial ordering,
the forcing effectively holds a lottery among all the
partial orderings in $ A $.
The generic object chooses the ``winning''
partial ordering $ \FQ $ and then forces with it.
We turn now to the proof of Theorem \ref{levelbylevelindest}.
\begin{proof}
Let $V$ be as in the hypotheses for Theorem \ref{levelbylevelindest}.
Without loss of generality, by the results of \cite{apter-shelah},
we assume as well that
$V \models \GCH$ + ``Level by level equivalence between
strong compactness and supercompactness holds.''
Let $ S = \{ \gd<\gk_0:\gd $ is a measurable limit of strong cardinals
in $V$\}. By Lemma 2.1 of \cite{aptercummings-IC2},
since $\gk_0$ is supercompact, $S$ is unbounded in $\gk_0$.
Let $\ga^\prime$ be the least $V$-strong cardinal above $\ga$.
% and let $\ga^{\prime\prime}$ be the next strong cardinal after $\ga^\prime$.
$\FP=\langle \langle \FP_\ga, \dot \Q_\ga \rangle : \ga \in \gk_0 +1 \rangle$ will be an Easton support iteration of length $\gk_0 +1$ .
We start with $\FP_0=\add(\go,1)$. For $\ga = \gk_0$,
$ \dot\FQ_{\ga} $ will be a term for Add($ \gk_0,1) $.
Otherwise, for $\ga < \gk_0$,
$ \dot\q_\ga $ will be a term for trivial forcing
unless $ \ga \in S $.
If $ \ga \in S $, let
$ \FP_{\ga + 1} = \FP_\ga*\dot\add (\ga, 1)*\dot\q*
\dot\FS_{\ga,\ga^\prime}(\dot X)*\dot\FP_{\go,\gg_{\ga}} $,
where
$\dot \FQ$ is a term for the lottery sum of all $\ga$-directed
closed partial orderings of rank less than $\ga^\prime$,
$ \dot X $ is the name of the generic subset of $ \ga $
added by Add$ (\ga,1) $,
$ \dot\FS_{\ga,\ga^\prime}(\dot X) $ is a term for the forcing
in Definition \ref{D:codeA wo gch1},
$\gg_{\ga}$ is the least inaccesible limit of strong cardinals
greater than $\ga$,
and $\dot \FP_{\go,{\gg_\ga}}$ is
a term for the partial ordering to add a nonreflecting
stationary subset of cofinality $\go$ to $\gg_\ga$.
\begin{Sublemma}\label{L:kinKsc1} $ V^{\FP} \models `` \gk_0 $ is a supercompact cardinal whose supercompactness is indestructible under $ \gk_0 $-directed closed forcing."
\end{Sublemma}
\begin{proof}
%add p-gamma kappa to proof-done 2/13/13
It suffices to show that $ \gk_0 $ can be made indestructibly supercompact by forcing with $ \FP_{\gk_0}*\dot\add (\gk_0,1) $.
Let $ G*g \subseteq \FP_{\gk_0}*\dot\add(\gk_0,1)$ be $ V $-generic.
Now we want to show that $ V[G][g] \models ``\gk_0$ is\; supercompact" and
$V[G][g] \models $ ``$\gk_0 $'s supercompactness is indestructible by $ \gk_0 $-directed closed forcing."
%****take out all mention of tail forcing
Fix any $\Q \in V[G][g] $ which is $ \gk_0$-directed closed. Fix any $\dot\Q$, a name for $\Q$, for which 1 $ \forces ``\dot\Q $ is $\gk_0$-directed closed.''
Let $g^* \subseteq \Q$ be $V[G][g]$-generic.
We want to show $V[G][g][g^*] \models $ ``$ \gk_0$ is $ \lambda $-supercompact" for arbitrary cardinals $ \lambda > \kappa_0 $.
