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\title{$L$-like Combinatorial Principles
and Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
level by level equivalence between strong
compactness and supercompactness,
diamond, square,
morass, locally defined well-ordering.}}
\author{Arthur W.~Apter\thanks{The
author's research was
partially supported by
PSC-CUNY grants and CUNY
Collaborative Incentive grants.}
\thanks{The author wishes to thank the
referee for helpful comments,
suggestions, and corrections 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 2, 2009\\
(revised March 9, 2009)}
\begin{document}
\maketitle
\begin{abstract}
We force and construct
a model in which GCH and
level by level
equivalence between strong
compactness and supercompactness
hold, along with certain
additional ``$L$-like'' combinatorial
principles.
In particular, this model
satisfies the following properties:
\begin{enumerate}
\item\label{i1a} $\diamondsuit_\gd$ holds for
every successor and Mahlo cardinal
$\gd$.
\item\label{i2a} There is a stationary subset
$S$ of the least supercompact cardinal $\gk_0$
such that for every $\gd \in S$,
$\square_\gd$ holds and $\gd$
carries a gap 1 morass.
\item\label{i3a} A weak version of
$\square_\gd$ holds for every infinite
cardinal $\gd$.
\item\label{i4a} There is a locally defined well-ordering
of the universe ${\cal W}$, i.e., for all
$\gk \ge \ha_2$ a regular cardinal,
${\cal W} \rest H(\gk^+)$ is definable
over the structure $\la H(\gk^+), \in \ra$
by a parameter free formula.
\end{enumerate}
The model constructed amalgamates
and synthesizes results due to the
author, the author and Cummings, and
Asper\'o and Sy Friedman. It has no restrictions
on the structure of its class
of supercompact cardinals %its large cardinal structure
and may be considered
as part of Friedman's ``outer model programme''.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
In \cite{F07}, Sy Friedman introduced the
{\em outer model programme}, whose goal is to
construct ``outer models'', i.e., forcing extensions,
of models of ZFC containing very large cardinals
in which $L$-like principles hold.
Examples of papers falling under this rubric
include, but are not necessarily limited to,
\cite{A05}, \cite{AC08}, \cite{F07}, \cite{F08},
\cite{A09}, \cite{AF}, \cite{B},
\cite{BF}, and \cite{CS}.
The purpose of this paper is to produce
another contribution to Friedman's
outer model programme.
We show that the
existence of gap 1 morasses on a
stationary subset of the least
supercompact cardinal $\gk_0$ and
a locally defined well-ordering of
the universe is consistent with
level by level equivalence between
strong compactness and supercompactness,
together with certain instances of
diamond and square.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + GCH +
$\K \neq \emptyset$ is the class of supercompact cardinals''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH +
$\K$ is the class of supercompact cardinals''.
In $V^\FP$, level by level equivalence between
strong compactness and supercompactness holds.
In addition, $V^\FP$ satisfies
the following combinatorial properties:
\begin{enumerate}
\item\label{j1} $\diamondsuit_\gd$ holds for
every successor and Mahlo cardinal
$\gd$.\footnote{Of course, the fact that $\diamondsuit_\gd$
holds for every successor cardinal $\gd > \ha_1$
follows from GCH, by Shelah's famous result \cite{Sh1}
on diamonds.}
\item\label{j2} There is a stationary subset
$S$ of the least supercompact cardinal $\gk_0$
such that for every $\gd \in S$,
$\square_\gd$ holds and $\gd$
carries a gap 1 morass.
\item\label{j3} A weak version of
$\square_\gd$ holds for every infinite cardinal $\gd$.
In particular, $\square^T_\gd$ holds for
every infinite cardinal $\gd$, where
$T = {\rm Safe}(\gd)$ is a certain final
segment of regular cardinals less than or
equal to $\gd$.\footnote{The exact definition of
Safe$(\gd)$ will be given in Definition \ref{d1}.}
\item\label{j4} There is a locally defined well-ordering
of the universe ${\cal W}$, i.e., for all
$\gk \ge \ha_2$ a regular cardinal,
${\cal W} \rest H(\gk^+)$ is definable
over the structure $\la H(\gk^+), \in \ra$
by a parameter free formula.
