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\begin{document}
\title{Coding into HOD via Normal Measures with some Applications}
\author{Arthur W. Apter}
\address{A.W. Apter, 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}
\email{awapter@alum.mit.edu}
\urladdr{http://faculty.baruch.cuny.edu/apter}
\author{Shoshana Friedman}
\address{Sh. Friedman, Department of Mathematics and Computer Science,
Kingsborough Community College-CUNY, 2001 Oriental Boulevard,
Brooklyn, New York 11235 USA}
\email{sf8dw@verizon.net}
\date{March 14, 2010 (revised September 8, 2010)}
\keywords{Measurable cardinal, supercompact cardinal,
normal measure, HOD}
\subjclass[2000]{ 03E35, 03E45, 03E55}
\thanks{The authors would like to thank the referee and
Paul Corazza, for helpful comments and suggestions which
have been incorporated into the current version of the paper.}
\thanks{The research of the first author was partially
supported by PSC-CUNY grants.}
\thanks{The results obtained in this paper form 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.}
\maketitle
%\begin{center}
%\end{center}
\begin{abstract}
We develop a new method for coding sets while preserving $ \GCH $ in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset $ A $ of $ \gk $, we require that our model contain $ \gk $ many measurable cardinals above $ \gk $. Additionally we will describe some of the applications of this result.
\end{abstract}
\section{Introduction}
The model $\HOD $ has long been of interest to set theorists. Philosophically, large cardinal consistency with an inner model such as $\HOD$ can be seen as further verification of the existence of large cardinals. One approach to achieve this compatibility has been to construct a canonical inner model with a particular large cardinal property.
%*, however, thus far, this approach seems to have an upper bound.
Another approach to this problem has been to start with a model that
exhibits the desired large cardinal property, then force over it to get
a model with some inner model like properties, particularly \textsc{V=hod}.
To that end, if one wishes to obtain a model of \textsc{V=hod} via forcing or
to code a generic subset into $ \HOD $ while preserving large cardinals,
the standard method available is to use the continuum function as an oracle,
which requires significant failures of $ \GCH $. Recently,
a method for forcing \textsc{V=hod} while preserving certain large
cardinals and $ \GCH $
was developed by Brooke-Taylor \cite{BrookeTaylor:diamondstar}.
His method involves using whether $ \diamondsuit^* $ holds as an oracle,
and his proofs do not indicate how to preserve supercompactness in general.
In this paper, we present an alternative method of coding a subset $ A $ of $ \gk $ while preserving $ \GCH $ and supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset $ A $ of $ \gk $, we require that our model contain $ \gk $ many measurable cardinals above $ \gk $. Additionally we will describe some of the applications of this result, where the original coding required violating $ \GCH $.
\section{The coding oracle}\label{coding}
Before describing this coding, we present some relevant definitions and lemmas. A poset $\FP$ is \textit{$\gk$-closed} if for every $ \gd\leq \gk $, given every 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{$\gk$-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 stages, player II has a strategy which ensures the
game
can always be continued. For $ \gk $ a regular cardinal and $ \gl $ an ordinal, Add($ \gk, \gl $) is the standard poset for adding $ \gl $ many Cohen subsets of $ \gk $.
We will 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 extensively 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 will rely on
the following consequence of the main result of \cite{Hamkins2003:ExtensionsWithApproximationAndCoverProperties}.
\begin{Thm}\label{T:gapforcing}
(\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 cardinals and supercompact cardinals of the forcing extension already existed in the ground model.
\begin{Def}\label{D:collapseforcing} Let $ \gs $ be a cardinal. Then {\rm Coll}$ (\gs^+,\gs^{++}) $ is the standard L\'evy collapse of $ \gs^{++} $ to $ \gs^+ $
%**
using partial functions from $ \gs^+ $ to $ \gs^{++}$ of cardinality less than $ \gs^+ $ .
\end{Def}
\begin{Def}\label{D:blowupnumnormalmeasures} Let $ \ga < \gg. $ Let $ M_{\ga,\gg} $ be the reverse Easton support iteration using $ {\rm Add} (\gd^+,1) $ at every inaccessible cardinal $ \gd \in (\ga,\gg) $ and trivial forcing at all other stages.
\end{Def}
Standard arguments show that this forcing preserves $\GCH$. In addition, standard arguments (see the first paragraph of the proof of the main theorem of \cite{ACH}) together with Theorem \ref{Theorem.ClosurePoint} show that no new measurable cardinals are created, all ground model measurable cardinals are preserved, and every measurable cardinal $ \gd $ in $ (\ga,\gg) $ carries the maximum number of normal measures, that is, $ 2^{2^{\gd}} = \gd^{++}$.
The basic strategy of our coding is if we would like to code a subset $ A $ of $ \gk $, we take $ \gk $ many measurable cardinals above $ \gk $, blow up the number of normal measures on those $ \gk $ measurable cardinals to the maximum number, then force to reduce the number of normal measures on the $ \rm{\ga^{th}} $ measurable cardinal above $ \gk $ to fewer than the maximum number, according to whether or not $ \ga $ is in $ A $.