Fix any $ \lambda > \kappa_0, \;\gl $ a cardinal such that
$\dot\Q\in H_{\lambda^{+}}$ and let $ \theta >> 2 ^{\lambda^{< \kappa_0}} $.
Fix in $V$ a $ \theta $-supercompactness embedding $j: V \longrightarrow M$ .
%Let $\gth^\prime$ be the least $V$-strong cardinal greater than $\gth$.
Since $\gth\geq 2^{\gk_0}, \; M\models``\gk_0$ is measurable''. Also, if $V\models``\gd<\gk_0$ is a strong cardinal'', $M\models``j(\gd)=\gd$ is a strong cardinal''. Thus, $\gk_0\in j(S)$, so $\gk_0$ is a nontrivial stage of forcing in $M$. Therefore, below a condition in $M$ which opts for $\dot{\FQ}$ in the stage $\gk_0$ lottery, we have that $j(\FP_{\gk_0}*\dot\add (\gk_0,1)*\dot \FQ) $ is forcing equivalent to
$\FP_{\gk_0}*\dot\add (\gk_0,1)*\dot\q*\dot\FS_{{\gk_0},{{\gk_0}^\prime}}(\dot X)*\dot\FP_{\go,\gg_{\gk_0}}*\dot\FP_{tail}*\dot\add(j(\gk_0),1) *j(\dot\FQ)$, with $ \dot X $ being the name for the generic subset added by Add$ (\gk_0,1) $, $\dot\FP_{tail} $ being a name for the forcing from $ (\gg_{\gk_0},j(\gk_0)) $, and the first nontrivial stage of forcing in $\dot\FS_{{\gk_0},{\gk_0}^\prime}(\dot X)*\dot\FP_{\go,\gg_{\gk_0}}*\dot\FP_{tail}$ taking place well beyond $\gth$.
This is since in $M$, there are no strong cardinals in $(\gk_0,\gth]$, because if there were, $M\models ``\gk_0$ is supercompact up to a strong cardinal."
By the proof of Lemma 2.4 of \cite{aptercummings-IC2},
this implies that $M\models ``\gk_0$ is fully supercompact." Therefore, by reflection in $V$, $\gk_0$ is a limit of supercompact cardinals, contradicting that $\gk_0$ is the least supercompact cardinal in $V$.
Force to add $ G^* \subseteq\;\dot\FS_{\gk_0,{\gk_0^{\prime}}}(\dot X)*\dot\FP_{\go,\gg_{\gk_0}}*\dot\FP_{tail} $, a $ V[G][g][g^*] $-generic object.
Then $ j $ lifts to $j: V[G] \longrightarrow M[j(G)]$ in $ V[G][g][g^*][G^*]$ and $ j(G) = G*g*g^* *G^*$.
In order to complete the proof of the lemma we need to lift again. We can, if $ j''(g*g^*) \in M[j(G)] $. It is by the usual argument, since $ g*g^* \in M[j(G)]$ and $ M[j(G)]^\gth\subseteq M[j(G)] $ in $ V[G][g][g^*][G^*]$. The fact that $ M[j(G)]^\gth\subseteq M[j(G)] $ follows since $\FP_{\gk_0}*\dot\add (\gk_0,1)*\dot\q$ is $\gth$-c.c. (because $|\FP_{\gk_0}*\dot\add (\gk_0,1)*\dot\q|<\gth)$ and $\dot\FS_{\gk_0,{\gk_0^{\prime}}}(\dot X)*\dot\FP_{\go,\gg_{\gk_0}}*\dot\FP_{tail}$ is $\gth$-strategically closed in both $M[G][g][g^*]$ and $V[G][g][g^*]$ (because the first stage of nontrivial forcing in $\dot\FS_{\gk_0,{\gk_0^{\prime}}}(\dot X)*\dot\FP_{\go,\gg_{\gk_0}}*\dot\FP_{tail}$ takes place well beyond $\gth$).
