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\title{{An $L$-like model containing very large cardinals}
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
diamond, diamond primed, diamond star,
diamond plus, square,
level by level equivalence between strong
compactness and supercompactness.}}
\author{Arthur W.~Apter\thanks{The
first author's research was
partially supported by
PSC-CUNY Grants and
CUNY Collaborative
Incentive Grants.} \\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu\\
\\
James Cummings\thanks{The
second author's research was
partially supported by NSF
Grant DMS-0400982.} \\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
Pittsburgh, Pennsylvania 15213 USA\\
http://www.math.cmu.edu/users/jcumming \\
jcumming@andrew.cmu.edu}
\date{February 5, 2007}
\begin{document}
\maketitle
\begin{abstract}
We force and construct
a model
in which level by level
equivalence between strong
compactness and supercompactness
holds, along with a strong
form of diamond and a version
of square consistent with supercompactness.
This generalises a result due to
the first author.
There are no restrictions in
our model
%ground models or our generic extensions
on the structure of the
class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
%{Comments are in green, alterations to the text are in red.}
\section{Introduction and preliminaries}\label{s1}
In \cite{A05}, the first author
proved the following theorem.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\K$ is
the class of supercompact cardinals
(so $\gk$ is the least
supercompact cardinal). In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds. In addition,
in $V^\FP$, for every $\gd \in A$ where $A$
is a certain stationary subset of $\gk$,
$\square_\gd$ holds, and for
every regular uncountable cardinal $\gd$,
$\diamondsuit_\gd$ holds''.
\end{theorem}
In terminology used by Woodin,
this theorem
%to describe the main theorem of \cite{AS97a},
can be classified as an
``inner model theorem proven via forcing.''
This is since the model constructed
satisfies {pleasant properties}
one usually associates with an
inner model, namely {GCH and many instances of square and diamond,}
along with a property one might
perhaps expect if a ``nice'' inner model
containing supercompact
cardinals ever were to be
constructed, namely
level by level equivalence
between strong compactness
and supercompactness.
The purpose of this paper is to
extend and generalise Theorem \ref{t1},
in order to construct a model for
level by level equivalence between strong
compactness and supercompactness in which
a version of square consistent with
supercompactness holds on the class
of all infinite cardinals and in which a strong form
of diamond holds on a proper class of regular cardinals.
Our model for level by level equivalence
between strong compactness and supercompactness
consequently becomes, in a sense, even more
``inner model like'' than the one for Theorem \ref{t1}.
Specifically, we prove the following theorem.
\begin{theorem}\label{t2}
Let
$V \models ``$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 +
Level by level equivalence
between strong compactness and
supercompactness holds''.
In $V^\FP$, $\square^S_\gg$ holds
for every infinite cardinal $\gg$,
where $S = {\rm Safe}(\gg)$.
In addition, in $V^\FP$, $\diamondsuit_\mu$
holds for every $\mu$ which is
inaccessible or the successor of
a singular cardinal, and
$\diamondsuit^+_\mu$ holds for every
$\mu$ which is the successor
of a regular cardinal.
\end{theorem}
Pertinent definitions are presented
at various junctures throughout
the course of the paper.
In particular, we will give the
definitions of $S = {\rm Safe}(\gg)$
(Definition \ref{d1})
and $\square^S_\gg$ in Section \ref{s2},
and the definitions of our various
diamond principles in Section \ref{s3}.
We do, however, take this opportunity to
mention that for $\gk$ a regular cardinal
and $\ga$ an ordinal, $\add(\gk, \ga)$
is the standard Cohen poset for adding
$\ga$ many new subsets of $\gk$.
The overall structure of this paper
is as follows.
In Section \ref{s1}, we provide
a brief introduction.
In Section \ref{s2},
we discuss forcing the relevant
version of square.
In Section \ref{s3},
we discuss forcing a strong
form of diamond.
In Section \ref{s4},
we give a proof of Theorem \ref{t2}.
Before continuing, we do wish
now to take the opportunity
to state
a result which will be used in the proof
of Theorem \ref{t2}. This is
a corollary of Theorems 3 and 31
and Corollary 14 of
Hamkins' paper \cite{H5}.
This theorem is a generalisation 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
$(\gd + 1)$-strategically closed''
(meaning that there is a winning strategy
for player II in the game having
length $\gd + 1$).
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{t3}
%(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, if $\FP$ is mild with
respect to $\gk$ and $\gk$ is
$\gl$-strongly compact in $V[G]$
for any $\gl \ge \gk$, then
$\gk$ is $\gl$-strongly compact in $V$.
\end{theorem}
%{Theorem just seems to be talking about strong compactness, don't we
% need a version for supercompactness to justify the claims below?}
It immediately follows from Theorem \ref{t3}
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 $\go$)
must also be $\gl$-supercompact
in the ground model.
In particular, if $\ov V$ is a forcing extension
of $V$ by a poset that admits a closure point
%below the least supercompact cardinal
at $\go$
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}.
\section{Forcing a weak version of $\square$}\label{s2}
\subsection{Partial squares and the basic forcing} \label{s2-1}
We state a partial version of $\square$ compatible with supercompact
cardinals. { Square sequences of this kind were first shown to be
consistent with supercompactness by Foreman and Magidor \cite[p.~191]{VWS}, using techniques
of Baumgartner. In the notation of Definition \ref{funnysquare} they
showed that $\square_{\kappa^{+\omega}}^{ \{ \kappa^{+n} : n < \omega \}
}$ is consistent with $\kappa$ being supercompact. }
Given a set $S$ of regular cardinals, we denote by ${\rm cof}(S)$
the class of ordinals $\alpha$ such that ${\rm cf}(\alpha) \in S$.
\begin{definition} \label{funnysquare} Let $\gamma$ be an infinite cardinal and let $S$ be
a set of regular cardinals which are less than or equal to $\gamma$.
Then a {\em $\square_\gamma^S$-sequence} is a sequence
$\langle C_\alpha: \alpha \in \gamma^+ \cap {\rm cof}(S) \rangle$
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 $\square_\gamma^S$ holds} if and only if there is a
{\em $\square_\gamma^S$-sequence}.
\end{definition}
We note that if $T = (\gamma^+ \cap {\rm cof}(S)) \cup
\{ \eta : \exists \beta \in
\gg^+ \cap {\rm cof}(S) \; \eta \in
\lim(C_\beta) \}$ 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
$\eta \in \lim(C_\beta)$.
We define a forcing poset $\FQ^\square(\gamma, S)$ to add such a sequence.
Elements of $\FQ^\square(\gamma, S)$
are all pairs $(q, \eta)$ such that $\eta < \gamma^+$, and
$q$ is a partial function on $\gamma^+$ such that
\begin{enumerate}
\item $(\eta + 1) \cap {\rm cof}(S) \subseteq \dom(q) \subseteq \eta + 1$.
\item For all $\alpha \in \dom(q)$
\begin{enumerate}
\item $q(\alpha)$ is club in $\alpha$, and ${\rm ot}( q(\alpha) ) \le \gamma$.