Now let us define our coding and the notation $ \gs_\ga $ and $ \gg_\gs $, which will be used throughout the rest of the paper.
\begin{Def}\label{D:numnormmeasCoding} Let $\gs$ be a cardinal. Let $ \gg_{\gs}$ be the supremum of the first $ \gs $ many measurable cardinals beyond $ \gs $. Suppose that $\; A \subseteq \gs $.
For every $ \ga < \gs $, let $ \gs_\ga $ be the $ (\ga +1 )^{\rm{st}} $ measurable cardinal beyond $ \gs $ and then let $ \mathbb C \rm {oll}_{\gs,\gg_{\gs}}(A) $ be the reverse Easton support iteration which forces with {\rm Coll}$ (\gs_\ga^{+},\gs_\ga^{++}) $ for every $ \ga \in A $, and does trivial forcing otherwise.
Now we define $ \bar{\FS}_{\gs,\gg_\gs}(A) = {\rm Add} (\gs^+,1)* \dot\Fcoll_{\gs,\gg_{\gs}}(A).$
\end{Def}
Let $\bar V = V ^{M_{\gs,\gg_\gs}} $ be our ground model.
\begin{Sublemma}Suppose $ A\subseteq\gs, A\in \bar V $. Forcing over $ \bar V $ with $ \bar{\FS}_{\gs,\gg_\gs}(A)$ will give us the required number of normal measures in our forcing extension, that is, in $ \bar V^{ \bar{\FS}_{\gs,\gg_\gs}(A)} $\begin{description}
\item[(i)] if $ \ga \in A,\; \gs_\ga $ carries fewer than the maximum number of normal measures, specifically, $ \gs_\ga^+ $ many normal measures.
\item[(ii)] if $ \ga \notin A,\; \gs_\ga $ carries the maximum number of normal measures, that is, $2^{2^{\gs_\ga}} = \gs_\ga^{ ++} $ many normal measures.
\end{description}
\end{Sublemma}
\begin{proof} We begin by observing that standard arguments show that $ \GCH $ is preserved to $ \bar V^{ \bar{\FS}_{\gs,\gg_\gs}(A)}.$ Also, by Theorem \ref{Theorem.ClosurePoint}, no new measurable cardinals in the open interval $ (\gs,\gg_\gs) $ are created by forcing with $\bar{\FS}_{\gs,\gg_\gs}(A) $. Since our proof will show that all $ \bar V$-measurable cardinals in $ (\gs,\gg_\gs) $ are preserved when forcing with $\bar{\FS}_{\gs,\gg_\gs}(A) $, we will write $ \gs_\ga $ without fear of ambiguity.
\begin{description}
\item[(i)] If $ \ga \in A$, then we force with Coll$ (\gs_\ga^{+},\gs_\ga^{++}) $ at stage $ \gs_\ga $. Let $ \FP_0 $ be the forcing up until stage $ \gs_\ga $, let $ \FP_1 = \Coll (\gs_\ga^{+},\gs_\ga^{++}) $ be the forcing at stage $ \gs_\ga $, and let $ \FP_2 $ be the forcing beyond stage $ \gs_\ga $. Since $| \FP_0 |< \gs_\ga $, and the next forcing is Coll$ (\gs_\ga^{+},\gs_\ga^{++}) $, the arguments of \cite{ACH} hold (Main Theorem, paragraph 2). Namely, $ \bar V^{\FP_0*\dot\FP_1}\models``\gs_\ga $ carries $ \gs_\ga^+ $ many normal measures."
Additionally, any nontrivial forcing beyond $ \gs_\ga $ will be sufficiently closed so as not to add any new normal measures to $ \gs_\ga $. So in $\bar V^{ \bs_{\gs,\gg_\gs}(A)}= \bar V^{\FP_0*\dot\FP_1*\dot\FP_2} $, there are exactly $ \gs_\ga^+ $ many normal measures on $ \gs_\ga $, as desired.
\item[(ii)] If $ \ga \notin A$, then no nontrivial forcing occurs at $ \gs_\ga $. Any nontrivial forcing which occurs below $ \gs_\ga $ will be small with respect to it, and so by \cite{LevySolovay} will not affect the number of normal measures on $ \gs_\ga $. In addition, any nontrivial forcing beyond $ \gs_\ga $ will be sufficiently closed so as not to add any new normal measures to $ \gs_\ga $. So in $\bar V^{ \bs_{\gs,\gg_\gs}(A)}=\bar V^{\FP_0*\dot\FP_1*\dot\FP_2} $, there are exactly $ \gs_\ga^{++}=2^{2^{\gs_\ga}} $ many normal measures on $ \gs_\ga $, as desired.
\end{description}
\end{proof}
In the subsequent sections, we will describe various results that arise from this method.
\section{Extending a property concerning
\textsc{hod}-supercompactness with \textsc{gch}}
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 the following recent result of Sargsyan \cite{Sargsyan}, which answers Woodin's question with the following theorem:
\begin{Thm}\label{T:Sargsyan}(\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 convention and abuse notation by 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\longrightarrow j(\HOD) $ is fully elementary.