Since $ j''(g*g^*) \in M[j(G)] $, there is a master condition $ p^* $ below $ j''(g*g^*) $. $p^*$ exists because $j(\gk_0)>\gth$, $ j(\add(\gk_0,1)*\dot\q) $ is $ j(\gk_0) $-directed closed, and $M[j(G)]^\gth\subseteq M[j(G)]$. We can therefore force to add an $ H $ which is $ V[G][g][g^*][G^*] $-generic such that $ p^*\in H $.
Now we can lift $ j$ to $j:V[G][g][g^*]\longrightarrow M[j(G)][H] $ in $ V[G][g][g^*][G^*][H] $.
Since we chose $ \gth $ to be sufficiently large, $ G^**H $ is $V[G][g][g^*]$-generic over a partial ordering which does not add any ultrafilters over $ P_{\gk_0}(\gl) $. We thus have $V[G][g][g^*] \models $ ``$ \kappa_0 $ is $ \lambda $-supercompact."
\end{proof}
\begin{Sublemma} \label{Lemma:k0sprdestr}
In $V^\FP$, for any $\gk\in K$,
$\gk>\gk_0,\;\gk$ is superdestructible.
\end{Sublemma}
\begin{proof}
%***** add definition of superdestructibility (somewhere)\\
Since for any $\gk\in K$ with $\gk>\gk_0$, $|\FP|<\gk$,
by the Main Theorem of \cite{Hamkins98:SmallForcing},
$\gk$ is superdestructible in $V^\FP$.
\end{proof}
\begin{Sublemma}\label{L:KremainsK1} $K$ is the class of supercompact cardinals in $ V^{\FP}. $\end{Sublemma}
\begin{proof}
Since $\FP$ can be defined to have cardinality $\gk_0$,
by the L\'evy-Solovay results \cite{LevySolovay},
the class of supercompact cardinals above $\gk_0$ remains the same in $V$ and $V^\FP$. By Lemma \ref{L:kinKsc1}, $\gk_0$ remains supercompact. It thus suffices to show that no new supercompact cardinals are created by $ \FP $.
Since $ \FP $ admits a closure point at $ \go $, by an application of Theorem \ref{Theorem.ClosurePoint}, no new supercompact cardinals were created by $ \FP $. Hence $K$ is the class of supercompact cardinals in $V^\FP$.
\end{proof}
\begin{Sublemma}\label{L:kappaisnotHODsc} $V^{\FP}\models ``\gk_0$ is not \HOD -supercompact.''
\end{Sublemma}
%note: change kappa to kappa 0
\begin{proof}
This proof follows closely the proof of Lemma 2.2 in \cite{Sargsyan}. Let $ G\subseteq\FP $ be $ V $-generic.
Factor $ \FP = \FP_{\gk_0}*\dot\FQ_{\gk_0}$.
%where $ \FP_{\gk_0} $ is the forcing up to stage $ {\gk_0} $,
%and $ \dot\FQ_{\gk_0}=\dot\add({\gk_0},1) $.
Let $ G=G_{\gk_0}*g$ be the corresponding generic objects. Assume $ {\gk_0} $ is \HOD -supercompact in $ V[G] = W$ and let $\HOD=\HOD^{W}$.
Fix a strong limit cardinal
$ \gl > \gk_0$ such that $\HOD^{ W_\gl }= \HOD \cap\; W_\gl $. Let $ j : W \longrightarrow M$ be a $\gl $-supercompactness embedding such that $ j(\HOD) \cap W_\gl = \HOD\cap \; W_\gl = \HOD^{ W_\gl }$. By Theorem \ref{Theorem.ClosurePoint}, $ i = j\upharpoonright V $ is definable in $ V $,
$ j $ is the lift of $ i $, and $i$ is a $\gl$-supercompactness embedding.
Let $ N = i(V)= \bigcup_{\ga <\textsc{ord}}i(V_\ga) $. If $ H $ is the $ N$-generic object for $ i(\FP) $, then $ M = N[H] = N[j(G)] $. We also have that $ H \cap \FP_{\gk_0} = G_{\gk_0}$.