\item For all $\beta \in \lim q(\alpha)$, $\beta \in \dom(q)$ and
$q(\alpha) \cap \beta = q(\beta)$.
\end{enumerate}
\end{enumerate}
$(q, \theta)$ extends $(p, \eta)$ if and only if $\theta \ge \eta$, $\dom(q) \cap (\eta +1) = \dom(p)$
and $q \restriction \dom(p) = p$.
Routine arguments \cite[Lemma 6.1]{CFM} show that the poset is $(\gamma+1)$-strategically
closed, in particular it adds no $\gamma$-sequences. If $2^\gamma =
\gamma^+$ then the forcing contains only $\gamma^+$ many conditions, so
trivially has $\gamma^{++}$-cc.
The key property is that when we form the union of a decreasing chain of
conditions of length less than ${\rm min}(S)$, we can obtain a condition, because
we are only {\em obliged} to put points of ${\rm cof}(S)$ into the support of a
condition.
\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 note 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{t3}, ${\rm Safe}(\gg)$
is upwards absolute to any cardinal and cofinality
preserving forcing extension by a poset admitting
a closure point at $\go$.
The following easy lemma (essentially due to Solovay) motivates the
definition of the safe set.
\begin{lemma} If $S$ is a final segment of ${\rm REG} \cap (\gamma+1)$ and
$\square^S_\gamma$ holds then $S \subseteq {\rm Safe}(\gamma)$.
\end{lemma}
\begin{proof} Suppose not, then by definition there is $\mu \in S$ such that
$\mu < \lambda \le \gamma$ and $\lambda$ is $\gamma^+$-supercompact.
Fix a sequence $\langle C_\alpha: \alpha \in \gamma^+ \cap {\rm cof}(S) \rangle$
witnessing the square.
{
In particular $C_\alpha$ is defined for every $\alpha$ in the stationary
set $\gamma^+ \cap {\rm cof}(S)$, and $\ot(C_\alpha) \le \gamma$,
so by Fodor's lemma we may find a stationary
set $T \subseteq \gg^+ \cap {\rm cof}(S)$
and an ordinal $\eta$ such that $\ot(C_\alpha) = \eta$ for all
$\alpha \in T$. It follows
easily from the coherence of the clubs $C_\beta$ that
$T \cap \lim(C_\beta)$
has size at most $1$ for all $\beta \in \gamma^+ \cap {\rm cof}(S)$.
In particular $T$ cannot reflect at any point of $\gamma^+ \cap {\rm cof}(S)$.
}
{
But the $\gamma^+$-strong compactness of $\lambda$ \cite[Theorem 4.8]{SRK}
implies that
$T$ must reflect at some point $\zeta$ with $\mu < {\rm cf}(\zeta) < \gamma$.
This is a contradiction since ${\rm cf}(\zeta) \in S$.
}\end{proof}
\subsection{Iteration and preservation}
We assume that $V$ satisfies GCH. In the cases of interest $V$ will contain
some supercompact cardinals.
We define an Easton support iteration $\FP^\square_\infty$ of length ${\rm ON}$
(the superscript is intended to distinguish this iteration of length
$\infty$ from some other ones appearing later in the paper).
As usual $\FP^\square_\gamma$ is the forcing up to stage $\gamma$ and
$\dot \FQ^\square_\gamma$ is a $\FP^\square_\gamma$-name for the forcing poset to be used at
stage $\gamma$.
$\FQ^\square_\gamma$ is trivial unless $\gamma$ is zero or an infinite cardinal,
and $\FQ^\square_0 = \add(\go, 1)$. At each stage
$\gamma$ where $\gamma$ is an infinite cardinal, we will set
$\FQ^\square_\gamma = {\FQ^\square(\gamma, S)}^{V^{\FP^\square_\gamma}}$ where $S = {\rm Safe}(\gamma)^V$
and $\FQ^\square(\gamma, S)$ is the forcing poset
defined in Section \ref{s2-1}.
At all other stages, the forcing is trivial.
Routine arguments show that this iteration preserves all cardinals and
cofinalities, together with GCH and the fact that
$V^{\FP^\square_\infty}$ is a model of ZFC. In addition, it is
easily verified that for every infinite cardinal $\gg$,
$\square^S_\gg$ holds in $V^{\FP^\square_\infty}$.
{As we mentioned above, the following result is a descendant of results
by Foreman and Magidor \cite[p.~191]{VWS}. The trick in
the master condition argument first appeared in unpublished work
of Baumgartner.}
\begin{theorem}\label{sqth} If $\kappa$ is supercompact in $V$, then the supercompactness
of $\kappa$ is preserved in the extension by $\FP^\square_\infty$.
\end{theorem}
\begin{proof} Let $\gamma > \kappa$ be regular and let $U$ be a
supercompactness measure on $P_\kappa \gamma$. Let $j : V \rightarrow M$ be
the ultrapower map. As usual ${\rm crit}(j) = \kappa$,
and $\gamma^+ < j(\kappa) < j(\gamma) < \gamma^{++}$.
It will suffice to show that $\kappa$ is $\gamma$-supercompact
in the extension by $\FP^\square_{\gamma+1}$, since the rest of the
iteration adds no new subsets of $P_\kappa \gamma$.
As usual the resemblance between $V$ and $M$ implies that
$\FP^\square_{\gamma+1}$ is an initial segment of $j(\FP^\square_{\gamma
+1})$.
We will break up the generic object for $\FP^\square_{\gamma+1}$ as
$G * g * H$ where $G$ is generic for $\FP^\square_\kappa$,
$g$ is generic for the part of the iteration
in the interval $[\kappa, \gamma)$, and $H$ is generic for the forcing at $\gamma$.
It is easy to see that $\FP^\square_\gamma$ has cardinality at most
$\gamma$, so in particular
$V[G * g] \models {}^\gamma M[G * g] \subseteq M[G * g]$.
Let $\mathbb R \in M[G * g * H]$ be the usual forcing for prolonging
$G * g * H$ to a generic filter for $j(\FP^\square_\kappa)$. By the usual arguments,
from the point of view of $V[G * g * H]$ this poset has cardinality
$\gamma^+$ and is $\gamma^+$-closed, so we may build a generic filter $h \in
V[G* g * H]$
for it and extend $j$ to get
$j: V[G] \rightarrow M[j(G)]$, where $j(G) = G * g * H * h$.
We note that since $\mathbb R$ is sufficiently closed,
$V[G * g * H] \models {}^{\gamma} M[j(G)] \subseteq M[j(G)]$.
Let $\mathbb S \in M[j(G)]$ be the natural forcing for prolonging $j(G)$ to
a $j(\FP^\square_\gamma)$-generic filter. Again $\mathbb S$ has cardinality
$\gamma^+$ and is $\gamma^+$-closed from the point of view
of $V[G * g * H]$, but to lift the embedding we need a generic filter
which contains $j `` g$.