%We will say that a supercompact cardinal $ \gk $, as in Theorem \ref{T:Sargsyan}, satisfies the \textit{Sargsyan property}.
In a previous paper, two of the natural questions which arise as a result of this theorem were answered in the affirmative, namely:
\begin{enumerate}
\item \label{Ques:Sargsyan extend to K?} \textit{Can this result be extended to the class of supercompact cardinals, K, assuming K has more than one member?
}\item \label{Ques: Sarg w/o gch?}\textit{Can this result be obtained by doing set forcing over a model of $\ZFC$ that does not satisfy $ \GCH $?}
\end{enumerate}
See \cite{mine1} for more details.
We now can answer another natural question that arises from this theorem, that is, \textit{can this result hold in a forcing extension which still satisfies $ \GCH $}?
Using the coding described in the previous section, we can answer the question in the affirmative with the following theorem:
\begin{Thm}\label{T:Sarg preservingGCH}
Let $ V \models \ZFC + \GCH +`` \gk$ is a supercompact cardinal.'' Then there is a forcing extension $V^\FP$ such that
$V^\FP\models \GCH+ ``\kappa $ is supercompact, but not \HOD -supercompact.''
\end{Thm}
\begin{proof}Let $ f $ be a Laver function for $\gk $ \cite{Laver:Indest}.
Let $ S = \{ \gd:\gd $ is a measurable limit of measurable cardinals, $f``\gd \subseteq \gd $ and $f(\gd)>\gd\}$. Our partial ordering $ \FP $ will be a length $ \gk+1 $ reverse Easton support iteration with $ \FP = \FP_{\gk}*\dot{\rm Add}(\gk,1)$.
Let $\FP_{\gk} = \langle \langle\FP_\ga,\dot\q_\ga\rangle : \ga < \gk\rangle $ with $ \FP_0={\rm Add}(\go,1) $. Let $ \dot\q_\ga $ be a term for trivial forcing, unless $ \ga \in S $.
If $ \ga \in S $, let
$\dot \q_\ga = \dot{\rm Add}(\ga,1)*\dot M_{\ga^*,\gg_{\ga^*}}*\bar\FS_{\ga^*,\gg_{\ga^*}}(\dot X) $, where $ \dot X $ is the name of the generic subset of $ \ga $ added by Add$ (\ga,1) $ and $ \ga^* $ is the least measurable cardinal greater than $ f(\ga)$. Thus, forcing with $ \q_\ga $ introduces a new subset $X$ of $ \ga $ and codes $X$ beyond $ f(\ga) $ by first forcing to blow up the number of normal measures on the first $ \ga^* $ many consecutive measurable cardinals beyond $ \ga^* $ and then forcing to reduce the number of normal measures according to the subset added to $ \ga $.
So $ \eta \in X \leftrightarrow \gs_\eta $ carries $ \gs_\eta^{+} $ many normal measures.
Let $ G \subseteq \FP_\gk$ be $ V $-generic, and let $ g $ be $ V[G] $-generic for $ ({\rm Add}(\gk,1))^{V[G]} $. Standard arguments show that $ V[G][g]\models \GCH $.
We follow the proof of \cite[Lemma 2.1]{Sargsyan}.
\begin{Sublemma}\label{L:kissc2} $ V[G][g] \models ``\gk $ is supercompact.''\end{Sublemma}
\begin{proof}Let $ \gl > \gk $ be an arbitrary strong limit cardinal,
and let $ j: V \longrightarrow M $ be a $ \gl$-supercompactness embedding
with $ j(f)(\gk) = \gl $. Since $\gk \in j(S) $, the stage $ \gk $ forcing
in $ M^{\FP_{\gk}} $ is nontrivial, and we have that
$ j(\FP) = \FP_{\gk}*\dot{\rm Add}(\gk,1)*\dot M_{\gk^*,\gg_{\gk^*}}*
\bar{\dot \FS}_{\gk^*,\gg_{\gk^*}} (\dot X)*\dot \FP_{tail}$,
with $ \dot X $ the name for the generic subset added by Add$ (\gk,1) $, $ \gk^* $ the least measurable cardinal greater than $ \gl $ in $ M $ and $\dot\FP_{tail} $ a term for the forcing defined in the half-open interval $ (\gg_{\gk^*},j(\gk)] .$ Since the first stage of nontrivial forcing in $ M_{\gk^*,\gg_{\gk^*}}*\dot{\bar \FS}_{\gk^*,\gg_{\gk^*}}(\dot X) $ is beyond $ \gl $,
we may write $ j(\FP)$ as $ \FP_{\gk}*\dot{\rm{Add}} (\gk,1)*\dot\FP_{tail} $, where the first stage of nontrivial forcing in $ \dot\FP_{tail} $ is beyond $ \gl $. Standard arguments show (see \cite[Lemma 4.1]{mine1} and \cite{Laver:Indest}) that for any cardinal $\gg <\gl ,\; V[G][g] \models `` \gk$ is $ \gg $-supercompact". Since $ \gl $ was arbitrary, this completes the proof of Lemma \ref{L:kissc2}.