In addition, for any $\gd<\gk_0$ such that $V\models``\gd$ is a strong cardinal", $N\models ``i(\gd)=\gd$ is a strong cardinal". Since $\gl>2^{\gk_0}$ and $N^\gl\subseteq N,\;N\models ``\gk_0 $ is measurable." Thus, in $N$, $\gk_0$ is a nontrivial stage of forcing. This means that we can let $ g^{\prime} $ be the generic for Add$ (\gk_0,1) $ given by $ H $. Then in $ N[H] = M,\; g^{\prime} $ is ordinal definable.
But because $ \gk_0 $ is \HOD -supercompact,
$ j(\HOD) \cap W_\gl = \HOD\cap W_\gl = \HOD^{ W_\gl }.$ This implies $g^{\prime}\in \HOD^{ W_\gl }$. Thus $ g^{\prime} $ is ordinal definable in $ W_\gl = V_\gl^{V[G]}=V_{\gl}^{V[G_{\gk_0}][g]} $.
%Since $\FP_{tail} $ is $ \gk $-closed, $ g^\prime $ could not have been added by $\FP_{tail} $.
Thus $ g^\prime $ had to have been added over $ V_\gl[G_{\gk_0}]$, and more particularly, $ g^\prime $ is added over $ V_\gl[G_{\gk_0}] $ by homogeneous forcing. This fact, along with $ g^\prime $ being ordinal definable in $ W_\gl $,
%along with the fact that $ g^{\prime} $ is added by homogenous forcing over $ V_\gl $ imply Note: key part of the proof is that we will add a set in V to kappa but will only be coded in M, bec in M, stage kappa forcing is nontrivial and eventually the kappa set will be coded in V by a density argument, but not below lamda where weve cut off the universe-i think
implies that $ g^{\prime} $ is in $ V_\gl[G_{\gk_0}] $. This is impossible as $ g^{\prime} $ is a $ V [G_{\gk_0}]$-generic object for Add$ ({\gk_0},1) $. Therefore $ \gk_0 $ is not \HOD -supercompact.
\end{proof}
\begin{Sublemma}\label{L:stglycptareinKorlimpts1}
Suppose $V\models ``\gd>\gk_0 $ is $\gd^+$-supercompact." Then $V^{\FP} \models$ ``Level by level equivalence between strong compactness and supercompactness holds above $\gk_0$ but fails below $\gk_0$."
\end{Sublemma}
\begin{proof}
Since $\FP$ can be defined to have cardinality $\gk_0$,
and level by level equivalence holds in $V$,
by the results of \cite{LevySolovay},
level by level equivalence holds above $\gk_0$ in $V^\FP$.
Further, by the results of \cite{LevySolovay},
$V^\FP \models ``\gd$ is $\gd^+$-supercompact.''
Level by level equivalence does not hold below $\gk_0$ in $V^\FP$
because $\gk_0$ is indestructibly supercompact and there exists
a cardinal $\gd>\gk_0$ which is $\gd^+ $-supercompact.
Thus, by Theorem 5 of \cite{ApterHamkins2002:LevelByLevel},
$\{\gd<\gk_0:\gd$ is $\gd^+$-strongly compact but $\gd$ isn't $\gd^+$-supercompact\} is unbounded in $\gk_0$.
This proves Lemma \ref{L:stglycptareinKorlimpts1}.
\end{proof}
\begin{Sublemma} \label{L:stgcptnessindest1} $ V^{\FP}\models $ ``$\gk_0$ is the least strongly compact cardinal.''
\end{Sublemma}
\begin{proof}
By Lemma \ref{L:kinKsc1}, it suffices to show that $ V^{\FP}\models $ ``No cardinal $\gd<\gk_0$ is strongly compact.'' By the definition of $\FP$, unboundedly many $\gg<\gk_0$ contain nonreflecting stationary sets of ordinals of cofinality $\go$. By Theorem 4.8 of \cite{SRK} and the succeeding remarks, no cardinal $\gd<\gk_0$ is strongly compact in $V^{\FP}$.