\end{enumerate}
\end{theorem}
The model witnessing the conclusions of
Theorem \ref{t1} amalgamates and synthesizes
the results of \cite{A05}, \cite{AC08}, and \cite{AF}.
Since our ground model has no restrictions
on the structure of its class of supercompact
cardinals, neither does the resulting generic extension.
It is produced by forcing over the model of
\cite{AC08} with a modified version of one
of the partial orderings used in \cite{A05}
(which will add gap 1 morasses and square
sequences to each member of a certain
stationary subset of the least
supercompact cardinal) and the partial
ordering of \cite{AF}.
The key step will be to show that
the forcing of \cite{AF} preserves the
combinatorial properties previously added,
together with level by level equivalence between
strong compactness and supercompactness.
Before beginning the proof of Theorem \ref{t1},
we give the most pertinent definitions
and facts, many of which are taken from
\cite{A05} or \cite{AC08}. When forcing,
$q \ge p$ means that $q$ {\em is stronger than} $p$.
Let $\gk$ be a cardinal.
We abuse notation slightly by using both
$V[G]$ and $V^\FP$ to denote the generic
extension of $V$ by $\FP$, assuming that
$G$ is $V$-generic over $\FP$.
The partial ordering $\FP$ is {\em $\gk$-directed closed}
if every directed set of elements of $\FP$ of size
less than $\gk$ has an upper bound.
$\FP$ is {\em $\gk$-closed} if every increasing
chain of elements of $\FP$ of size less than
$\gk$ has an upper bound.
%Observe that if $\FP$ is $\gk$-closed, then $\FP$ is actually
%{\em ${\prec}\gk$-closed}, i.e., it is possible to
%construct an increasing chain $\la p_\ga \mid \ga < \gk \ra$
%of elements of $\FP$.
$\FP$ is {\em $\gk$-strategically closed} if in the
two person game in which the players construct an
increasing sequence $\la p_\ga \mid \ga \le \gk \ra$
of elements of $\FP$, where player I plays odd
stages and player II plays even stages
(choosing the trivial condition at stage $0$),
player II has a strategy ensuring the game can always be continued.
$\FP$ is {\em ${\prec}\gk$-strategically closed} if in the
two person game in which the players construct an
increasing sequence $\la p_\ga \mid \ga < \gk \ra$
of elements of $\FP$, where player I plays odd
stages and player II plays even stages
(choosing the trivial condition at stage $0$),
player II has a strategy ensuring the game can always be continued.
%Note that if $\FP$ is $\gk$-directed closed, then
%$\FP$ is both $\gk$-closed and ${\prec}\gk$-strategically closed.
Note that for any partial ordering $\FP$,
we have the chain of implications
$\gk$-directed closed $\implies$ $\gk$-closed
$\implies$ ${\prec}\gk$-strategically closed.
In addition, if $\FP$ is $\gk$-strategically
closed and $f : \gk \to V$ is a function in
$V^\FP$, then $f \in V$.
Next, we state
a result which will be used in the proof
of Theorem \ref{t1}. This is
a corollary of Theorems 3 and 31
and Corollary 14 of
Hamkins' paper \cite{H5}.
This theorem is a generalization of
Hamkins' Gap Forcing Theorem and
Corollary 16 of
\cite{H2} and \cite{H3}
(and we refer readers to \cite{H2},
\cite{H3}, and \cite{H5} for
further details).
We therefore state the theorem
we will be using now, along
with some associated terminology.
%quoting freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} \le \gd$,
$\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H5},
$\FP$ {\em admits a closure point at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{H3},
$\FP$ is {\em mild
with respect to a cardinal $\gk$}
iff every set of ordinals $x$ in
$V^\FP$ of size below $\gk$ has
a ``nice'' name $\tau$
in $V$ of size below $\gk$,
i.e., there is a set $y$ in $V$,
$|y| <\gk$, such that any ordinal
forced by a condition in $\FP$
to be in $\tau$ is an element of $y$.
Also, as in the terminology of
\cite{H2}, \cite{H3}, \cite{H5},
and elsewhere,
an embedding
$j : \ov V \to \ov M$ is
{\em amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific corollary of
Theorems 3 and 31
and Corollary 14 of
\cite{H5} we will be using
is then the following.
\begin{theorem}\label{t2}
%(Hamkins' Gap Forcing Theorem)
{\bf(Hamkins)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing that
admits a closure point
at some regular $\gd < \gk$.