\end{proof}
\begin{Sublemma}\label{L:kisnotHODsc} $ V[G][g] \models``\gk$ is not \HOD -supercompact.''
\end{Sublemma}
\begin{proof}
Since $\FP$ admits a closure point at $ \go $, Sargsyan's argument of \cite[Lemma 2.2]{Sargsyan} shows that $ \gk $ is not $ \HOD $-supercompact in $ V[G][g] $.
%This proof follows closely the proof of Lemma 2.2 in Sargsyan \cite{Sargsyan}.
Namely, let $ G\subseteq\FP $ be $ V $-generic. Factor $ \FP$ as $ \FP_\gk*\dot\FQ_\gk*\dot\FP_{tail}$, where $ \FP_\gk $ is the forcing up to stage $ \gk $, $ \dot\FQ_\gk=\dot{\rm Add}(\gk,1) $ and $\dot \FP_{tail} $ is a term for the forcing beyond $ \gk. $ Let $ G=G_\gk*g*G_{tail}$ be the corresponding factorization of the generic. Assume $ \gk $ is \HOD -supercompact in $ V[G] = W$ and let $\HOD=\HOD^{W}$. Fix a strong limit cardinal $ \gl $ 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 $ and $ j $ is the lift of $ i $.
Let $ N = j(V)= \bigcup_{\ga <\textsc{ord}}i(V_\ga) $. If $ H $ is the $ N$-generic for $ i(\FP) $, then $ M = N[H] = N[j(G)] $. We also have that $ H \cap \FP_{\gk} = G_{\gk}$. Let $ g^{\prime} $ be the generic for Add$ (\gk,1) $ given by $ H $. Then in $ N[H] = M,\; g^{\prime} $ is ordinal definable.
But because $ \gk $ 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][g][G_{tail}]} $. Since $\FP_{tail} $ is $ \gk $-closed, $ g^\prime $ could not have been added by $\FP_{tail} $. So $ g^\prime $ had to have been added over $ V_\gl[G_\gk]$, and more particularly, $ g^\prime $ is added over $ V_\gl[G_\gk] $ 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] $. This is impossible, as $ g^{\prime} $ is a $ V [G_\gk]$-generic object for Add$ (\gk,1) $. Therefore $ \gk $ is not \HOD -supercompact.
\end{proof}
Lemmas \ref{L:kissc2}-\ref{L:kisnotHODsc} prove Theorem \ref{T:Sarg preservingGCH}.
\end{proof}
\section{\textsc{mca} and a proper class of supercompact cardinals}
Another application of this result is forcing \textsc{V=hod} in the presence of a proper class of supercompact cardinals while preserving $ \GCH $. Note that in order to carry out the coding it is only necessary to have a proper class of measurable cardinals.
\begin{Def} The {\rm Measurable Cardinals Coding Axiom (}\textsc{mca}{\rm )} is the assertion that for every cardinal $ \gd $, and
for every $ A \subseteq \gd, \;\exists \gs >\gd$ such that for every $ \ga < \gd ,\; \ga \in A \leftrightarrow \gs_\ga $ carries $ \gs_\ga^{+} $ many normal measures.
\end{Def}
Note that \textsc{mca} implies the existence of a proper class of measurable cardinals.
Following the strategy found in Reitz \cite{ReitzDissertation}, \cite{Reitz:GroundAxiom}, who forces a similar axiom he calls the \textit{Continuum Coding Axiom} ($\textsc{cca}$), we will force the $ \textsc{mca} $,
which we note is a strong form of \textsc{V=hod}.
%$ V=\HOD$.
%Let $ V\models \ZFC + \GCH + $``There exists a proper class of measurable cardinals."
We define a building block of our forcing, which we call the lottery sum after Hamkins \cite{Hamkins2000:LotteryPreparation}. Specifically, the {\em lottery sum} of a collection $ A $ of posets 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 $. Since all compatible conditions must be in the same $ \FQ $, the forcing effectively holds a lottery of all the posets in $ A $, and the generic chooses the ``winning'' poset $ \FQ $ and then forces with it.
We now present the following theorem:
\begin{Thm}\label{T:v=hod+gch+K}
Let $ V\models \ZFC + \GCH + ``$There is a (proper) class of supercompact cardinals $ K $.''
Then there is a class forcing $ \FQ\subseteq V $ such that
$ V^\FQ \models \ZFC + \GCH + \textsc{V=hod} + `` K $ is the class of supercompact cardinals."
\end{Thm}
\begin{proof}
Our proof will combine the methods we have introduced with a technique
due to Brooke-Taylor, which was given by
him in \cite{BrookeTaylor:diamondstar}.
%of Sy Friedman which was told to us by Joel Hamkins.
Let $ \FM $ be the reverse Easton support class
iteration using $ {\rm Add} (\gd^+,1) $ at every
inaccessible cardinal $ \gd $ and trivial forcing at all other stages.
