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%
% ------------------------------------------------------------------------------
%
\title{Stationary Reflection
%Ramseyness,
and Level by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal,
%strong cardinal, diamond, square,
level by level equivalence between strong
compactness and supercompactness, weakly
compact cardinal, Jonsson cardinal,
Ramsey cardinal, non-reflecting
stationary set of ordinals, diamond, square.}}
\author{Arthur W.~Apter\thanks{The
author's research was
partially supported by
PSC-CUNY Grants and CUNY
Collaborative Incentive Grants.
In addition, the author
wishes to
thank both Joel Hamkins and
Mirna D\v zamonja for helpful
correspondence concerning forcing
and the preservation of diamond,
and further thank Hamkins for helpful
conversations on the subject matter
of this paper. The author also
wishes to thank James Cummings for
having told him Theorem \ref{tnewb}
and its proof and having suggested
the use of Kunen's methods in the
proof of Theorem \ref{tnew}.
Finally, and most importantly, the
author offers his deepest thanks to
Grigor Sargsyan, for helpful conversations
and correspondence that led to the inner
model nature of this work, in particular,
for having supplied
the proof that every regular Jonsson
cardinal is weakly compact in
Theorem \ref{t1} and its generalizations.} \\
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 23, 2008\\
(revised August 30, 2008)}
\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 certain
additional ``inner model like''
properties.
In particular, in this model,
the class of Mahlo cardinals
reflecting stationary sets
is the same as the class of
weakly compact cardinals,
and every regular Jonsson cardinal
is weakly compact.
%and Ramsey in an inner model for the
%level by level equivalence between
%strong compactness and supercompactness definable with parameters.
On the other hand, we
force and construct a model for the
level by level equivalence between strong
compactness and supercompactness in which
on a stationary subset of the least
supercompact cardinal $\gk$,
there are non-weakly compact Mahlo
cardinals which reflect stationary sets.
We also examine some extensions and limitations
on what is possible in our theorems.
Finally, we indicate how to ensure in our models that
%In addition, we show in our models how to ensure that
$\diamondsuit_\gd$ holds for
%every regular uncountable cardinal $\gd$ (including inaccessible
%cardinals), and
every successor and Mahlo cardinal $\gd$, and
below the least supercompact
cardinal $\gk$, $\square_\gd$
%holds on a set having measure
%$1$ with respect to a certain
%normal measure over $\gk$.
holds on a stationary subset of $\gk$.
There are no restrictions in
our main models
%ground models or our generic extensions
on the structure of the
class of supercompact cardinals.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
It is well-known that
in canonical inner models,
many large cardinals
exhibit regularity properties
that allow for precise characterizations.
For instance, in $L$ and
higher inner models (see
\cite[Theorem 1.3, page 304]{D}
and \cite{Z}), the weakly compact
cardinals are exactly the class
of Mahlo cardinals admitting
stationary reflection\footnote{In
fact, in $L$ and higher inner
models, the weakly compact cardinals
are exactly the class of inaccessible
cardinals admitting stationary reflection.
We will come back to this point at the
end of the paper.}, and in
higher inner models (see \cite{Ku2},
\cite[Theorem 20.23]{K}, and \cite{Mi}),
the regular Jonsson cardinals
are precisely the Ramsey cardinals.
These results are of course obtained by
an analysis using standard inner
model techniques.
%of the relevant fine structure properties.
%of the inner models under consideration.
The purpose of this paper is not only to
construct via forcing a model
for the level by level equivalence
between strong compactness and
supercompactness in which analogues
of the above properties hold,
but also to construct via forcing a
model for the level by level equivalence
between strong compactness and supercompactness
in which stationary reflection can
occur on a stationary subset of the
least supercompact cardinal $\gk$
composed of non-weakly compact Mahlo cardinals.
Specifically, we prove the
following as our main theorems.
\begin{theorem}\label{t1}
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''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds.
Further, in $V^\FP$, the Mahlo
cardinals reflecting stationary
sets are precisely the weakly
compact cardinals.
Finally, every regular Jonsson
cardinal in $V^\FP$ is
weakly compact.
%in $V^\FP$ and Ramsey in an inner model of $V^\FP$.
\end{theorem}
\begin{theorem}\label{tnew}
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 +
$\gk$ is the least supercompact cardinal''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds.
Further, in $V^\FP$, there is a stationary
subset $A \subseteq \gk$
%of the least supercompact cardinal $\gk$
composed of non-weakly compact Mahlo cardinals which
reflect stationary sets.
\end{theorem}
At the end of the paper, we will briefly discuss
how to prove a generalized version of
Theorem \ref{t1} in a universe containing
relatively few large cardinals,
and also mention why some restrictions are necessary.
We will in addition indicate how to augment
Theorems \ref{t1}, \ref{tnew}, and the
generalized version of Theorem \ref{t1}
just mentioned to obtain models
%our theorems to obtain models
witnessing the same conclusions in which
$\diamondsuit_\gd$ holds for
%every regular uncountable cardinal $\gd$
%(including inaccessible cardinals), and
every successor and Mahlo cardinal $\gd$, and
$\square_\gd$ holds
below the least supercompact cardinal $\gk$
on a stationary set.