\end{proof}
\begin{Sublemma} \label{L:stgcptnessindest} $ V^{\FP}\models $ ``The only strongly compact cardinals are the elements of $K$ or their measurable limit points.''
\end{Sublemma}
\begin{proof}
%************
By Lemma \ref{L:stgcptnessindest1}, in $V^{\FP}$, $\gk_0$ is the least strongly compact cardinal. By level by level equivalence between strong compactness and supercompactness, in $V$, the only strongly compact cardinals are the elements of $K$ or their measurable limit points. Since $|\FP|=\gk_0$, by the results of \cite{LevySolovay}, in $V^{\FP}$, the only strongly compact cardinals are the elements of $K$ or their measurable limit points.
\end{proof}
%By [], since $|\FP|=\gk_0$, for every $\gk\in K, \gk>\gk_0$, $\gk$ is superdestructible.
Lemmas \ref{L:kinKsc1}-\ref{L:stgcptnessindest} prove Theorem \ref{levelbylevelindest}.
\end{proof}
If there is no $\gd>\gk_0$ in $V$ such that $\gd$ is $\gd^+$-supercompact,
then Theorem \ref{levelbylevelindest} is still true.
This is shown by modifying the definition of $\FP$ to force
a failure of level by level equivalence between strong compactness
and supercompactness below $\gk_0$
(e.g., by keeping the definition of $\FP$ as it was originally,
except that after forcing with $\add(\go,1)$, we
iteratively force to add a nonreflecting stationary
set of ordinals of cofinality $\go$ to each measurable cardinal
below the least cardinal $\gd<\gk_0$ which is $\gd^+$-supercompact).
The other clauses in the statement of Theorem \ref{levelbylevelindest}
remain true with the same proofs as before
(although some may only be vacuously true).
\section{Level by Level equivalence and hod-supercompactness}\label{s4}
We now come to Question \ref{ques:lblequiv},
namely \textit{Can Sargsyan's result hold
in a model containing supercompact cardinals
which also satisfies level by level equivalence
between strong compactness and supercompactness}?
We answer this question in the affirmative with the following theorem.
\begin{Thm} \label{T:lblequivandSarg} Let $ V \models \ZFC +
``K$ is the class of supercompact cardinals'' + ``$\gk_0$ is the least supercompact cardinal." Then there is an Easton support iteration of length $\gk_0+1$, $ \FP \in V$,
%and a model $ V \subseteq \FV $, with $K, \;\FP \subseteq V and\; V \models \ZFC$
such that
%\item[(1)]$ V^{\FP} \models \ZFC + $``If $ \gk \in K $, then $ \gk $ is a supercompact cardinal whose supercompactness is indestructible under $ \gk $-directed closed forcing."
$ V^{\FP} \models \ZFC + ``K $ is the class of supercompact cardinals" +
``Level by level equivalence between strong compactness and supercompactness holds" + ``$ \gk_0 $ is supercompact, but not \HOD -supercompact."
\end{Thm}
%begin paste
%**********begin proof\\
%We now turn to the proof of Theorem \ref{T:lblequivandSarg}.
\begin{proof}
Without loss of generality,
as in the proof of Theorem \ref{levelbylevelindest},
we assume as well that $V\models\GCH+$ ``Level by level equivalence between strong compactness and supercompactness holds."
As in the proof of Theorem \ref{levelbylevelindest}, let $ S = \{ \gd<\gk_0:\gd $ is a measurable limit of strong cardinals in $V$\}.
$\FP=\langle \langle \FP_\ga, \dot \Q_\ga \rangle : \ga \in \gk_0 +1 \rangle$ will be an Easton support iteration of length $\gk_0 +1$ .