Suppose further that
$j: V[G] \to M[j(G)]$ is an embedding
with critical point $\gk$ for which
$M[j(G)] \subseteq V[G]$ and
${M[j(G)]}^\gd \subseteq M[j(G)]$ in $V[G]$.
Then $M \subseteq V$; indeed,
$M = V \cap M[j(G)]$. If the full embedding
$j$ is amenable to $V[G]$, then the
restricted embedding
$j \rest V : V \to M$ is amenable to $V$.
If $j$ is definable from parameters
(such as a measure or extender) in $V[G]$,
then the restricted embedding
$j \rest V$ is definable from the names
of those parameters in $V$.
Finally, for any pair of cardinals $\gl \ge \gk$,
if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$ strongly compact in $V[G]$, then
$\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
It immediately follows from Theorem \ref{t2}
that any cardinal $\gk$ which is $\gl$ supercompact
in a generic extension obtained
by forcing that admits a closure point
below $\gk$ (such as at or less than the
least inaccessible cardinal) %at $\go$)
must also be $\gl$ supercompact
in the ground model.
In particular, if $\ov V$ is a forcing extension
of $V$ by a partial ordering that admits a closure point
%below the least supercompact cardinal
at or less than the least inaccessible cardinal %$\go$
in which each supercompact cardinal is preserved,
the class of supercompact cardinals in $\ov V$ remains
the same as in $V$.
%We conclude Section \ref{s1} with a short discussion of some
%important terminology.
Suppose $V$ is a model of ZFC
in which for all 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.
Such a model will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
%We will also say that {\em $\gk$ is a witness
%to level by level equivalence between
%strong compactness and supercompactness}
%iff for every regular cardinal $\gl > \gk$,
%$\gk$ is $\gl$ strongly compact iff
%$\gk$ is $\gl$ supercompact.
Note that the exception
is provided by a theorem of Menas \cite{Me},
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.
%When this situation occurs, the
%terminology we will henceforth
%use is that $\gk$ is a witness
%to the Menas exception at $\gl$.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
We presume familiarity with the
combinatorial notions of
$\square$, $\diamondsuit$ and its
variants, and morasses. We refer
readers to \cite{A05}, \cite{AC08}, and \cite{BF}
for further information.
Since it is somewhat less familiar,
however, we do state
a weak version of $\square_\gg$,
$\square^T_\gg$,
compatible with supercompact
cardinals. Square sequences of this kind were first shown to be
consistent with supercompactness by Foreman and Magidor \cite[p.~191]{FM},
using techniques
of Baumgartner. In the notation of
Definition \ref{funnysquare}, Foreman and Magidor
showed that $\square_{\kappa^{+\omega}}^{ \{ \kappa^{+n} \mid
n < \omega \}}$
is consistent with $\kappa$ being supercompact.
Given a set $T$ of regular cardinals, we denote by ${\rm cof}(T)$
the class of ordinals $\alpha$ such that ${\rm cf}(\alpha) \in T$.
\begin{definition} \label{funnysquare} Let $\gamma$ be an infinite
cardinal, and let $T$ be
a set of regular cardinals which are less than or equal to $\gamma$.
Then a {\em $\square_\gamma^T$sequence} is a sequence
$\langle C_\alpha\mid \alpha \in \gamma^+ \cap {\rm cof}(T) \rangle$
such that
\begin{enumerate}
\item $C_\alpha$ is club in $\alpha$, and ${\rm ot}(C_\alpha) \le \gamma$.
\item If $\beta \in \lim(C_\alpha) \cap \lim(C_{\alpha'})$, then
$C_\alpha \cap \beta = C_{\alpha'} \cap \beta$.
\end{enumerate}
{\em $\square_\gamma^T$ holds} if and only if there is a
{\em $\square_\gamma^T$sequence}.
\end{definition}
%We note that if $T^* = (\gamma^+ \cap {\rm cof}(T)) \cup
%\{ \eta \mid \exists \beta \in
%\gg^+ \cap {\rm cof}(T) [\eta \in
%\lim(C_\beta)] \}$, then we can trivially
%extend the domain of the sequence
%to $T^*$, by defining $C_\eta = C_\beta \cap \eta$
%for some (any) $\beta$ with $\eta \in \lim(C_\beta)$.
\begin{definition}\label{d1}
For each infinite cardinal $\gamma$, a regular cardinal
$\mu$ is {\em safe for $\gamma$} if and only if
\begin{enumerate}
\item $\mu \le \gamma$.