Theorem \ref{t1} and its generalizations
in which the models constructed satisfy
$L$-like properties may be considered to
follow the ``outer model programme'' as set
forth by S$.$ Friedman in \cite{F}.
Theorem \ref{t1} may be classified,
in Woodin's phrase, as an
``inner model theorem proven via forcing''.
This is because it satisfies properties
analogous to those mentioned in the
first paragraph above,
along with a property one might expect
in a ``nice'' inner model containing
supercompact cardinals, namely GCH and the
level by level equivalence between
strong compactness and supercompactness.
On the other hand, the model constructed
for Theorem \ref{tnew} doesn't have the
properties one might necessarily expect
in an inner model for a supercompact cardinal.
Of course,
%due to the lack of technology for constructing any sort of
it is presently unknown how to build any sort of
inner model for supercompact
cardinals along the lines
of the inner models currently
known\footnote{Woodin
has announced he can construct
an inner model of ZFC
containing supercompact
cardinals. His construction, however,
yields a model without covering
and indiscernibility; in particular,
his model does not have sharps.
This makes it quite different from
the usual kind of inner model,
and means its structural properties
are far more difficult to analyze.}.
Thus, it is necessary to use forcing to
produce models such as those
given in Theorems \ref{t1} and \ref{tnew},
and it is completely unclear what to expect
in terms of combinatorial properties
in a ``reasonable'' inner model containing
supercompact cardinals.
Before presenting the proofs
of our theorems, we briefly
mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals,
$[\ga, \gb], [\ga, \gb), (\ga, \gb]$, and $(\ga,
\gb)$ are as in the usual interval notation.
The regular cardinal $\gk$ will be said to
reflect stationary sets (or equivalently,
to admit stationary reflection) if for every
stationary $S \subseteq \gk$, there is a
limit ordinal $\gd < \gk$ such that
$S \cap \gd$ is a stationary subset of $\gd$.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
If $G$ is $V$-generic over $\FP$, we will
abuse notation somewhat and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
%If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
%If we also have that $\gk$ is inaccessible and
%$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra : \ga < \gk \ra$
%is a reverse Easton iteration of length $\gk$
%so that at stage $\ga$, a nontrivial forcing is done
%adding a subset of some ordinal
%$\gd_\ga$, then we will say that
%$\gd_\ga$ is in the field of $\FP$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$ or
$\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
Let $\gk$ be a regular cardinal.
$\add(\gk, 1)$ is the standard partial
ordering for adding a Cohen subset of $\gk$.
The partial ordering
$\FP$ is $\gk$-directed closed if for every cardinal $\delta < \gk$
and every directed
set $\langle p_\ga : \ga < \delta \rangle $ of elements of $\FP$,
%(where $\langle p_\alpha : \alpha < \delta \rangle$ is directed if
%for every two distinct elements $p_\rho, p_\nu \in
%\langle p_\alpha : \alpha < \delta \rangle$, $p_\rho$ and
%$p_\nu$ have a common upper bound of the form $p_\sigma$)
there is an upper bound $p \in \FP$.
$\FP$ is $\gk$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\langle p_\ga: \ga \le\gk\rangle$, where player I plays odd
stages and player
II plays even and limit stages
(choosing the trivial condition at stage 0),
player II has a strategy which
ensures the game can always be continued.
%$ \FP$ is ${<}\gk$-strategically closed if $\FP$ is $\delta$-strategically
%closed for all cardinals $\delta < \gk$.
$\FP$ is ${\prec}\gk$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\langle p_\alpha : \alpha < \gk \rangle$, where
player I plays odd stages and player II
plays even and limit stages
(again choosing the trivial condition at stage 0),
player II has a strategy which ensures the game
can always be continued.
%Note that trivially, if $\FP$ is ${<}\gk$-closed, then $\FP$ is
%${<}\gk$-strategically
%closed and ${\prec}\gk $-strategically closed. The converse of
%both of these facts is false.
Note that if $\FP$ is $\gk^+$-directed
closed, then $\FP$ is $\gk$-strategically closed.
In addition, if $\FP$ is $\gk$-strategically closed and
$f : \gk \to V$ is a function in $V^\FP$, then $f \in V$.
Suppose now that $\gk$ is a Mahlo cardinal.
A partial ordering $\FP(\gk)$
whose use will be critical in the
proof of Theorem \ref{t1} is the partial
ordering for adding a non-reflecting
stationary set of ordinals of a certain
type to $\gk$.
Specifically, $\FP(\gk) =
\{ p$ : For some
$\ga < \gk$, $p : \ga \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\ga$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that if $\gb < \sup(S_p)$ is inaccessible,
then $S_p - (S_p \cap \gb)$ is composed of
ordinals of cofinality at least $\gb\}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is shown in \cite{A01} that
forcing with $\FP(\gk)$ adds a
non-reflecting stationary set of
ordinals to $\gk$ and that
$\FP(\gk)$ is ${\prec}\gk$-strategically
closed.