Let $ \bar V=V^\FM $.
In analogy to what we discussed in the paragraph immediately following Definition \ref{D:blowupnumnormalmeasures}, $\bar V\models \GCH+
$ ``Every measurable cardinal carries the
maximum number of normal measures." In addition, standard arguments show
that for every $ \gk\in K,\;\bar V\,\models ``\gk$ is supercompact."
Therefore, by our discussion in the paragraph immediately following the statement of Theorem \ref{T:gapforcing},
$ \bar V\models``K $ is the class of supercompact cardinals."
%Standard arguments show that this forcing preserves \GCH. In addition
%standard arguments (see the first paragraph of the proof of the
%main theorem of \cite{ACH}) show that every measurable cardinal
%$ \gs$ carries the
%maximum number of normal measures.
%,that is, $ 2^{2^{\gs}} = \gs^{++}$.
%$ V^\FM\models $ if $ \gs $ is measurable, $ \gs $ carries the
%maximum number of normal measures.,that is,$ 2^{2^{\gs}} = \gs^{++}$.
We now work in $ \bar V $.
Let $ \mathbb C_\gs= \rm {Coll}(\gs^+,\gs^{++}) $ and let $ \mathbb B_\gs = \{\emptyset\} $.
Let $ \FP $ be the reverse Easton support class iteration defined as $ \FP= {\rm Add} (\go,1) *\langle \dot\q_\gs : \gs \in \textsc{ord}\rangle $.
Then $\dot \q_\gs $ will be taken as a term for trivial forcing unless $ \gs\in \bar V $ is a ``successor" measurable cardinal, that is, $ \gs $ is not a measurable limit of measurable cardinals in $ \bar V $.
At a nontrivial stage of forcing $ \gs $, we take $\dot \q_\gs $ as a term for the lottery sum between the
collapse forcing $ \mathbb C_\gs$ and trivial forcing, that is,
$\dot\q_\gs= \oplus\{\dot{\mathbb C}_\gs, \dot{\mathbb B}_\gs \}=\oplus\{\rm {\dot {Coll}}(\gs^+,\gs^{++}), \{\emptyset\}\}$.
We let the ``generic decide" which ``bit" of information will be coded.
\begin{Sublemma}\label{L:extensionmodelsv=hod}
$ \bar V^\FP\models $ \textsc{mca}. In particular, in $ \bar V^{\FP} $,
every set of ordinals is ordinal definable, that is, $ \bar V^\FP\models$
\textsc{V=hod}.
\end{Sublemma}
\begin{proof}
First of all, we need a lemma to ensure that our coding is the same in $ \bar V^\FP $ as in $ \bar V $.
\begin{Sublemma}\label{L:successormeasaresameinext}
$ \bar V^\FP \models ``\gd $ is a successor measurable cardinal'' $ \leftrightarrow \bar V \models ``\gd$ is a successor measurable cardinal.''
\end{Sublemma}
\begin{proof}
Since $ \FP $ admits a closure point at $ \go $, by Theorem \ref{Theorem.ClosurePoint}, no new measurable cardinals were created by $ \FP $.
\begin{description}
\item[(i)]$ \Longleftarrow $ Let $ \gd $ be a successor measurable cardinal in $ \bar V $. Let $ \FP=\FP_\gd*\dot\FP^\gd $, where $ \FP_\gd $ is the forcing up to stage $ \gd $.
% and $ \FP^\gd $ is the forcing at stage $ \gd $ and beyond.
Since $ \gd $ is a successor measurable cardinal in $ \bar V $, $ |\FP_\gd|<\gd $. By the results of \cite{LevySolovay}, forcing with $ \FP_\gd $ will therefore not affect the measurability of $ \gd $, and the remnant of the forcing is sufficiently closed so as not to affect $ \gd $'s measurability. So $ \gd $ is measurable in $ \bar V^\FP $.
Suppose that $ \gd $ is now a measurable limit of measurable cardinals in $\bar V^\FP $. If that were the case, then many new measurable cardinals would have to have been created by $ \FP $. As before, by Theorem \ref{Theorem.ClosurePoint}, no new measurable cardinals were created. So successor measurable cardinals are preserved to $ \bar V^\FP $. In particular, $ \gd $ is a successor measurable cardinal in $ \bar V^\FP $.
\item[(ii)]$ \Longrightarrow $ Let $ \gd $ be a successor measurable cardinal in $ \bar V^\FP $. Since no new measurable cardinals were created by $ \FP $, we know that $ \gd $ was measurable in $ \bar V $. Suppose that $ \gd $ were a measurable limit of measurable cardinals in $ \bar V $. In particular, it was a measurable limit of successor measurable cardinals. But the successor measurable cardinals are preserved by $ \FP $ (see (i)), so $ \gd $ would remain a measurable limit of measurable cardinals in $ \bar V^\FP $. This is a contradiction, so $ \gd $ was a successor measurable cardinal in $ \bar V $. \end{description}
\end{proof}
We return to the proof of Lemma \ref{L:extensionmodelsv=hod}.