We start with $\FP_0=\add(\go,1)$. For $ \gk_0$, $ \dot\FQ_{\gk_0} $ will be a term for Add($ \gk_0,1) $. Otherwise $ \dot\q_\ga $ will be a term for trivial forcing unless $ \ga \in S $.
If $ \ga \in S $, let
$ \FP_{\ga + 1} = \FP_\ga*\dot\add (\ga, 1)* \dot\FS_{\ga,\ga'}(\dot X) $,
where $ \dot X $ is the name of the generic subset of $ \ga $ added by Add$ (\ga,1) $.
\begin{Sublemma}\label{L:kinKsc} $ V^{\FP} \models `` \gk_0 $ is a supercompact cardinal.''
\end{Sublemma}
\begin{proof}Let $ \gl > \gk_0 $ be an arbitrary cardinal,
and let $ j: V \longrightarrow M $ be a \break
$ \gth$-supercompactness embedding with $\gth>>2^{\gl^{<\gk_0}}$.
Since as in the proof of Lemma \ref{L:kinKsc1},
$\gk_0 $ is a measurable limit of strong cardinals in $M$,
the stage $ \gk_0 $ forcing in $ M^{\FP_{\gk_0}} $ is nontrivial,
and we have that $ j(\FP) = \FP_{\gk_0}*\dot\add (\gk_0,1)*
\dot\FS_{\gk_0,\gk'_0}(\dot X)*\dot\FP_{tail} ,$
with $ \dot X $ the name for the generic subset added by
Add$ (\gk_0,1) $, and $\dot\FP_{tail} $
a term for the rest of the forcing up to and including $j(\gk_0)$.
Since as in the proof of Lemma \ref{L:kinKsc1},
the first nontrivial stage of forcing in
$\dot\FS_{\gk_0,\gk'_0}(\dot X)*\dot\FP_{tail} $ is beyond max$(\gl,\gth)=\gth $, we may abuse notation and write $ j(\FP)$ as $ \FP_{\gk_0}*\dot\add (\gk_0,1)*\dot\FP_{tail} $, where the first nontrivial stage of forcing in $ \dot\FP_{tail} $ is beyond $ \gth $. The arguments of Lemma \ref{L:kinKsc1} now show that $V^{\FP}\models ``\gk_0$ is $\gl$-supercompact." Since $ \gl $ was arbitrary, this completes the proof of Lemma \ref{L:kinKsc}.
\end{proof}
\begin{Sublemma}\label{L:KremainsK} $K$ is the class of supercompact cardinals in $ V^{\FP} $.\end{Sublemma}
\begin{proof}
Since $\FP$ can be defined to have cardinality $\gk_0$, the proof given in Lemma \ref{L:KremainsK1} remains valid and shows that $K$ is the class of supercompact cardinals in $V^\FP$.
\end{proof}
\begin{Sublemma}\label{kislambdasc}
Suppose $\gd <\gk_0$ and $\gl>\gd$ is regular.
If $V\models ``\gd$ is $\gl$-supercompact",
then $V^\FP\models ``\gd$ is $\gl$-supercompact."
\end{Sublemma}
%\begin{sublemma} Suppose $\gd <\gl<\gd_0$. If $V\models "`\gd$ is $\gl$ supercompact" then $V^\FP\models "\gd$ is $\gl$ supercompact", where $\gd$ is a non-trivial stage of forcing.
%\end{Sublemma}
\begin{proof} We divide into two cases.\\
Case 1: Suppose $\gd$ is a limit of strong cardinals.
%and $V\models ``\gd$ is $\gl$-supercompact.''
Then $\gl$ is less than the least $V$-strong cardinal
$\gd^\prime$ above $\gd$. This is
because otherwise,
$\gd$ would be supercompact up to a strong cardinal.
As we observed in the proof of
Lemma \ref{L:kinKsc1},
this means that $\gd$ would be fully supercompact,
a contradiction to the fact that $\gk_0$
is the least supercompact cardinal.