This strategic closure property of
$\FP(\gk)$, together with the
fact that $\card{\FP(\gk)} = \gk$,
consequently imply that forcing
with $\FP(\gk)$ over a model of
GCH preserves GCH.
It is further shown in \cite{A01}
that for any inaccessible cardinal
$\gd < \gk$, there is a partial
ordering $\FP(\gk / \gd)$ dense in
$\FP(\gk)$ which is $\gd$-directed closed.
We take this opportunity to recall briefly
the combinatorial notions of diamond
and square.
%which were introduced by Jensen in \cite{Je}.
If $\gk$ is a regular uncountable
cardinal, $\diamondsuit_\gk$ is
the principle stating that
there exists a sequence of sets
$\la S_\ga : \ga < \gk \ra$ such that
$S_\ga \subseteq \ga$,
with the additional property that
for every $X \subseteq \gk$,
$\{\ga < \gk : X \cap \ga = S_\ga\}$ is
a stationary subset of $\gk$.
If $\gk$ is an arbitrary
uncountable cardinal,
$\square_\gk$ is the principle
stating that there exists
a sequence of sets
$\la C_\ga : \ga < \gk^+$ and
$\ga$ is a limit ordinal$\ra$ such that
$C_\ga$ is a closed, unbounded subset
of $\ga$ so that
if ${\rm cof}(\ga) < \gk$,
then $C_\ga$ has order type below $\gk$,
with the additional property that
for any limit point
$\gb \in C_\ga$, $C_\ga \cap \gb = C_\gb$.
For $\gk$ a regular uncountable cardinal,
it is possible to preserve
$\diamondsuit_\gk$ via certain forcing notions.
For a general treatment of this topic,
we refer readers to \cite{Sh2}.
For our purposes, we will need the
following two simple folklore facts.
%whose proofs we sketch.
\begin{fact}\label{f1}
Suppose
$V \models ``\gk$ is a regular uncountable
cardinal for which $\diamondsuit_\gk$ holds''
and $\FP \subseteq V$ is
${\prec}\gk$-strategically closed. Then
$V^\FP \models ``\diamondsuit_\gk$ holds''.
\end{fact}
\begin{proof}
Suppose
$V \models ``S = \la S_\ga : \ga < \gk \ra$
is a diamond sequence for $\gk$'' and
$\FP \subseteq V$ is ${\prec}\gk$-strategically
closed. Assume
$p \forces ``\dot X \subseteq \gk$
and $\dot C \subseteq \gk$ is club''.
Consider the game of length $\gk$
in which players I and II construct an
increasing sequence of conditions.
The game begins with
player II choosing the trivial condition and
player I choosing a condition
extending $p$ which decides the statements
``$0 \in \dot X$'' and ``$0 \in \dot C$''.
At non-limit even stages
$2\ga > 0$, player II must choose a
condition deciding the statements
``$\ga \in \dot X$'' and ``$\ga \in \dot C$''.
By the ${\prec}\gk$-strategic closure of $\FP$,
player II has a winning strategy for this game.
We may thus assume that
$\la p_\ga : \ga < \gk \ra$ is an increasing
sequence of conditions extending $p$ such that
$p_\ga$ completely determines both
$\dot X \cap \ga$ and $\dot C \cap \ga$,
sets in $V$ which we denote as
$X_\ga$ and $C_\ga$ respectively.
Let $X' = \bigcup_{\ga < \gk} X_\ga$ and
$C' = \bigcup_{\ga < \gk} C_\ga$.
Both $X'$ and $C'$ are members of $V$,
and $C'$ is a club subset of $\gk$.
Hence, since $S$ is a $\diamondsuit_\gk$
sequence in $V$, let $\gb \in C'$ be such that
$X' \cap \gb = S_\gb$.
It is then the case that there is $\gg > \gb$ with
$p_{\gg} \forces ``\gb \in \dot C$ and
$\dot X \cap \gb = S_\gb$'', which means that
$S$ remains in $V^\FP$ a $\diamondsuit_\gk$
sequence for $\gk$.
This completes the proof of Fact \ref{f1}.
\end{proof}
\begin{fact}\label{f2}
Suppose
$V \models ``\gk$ is a regular uncountable
cardinal for which $\diamondsuit_\gk$ holds''
and $\FP \in V$ is
$\gk$-c.c$.$ and has cardinality $\gk$. Then
$V^\FP \models ``\diamondsuit_\gk$ holds''.
\end{fact}
\begin{sketch}
We give a proof sketch which was essentially
told to us by Joel Hamkins.
We quote liberally from his presentation.
Suppose that $\la A_\ga : \ga < \gk \ra$
is a $\diamondsuit_\gk$ sequence, $G$ is $V$-generic over
$\FP$, and $\FP$ has cardinality $\gk$ and satisfies
$\gk$-c.c.
Let $B_\ga=i_G(A_\ga)$, provided that
$A_\ga$ codes a $\FP$-name, and
$B_\ga = \emptyset$ otherwise.