We will build a suitable master condition. Let $\delta$ be such that in $M$,
$\delta$ is a cardinal with $j(\kappa) \le \delta < j(\gamma)$. The key
point is that by elementarity $j(\kappa)$ is $j(\gamma)$-supercompact in
$M$, in particular $j(\kappa)$ is $\delta^+$-supercompact: so the set
${\rm Safe}^M(\delta)$ contains only cardinals greater than or equal
to $j(\kappa)$, and in particular since $\gamma < j(\kappa)$ we have
$\gamma < {\rm min}( {\rm Safe}^M(\delta) )$.
Now we consider the partial function $r$ defined as follows: the domain of
$r$ is $\bigcup_{s \in g} \dom(j(s))$, and for each $\delta \in \dom(r)$,
$r(\delta) = \bigcup_{s \in g} j(s)(\delta)$. We claim that this is a condition
in $\mathbb S$. The key point is that $\vert g \vert \le \gamma <
{\rm min}({\rm Safe}^M(\delta) )$, so that $r(\delta)$ names the union of a rather short chain of
conditions and hence is a name for a condition.
So we may lift $j$ once again to obtain $j:V[G * g] \rightarrow M[j(G * g)]$.
To finish we just note that $\FQ^\square_\gamma$ adds no $\gamma$-sequences
and $\vert P_\kappa \gamma \vert = \gamma$. We may therefore transfer
the generic object $H$ along $j$ to obtain
$j:V[G * g * H] \rightarrow M[j(G * g * H)]$.
\end{proof}
\section{Forcing $\diamondsuit^+_{\lambda^+}$}\label{s3}
\subsection{Strong diamond and the basic forcing}
\label{diamond_forcing}
We recall that
\begin{enumerate}
\item $\diamondsuit'_{\lambda^+}$ is the assertion that there exists
a sequence $\langle {\mathcal S}_\alpha: \alpha < \lambda^+ \rangle$
such that
\begin{enumerate}
\item For every $\alpha$, ${\mathcal S}_\alpha$ is a family of subsets
of $\alpha$ with $\vert {\mathcal S}_\alpha \vert \le \lambda$.
\item For every $X \subseteq \lambda^+$, the set
$\{ \alpha < \lambda^+ : X \cap \alpha \in {\mathcal S}_\alpha \}$ is
stationary in $\alpha$.
\end{enumerate}
\item $\diamondsuit^*_{\lambda^+}$ is the assertion that there exists
a sequence $\langle {\mathcal S}_\alpha: \alpha < \lambda^+ \rangle$
such that
\begin{enumerate}
\item For every $\alpha$, ${\mathcal S}_\alpha$ is a family of subsets
of $\alpha$ with $\vert {\mathcal S}_\alpha \vert \le \lambda$.
\item For every $X \subseteq \lambda^+$, there is $C \subseteq \lambda^+$ a
club set such that
$\forall \alpha \in C \; X \cap \alpha \in {\mathcal S}_\alpha$.
\end{enumerate}
\item $\diamondsuit^+_{\lambda^+}$ is the assertion that there exists
a sequence $\langle {\mathcal S}_\alpha: \alpha < \lambda^+ \rangle$
such that
\begin{enumerate}
\item For every $\alpha$, ${\mathcal S}_\alpha$ is a family of subsets
of $\alpha$ with $\vert {\mathcal S}_\alpha \vert \le \lambda$.
\item For every $X \subseteq \lambda^+$, there is $C \subseteq \lambda^+$ a
club set such
$\forall \alpha \in C \; X \cap \alpha, C \cap \alpha \in {\mathcal S}_\alpha$.
\end{enumerate}
\end{enumerate}
{
Kunen \cite[Theorem 2]{DJ} showed that $\diamondsuit'_{\lambda^+}$ is equivalent to
$\diamondsuit_{\lambda^+}$.
In unpublished work Jensen showed that in general $\diamondsuit^*_{\lambda^+}$ is
stronger than $\diamondsuit_{\lambda^+}$ and $\diamondsuit^+_{\lambda^+}$
is stronger than $\diamondsuit^*_{\lambda^+}$.
}
Jensen showed \cite{Devlin}
that $\diamondsuit^+_{\lambda^+}$ holds in $L$,
and that a $\diamondsuit^*_{\omega_1}$-sequence
can be added by countably closed forcing \cite[Lemma 8.3]{DJ}
when $\lambda=\omega$.
It is probably possible to adapt that argument to show that a
$\diamondsuit^+_{\lambda^+}$-sequence can be added by
$\lambda^+$-directed-closed forcing; it is not clear to us whether
such an adapted poset would work for our results, since we will be preserving
large cardinals by something more elaborate than a straightforward
master condition argument.
We will use a poset constructed by Cummings, Foreman and Magidor
(see \cite[Section 12]{CFM}).
We give a fairly detailed exposition here to make this paper reasonably
self-contained, and to stress that there is some extra flexibility in
computing lower bounds in the poset which will be useful later.
We fix $\lambda$ a cardinal with
$2^\lambda = \lambda^+$. We will define a poset $\FQ^\diamondsuit(\lambda^+)$ such that
$\FQ^\diamondsuit(\lambda^+)$ adds $\diamondsuit^+_{\lambda^+}$, where
$\FQ^\diamondsuit(\lambda^+)$ is $\lambda^+$-directed closed and $\lambda^{++}$-cc.
The main idea is that we will add a $\diamondsuit'_{\lambda^+}$-sequence,
and then iterate in length $\lambda^{++}$ by adding club subsets of $\lambda^+$
so as to make this sequence into a $\diamondsuit^+_{\lambda^+}$-sequence.
We will then let $\FQ^\diamondsuit(\lambda^+)$ be the result $\FP_{\lambda^{++}}$
of this iteration, which will turn out to have $\lambda^{++}$-cc.
We start by defining a poset $\mathbb Q_0$ to add the
$\diamondsuit'_{\lambda^+}$-sequence. $\mathbb Q_0$ is the set
of those $q$ such that $q = \langle {\mathcal S}_\alpha: \alpha \le \beta \rangle$ where
\begin{enumerate}
\item $\beta < \lambda^+$.
\item For every $\alpha \le \beta$
\begin{enumerate}
\item ${\mathcal S}_\alpha$ is a family of subsets of $\alpha$.
\item $\vert {\mathcal S}_\alpha \vert \le \lambda$.
\end{enumerate}
\end{enumerate}
The ordering is by end-extension. One can check by standard arguments that
$\diamondsuit'_{\lambda^+}$ holds in the extension by $\mathbb Q_0$, but
we will not do this since it follows from our later analysis.
For $\alpha > 0$ we will choose (by some bookkeeping scheme) $\dot X_\alpha$ a $\FP_\alpha$-name
for a subset of $\lambda^+$, and then define $\mathbb Q_\alpha$ in
$V^{ \FP_\alpha}$ to be the
set of $c$ such that
\begin{enumerate}
\item $c$ is closed and bounded in $\lambda^+$.
\item $\forall \beta \in \lim(c) \; X_\alpha \cap \beta, c \cap \beta \in {\mathcal
S}_\beta$.