Factor $\FP=\FP_\gd*\dot\add(\gd,1)*\dot\FS_{\gd,\gd'} (\dot X)*
\dot \FP^{\gd+1}$. Since $\gl<\gd^\prime$,
the first stage of nontrivial forcing in
$\dot\FS_{\gd,\gd'} (\dot X)*\dot \FP^{\gd+1}$ is beyond $\gl$.
Thus, $V^{\FP}\models``\gd $ is $\gl$-supercompact" iff
$V^{\FP_\gd*\dot\add(\gd,1)} \models``\gd$ is $\gl$-supercompact."
%Fix a $\gl$-supercompactness embedding generated over $\FP_\gd(\gl)$, $j:V\longrightarrow M$ such that $\gd$ is not $\gl$-supercompact in $M$ (otherwise $\gd$ would be supercompact up to a strong cardinal in $M$ so it would be fully supercompact in $M$).
Let $U$ be a normal, fine ultrafilter over $P_\gd(\gl)$
such that for $j:V\longrightarrow M$ the associated elementary embedding,
$M\models ``\gd$ is not $\gl$-supercompact."
Note that $M\models$ ``No cardinal $\gg\in(\gd,\gl]$ is strong."
This is
since otherwise, $M\models ``\gd$ is supercompact
up to a strong cardinal and hence is fully supercompact'',
which contradicts that
$M\models ``\gd$ is not $\gl$-supercompact."
As in the proof of Lemma \ref{L:kinKsc1}, $\gd$ is a nontrivial stage of forcing in $M$.
This means that $j(\FP_\gd*\dot \add(\gd,1))=
\FP_\gd*\dot\add(\gd,1)*\dot \FQ * \dot \add (j(\gd), 1)$,
where the first stage of nontrivial forcing in $\dot\FQ$ is above $\gl$.
Let $G * g \subseteq \FP_\gd*\dot \add(\gd,1)$ be $V$-generic.
We now argue as in the proof of Lemma 1.2 of \cite{Apter:AML05},
from which we quote freely.
The usual diagonalization argument
(as given, e.g., in the construction of
the generic object $G_1$ in Lemma 2.4 of \cite{aptercummings-IC2})
may be used to build in $V[G][g]$ an $M[G][g]$-generic object $g^*$
over $\FQ$ and lift $j$ to $j : V[G] \longrightarrow M[G][g][g^*]$.
(This is since $\GCH$ in both $V[G][g]$ and $M[G][g]$ at and above $\gd$
combined with the fact that $\FQ$ is $\gl^+$-directed
closed in both $V[G][g]$ and $M[G][g]$ allow us to let
$\la D_\ga : \ga < \gl^+ \ra \in V[G][g]$ be an enumeration of the
dense open subsets of $\FQ$ present in $M[G][g]$ and then construct
$g^*$ in $V[G][g]$ to meet each $D_\ga$. Full details may be found in
\cite{aptercummings-IC2}.) A master condition for $j''g$ may now be
constructed in $V[G][g]$ as in the proof of Lemma \ref{L:kinKsc1}.
The diagonalization argument just mentioned then once again
allows us to construct in $V[G][g]$ an $M[G][g][g^*]$-generic object $H$
over $\add(j(\gd), 1)$ containing this master condition and fully lift $j$
in $V[G][g]$ to $j : V[G][g] \longrightarrow M[G][g][g^*][H]$.
% The argument will now be identical to that in Lemma \ref{L:kinKsc1}.
%That is, standard arguments will show (see \cite{Laver:Indest}) that for any cardinal $\gg <\gl ,\; V^\FP\models "\gd$ is $\gl$ supercompact." Since $ \gl $ was arbitrary, this completes the proof of Case 1.