%(e.g. $A_\ga$ codes a relation on $\ga$, which when Mostowski collapsed,
%turns into a $\FP$-name).
If $C$ is any subset of $\gk$
in $V[G]$, then let $\dot C$ be a name for $C$.
Since $\card{\FP} = \gk$ and $\FP$ satsifies
$\gk$-c.c., we may
assume that $\dot C$ is hereditarily of
cardinality at most $\gk$.
Therefore, we may let
$C^* \subseteq \gk$, $C^* \in V$ code $\dot C$.
%in the way I mentioned.
In $V$, $C^*$ is anticipated on a stationary set, i.e.,
$S = \{\ga < \gk : C^* \cap \ga = A_\ga\}$ is
stationary.
%This gives rise to a stationary set in $V$ such that
%$A_\ga$ is $C^* \cap \ga$, and this is a $\FP$-name.
Further, $S$ is a $\FP$-name.
By \cite[Exercise H2, page 247]{Ku1},
since $\FP$ is $\gk$-c.c.,
$S$ remains stationary in $V[G]$.
And, on a club, $i_G(C^* \cap
\ga) = C \cap \ga$.
Thus, $C$ is anticipated on a stationary subset by
$\la B_\ga : \ga < \gk \ra$.
This completes the proof sketch of Fact \ref{f2}.
\end{sketch}
The notion
of level by level equivalence
between strong compactness
and supercompactness
was introduced by Shelah and
the author in \cite{AS97a}.
In this paper, the following
theorem was proven.
\begin{theorem}\label{t2}
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}
We will say that
any model witnessing the conclusions of
Theorem \ref{t2} is a model for the
level by level equivalence between
strong compactness and supercompactness.
Note that the exception in Theorem \ref{t2}
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.
Observe also that Theorem \ref{t2} 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.
A result which will be key in the proof
of Theorem \ref{t1} is an
amalgamation of Hamkins'
Gap Forcing Theorem of \cite{H2, H3}
together with \cite[Corollary 16]{H2, H3}.
We therefore state the theorem
we will be using now, along
with some associated terminology, quoting
freely from \cite{H2, H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| < \gd$, $\FQ$ is nontrivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2, H3},
$\FP$ {\rm admits a gap at $\gd$}.
In Hamkins' terminology of \cite{H2, H3},
$\FP$ is {\rm 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, H3} (and elsewhere),
an embedding
$j : \ov V \to \ov M$ is
{\rm amenable to $\ov V$} when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The specific theorem we will be
using is then the following.
\begin{theorem}\label{t3}
{\bf (Hamkins)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing
with a partial ordering $\FP$ that
admits a gap at some $\gd < \gk$ and
$j: V[G] \to M[j(G)]$ 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 and $\gk$ is $\gl$ strongly
compact in $V[G]$ for any $\gl \ge \gk$,
then $\gk$ is $\gl$ strongly compact in $V$.
\end{theorem}
Finally, we mention that we are assuming
familiarity with standard large
cardinal notions.
%the large cardinal notions of weak compactness, Jonsonness,
%Ramseyness, measurability, strong compactness, and supercompactness.
Interested readers may consult
\cite{J} or \cite{K}
for further details.
We note only that the cardinal
$\gk$ is supercompact up to the cardinal $\gl$
if $\gk$ is $\gd$ supercompact for every $\gd < \gl$.
\section{The Proofs of Theorems \ref{t1}
and \ref{tnew}}\label{s1}
We turn now to the proofs of
Theorems \ref{t1} and \ref{tnew}.
\begin{proof}
Let
$V \models ``$ZFC + $\K \neq
\emptyset$ is the class of
supercompact cardinals''.
Without loss of generality, by first doing
a preliminary forcing if necessary, we may
also assume
that $V$ is as in Theorem \ref{t2}, i.e.,
that GCH and level by level equivalence between
strong compactness and supercompactness
hold in $V$.
This allows us to define the partial
ordering $\FP$ used in the proof
of Theorem \ref{t1}
as the (possibly proper
class) reverse Easton iteration
which begins by forcing with $\add(\go, 1)$
%adding a Cohen subset of $\go$
and then is trivial forcing, except
at cardinals which are in $V$ both
non-Ramsey and Mahlo.
At such a cardinal $\gk$, we
force with the partial ordering
$\FP(\gk)$.
Standard arguments (see \cite{J}) then
show that for $\FQ$ any initial
segment (proper or improper) of $\FP$,
$V^\FQ \models $ ZFC + GCH
and $V$ and $V^\FQ$ have
the same cardinals and cofinalities.
\begin{lemma}\label{l1}
If $V \models
``\gk < \gl$ are such that
$\gk$ is $\gl$ supercompact
and $\gl$ is regular'', then
$V^\FP \models ``\gk$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
Suppose that
$\gk$ and $\gl$ are as
in the hypotheses of Lemma \ref{l1}.