\end{enumerate}
Here $X_\alpha$ is the realisation of the term $\dot X_\alpha$, and
$\langle {\mathcal S}_\alpha : \alpha < \lambda^+ \rangle$
is the $\diamondsuit'_{\lambda^+}$-sequence added by $\mathbb Q_0$.
The ordering is end-extension.
The bookkeeping will arrange that after $\lambda^{++}$ steps in the
$\lambda^{++}$-cc iteration we have handled every subset of $\lambda^+$.
To complete the definition of our iteration, we specify that we will force
with supports of size at most $\lambda$; equivalently we will form
inverse limits at limit stages $\delta$ with ${\rm cf}(\delta) \le \lambda$,
and direct limits when ${\rm cf}(\delta) > \lambda$.
As usual when we are iterating forcing to shoot club sets, the key point
is to prove that there is a dense set of ``tame'' conditions.
\begin{definition} A condition $p \in \FP_\alpha$ is {\em
rectangular} if and only if there is a limit ordinal $\beta < \lambda^+$ such that
\begin{enumerate}
\item $p(0)$ has the form $\langle {\mathcal S_\gamma} : \gamma \le \beta \rangle$.
\item For all $\eta \in {\rm supp}(p)$ with $\eta > 0$
\begin{enumerate}
\item There exist $d_\eta, x_\eta \in V$ such that
$p \restriction \eta \Vdash ``p(\eta) = \check d_\eta, \dot X_\eta \cap
\beta = \check x_\eta$''.
\item $\max(d_\eta) = \beta$, $\beta \in \lim(d_\eta)$.
\end{enumerate}
\end{enumerate}
\end{definition}
NOTE: Since $p$ is a condition, it follows that $x_\eta, d_\eta \cap \gb \in
{\mathcal S}_\beta$.
In a harmless abuse of notation we will often assume that for $p$
rectangular, $p(\gamma)$ is literally a canonical name for an element of
$V$. We call the ordinal $\beta$ the {\em height} of $p$.
Let $\FP^{\rm rect}_\alpha$ be the set of rectangular conditions in
$\FP_\alpha$.
\begin{lemma} \label{rectangulardenseclosed}
Let $1 \le \alpha \le {\lambda^{++}}$. Then
\begin{enumerate}
\item $\FP^{\rm rect}_\alpha$ is ${\lambda^+}$-directed closed.
Moreover, if $\{ p_\gg : \gg < \mu \}$
is a directed set of rectangular conditions for some $\mu < {\lambda^+}$,
and the height of $p_\gg$ is $\sigma_\gamma$, then
there is a greatest lower
bound (in $\FP^{\rm rect}_\alpha$) $p$ which is given by
\begin{enumerate}
\item ${\rm supp}(p) = \bigcup_{\gamma < \mu} {\rm supp}(p_\gamma)$.
\item $\dom(p(0)) = \sigma + 1$, where $\sigma = \sup_{\gamma < \mu} \sigma_\gamma$.
\item $p(0) \restriction \sigma = \bigcup_{\gamma < \mu} p_\gamma(0)$.
\item For $\beta > 0$, $p(\beta) = \bigcup_{\gamma < \mu} \{ p_\gamma(\beta) : \beta \in
\dom(p_\gamma) \}
\cup
\{\sigma \}$.
\item $p(0)(\sigma) = \{ p(\beta) \cap \sigma : \beta \in \dom(p),
\gb > 0 \}
\cup \{x_\beta : \beta \in \dom(p), \gb > 0 \}$,
where $x_\beta$ is the subset of $\sigma$ such that $p \restriction \beta
\Vdash ``\dot X_\beta \cap \sigma = \check x_\beta$''.
\end{enumerate}
$p \in \FP^{\rm rect}_\alpha$ and $p$ has height $\sigma$.
\item $\FP^{\rm rect}_\alpha$ is dense in $\FP_\alpha$.
\end{enumerate}
\end{lemma}
\begin{proof} The first claim is easy to verify.
We prove the second claim by induction on $\alpha$. Suppose $\FP^{\rm rect}_{\bar\alpha}$
is dense in $\FP_{\bar\alpha}$ for all $\bar \alpha < \alpha$. We show $\FP^{\rm rect}_\alpha$
is dense in $\FP_\alpha$.
\medskip
\noindent Case 1: $\alpha = 1$, $\FP^{\rm rect}_1 = \FP_1 \simeq \FQ_0$
and there is nothing to prove.
\medskip
\noindent Case 2: $\alpha = \beta +1$.
Fix a condition $p \in \FP^{\rm rect}_{\beta+1}$.
By induction, we know that $\FP^{\rm rect}_\beta$ is ${\lambda^+}$-closed and dense in
$\FP_\beta$. In particular $\FP_\beta$ adds no bounded subsets of ${\lambda^+}$,
and so we may choose $p_0 \le p \restriction \beta$
such that $p_0$ decides the value
of $p(\beta)$, say that $p_0$ forces $p(\beta)$ to equal $c$ where
$\max(c) = \gamma$. As $\FP^{\rm rect}_\beta$ is dense, we may choose $p_0 \in \FP^{\rm rect}_\beta$
and by extending if necessary may also assume that the height of $p_0$
is greater than $\gamma$.
Now we argue in a similar vein to build a decreasing $\omega$-sequence of conditions
$p_0 > p_1 > p_2 > \ldots$ and an increasing $\omega$-sequence of ordinals
$\rho_0 < \rho_1 < \ldots$, where $p_n$ is
a condition in $\FP^{\rm rect}_\beta$ of height $\rho_n$, and $p_{n+1}$ decides
$\dot X_\beta \cap \rho_n$; say $p_{n+1} \Vdash ``\dot X_\beta \cap \rho_n = x_n$''.
By the first claim of the Lemma, we may form a greatest
lower bound for $\vec p$ which will be a condition $q \in \FP^{\rm rect}_\beta$
of height $\rho = \sup \rho_n$.
Now we define $q^+$ as follows. $\dom(q^+) = \beta + 1$,
and $q^+(\gamma) = q(\gamma)$ for $0 < \gamma < \beta$.
$q^+(\beta) = c \cup \{ \rho_0, \rho_1, \ldots, \rho \}$.
$q^+(0) \restriction \rho = q(0) \restriction \rho$, and
$q^+(0)(\rho) = q(0)(\rho) \cup \{ q^+(\beta), \bigcup_n x_n \}$.
It is now routine to check that $q^+ \in \FP_{\beta+1}$, $q^+$ is
rectangular of height $\rho$, and $q^+$ refines $p$.
\noindent Case 3: $\alpha$ is limit with ${\rm cf}(\alpha) \ge {\lambda^+}$. Fix $p \in \FP_\alpha$,
then the support of
$p$ is bounded by some $\beta < \alpha$. By induction we may find
$q \le p \restriction \beta$ with $q \in \FP^{\rm rect}_\beta$; if $q^+ \in \FP_\alpha$
is defined by $q^+ \restriction \beta =q$ and $q^+ \restriction [\beta, \alpha)
= 1$ then $q^+ \in \FP^{\rm rect}_\alpha$ and $q^+ \le p$, as required.
\medskip
\noindent Case 4: $\alpha$ is limit and ${\rm cf}(\alpha) \le \lambda$.