\item For every cardinal $\lambda \le \gamma$, if $\lambda$ is
$\gamma^+$ supercompact, then
$\lambda \le \mu$.
\end{enumerate}
${\rm Safe}(\gamma)$ is the set of safe regular cardinals for $\gamma$.
\end{definition}
We observe that the safe set is a final segment
of ${\rm REG} \cap (\gamma+1)$, and that
the safe set can only be empty when $\gamma$ is a singular
limit of cardinals which are $\gamma^+$ supercompact.
In addition, by the remarks immediately following
the statement of Theorem \ref{t2}, ${\rm Safe}(\gg)$
is upwards absolute to any cofinality (and hence cardinal)
preserving generic extension by a partial ordering admitting
a closure point at or below the
least inaccessible cardinal. %$\go$.
\section{The Proof of Theorem \ref{t1}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH +
$\K \neq \emptyset$ is the class of
supercompact cardinals''. By first
forcing with the partial ordering of
\cite{AC08},% and with a slight abuse of notation,
we assume in addition that
level by level equivalence between strong
compactness and supercompactness holds in $V$ and that
$V \models ``\diamondsuit_\gd$ holds for
every regular cardinal $\gd$\footnote{In fact,
forcing with the partial ordering of \cite{AC08}
actually allows us to assume that
$V \models ``\diamondsuit^+_\gd$ holds for
every $\gd$ which is the successor of a
regular cardinal''.} + $\square^T_\gd$
holds for every infinite cardinal $\gd$, where
$T = {\rm Safe}(\gd)$''.
Let $\gk_0$ be the least
supercompact cardinal, and
let $S = \{\gd < \gk_0 \mid \gd$ is a non-measurable
Mahlo limit of strong cardinals$\}$.
As in \cite{A05}, $S$ is stationary.
This allows us to define our first partial ordering
$\FP^* = \la \la \FP_\delta, \dot \FQ_\delta \ra \mid
\delta < \gk_0 \ra$ as the reverse Easton iteration
of length $\gk_0$ which begins by forcing with
$\add(\ha_1, 1)$, the partial ordering for adding a Cohen
subset of $\ha_1$.
For any stage $\delta > 0$, $\dot \FQ_\delta$ is
a term for the trivial partial ordering
$\{\emptyset\}$, except if $\delta \in S$.
Under these circumstances, $\dot \FQ_\delta =
\dot \FQ^0_\delta \ast \dot \FQ^1_\delta$, where
$\dot \FQ^0_\delta$ is a term for the partial
ordering of \cite{A05} which adds a
$\square_\delta$ sequence, and $\dot \FQ^1_\delta$
is a term for the partial ordering of
\cite{F07} (see also \cite{BF}) which
adds a gap 1 morass at $\delta$.\footnote{Quoting
\cite{A05}, the partial ordering %$\FQ^0_\delta$ consists
for adding a $\square_\delta$ sequence consists
of proper initial segments of $\square_\delta$ sequences,
ordered by end-extension. Quoting \cite{F07},
the partial ordering
for adding a gap 1 morass at $\delta$ consists of
proper initial segments of a gap 1 morass at
$\delta$ up to some top level, together with a
map of an initial segment of this top level into
$\delta^+$ which obeys the requirements of a
morass map. The ordering is by
end-extending the morass up to its
top level and requiring that the map
from the given initial segment of its top level
into $\delta^+$ factors as the composition of a
map into the top level of the stronger condition,
followed by the map given by the stronger
condition into $\delta^+$.}
Since $\FP^*$ is a reverse Easton
iteration of length $\gk_0$ and $\gk_0$
is Mahlo, $\FP^*$ is $\gk_0$-c.c.
Hence, it follows that
$V^{\FP^*} \models ``S$ is a stationary
subset of $\gk_0$''.