Fix $j : V \to M$ an elementary
embedding witnessing the $\gl$
supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$, and
write $\FP = \FP_{\gl + 1} \ast
\dot \FP^{\gl + 1}$. Since
$\forces_{\FP_{\gl + 1}} ``\dot
\FP^{\gl + 1}$ is $\gg$-strategically
closed for $\gg$ the least
inaccessible cardinal above $\gl$'',
it suffices to show that
$V^{\FP_{\gl + 1}} \models ``\gk$
is $\gl$ supercompact''.
To do this,
we use a variant of the
argument given in the
proof of \cite[Lemma 3.1]{A01}.
Write $\FP^{\gl + 1}$ as
$\FP^0 \ast \dot \FP^1$,
where $\FP^0$ is
$\FP^{\gl + 1}$ defined
through stage $\gk$,
i.e., $\FP^0 = \FP_\gk$, and
$\dot \FP^1$ is a term for
the rest of $\FP^{\gl + 1}$,
i.e., the portion acting on the
non-Ramsey Mahlo cardinals
in the half-open interval
$(\gk, \gl]$.
If it is not the case that
$V^{\FP_{\gl + 1}} \models ``\gk$
is $\gl$ supercompact'', then let
$p = \la p_0, \dot p_1 \ra \in \FP^0 \ast \dot \FP^1$
be such that
$p \forces ``\gk$ isn't $\gl$ supercompact''.
By our remarks in Section \ref{s0},
we assume without loss of generality
that each nontrivial coordinate of
$p_1$ is a term for a condition in the appropriate
$\FP(\gd / \gk)$.
Let $G_0$ be $V$-generic
over $\FP^0$ such that
$p_0 \in G_0$.
Since $\FP^0$ may be defined
so as to have cardinality $\gk$,
by the L\'evy-Solovay results \cite{LS},
we know that the set
$A = \{\gd \in (\gk, \gl] : \gd$
is a non-Ramsey Mahlo cardinal$\}$
is the same in both $V$ and
$V[G_0]$. Consequently,
working in $V[G_0]$ and once again using
our remarks from Section \ref{s0}, let $\FP^3$ be the reverse Easton
iteration of partial orderings which,
for every $\gd \in A$,
add non-reflecting
stationary sets of ordinals using
$\FP(\gd / \gk)$.
%Again without loss of generality, we assume that
%every element of $\FP^3$ extends $p_1$.
Note now that if $G_1$ is $V[G_0]$-generic over
$\FP^3$ and $p_1 \in G_1$, then $G_1$ must also
generate a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
To see this, it clearly suffices to show that
$G_1$ meets all dense open subsets of $\FP^1$
above $p_1$.
If $D$ is such a set,
then let $D_1 = \{q \in \FP^3 : q$ extends some
element of $D\}$.
$D_1$ is clearly open.
If $q \in \FP^3$, then $q \in \FP^1$,
so by density, there is $q' \ge q$, $q' \in D$.
By using our remarks from Section \ref{s0} if necessary to
find a term which is forced to extend
each term denoting a
nontrivial coordinate of $q'$ to
a term for an
element of the appropriate $\FP(\gd / \gk)$,
we obtain $q'' \ge q' \ge q$,
$q'' \in D_1$.
Thus, $G_1$ meets $D_1$ and hence meets $D$, so
$G_1$ generates a $V[G_0]$-generic filter $G^*_1$
over $\FP^1$.
By the definition of
$\FP$ and the closure properties of $M$,
$j(\FP^0 \ast \dot \FP^1) =
\FP^0 \ast \dot \FP^1 \ast \dot \FQ \ast \dot \FR$,
where $\dot \FQ$ is a term for the portion of
$j(\FP^0 \ast \dot \FP^1)$ acting on
ordinals in the open interval $(\gl, j(\gk))$, and
%defined in $M$ between stages $\gl$ and $j(\gk)$, and
$\dot \FR$ is a term for $j(\dot \FP^1)$, i.e.,
the portion of
$j(\FP^0 \ast \dot \FP^1)$
acting on ordinals in the interval $(j(\gk), j(\gl)]$.
%defined in $M$ between stages $j(\gk)$ and $j(\gl)$.
If $G_1$ is $V[G_0]$-generic over $\FP^3$ and
$p_1 \in G_1$, then
by the preceding paragraph,
$G_1$ generates a $V[G_0]$-generic
filter $G^*_1$ over $\FP^1$.
Since $\FP^1$ is $\gl^+$-c.c$.$
in $V[G_0]$, $M[G_0][G^*_1]$ remains
$\gl$-closed with respect to
$V[G_0][G^*_1]$. Consequently,
by GCH in $V[G_0][G^*_1]$,
the usual diagonalization argument
(as given, e.g., in the construction
of the generic object $G_1$ in
\cite[Lemma 2.4]{AC2})
may be used to build in $V[G_0][G^*_1]$
an $M[G_0][G^*_1]$-generic object
$G_2$ over $\FQ$.