Choose a sequence
$\langle \alpha_i : i < {\rm cf}(\alpha) \rangle$ which is increasing, continuous
and cofinal in $\alpha$.
Fix $p \in \FP_\alpha$.
We will define a decreasing sequence of conditions $\langle p_i : i \le
{\rm cf}(\alpha) \rangle$
such that $p_0 \le p$, $p_i \restriction \alpha_i \in \FP_{\alpha_i}^{\rm rect}$
for each $i$, and $p_{{\rm cf}(\alpha)} \in \FP^{\rm rect}_\alpha$. We let $\sigma_i$ denote the
height of $p_i \restriction \alpha_i$.
\medskip
\noindent $i = 0$. Let $q_0 \le p \restriction \alpha_0$, $q_0 \in \FP^{\rm rect}_{\alpha_0}$.
Let $p_0 \restriction \alpha_0 = q_0$,
$p_0 \restriction [\alpha_0, \alpha) = p \restriction [\alpha_0, \alpha)$.
\medskip
\noindent $i = j+1$. Let $q_i \le p_j \restriction \alpha_i$,
$q_i \in \FP^{\rm rect}_{\alpha_i}$. Now let
$p_i \restriction \alpha_i = q_i$,
and $p_i \restriction [\alpha_i, \alpha) = p \restriction [\alpha_i, \alpha)$.
\medskip
\noindent $i$ is limit. For each $j < i$ consider the sequence
$\langle p_k \restriction \alpha_j: j \le k < i \rangle$. This is a decreasing
sequence from $\FP^{\rm rect}_{\alpha_j}$ so by the first claim of the Lemma we can form a greatest
lower bound $r_j$, where $r_j \in \FP^{\rm rect}_{\alpha_j}$ and $r_j$ has height
$\sigma = \sup_{k < i} \sigma_k$.
It is easy to see that if $m < n$ then
$r_m \restriction (0, \alpha_m) = r_n \restriction (0, \alpha_m)$,
$r_m(0) \restriction \sigma = r_n(0) \restriction \sigma$,
and $r_m(0)(\sigma) \subseteq r_n(0)(\sigma)$.
We define $q_i$ such that $\dom(q_i) = \alpha_i$,
$q_i \restriction (0, \alpha_j) = r_j \restriction (0, \alpha_j)$ for all $j$,
$q_i(0) \restriction \sigma$ is the common value of $r_j(0) \restriction \sigma$,
and $q_i(0)(\sigma) = \bigcup_{j< i} r_j(0)(\sigma)$.
It is routine to check that $q_i \in \FP^{\rm rect}_{\alpha_i}$ and $q_i$ has height $\sigma$,
also that $q_i \le p_j \restriction \alpha_i$ for each $j < i$. Now let
$p_i \restriction \alpha_i = q_i$, $p_i \restriction [\alpha_i, \alpha) = p \restriction [\alpha_i, \alpha)$.
\medskip
The construction for the limit step also works for $i = {\rm cf}(\alpha)$, and produces
$p_{{\rm cf}(\alpha)}$ which is in $\FP^{\rm rect}_{\alpha}$ and refines $p$.
\end{proof}
It will be crucial later that we have a certain latitude when we are
forming a lower bound for a directed set of conditions. In particular
in the argument for Case 4 above we could extend $q_{{\rm cf}(\alpha)}(0)(\sigma)$ by
adding in $\lambda$ many additional subsets of $\sigma$, and still obtain
a lower bound.
\subsection{Iteration and preservation}
Let $V$ be a model of GCH.
We describe an iteration $\FP^\diamondsuit_\infty$ of length ${\rm ON}$ with Easton support to add
a $\diamondsuit_\mu$-sequence for every $\mu$ which is inaccessible or the successor of
a singular cardinal,
and a $\diamondsuit^+_\mu$-sequence for every $\mu$
which is the successor of a regular cardinal. To do this
we begin by forcing with $\FQ^\diamondsuit_0 = \add(\go, 1)$ and then let $\FQ^\diamondsuit_\mu$ be
\begin{enumerate}
\item The %Cohen
poset ${\rm Add}(\mu, 1)^{V^{\FP^\diamondsuit_\mu}}$ for $\mu$ inaccessible or $\mu$ the successor of a singular cardinal.
\item The poset $\FQ^\diamondsuit(\mu)^{V^{\FP^\diamondsuit_\mu}}$ as described in Section \ref{diamond_forcing} when
$\mu = \lambda^+$ for $\lambda$ regular.
\end{enumerate}
\noindent At all other stages, the forcing is trivial.
We need to show that this iteration $\FP^\diamondsuit_\infty$
preserves all regular instances of supercompactness.
Routine arguments show that the iteration preserves all cardinals and
cofinalities, together with GCH and the fact that
$V^{\FP^\diamondsuit_\infty}$ is a model of ZFC.
In addition, using the arguments of \cite[Lemma 1.1]{A05}
and \cite[Theorem 12.2]{CFM}, it is easily
verified that in $V^{\FP^\diamondsuit_\infty}$, $\diamondsuit_\mu$
holds for every $\mu$ which is
inaccessible or the successor of a singular cardinal, and
$\diamondsuit^+_\mu$ holds for every $\mu$ which is
the successor of a regular cardinal.
\begin{theorem}\label{diath}
If $\gamma$ is regular and $\kappa$ is $\gamma$-supercompact in $V$, then
$\kappa$ is $\gamma$-supercompact in $V^{\FP^\diamondsuit_\infty}$.
\end{theorem}
\begin{proof} It is enough to show that $\kappa$ is $\gamma$-supercompact in
$V^{\FP^\diamondsuit_{\gamma+1}}$, since the rest of the iteration does not change
$P_\kappa \gamma$. We will distinguish various cases.
\smallskip
\noindent {\bf $\gamma$ is inaccessible or the successor of
a singular cardinal.}
Here, we give an argument similar to the
one presented in \cite[Lemma 1.2]{A05}.
We fix as usual $U$ a
supercompactness measure on $P_\kappa \gamma$, and $j : V \rightarrow M$
the associated ultrapower map. By GCH we get that $\gamma ^+ < j(\kappa)
\le j(\gamma) < \gamma^{++}$.
We can factorise $\FP^\diamondsuit_{\gamma+1}$ as $\FP^\diamondsuit_\kappa * \dot {\mathbb
Q}$, where $\mathbb Q$ is the part of the iteration in the interval
$[\kappa, \gamma]$. $\mathbb Q$ is a $\kappa$-directed closed forcing of size
$\gamma$. Let $G * g$ be the corresponding factorisation of
a $\FP^\diamondsuit_{\gamma+1}$-generic filter.
We note that $V[G * g] \models {}^\gamma M[G * g] \subseteq M[G * g]$.
Since $\vert j(\gamma) \vert = \gamma^+$, by the usual arguments, working
in $V[G *g]$ we may prolong $G * g$ to $j(G) =
G * g * H$ which is $j(\FP^\diamondsuit_\kappa)$-generic, and lift
to get $j: V[G] \rightarrow M[G * g * H]$.