Now it will suffice to show that $ \forall A\subseteq $ \ORD,
$ A \in \bar V^\FP$ , $ D_A = \{p\in \FP: p \Vdash ``\dot A
\rm{\;is\; coded,\; as\; in\; the\; statement\; of\; \textsc{mca}"}\} $
is dense in $ \FP $.
Fix some $ A \in \bar V^\FP.$ There exists a $ \gd $ such that
$ A\in \bar V^{\FP_\gd} $.
Fix any $ p \in \FP $ and an ordinal $ \varrho $ such that $ p \Vdash ``\dot A \subseteq \varrho" $. Since $ \FP $ uses Easton support, support($ p) \subseteq \gg $ for some $ \gg $.
Let $ \gs $ be the least measurable cardinal $ > \rm{max}(\gg,\gd,\varrho) $. Since there exists a proper class of supercompact cardinals, in particular $ \gs $ exists,
and there are $ \varrho $ many measurable cardinals above $ \gs $.
This allows us to use Brooke-Taylor's coding technique of
\cite{BrookeTaylor:diamondstar}.
We now extend $ p $ by going to $\gs $ and then
if $ \ga\in A $, at stage $ \gs_\ga $, extend $ p $ to a condition
which forces that $\mathbb C_{\gs_\ga} $ will be chosen. If $ \ga \notin A $, we extend $ p $ to choose $\mathbb B_{\gs_\ga} $.
Therefore $ p $ can be extended to code $ A $ as in the statement of \textsc{mca} for any $ A\in \bar V^\FP,\; A\subseteq \textsc{ord} $.
Therefore $ \bar V^\FP\models $ \textsc{mca},
so $ \bar V^\FP\models$ \textsc{V=hod}.\end{proof}
\begin{Sublemma}\label{L:K is classofscinext} $ \bar V^\FP\models ``K$ is the class of supercompact cardinals.''
\end{Sublemma}
\begin{proof} Again, since $ \FP $ admits a closure point at $ \go $, by an application of Theorem \ref{Theorem.ClosurePoint}, no new supercompact cardinals were created by $ \FP $. So it suffices to show that all supercompact cardinals in $ \bar V $ are preserved to $ \bar V^\FP $.
Let $ \gk \in K $. Now it remains to show that $ \gk$ is supercompact in $ \bar V^\FP $.
Let $ G\subseteq \FP $ be $ \bar V $-generic.
Let $ \gl >\gk $ be such that $ \gl $ has cofinality $ \gk $ and is a limit of measurable cardinals. Note that $ \gl^{<\gk}=\gl $.
Let $ \FP = \FP_\gk*\dot\FP_{\gk,\gl}*\dot\FP_{tail} $ and let $ G = G_\gk*G_{\gk,\gl}*G_{tail} $,
where $ \FP_\gk $ is the forcing up to stage $ \gk$, $\dot \FP_{\gk,\gl} $ is a term for the forcing from $ \gk $ to $\gl$, and $ \dot\FP_{tail} $ is a term for the forcing beyond $ \gl $.
$\FP_{tail} $ will be sufficiently closed so that by the choice of $ \gl $, $ \bar V^{\FP_\gk*\dot\FP_{\gk,\gl}}\models ``\gk $ is $ \gl $-supercompact" $ \leftrightarrow \bar V^\FP \models ``\gk$ is $\gl$-supercompact."
Since $ \gl $ may be chosen to be arbitrarily large, it will suffice to show that $ \bar V^{\FP_\gk*\dot\FP_{\gk,\gl}}\models ``\gk $ is $ \gl $-supercompact."
To this end, let $ j: \bar V \longrightarrow M $ be a $ 2^{\gl}$-supercompactness embedding for $ \gk $. Since $ M $ and $ \bar V $ agree up to $ 2^{\gl} $, it follows that up to and including stage $ \gl $, this forcing is the same in $ M $ as it is in $ \bar V $ and that we may factor $j(\FP_\gk*\dot\FP_{\gk,\gl})$ as $\FP_\gk*\dot\FP_{\gk,\gl}*\dot\FP_{\gl,j(\gk)}*j(\dot\FP_{\gk,\gl}) $, where $ \dot\FP_{\gl,j(\gk)} $ is a term for the forcing defined in the open interval $( \gl,j(\gk)) $ and $ j(\dot\FP_{\gk,\gl}) $ is a term for the forcing defined in the closed interval $ [j(\gk),j(\gl)] $.
As in the proof of Lemma \ref{L:kissc2}, standard arguments
(see \cite[Lemma 4.1]{mine1} and \cite{Laver:Indest})
once again show that $\bar V^{\FP_\gk*\dot\FP_{\gk,\gl}}\models``\gk$ is $ \gl $-supercompact."
This proves Lemma \ref{L:K is classofscinext}.\end{proof}
Standard arguments show that $ \bar{V}^\FP \models \GCH$. With $ \FQ=\FM*\dot \FP $, Lemmas \ref{L:extensionmodelsv=hod}-\ref{L:K is classofscinext} prove Theorem \ref{T:v=hod+gch+K}.