(This argument uses the
${\prec}\gl^+$-strategic closure
of $\FQ$ in both $M[G_0][G^*_1]$ and
$V[G_0][G^*_1]$, together with the
fact that by GCH, there are only
$2^\gl = \gl^+$ many dense open
subsets of $\FQ$ present in
$M[G_0][G^*_1]$, to meet all of
the required sets.)
We may then lift $j$ in
$V[G_0][G^*_1]$ to
$j : V[G_0] \to M[G_0][G^*_1][G_2]$.
Since
$G_1 \subseteq G^*_1$ and $G_1$ is $V[G_0]$-generic over
a partial ordering ($\FP^3$) that is
$\gk$-directed closed in $V[G_0]$,
$j''G_1$ generates in $V[G_0][G^*_1][G_2]$ a
compatible set of conditions of cardinality
$\gl < j(\gk)$ in a partial ordering
($j(\FP^3)$) that is $j(\gk)$-directed
closed in $M[G_0][G^*_1][G_2]$.
Therefore, by the fact
$M[G_0][G^*_1][G_2]$ is
$\gl$-closed with respect to
$V[G_0][G^*_1][G_2] = V[G_0][G^*_1]$, we can let
$r$ be a master condition for
$j''G_1$ and once again use the
usual diagonalization argument
in $V[G_0][G^*_1]$
to build $G_3$ to be an
$M[G_0][G^*_1][G_2]$-generic object over
$j(\FP^3)$ containing $r$.
By elementarity, it will be the case that
$G_3$ generates an $M[G_0][G^*_1][G_2]$-generic
object $G^*_3$ over
$\FR = j(\FP^1)$ containing $r$.
As usual, we will then have that in
$V[G_0][G^*_1]$, $j$ lifts to
$j : V[G_0][G^*_1] \to M[G_0][G^*_1][G_2][G^*_3]$, so
$\gk$ is $\gl$ supercompact in $V[G_0][G^*_1]$.
This, however, contradicts that
$p = \la p_0, p_1 \ra \in G_0 \ast G^*_1$ and
$p \forces ``\gk$ isn't $\gl$ supercompact''.
This contradiction completes the proof of
Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$V^\FP \models ``$Level by level
equivalence between strong compactness
and supercompactness holds''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2}
follows closely
the proofs of \cite[Lemma 3.2]{A02}
and \cite[Lemma 1.3]{A05}.
Suppose
$V^\FP \models ``\gk < \gl$ are
regular cardinals such that
$\gk$ is $\gl$ strongly compact and
$\gk$ isn't a measurable limit of
cardinals $\gd$ which are
$\gl$ supercompact''.
%As we observed in the first
%paragraph of the proof of
%Theorem \ref{t1}, forcing with
%$\FP$ preserves all cardinals
%and cofinalities.
By Lemma \ref{l1}, any cardinal
$\gd$ such that $\gd$ is
$\gl$ supercompact in $V$ remains
$\gl$ supercompact in $V^\FP$.
This means that
$V \models ``\gk < \gl$ are
regular cardinals such that
%$\gk$ is $\gl$ strongly compact and
$\gk$ isn't a measurable limit of
cardinals $\gd$ which are
$\gl$ supercompact''.
Note that it is possible to
write $\FP =
\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$,
$\FQ$ is nontrivial, and
$\forces_{\FQ} ``\dot \FR$ is
$\go$-strategically closed''.
Further, by the definition of
$\FP$, it is easily seen that
$\FP$ is mild with respect to $\gk$.
Therefore, by Theorem \ref{t3},
$V \models ``\gk$ is $\gl$ strongly compact''.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$V \models ``\gk$ is $\gl$ supercompact'',
so another application of Lemma \ref{l1}
yields that
$V^\FP \models ``\gk$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``\K$ is the
class of supercompact cardinals''.
\end{lemma}
\begin{proof}
By Lemma \ref{l1}, if
$\gk$ is $\gl$ supercompact in $V$
for $\gl > \gk$ regular,
then $\gk$ is $\gl$ supercompact in $V^\FP$.
Further, by the factorization of
$\FP$ as $\FQ \ast \dot \FR$ given
in Lemma \ref{l2} and
an application of Theorem \ref{t3},
any cardinal $\gk$ which is $\gl$ supercompact in
$V^\FP$ had to have been $\gl$ supercompact
in $V$. Thus, $\K$ is precisely the
class of supercompact cardinals in $V^\FP$.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``$If $\gk$
is a Ramsey cardinal in $V$,
then $\gk$ is weakly compact''.
\end{lemma}
\begin{proof}
Suppose $V \models ``\gk$ is
a Ramsey cardinal''. Write
$\FP = \FP_\gk \ast \dot \FQ$.
By the definition of $\FP$,
$\forces_{\FP_\gk} ``\dot \FQ$ is
$\gk$-strategically closed''.
Thus, to prove Lemma \ref{l4},
it suffices to show that
$V^{\FP_\gk} \models ``\gk$
is weakly compact''.
To do this, we adapt an argument
from Theorem 1.4 of Hamkins' paper \cite{H4},
quoting liberally from his presentation.
By the definition of $\FP$, it is
clearly the case that $\gk$ remains
inaccessible in $V^{\FP_\gk}$.