Case 2:
Suppose $\gd$ is not a limit of strong cardinals. Note that $\gd$ is not itself a strong cardinal. This is since otherwise, by $\GCH$ and the fact that $\gl>\gd$, $\gd$ is both (at least) $2^\gd$-supercompact and strong. Hence, by Lemma 2.1 of \cite{aptercummings-IC2}, $\gd$ is a limit of strong cardinals, contrary to our assumptions.
Let $\FP=\FP_\gd*\dot{\FP^\gd}$. The supremum of the strong cardinals less than or equal to $\gd$ is below $\gd$, since $\gd$ is not a strong cardinal. Hence, $|\FP_\gd|<\gd$ by the definition of the forcing. By the results of \cite{LevySolovay}, $V^{\FP_\gd}\models ``\gd $ is $\gl$-supercompact." No nontrivial forcing is done at stage $\gd$, since $\gd$ is neither a strong cardinal nor a limit of strong cardinals. The next stage of nontrivial forcing is beyond $\gl$, because otherwise $\gd$ would be supercompact up to a strong cardinal, and thus would be fully supercompact. Since the next stage of nontrivial forcing is beyond $\gl$, $\dot\FP^\gd$ is therefore sufficiently directed closed in $V^{\FP_\gd}$ so that $\gd$ will remain $\gl$-supercompact in $V^{\FP_\gd *\dot\FP^\gd} =V^\FP$.
\end{proof}
\begin{Sublemma}
\label{L:stglycptareinKorlimpts}
$V^{\FP} \models$ ``Level by level equivalence between strong compactness and supercompactness holds."
% holds above $\gk_0$ but fails below $\gk_0$"
\end{Sublemma}
\begin{proof}
Since level by level equivalence between strong compactness and supercompactness holds in $V$ and $|\FP|=\gk_0$, by the results of \cite{LevySolovay}, $V^\FP\models$ ``Level by level equivalence between strong compactness and supercompactness holds above $\gk_0$.'' By Lemma \ref{L:kinKsc}, $V^\FP\models ``\gk_0$ is supercompact.'' It thus suffices to show that $V^\FP\models ``$Level by level equivalence between strong compactness and supercompactness holds below $\gk_0$'' to prove this lemma.
Towards this end, let $\gd<\gk_0$ and $\gl>\gd$ be such that $V^\FP\models ``\gd$ is $\gl$ strongly compact and $\gl$ is regular.''
%Since $\FP$ is a mild gap forcing,
By the definition of $\FP$
and the results of \cite{Hamkins:GenGapForcing},
\cite{Hamkins:GapForcing},
and \cite{Hamkins2003:ExtensionsWithApproximationAndCoverProperties},
$V\models ``\gd$ is $\gl$-strongly compact.''
Because level by level equivalence between strong compactness and supercompactness holds in $V$, either $V\models``\gd$ is $\gl$-supercompact'',
or $V\models ``\gd$ is a measurable
limit of cardinals which are $\gl$-supercompact.''
By Lemma \ref{kislambdasc},
either $V^\FP\models``\gd$ is $\gl$-supercompact'',
or $V^\FP\models ``\gd$ is a measurable
limit of cardinals which are $\gl$-supercompact.'' Therefore, $V^\FP\models ``$Level by level equivalence between strong compactness and supercompactness holds below $\gk_0$.'' This proves Lemma \ref{L:stglycptareinKorlimpts}.
\end{proof}
We note that in Lemma \ref{L:stglycptareinKorlimpts},
$\gl<\gd'$ as well. This is since otherwise,
some cardinal $\gg<\gk_0$ is supercompact up to a strong cardinal
and hence is fully supercompact.
The proof that $V^\FP\models``\gk_0$ is not $\HOD$-supercompact'' is the same as the proof of Lemma \ref{L:kappaisnotHODsc}. Thus, Lemmas \ref{L:kinKsc}-\ref{L:stglycptareinKorlimpts} complete the proof of Theorem \ref{T:lblequivandSarg}.
\end{proof}
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