Since $V[G * g] \models {}^\gamma M[G * g * H] \subseteq M[G * g * H]$,
and $j( \FQ )$ is $j(\kappa)$-directed closed, we may find
a lower bound for $j `` g$ in $j( \FQ )$ and use this as a master
condition. Since $\vert j(\gamma) \vert = \gamma^+$ we may build in
$V[G * g]$ a generic $h$ with $j`` g \subseteq h$, and finish by lifting
to $j: V [ G * g ] \rightarrow M[ G * g * H * h]$.
\smallskip
\noindent {\bf $\gamma = \delta^+$ for $\delta$ regular.} We fix a supercompactness measure
$U$ on $P_\kappa \delta^+$, and let $j : V \rightarrow M$ be the
ultrapower map. As usual
$\delta^{++} < j(\kappa) < j(\delta^+) < \delta^{+++}$.
{Also $j$ is continuous at $\gamma^+ = \gd^{++}$.}
The last step in $\FP^\diamondsuit_{\gamma+1}$ is $\FQ^\diamondsuit_\gamma$, the
forcing for adding a $\diamondsuit^+_{\delta^+}$-sequence.
We will break up $\FP^\diamondsuit_{\gamma+1}$ as $\FP^\diamondsuit_\kappa *
\dot \FQ * \dot \FQ^\diamondsuit_\gamma$, where $\mathbb Q$ is the iteration in
the interval $[\kappa, \gamma)$. We know that $\mathbb Q$ is
$\kappa$-directed closed and has size at most $\gamma$.
$\mathbb Q^\diamondsuit_\gamma$ is $\gamma$-directed closed and $\gamma^+$-cc forcing
of size $\gamma^+$.
Let $G * g * H$ be the corresponding factorisation of
a $\FP^\diamondsuit_{\gamma+1}$-generic filter.
Since $G * g$ is generic for forcing of size $\gamma$,
$V[G * g] \models {}^\gamma M[G * g] \subseteq M[G * g]$.
Since $H$ is generic for $\gamma^+$-cc forcing,
$V[G * g *H] \models {}^\gamma M[G * g *H] \subseteq M[G * g *H]$.
Using this closure, we may as usual prolong $G *g * H$ to
$j(G) = G * g * H * h$ which is $j( \FP^\diamondsuit_\kappa)$-generic
over $M$, and lift to get $j : V[G] \rightarrow M[ j(G) ]$.
We have that $V[G * g *H] \models {}^\gamma M[ j(G) ] \subseteq M[ j(G) ]$.
Since $\vert \FQ \vert = \gamma$ and $\mathbb Q$ is
$\kappa$-directed closed, we may argue just as before to
produce $K \supseteq j`` g$ which is $j(\FQ)$-generic
and extend once more to
$j: V[G * g] \rightarrow M[j(G) * K]$, where
$V[G * g *H] \models {}^\gamma M[ j(G) * K ] \subseteq M[ j(G) * K ]$.
Thus far the argument was fairly routine, but now we need a new idea
because $\FQ^\diamondsuit_\gamma$ has size $\gamma^+$ and is a complicated
forcing. We use a version of an idea of Magidor \cite{Ma}, \cite{JMMP} with an added
twist. We digress briefly to explain how Magidor's technique works in a
simpler setting. A more complete explanation of
Magidor's method may be found in \cite[Corollary 10,
pp$.$ 832--833]{AH02} (see also \cite[Lemma 9,
pp$.$ 119--120]{AS97a}).
{
Suppose for a moment that $\FQ_\gamma$ were the %Cohen
poset
${\rm Add}(\gamma, \gamma^+)$. Conditions are partial functions from
$\gamma \times \gamma^+$ to $2$ of cardinality less than $\gamma$,
ordered by $p \le q$ if and only if $q \supseteq p$.
This is quite similar in the sense that it is $\gamma^+$-cc,
adds $\gamma^+$ subsets of $\gamma$ and can be seen as an iteration
with ${<}\gamma$-support.
}
{
As usual if $H$ is a generic filter and
$\eta < \gamma^+$ we may form a restricted filter
$H \restriction \eta$ which is generic for
$\FQ_\gg \rest \eta = {\rm Add}(\gamma, \eta)$.
For each $\eta < \gg^+$ we can form
a ``partial master condition'' $q_\eta =_{\rm def} \bigcup j`` H \restriction \eta$,
and the $q_\eta$ form a decreasing sequence.
}
{
By a standard chain condition
argument, if $A$ is a maximal antichain in $\FQ_\gamma$
then there is $\sigma < \gamma^+$ such that
$A \subseteq \FQ_\gg \rest \eta$ for
every $\eta \ge \gs$.
% for any condition
% $r \in \FQ_\gamma$ there is $r' \le r \restriction (\gamma \times \sigma)$
% such that $r' \cup r \in D$; say in this case that {\em $\sigma$ captures $D$.}
We may now proceed to build a generic filter
for $j(\FQ_\gamma)$ which is compatible with $H$.
We enumerate the maximal antichains
in $j(\FQ_\gamma)$ in order type $\gamma^+$,
say as $\la A_i : i < \gamma^+ \ra$.
% , and fix for each $D_i$ some
% $\sigma_i < j(\gamma^+)$ such that $\sigma_i$ captures $D_i$.
}
{
We now construct a decreasing sequence of conditions $\la r_i : i < \gamma^+ \ra$,
maintaining the hypotheses that $r_{i+1} \in A_i$ and $r_i$ is compatible with
all the $q_\eta$. We set $r_0$ to be the trivial condition, and take
unions at limits. Given $r_i$ we first find $\eta_i$ such that
% $j(\eta_i) > \sigma_i$,
$A_i \subseteq (j(\FQ_\gg) \rest j(\eta_i))$
and $\dom(r_i) \subseteq j(\gamma) \times j(\eta_i)$, which is
possible as $j$ is continuous at $\gamma^+$. We form $r_i \cup
q_{\eta_i}$, which is a condition because $r_i$ is compatible with
$q_{\eta_i}$, and then use the fact that $A_i \subseteq (j(\FQ_\gg) \rest j(\eta_i))$
to find $r_{i+1} \le r_i \cup q_{\eta_i}$
such that $r_{i + 1}$ meets $A_i$ and
$\dom(r_{i+1}) \subseteq j(\gamma) \times j(\eta_i)$. The key
point is that since $r_{i+1}$ extends $q_{\eta_i}$, and $\dom(r_{i+1})
\subseteq j(\gamma) \times j(\eta_i)$, $r_{i+1}$ is compatible with
$q_\eta$ for all $\eta$.
}
Returning to the case at hand, we would like to apply the same idea.
But there is a subtle problem, which is that when we form
the ``partial master conditions'' $q_\eta$ in the natural way
they do not form a decreasing sequence. We now give a detailed analysis
of this issue.