Suppose $\delta \in S$. It is
inductively the case that
forcing with either
$\FP_\delta$ or $\FP_\delta \ast \dot \FQ^0_\delta$
%and $\FP_\delta \ast \dot \FQ^0_\delta \ast \dot \FQ^1_\delta$ all
preserves GCH. It therefore is true (see \cite{F07}) that
$\forces_{\FP_\delta \ast \dot \FQ^0_\delta} ``\dot \FQ^1_\delta$
is $\delta$-closed and $\delta^+$-c.c.''.
%, and in fact, that the stronger property that
%$\forces_{\FP_\delta \ast \dot \FQ^0_\delta}
%``\dot \FQ^1_\delta$ is ${\prec}\delta$-closed'' holds as well.
This of course means that
$\forces_{\FP_\delta \ast \dot \FQ^0_\delta} ``$Forcing
with $\dot \FQ^1_\delta$ preserves all cofinalities''.
It is also the case
(because of the way in which $\FQ^1_\gd$ is defined and because
$\forces_{\FP_\delta \ast \dot \FQ^0_\delta} ``\dot \FQ^1_\delta$
is $\delta$-closed and $\delta^+$-c.c.'') that
$\forces_{\FP_\delta \ast \dot \FQ^0_\delta} ``$Forcing
with $\dot \FQ^1_\delta$ preserves GCH''.
Because $\square_\delta$ is upwards absolute to a
cofinality preserving generic extension,
we may thus infer that
$\forces_{\FP_\delta \ast \dot \FQ^0_\delta \ast \dot \FQ^1_\delta}
``$There is both a $\square_\delta$ sequence and
a gap 1 morass at $\delta$''. In addition, since
$\forces_{\FP_\delta} ``\dot \FQ^0_\delta$ is
$\delta$-strategically closed'',
$\forces_{\FP_\delta} ``\dot \FQ_\delta =
\dot \FQ^0_\delta \ast \dot \FQ^1_\delta$
is ${\prec}\delta$-strategically closed''.
We may consequently conclude that
$V^{\FP^*} \models ``S$ is a stationary subset
of $\gk_0$ such that for every $\gd \in S$,
$\square_\gd$ holds and $\gd$ carries a gap 1 morass''.
Further, $\FP^*$ admits a closure point at $\ha_1$, and
inductively, forcing with $\FP^*$ preserves
cofinalities. Thus, by our remarks
immediately following Definition \ref{d1},
${\rm Safe}(\gd)$ is upwards absolute to $V^{\FP^*}$
for every infinite cardinal $\gd$.
This means that $V^{\FP^*} \models ``\square^T_\gd$
holds for every infinite cardinal $\gd$, where
$T = {\rm Safe}(\gd)$''.
By exactly the same arguments as in
\cite[Lemma 1.7]{A05}, we have that
if $V \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact and $\gl$
is regular'', then $V^{\FP^*} \models ``\gd$
is $\gl$ supercompact''.\footnote{An outline
of the proof is as follows. By the
L\'evy-Solovay results \cite{LS},
this is certainly true if $\gd > \gk_0$.
If $\gd = \gk_0$ and $\gl > \gk_0$ is a
regular cardinal, let $j : V \to M$ be
an elementary embedding witnessing the
$\gl$ supercompactness of $\gk_0$ such
that $M \models ``\gk_0$ is not $\gl$
supercompact''. We may then
write $j(\FP^*) = \FP^* \ast \dot \FQ$,
where the first ordinal at which
$\dot \FQ$ is forced to do nontrivial
forcing is well above $\gl$.
A standard lifting argument now shows that
$V^{\FP^*} \models ``\gk_0$ is $\gl$ supercompact''.
Finally, if $\gd < \gk_0$ and
$V \models ``\gd$ is $\gl$ supercompact
and $\gl$ is regular'', it follows as in
\cite{A05} that $\gl$ is below the least
$V$-strong cardinal above $\gd$. If we
write $\FP^* = \FP_\gd \ast \dot \FP^\gd$, then
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is $\eta$-strategically
closed for $\eta$ the least inaccessible cardinal
above $\gl$''. Therefore, to show that
$V^{\FP^*} \models ``\gd$ is $\gl$ supercompact'',
it suffices to show that
$V^{\FP_\gd} \models ``\gd$ is $\gl$ supercompact''.