It therefore is enough to show that
$\gk$ has the tree property in $V^{\FP_\gk}$.
Suppose as a consequence that
$\dot T$ is a name for a $\gk$-tree in
$V^{\FP_\gk}$. In $V$, let
$N$ be a transitive elementary
substructure of $H(\gk^+)$ of size
$\gk$ containing $\FP_\gk$ and
$\dot T$ which is closed under
${<}\gk$ sequences.
Since $\gk$, being Ramsey in $V$,
is also weakly compact in $V$,
there is an elementary embedding
$j : N \to M$ having critical point $\gk$.
As in \cite[Theorem 1.4]{H4}, we
may also assume that $\card{M} = \gk$ and
$V \models ``M^{< \gk} \subseteq M$''.
Write $j(\FP_\gk) = \FP_\gk \ast \dot \FR$.
For any $V$-generic object $G$ over
$\FP_\gk$, the fact $\FP_\gk$ is
$\gk$-c.c$.$ allows us to infer that
$V[G] \models ``M[G]^{< \gk} \subseteq
M[G]$''. Further,
regardless if $\FR$ acts nontrivially on $\gk$,
%$\gk \in {\rm field}(\FR)$,
it is the case that
$M[G] \models ``\FR$ is
${\prec}\gk$-strategically closed''.
Therefore, since by the fact
$V[G] \models ``\card{M[G]} = \gk$'',
there are only $\gk$ many dense open
subsets of $\FR$ present in $M[G]$,
and since
$V[G] \models ``M[G]^{< \gk} \subseteq
M[G]$'',
we may use the diagonalization argument
mentioned in the proof of Lemma \ref{l1}
to meet the $\gk$ many dense open
subsets of $\FR$ and
construct in $V[G]$ an
$M[G]$-generic object $H$ for $\FR$.
$j$ then lifts in $V[G]$ to an elementary embedding
$j : N[G] \to M[G][H]$.
Because $\dot T \in N$,
$T \in N[G]$.
Since $T$ is a $\gk$-tree in both
$V[G]$ and $N[G]$, by elementarity,
$j(T)$ is a $j(\gk)$-tree in $M[G][H]$.
Any element on the $\gk^{\rm th}$ level of
$j(T)$ gives a branch of length
$\gk$ through $T$.
This means that $\gk$ has the tree
property in $V[G]$, as desired.
This completes the proof of Lemma \ref{l4}.
\end{proof}
\begin{lemma}\label{l5}
In $V^\FP$, the Mahlo
cardinals reflecting stationary sets
are precisely the weakly
compact cardinals.
\end{lemma}
\begin{proof}
If $\gk$ is weakly compact,
then clearly, $\gk$ is both
Mahlo and reflects stationary sets.
For the reverse direction, suppose
$V^\FP \models ``\gk$ is Mahlo and reflects
stationary sets''.
Let $\gd$ be a $V$-Mahlo non-Ramsey
cardinal, and write
$\FP = \FP_{\gd + 1} \ast \dot \FQ$.
Since
$\forces_{\FP_{\gd + 1}} ``\gd$
contains a non-reflecting stationary
set of ordinals and $\dot \FQ$ is
$\gd$-strategically closed'',
$V^\FP \models ``\gd$ contains
a non-reflecting stationary set of ordinals''.
Since any cardinal Mahlo in $V^\FP$
had to have been Mahlo in $V$,
$\gk$ had to have been a Ramsey cardinal in $V$.
By Lemma \ref{l4},
$V^\FP \models ``\gk$ is weakly compact''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
Every regular Jonsson cardinal in
$V^\FP$ is weakly compact.
%in $V^\FP$ and Ramsey in an inner model of $V^\FP$.
%definable from a parameter.
\end{lemma}
\begin{proof}
Suppose
$V^\FP \models ``\gk$ is
a regular Jonsson cardinal''.
Since $V^\FP \models {\rm GCH}$,
by \cite[Chapters 3 and 4]{Sh1},
$\gk$ must also be a Mahlo cardinal.
By a result of Tryba \cite{Tr}
(also independently due
\break to Woodin --- see
\cite[Proposition 8.17, page 96]{K}), $\gk$ must reflect
stationary sets at some limit
ordinal $\gl < \gk$.
Hence, by Lemma \ref{l5},
%and its proof,
$\gk$ is weakly compact in $V^\FP$.
%and is Ramsey in $V$, which is of course an inner model of $V^\FP$.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Since $V$ is of course an inner model
of $V^\FP$, Lemma \ref{l5} and
its proof and Lemma \ref{l6} easily
imply that any regular Jonsson
cardinal in $V^\FP$ is Ramsey in an
inner model (namely $V$).
We note that by \cite[Theorem 3]{L},
if the class of non-Ramsey Mahlo
cardinals is actually a set, then
$V$ is definable within $V^\FP$
using a certain set parameter.
This tells us that this inner model
%of Lemma \ref{l6} (namely $V$)
is in a certain sense definable
within $V^\FP$, thereby enhancing
the analogy with the canonical
inner models mentioned in the
first paragraph of this paper.