We recall that $\FQ^\diamondsuit_\gamma$ is an iteration of length $\gamma^+$
with supports of size $\delta$. On coordinate zero we add a
$\diamondsuit'_\gamma$ sequence, and then we shoot clubs to make it into
a $\diamondsuit^+_\gamma$-sequence. For $\eta < \gamma^+$, let
$\FQ^\diamondsuit_\gamma \restriction \eta$ be the iteration up to stage
$\eta$ and let $H \restriction \eta$ be the corresponding generic
object.
Since $\FQ^\diamondsuit_\gamma \restriction \eta$ has size $\gamma$, we may
compute in $M[j(G) * K]$ the set $j`` (H \restriction \eta)$ and form its
``canonical'' lower bound $q_\eta$ as in
Lemma \ref{rectangulardenseclosed}. We need to analyse the
condition $q_\eta$ more closely. Let $\zeta = \sup j `` \gamma$, so that
as usual $\zeta < j(\gamma)$.
$q_\eta(0)$ is a sequence of length $\zeta +1$, and $q_\eta(0)
\restriction \zeta$ is just the union of $j(q)$ as $q$ runs through
$H \restriction 1$. $q_\eta \restriction [1, j(\eta))$ is a
partial function with support
\[
Z_\eta = \bigcup \{ j(x): x \in V[G * g], x \subseteq [1, \eta), \vert x
\vert \le \delta \}.
\]
For each $\alpha \in Z_\eta$, $q_\eta(\alpha)$ is a closed set with
maximum point $\zeta$, and
\[
q_\eta(\alpha) \cap \zeta = \bigcup \{ j(q)(\alpha): q \in H
\restriction \eta, \alpha \in j( {\rm supp}(q) ) \}.
\]
Recall that $\FQ^\diamondsuit_\gamma$ is an iteration in which at stage
$\alpha > 0$ we are given a set $X_\alpha \subseteq \gamma$, and
we shoot a club set through the set of places where it is correctly guessed by
the generic sequence added at stage $0$. Let
$j(\langle \dot X_\alpha : \alpha < \gamma^+ \ra) = \langle \dot Y_\beta : \beta <
j(\gamma^+) \rangle$.
For each $q \in H \restriction \eta$, if $q(0)$ has domain $\mu +1$
then for every $\alpha \in {\rm supp}(q)$, $\mu = \max q(\alpha)$ and
$q \restriction \alpha$ determines $\dot X_\alpha \cap \mu$.
It follows that for every $\alpha \in Z_\eta$, $q_\eta \restriction
\alpha$ determines $\dot Y_\alpha \cap \zeta$, say it forces it to be
$\check y_\alpha$. Then $q_\eta(0)(\zeta)$ consists of the sets
$y_\alpha$ and $q_\eta(\alpha) \cap \zeta$ for $\alpha \in Z_\eta$.
It is easy to see that if $\eta < \eta'$ then
\begin{enumerate}
\item $q_{\eta'} \restriction [1, j(\eta)) = q_\eta \restriction [1, j(\eta))$.
\item $q_{\eta'}(0) \restriction \zeta = q_{\eta}(0) \restriction \zeta$.
\end{enumerate}
So $q_{\eta'}$ is almost an extension of $q_\eta$, but not quite because
$q_{\eta'}(0)$ contains more sets at level $\zeta$.
We need a slightly more detailed analysis of the sets which can appear as
$y_\alpha$ and $q_\eta(\alpha) \cap \zeta$.
Fix some $\eta \in [\gamma, \gamma^+)$, and a bijection $\pi \in V$
between $\gamma$ and $\eta$. We may view $H \restriction [1, \eta)$ as
giving us a sequence $\langle C_\alpha : 0 < \alpha < \eta \rangle$
of club subsets of $\gamma$, which we may then code up as
$W \subseteq \eta \times \gamma$, where $(\alpha, i) \in W \iff i \in
C_\alpha$. Define $\bar W \subseteq \gamma \times \gamma$ by
$(\alpha, i) \in \bar W \iff (\pi(\alpha), i ) \in W$.
It is now routine to check that if we define
$W^* = \bigcup_{i < \gamma} j(\bar W \cap (i \times i))$,
then $W^* \subseteq \zeta \times \zeta$ and the sets which appear
in the form $\{ \nu < \zeta : (\alpha, \nu) \in W^* \}$ as $\alpha$
runs through $\zeta$ are precisely the sets $q_\eta(\alpha) \cap \zeta$ for $\alpha \in Z_\eta$.
A similar analysis works for the $y_\alpha$, with the sequence
$\langle X_\alpha : 0 < \alpha < \eta \rangle$ in place of
$\langle C_\alpha : 0 < \alpha < \eta \rangle$.
{
Now let $T \subseteq \gamma \times \gamma$ with $T \in V[G * g]$ being
arbitrary. Clearly,
since $V[G * g *H] \models {}^\gamma M[ j(G) * K ] \subseteq M[ j(G) * K ]$,
$T \in M[ j(G) * K ]$. We claim that the map which
takes such $T$ to $T^\dagger = \bigcup_{i < \gamma} j(T \cap (i \times
i))$ is in $M[ j(G) * K ]$.
To prove this claim we split into several cases depending on the nature of $\delta$.
}
{
\noindent
Case 1: $\delta$ is inaccessible. In this case $\FQ^\diamondsuit_\delta$ is
$\add(\gd, 1)$, so $\FP^\diamondsuit_\gamma$ is a forcing poset of size $\delta$.
So for each $i < \gamma$ the $\FP^\diamondsuit_\gamma$ names for subsets of $i
\times i$ are essentially subsets
of $\delta$, and we know that $j \restriction \wp(\delta) \in M$.
}
{
\noindent
Case 2: $\delta$ is the successor of a regular cardinal. Then $\FQ^\diamondsuit_\delta$
is an iteration of length $\delta^+$ with $\delta^+$-cc, so that all bounded
subsets of $\gamma$ appear in the extension by some initial segment, a forcing poset
of size $\delta$. So again all the relevant names are essentially subsets of
$\delta$ and we are done as in Case 1.
}
{
\noindent
Case 3: $\delta$ is the successor of a singular cardinal $\rho$. The cardinality of
$\FP^\diamondsuit_\rho$ is $\delta$, we are doing Cohen forcing at $\delta$, so
$\FP^\diamondsuit_\gamma$ has size $\delta$, and we are done as in Case 1.
}
{
So now we can compute in $M[j(G) * K]$ the set $P$ of all $T^\dagger$ as above.
This set has size bounded by $j(\delta)$ so we can augment each of the
$q_\eta$'s defined above by adding all elements of $P$ to coordinate zero at level
$\zeta$. This gives us a new sequence of partial master conditions $p_\eta^*$ with
the crucial extra property that the $p_\eta^*$ are literally decreasing.
We may now proceed with Magidor's argument exactly as above.
}\end{proof}
\section{The proof of the main theorem}\label{s4}
Having completed the discussion found in
Sections \ref{s1} -- \ref{s3}, we are
now ready to turn our attention
to the proof of Theorem \ref{t2}.
\begin{proof}
Let
$V \models ``$ZFC + GCH + $\K \neq \emptyset$
is the class of supercompact cardinals''.