If $\card{\FP_\gd} < \gd$, then this follows
from the results of \cite{LS}, and if
$\card{\FP_\gd} = \gd$, then this follows from
the same argument as when $\gd = \gk_0$.}
This means that each supercompact cardinal is
preserved from $V$ to $V^{\FP^*}$. Consequently,
since $\FP^*$ admits a closure point at
$\ha_1$, by our remarks immediately following
the statement of Theorem \ref{t2},
$V^{\FP^*} \models ``\K$
is the class of supercompact cardinals (so
$\gk_0$ is the least supercompact cardinal)''.
Further, $V^{\FP^*} \models ``$Level by level
equivalence between strong compactness and
supercompactness holds'',
by the identical argument to
the one given in the proof
of \cite[Lemma 1.3]{A05}.\footnote{This
argument is analogous to the one
found in the proof of Lemma \ref{l1}
of this paper. As such, more details will
be given when this lemma is proven.
The key facts used are that
if $V \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact and $\gl$ is regular'', then
$V^{\FP^*} \models ``\gd$ is $\gl$ supercompact'' and that
$\FP^*$ admits a closure point at $\ha_1$.
The main changes are then
that $V_1$ in Lemma \ref{l1}
is replaced by $V$, and
$\FP^{**}$ in Lemma \ref{l1}
is replaced by $\FP^*$.}
Since an inductive argument together
with the facts of the preceding
paragraph show that $V^{\FP^*} \models {\rm GCH}$,
Shelah's result of \cite{Sh1} immediately implies that
$V^{\FP^*} \models ``\diamondsuit_\gd$ holds for every
successor cardinal $\gd > \ha_1$''.
Because the first nontrivial stage of $\FP^*$
adds a Cohen subset of $\ha_1$, as in \cite{A05},
$V^{\FP^*} \models ``\diamondsuit_{\ha_1}$ holds'', i.e.,
$V^{\FP^*} \models ``\diamondsuit$ holds''.
Now, for $\gd$ such that
$V^{\FP^*} \models ``\gd$ is a Mahlo cardinal'', it is
the case that $V \models ``\gd$ is a Mahlo cardinal''.
If we write
$\FP^* = \FP_\gd \ast \dot \FP^\gd$, then
$\FP_\gd$ is $\gd$-c.c., $\card{\FP_\gd} \le \gd$, and
$\forces_{\FP_\gd} ``\dot \FP^\gd$ is
${\prec}\gd$-strategically closed''.\footnote{If
$\gd \ge \gk_0$, then $\FP_\gd$ is forcing
equivalent to $\FP_{\gk_0}$, and $\dot \FP^\gd$
may be taken as a term for trivial forcing.}
By \cite[Facts 1.1 and 1.2]{A09}, this means that
$\diamondsuit_\gd$ is preserved from
$V$ to $V^{\FP^*}$, so
$V^{\FP^*} \models ``\diamondsuit_\gd$ holds for
every successor and Mahlo cardinal $\gd$''.
Hence, $V^{\FP^*}$ satisfies properties
(\ref{j1}) -- (\ref{j3}) of Theorem \ref{t1},
together with GCH, level by level equivalence
between strong compactness and supercompactness,
and the fact that $\K$ is the class of
supercompact cardinals.
We complete the proof of Theorem \ref{t1}
by forcing over $V^{\FP^*}$ with a partial ordering
$\FP^{**} \subseteq V^{\FP^*}$ which preserves
everything mentioned in the last sentence of
the preceding paragraph
and also adds property (\ref{j4}) of Theorem \ref{t1}.
$\FP^{**} = \add(\ha_1, 1) \ast \dot \FW$,
where $\dot \FW$ is a term for Asper\'o and Friedman's
reverse Easton class iteration
$\FW$ of \cite{AF} for adding a locally defined well-ordering
${\cal W}$ of the universe, i.e., a well-ordering
satisfying the conditions mentioned earlier.
%such that for all $\gk \ge \ha_2$ a regular
%cardinal, ${\cal W} \rest H(\gk^+)$ is definable
%over the structure $\la H(\gk^+), \in \ra$ by a
%parameter free formula.
For the exact definition of $\FW$, which is
rather involved, we refer readers to \cite{AF}.
We mention only a few relevant facts
from \cite{AF}, which are as follows.
\begin{enumerate}
\item\label{i1} Forcing with $\FW$
(and hence forcing with $\FP^{**}$) preserves GCH and
all cofinalities.