\end{proof}
We note that the results of this section produce a model of set theory
that satisfies the Ground Axiom introduced by
Hamkins \cite{Hamkins2005:TheGroundAxiom} and
Reitz \cite{ReitzDissertation}, \cite{Reitz:GroundAxiom}.
\begin{Def}\label{D:GAformal}The \emph{Ground Axiom} ($\GA$) is the assertion that the universe of sets $V$ is not a forcing extension of any inner model $W\subseteq V$ by nontrivial set forcing $\FP\in W$.
\end{Def}
In particular, we can produce a model of set theory which
satisfies \textsc{V=hod} $ +\ \GA+ \GCH \;+$ ``There exists a proper
class of supercompact cardinals" with the following additional theorem.
%This theorem follows by Reitz's work \cite{ReitzDissertation}, \cite{Reitz:GroundAxiom}.
\begin{Thm}\label{T:MCAimpliesGA} The \textsc{mca} implies the \GA.
\end{Thm}
\begin{proof} We follow Reitz's proof of
\cite[Theorem 9]{ReitzDissertation} and \cite[Theorem 10]{Reitz:GroundAxiom}. Suppose $ V \models$ \textsc{mca.} Suppose further
that $V $ is a set forcing extension of an inner model $V = W[h]$, where $h$ is
$W$-generic for some poset $ \FQ \in W$. For $\gs > |\FQ|$, by the results of \cite{LevySolovay}, the models $W$ and $V$ will
agree on the properties ``$ \gs $ is a measurable cardinal" and ``$\gs $ carries $ \gs^+ $ many normal measures." Every set of ordinals $ A $ in $V$ is coded into the ``number of normal measures" function of $V$. The claim is that one such code for $ A $ must appear above $|\FQ|$. If $A \subseteq\gd$ and $|\FQ| = \gg$, let $ \gg^*=\rm{max}(\gg,\gd) $. Then by the \textsc{mca}, there exists a $ \gs>\gg^* $ such that $A $ is coded using the first $ \gd $ many measurable cardinals beyond $ \gs $. Thus $A$ is coded into the ``number of normal measures" function of $V$ above $|\FQ|$, and so the code appears also in $W$. Thus $A\in W$, and so every set of ordinals of $V$ is also in $W$. This shows that $V = W$, and so the
forcing $\FQ$ was trivial. Thus $V\models $ \GA.
\end{proof}
Hence, the model constructed in Theorem \ref{T:v=hod+gch+K} is the witnessing model.
\section{The Wholeness Axiom, \textsc{wa}$ _0 $, with \textsc{V=hod},
\textsc{ga} and \textsc{gch}}
%and \GCH}
Using the methods of this paper,
we can extend a result of Hamkins \cite{Hamkins2001:WholenessAxiom} involving the hierarchy of Wholeness Axioms
and \textsc{V=hod}, which are derived from the Wholeness Axiom, proposed by Paul Corazza \cite{Corazza}. The Wholeness Axiom is intended as a
slight weakening of Kunen's famous inconsistency result \cite{Kunen}
concerning the nonexistence of a nontrivial elementary embedding
from the universe to itself.
The hierarchy of Wholeness Axioms is
formalized in the language $ \{\in,\textbf{j}\} $,
augmenting the usual language of set theory $ \{\in\} $
with an additional unary function symbol $ \textbf{j}$ to
represent the embedding.
We begin with the following definitions.
\begin{Def} The Wholeness Axiom \textsc{wa}$ _n $, where $ n$ is among $ 0,1,...,\infty $, consists of the following:
\begin{enumerate}
\item (Elementarity) All instances of $ \varphi(x)\leftrightarrow \varphi(\rm{\textbf{j}}(x)) $ for $ \varphi $ in the language $ \{\in\} $.
\item (Separation) All instances of the $\sum_n$-Separation Axioms for formulae in the full language $ \{\in, \rm{\textbf{j}}\} $.
\item (Nontriviality) The axiom $ \exists x (\rm{\textbf{j}}(x)\neq x) $.
\end{enumerate}
\end{Def}
The following theorem generalizes the Main Theorem of \cite{Hamkins2001:WholenessAxiom}.
\begin{Thm}\label{wholeness} If the Wholeness Axiom \textsc{wa}$_0 $
is itself consistent,
then it is consistent with \textsc{V=hod} + \GA + \GCH.
\end{Thm}
\begin{proof}
Suppose that $ V\models$ \textsc{wa}$_0$. As in \cite{Hamkins2001:WholenessAxiom},
every model of one of the Wholeness Axioms has the
form $ \langle V,\in,j \rangle $, where $\langle V,\in \rangle$
satisfies $ \ZFC $ and $ j:V\longrightarrow V $
is a nontrivial amenable elementary embedding.