We begin by forcing with the poset
$\FP^\square_\infty$ of Section \ref{s2}, to
obtain the extension $V_1 = V^{\FP^\square_\infty}$.
By Theorem \ref{sqth},
we know that all $V$-supercompact cardinals
are preserved to $V_1$. In addition, as in
Section \ref{s2}, we may write
$\FP^\square_\infty = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$, $\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is $(\go + 1)$-strategically
closed''. By Theorem \ref{t3}, this means that
any cardinal supercompact in $V_1$ had to have been
supercompact in $V$. Thus, $\K$ remains the
class of supercompact cardinals in $V_1$.
Also, by the work of Section \ref{s2}, we know that
$\square^S_\gg$ holds in $V_1$, for every
infinite cardinal $\gg$ and $S =
{\rm Safe}(\gg)$, and that GCH
is preserved to $V_1$ as well.
We force now over $V_1$ with the poset
of \cite{AS97a}, which for convenience
we denote here by $\FP^{LBL}_\infty$. This preserves all
$V_1$-instances of supercompactness, GCH,
and all cardinals and cofinalities, and
in addition forces
level by level equivalence between
strong compactness and supercompactness.
Call the resulting extension $V_2$.
Since $\FP^{LBL}_\infty$ may be defined so as to
admit a closure point at $\go$
%below the least $V_1$-supercompact cardinal
(see \cite{AS97a} for further details),
Theorem \ref{t3} and the remarks immediately
following imply that $\K$
remains the class of supercompact
cardinals in $V_2$.
Further, by the upwards absoluteness of
any form of square in a cardinal preserving
forcing extension (see the discussion
given in the proof of Theorem 1 of \cite{A05})
and the remarks in the paragraph following
the statement of Definition \ref{d1},
$\square^S_\gg$ for $S = {\rm Safe}(\gg)$
remains true in $V_2$ for all infinite cardinals $\gg$.
Finally, we force over $V_2$
with the poset $\FP^\diamondsuit_\infty$ of Section \ref{s3},
to obtain the extension $V_3$.
By the work of Section \ref{s3}, we know that
in $V_3$, $\diamondsuit_\mu$ holds
for every $\mu$ which is inaccessible or
the successor of a singular cardinal, and
$\diamondsuit^+_\mu$ holds for every
$\mu$ which is the successor of a regular cardinal.
GCH is preserved to $V_3$ as well. By Theorem \ref{diath},
we also know that all $V_2$-supercompact cardinals
are preserved to $V_3$. In addition,
we may write
$\FP^\diamondsuit_\infty = \FQ' \ast \dot \FR'$, where
$\card{\FQ'} = \go$, $\FQ'$ is nontrivial, and
$\forces_{\FQ'} ``\dot \FR'$ is $(\go + 1)$-strategically
closed''. By Theorem \ref{t3}, this means that
any cardinal supercompact in $V_3$ had to have been
supercompact in $V_2$. Thus, $\K$ remains the
class of supercompact cardinals in $V_3$.
Also, as in the last sentence of the
preceding paragraph, $\square^S_\gg$ for
$S = {\rm Safe}(\gg)$ remains true in
$V_3$ for all infinite cardinals $\gg$.
The proof of Theorem \ref{t2} is now
completed by the following lemma.
\begin{lemma}\label{lble}
$V_3 \models ``$Level by level equivalence
between strong compactness and supercompactness holds''.
\end{lemma}
\begin{proof}
We mimic the proof of Lemma 1.3 of \cite{A05}. Suppose
$V_3 \models ``\gk < \gl$ are regular cardinals such that
$\gk$ is $\gl$-strongly compact and $\gk$ isn't a
measurable limit of cardinals $\gd$ which are $\gl$-supercompact''.
By Theorem \ref{diath}, any cardinal $\gd$ such that
$\gd$ is $\gl$-supercompact in $V_2$ remains
$\gl$-supercompact in $V_3$. We may therefore infer that
$V_2 \models ``\gk < \gl$ are regular cardinals such that
$\gk$ isn't a measurable limit of cardinals $\gd$
which are $\gl$-supercompact''.
By the definition of $\FP^\diamondsuit_\infty$, it is easily seen that
$\FP^\diamondsuit_\infty$ is mild with respect to $\gk$. Hence, by
the factorization of $\FP^\diamondsuit_\infty$ given above and
Theorem \ref{t3}, $V_2 \models ``\gk$ is
$\gl$-strongly compact''. Consequently, by level by level
equivalence between strong compactness and supercompactness
in $V_2$, $V_2 \models ``\gk$ is $\gl$-supercompact'', so
another application of Theorem \ref{diath} yields that
$V_3 \models ``\gk$ is $\gl$-supercompact''.
This completes the proof of Lemma \ref{lble}.
\end{proof}
By defining $\FP = \FP^\square_\infty \ast \dot \FP^{LBL}_\infty
\ast \dot \FP^\diamondsuit_\infty$, the proof of Theorem \ref{t2}
is now complete.
\end{proof}
We conclude by asking if the results of Theorem \ref{t2}
can be generalised further. In particular, is it possible
to extend the above techniques so that
$\diamondsuit^+_\mu$ holds for every successor
cardinal $\mu$, and not just successors of
regular cardinals?
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which begins by adding a
Cohen subset of $\gk_1$
and then is defined exactly
as in Theorem 2 of \cite{A02}
using the supercompact
cardinals of $\ov V$,
which are the supercompact cardinals
of $V$ except for $\gk_1$.
In addition, the definition of $\gt$ tells us that if
$p \in j(\Q)$ is such that
$p \forces ``\gs \in \dot H$'', then
$p \forces ``\gt$ extends $\gs$''.
By elementarity, this means that
In \cite{AS97a}, Shelah and the
author introduced the notion
of level by level equivalence
between strong compactness
and supercompactness by proving
the following theorem.
\begin{theorem}\label{t0}
Let
$V \models ``$ZFC + $\K \neq
\emptyset$ 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 +
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{theorem}
%As was mentioned in \cite{A02},
We will say that
any model witnessing the conclusions of
Theorem \ref{t0} is a model for
level by level equivalence between
strong compactness and supercompactness.
%We will also say that $\gk$ is a witness
%to level by level equivalence between
%strong compactness and supercompactness
%iff for every regular $\gl > \gk$,
%$\gk$ is $\gl$ strongly compact iff
%$\gk$ is $\gl$ supercompact.
Note that the exception in Theorem \ref{t0}
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 $\gl$.
Observe also that Theorem \ref{t0} is a
strengthening of the result of Kimchi and
Magidor \cite{KM}, who showed it is
consistent for the classes of strongly
compact and supercompact cardinals to
coincide precisely, except at measurable
limit points.
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{\bf Studies in Logic and the Foundations of
Mathematics 102}, North-Holland, Amsterdam
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Mathematical Logic 10}, 1976, 33--57.
%\bibitem{Ma2} M.~Magidor, ``On the Existence of
%Nonregular Ultrafilters and the Cardinality of
%Ultrapowers'', {\it Transactions of the American
%Mathematical Society 249}, 1979, 97--111.