\item\label{i2} $\forces_{\add(\ha_1, 1)} ``\dot \FW$
is $\ha_2$-directed closed''.
\item\label{i3} $\FW = \FB \ast \dot \FC$, where
both $\FB$ and $\FC$ are direct
limits of proper class reverse
Easton iterations.
\item\label{i4} For each Mahlo cardinal
$\gd$, $\FP^{**} = \add(\ha_1, 1)
\ast \dot \FW = \add(\ha_1, 1) \ast \dot
\FB_\gd \ast \dot \FB^\gd
\ast \dot \FC_\gd \ast \dot \FC^\gd$,
where $\forces_{\add(\ha_1, 1)} `` \dot
\FB_\gd$ is $\gd$-c.c$.$'',
$\forces_{\add(\ha_1, 1)} `` \card{\dot \FB_\gd} = \gd$'',
$\forces_{\add(\ha_1, 1) \ast \dot \FB_\gd}
``\dot \FB^\gd$ is $\gd^+$-directed closed'',
$\forces_{\add(\ha_1, 1) \ast \dot \FB_\gd
\ast \dot \FB^\gd} ``\dot \FC_\gd$ is
$\gd$-c.c.'', $\forces_{\add(\ha_1, 1) \ast
\dot \FB_\gd \ast \dot
\FB^\gd} ``\card{\dot \FC_\gd} = \gd$'',
and $\forces_{\add(\ha_1, 1) \ast \dot \FB_\gd \ast
\dot \FB^\gd \ast \dot \FC_\gd}
``\dot \FC^\gd$ is $\gd$-directed closed''.
\item\label{i5} Forcing with $\FW$
%(and hence forcing with $\FP^{**}$)
preserves the
$\gl$ supercompactness of $\gd$, if $\gl \ge \gd$
are such that $\gd$ is $\gl$ supercompact
and $\gl$ is regular.
\end{enumerate}
Let $\FP = \FP^* \ast \dot \FP^{**}$. By property
(\ref{i1}) above, $V^\FP =
V^{\FP^* \ast \dot \FP^{**}} \models {\rm GCH}$.
The following lemma is key to the proof of Theorem \ref{t1}.
\begin{lemma}\label{l1}
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models ``$Level
by level equivalence between strong
compactness and supercompactness holds''.
\end{lemma}
\begin{proof}
We follow the proof of \cite[Lemma 1.3]{A05}. Let
$V_1 = V^{\FP^*}$, and suppose
$V_1^{\FP^{**}} \models ``\gk < \gl$ are regular
cardinals such that $\gk$ is $\gl$ strongly compact
and $\gk$ is not a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact''.
%Write $\FP^{**} = \add(\ha_1, 1) \ast \dot \FW$.
Suppose $V_1 \models ``\gd$ is $\gl$ supercompact''.
By the results of \cite{LS}, $V_1^{\add(\ha_1, 1)} \models
``\gd$ is $\gl$ supercompact'', and
by property (\ref{i5}) above,
$V_1^{\add(\ha_1, 1) \ast \dot \FW} = V_1^{\FP^{**}} \models
``\gd$ is $\gl$ supercompact'' as well. This means that
$V_1 \models ``\gk < \gl$ are regular cardinals such that
$\gk$ is not a measurable limit of cardinals
$\gd$ which are $\gl$ supercompact''.
By property (\ref{i4}) above, $\FP^{**}$ is mild
with respect to $\gk$. Therefore, by the fact
$\FP^{**} = \add(\ha_1, 1) \ast
\dot \FW$ (so $\FP^{**}$ admits a closure point
at $\ha_1$) and
Theorem \ref{t2}, $V_1 \models ``\gk$ is $\gl$
strongly compact''. Hence, by level by level equivalence
between strong compactness and supercompactness in $V_1$,
$V_1 \models ``\gk$ is $\gl$ supercompact''.
Consequently, as in the previous paragraph,
$V_1^{\FP^{**}} = V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\gk$ is $\gl$ supercompact'' as well.
This completes the proof of Lemma \ref{l1}.
\end{proof}
As we have just shown, if
$V^{\FP^*} \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact and $\gl$ is regular'', then
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\gd$ is $\gl$ supercompact''. Since $\FP^{**}$
admits a closure point at $\ha_1$, as before,
it follows that
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\K$ is the class of supercompact cardinals
(so $\gk_0$ is the least supercompact cardinal)''.