We now have the following easy lemma.
\begin{Sublemma}\label{wholenessaxiom}
If $ V\models$ \textsc{wa}$_0$, then $ V \models
$``There exists a proper class of measurable cardinals."\end{Sublemma}
\begin{proof}
Let $ j:V\longrightarrow V $ be the witnessing embedding,
with critical sequence $ \{\gk_n : n\in \go\} $,
defined by $ \gk_0=\gk= \rm{cp}(j) $ and $ \gk_{n+1}=j(\gk_n)$.
Since $ j\upharpoonright V_\gk $ is the identity function,
it follows that $ V_\gk\prec V_{\gk_1} $.
As in \cite{Hamkins2001:WholenessAxiom}, by iteratively applying
$ j $ to this fact, one easily concludes that
\begin{center}
$ V_{\gk_0}\prec V_{\gk_1}\prec V_{\gk_2}\prec \cdots \prec V$.
\end{center}
By \cite{Hamkins2001:WholenessAxiom}, $ V\models ``\gk$ is supercompact."
Hence, $ V_\gk\models $ ``There exists a proper class
of measurable cardinals." But since $ V_{\gk}\prec V$,
$ V\models $ ``There exists a proper class of
measurable cardinals."\end{proof}
We take this opportunity to make several remarks
concerning Lemma \ref{wholenessaxiom}.
First, as pointed out to us by Paul Corazza, \cite{Corazza:elemembedings}
shows that $ \ZFC +$ \textsc{wa}$_0$ implies the existence of a proper class of cardinals that are super-$n$-huge for any particular $ n\in \go $ (and more). Thus, $ V_\gk $ has an incredibly rich large cardinal structure.
In addition, as Corazza has pointed out, our proof of Lemma \ref{wholenessaxiom} skirts an ambiguity (which poses no difficulties in our case) that is addressed more precisely in
\cite[Section 8]{Corazza:elemembedings}.
Also, as the referee has observed,
it is possible to prove Lemma \ref{wholenessaxiom} as follows.
Since $ \gk_0= {\rm cp}(j), \;\gk_0 $ is measurable, so by elementarity, each member of the critical sequence $ \{\gk_n : n\in \go\} $ is also measurable.
By Kunen's original proof of \cite{Kunen}, the critical sequence $ \{\gk_n : n\in \go\} $ is cofinal in $ V $.
Therefore $ V\models ``\forall \ga \exists \gb >\ga (\gb $ is a measurable
cardinal)", so $ V\models ``$There exists a proper class of
measurable cardinals."
Returning to the proof of Theorem \ref{wholeness}, as in \cite{Hamkins2001:WholenessAxiom}, we force with the usual reverse
Easton support class iteration that forces \GCH.
Thus, at any cardinal $ \gg $, we force with
${\rm Add} (\gg^+,1) $.
By \cite{Hamkins2001:WholenessAxiom},
the resulting forcing extension $\bar V$
satisfies \textsc{wa}$ _0 $ and \GCH.
Furthermore, as in \cite{ACH}, this forcing ensures
that every measurable cardinal $ \gs$ carries
the maximum number of normal measures.
%, that is, $\gs$ carries $ 2^{2^{\gs}} = \gs^{++}$ many normal measures.
We define now in $\bar V$ the poset $\FP_\gk$
used in the construction of the witnessing model
for Theorem \ref{wholeness}.
$ \FP_\gk $ is a reverse Easton support $ \gk $-iteration, defined as
% Let $ \mathbb C_\gs= \rm {Coll}(\gs^+,\gs^{++}) $. Let $ \mathbb B_\gs = \{\emptyset\} $.\\
$ \FP_\gk= {\rm Add} (\go,1) * \langle \dot \q_\gs : \gs <\gk\rangle $.
Here, $\dot \q_\gs $ is
a term for trivial forcing unless $ \gs $ is a
``successor" measurable cardinal.
When this is true, $\dot \q_\gs $ is
a term for the poset defined as in Theorem \ref{T:v=hod+gch+K}.
% At a nontrival stage of forcing, $ \gs $, we let $ \q_\gs $ be the lottery sum between the collapse forcing $ \mathbb C_\gs$ and trivial forcing. ,that is, $\q_\gs= \oplus\{\mathbb C_\gs, \mathbb B_\gs \}=\oplus\{\rm {Coll}(\gs^+,\gs^{++}), \{\emptyset\}\}$.
% We let the "generic decide" which "bit" of information will be coded.\\
As we have observed previously, this iteration preserves \GCH.
%and codes sets of $ \bar V_\gk $ unboundedly often below $\kappa$.
Our earlier arguments also show that this iteration forces
\textsc{V=hod} in $ \bar V_\gk^{\FP_\gk} $. In addition, our earlier arguments
also show that $ \bar V_\gk^{\FP_\gk}\models$ \textsc{ga}.
The same lifting arguments as given in \cite{Hamkins2001:WholenessAxiom}
(literally presented unchanged) now show that
$ \bar V_\gk^{\FP_\gk}\models$ \textsc{wa}$_0$.
%$ \bar V_\gk^{\FP_\gk} $ satisfies \textsc{wa$_0$}.
\end{proof}
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