Therefore, by Lemma \ref{l1},
the proof of Theorem \ref{t1} will
be complete once we have shown that
properties (\ref{j1}) -- (\ref{j4})
of its statement hold in $V^\FP$.
By the definition of $\FP^{**}$, property (\ref{j4})
of the statement of Theorem \ref{t1}
holds in $V^{\FP^* \ast \dot \FP^{**}} = V^\FP$.
Because forcing with $\FP^{**}$ adds a Cohen subset
of $\ha_1$ and preserves GCH, as earlier,
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\diamondsuit _\gd$ holds for every successor
cardinal $\gd$''. For $\gd$ such that
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\gd$ is a Mahlo cardinal'', it is of course
once again the case that
$V^{\FP^*} \models ``\gd$ is a Mahlo cardinal''.
By successively applying the factorization of
$\FP^{**}$ found in property (\ref{i4}) above and
\cite[Facts 1.1 and 1.2]{A09}, we yet again have that
$\diamondsuit_\gd$ is preserved from
$V^{\FP^*}$ to $V^{\FP^* \ast \dot \FP^{**}} = V^\FP$, so
as previously, $V^\FP \models ``\diamondsuit_\gd$ holds
for every successor and Mahlo cardinal $\gd$''.
Thus, property (\ref{j1}) of the statement of
Theorem \ref{t1} holds in $V^\FP$.
To show that properties (\ref{j2}) and (\ref{j3})
of the statement of Theorem \ref{t1} hold in $V^\FP$,
we begin by assuming that $\gd \in S$.
Because both $\square_\gd$ and the existence of a
gap 1 morass at $\gd$ are upwards absolute to a
cofinality preserving generic extension,
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\square_\gd$ holds, and $\gd$ carries a gap 1 morass''.
Once more, by successively applying the
factorization of $\FP^{**}$ given in property (\ref{i4})
above, together with its chain condition and
directed closure properties, we may infer that
%the arguments found in the
%proof of \cite[Theorem 1]{A05} allow us to infer that
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``S$ is a stationary subset of $\gk_0$''.
Thus, property (\ref{i2}) of the statement
of Theorem \ref{t1} holds in $V^\FP$. Then,
since $\FP^{**}$ admits a closure point at
$\ha_1$ and forcing with $\FP^{**}$ preserves
cofinalities, we have as earlier that
${\rm Safe}(\gd)$ is upwards absolute to
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP$ for every
infinite cardinal $\gd$. By
another application of upwards absoluteness,
we have as before that
$V^{\FP^* \ast \dot \FP^{**}} = V^\FP \models
``\square^T_\gd$ holds for every infinite cardinal $\gd$,
where $T = {\rm Safe}(\gd)$''.
Hence, property (\ref{i3}) of the statement
of Theorem \ref{t1} holds in $V^\FP$.
This completes the proof of Theorem \ref{t1}.
\end{proof}
We conclude by asking the general question of
which additional $L$-like principles are
consistent with GCH and level by level equivalence
between strong compactness and supercompactness.
%those holding in the model
%witnessing the conclusions of Theorem \ref{t1}.
In particular, as is true of the model constructed
in \cite{AC08}, is it possible to infer that
$\diamondsuit^+_\gd$ holds for every $\gd$ which
is the successor of a regular cardinal in the
model of Theorem \ref{t1}? Is it possible to
infer that $\diamondsuit_\gd$ holds in this model
for any non-Mahlo inaccessible cardinal $\gd$?
Is it possible to obtain further instances of morasses?
By \cite[Proposition 5]{B}, $\diamondsuit^+_\gd$
for $\gd$ a regular cardinal is destroyed
by forcing with $\add(\gd, \gd^+)$ (the standard
partial ordering for adding $\gd^+$ many Cohen
subsets of $\gd$). In addition, the preservation
theorems for $\diamondsuit_\gd$ when $\gd$ is
inaccessible seem to require a chain condition
which is automatically true only when $\gd$
is also a Mahlo cardinal. It is therefore
unclear at the moment which partial orderings
preserve the various forms of
$\diamondsuit$. %$\diamondsuit^+_\gd$.
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\end{document}