Lemmas \ref{l1} -- \ref{l6} complete
the proof of Theorem \ref{t1}.
\end{proof}
Having finished the proof of
Theorem \ref{t1},
%along with the ensuing discussion,
we turn now to the proof of Theorem \ref{tnew}.
\begin{proof}
Let $V \models ``$ZFC + $\K$ is the
class of supercompact cardinals + $\gk$
is the least supercompact cardinal''.
Without loss of generality, as in
the proof of Theorem \ref{t1}, we
assume in addition that $V \models ``$GCH +
Level by level equivalence between strong
compactness and supercompactness holds''.
We define now a certain stationary
subset $A \subseteq \gk$. By
\cite[Lemma 2.1]{AC2} and the succeeding
remarks, $\gk$ is a limit of strong cardinals,
since $\gk$ is (at least) $2^\gk$
supercompact and strong.
Let $\mu$ be a normal measure over $\gk$
having Mitchell rank 1.\footnote{Relevant
facts and definitions concerning the
Mitchell ordering of normal measures
and supercompact cardinals may be
found in \cite{J}.} In the ultrapower
$M = V^\gk/\mu$, the statement
``$\gk$ is a measurable cardinal having
trivial Mitchell rank which is a limit
of strong cardinals'' is true, since
$\gk$ is the critical point of the
elementary embedding generated by $\mu$.
Therefore, by reflection,
$A = \{\gd < \gk : \gd$ is a measurable
limit of strong cardinals having trivial
Mitchell rank$\} \in \mu$, which automatically
implies that $A$ is a stationary subset of $\gk$.
Given the set $A$, we are now ready to
present the partial ordering $\FP$ used
in the proof of Theorem \ref{tnew}. Let
$B = \{\gd < \gk : \gd$ is a
strong cardinal which isn't a limit
of strong cardinals$\}$.
Easily, $A \cap B = \emptyset$.
Define $\FP$ as the reverse Easton iteration
having length $\gk$ which begins by
forcing with $\add (\go, 1)$ and then does trivial
forcing except when $\gd \in A \cup B$.
If $\gd \in B$, then we force with $\add(\gd, 1)$.
If $\gd \in A$, then
we force with the partial ordering
$\FQ_\gd$ of \cite[page 69]{Ku1a} adding a
$\gd$-Souslin tree (via homogeneous trees
of successor height less than $\gd$, ordered
by end-extension).
Because $\FP$ is $\gk$-c.c.,
an application of \cite[Exercise H2, page 247]{Ku1}
tells us that $A$ remains stationary after forcing
with $\FP$. In addition, the arguments of
\cite[pages 68--71]{Ku1a} tell us that
for $\gd \in A$, $V^{\FP_\gd \ast \dot \FQ_\gd} =
V^{\FP_{\gd + 1}} \models ``\gd$ is a non-weakly
compact Mahlo cardinal which reflects stationary sets''.
Since by \cite[page 70]{Ku1a}, for $\gd \in A$,
it is the case that $\FQ_\gd$ is
${\prec}\gd$-strategically closed,
we may now infer that
$V^\FP \models ``$A is a stationary subset of $\gk$
composed of non-weakly compact Mahlo
cardinals which reflect stationary sets''.
The following is the natural analogue of Lemma \ref{l1}.
\begin{lemma}\label{l7}
If $V \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact and $\gl$ is regular'', then
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
\end{lemma}
\begin{proof}
If $\gd > \gk$, then Lemma \ref{l7} easily
follows by the results of \cite{LS}.
We consequently assume for the remainder
of the proof of Lemma \ref{l7} that $\gd \le \gk$.
It is clear that $\gd \not\in A$, since by GCH
and the fact $\gl > \gd$, $\gd$ is at least
$2^\gd$ supercompact and hence has nontrivial
Mitchell rank. In addition, as we just mentioned,
\cite[Lemma 2.1]{AC2} and the succeeding remarks
show that if $\gd$ is (at least) $2^\gd$ supercompact
and strong, then $\gd$ is a limit of strong cardinals.
From this, it immediately follows that
$\gd \not\in B$, so $\gd$ must be a
trivial stage of forcing.
Let $\gg = \sup(\{\ga < \gd : \ga$
is a nontrivial stage of forcing$\})$, and
write $\FP = \FP_\gg \ast \dot \FQ$.
If $\gd = \gk$, then $\dot \FQ$
is a term for trivial forcing.
If $\gd < \gk$, then since
$\gd$ is a trivial stage of forcing, $\gb$,
the first member
on which $\dot \FQ$ is forced to act nontrivially,
%of the field of $\dot \FQ$,
must be above $\gd$. Further,
it is the case that $\gl < \gb$.
This is since otherwise,
$V \models ``\gd$ is $\ga$ supercompact
for every $\ga < \gb$ and $\gb$ is strong''.
Thus, as mentioned in the proof of
\cite[Lemma 2.4]{AC2}, $\gd$ must be
supercompact, which contradicts that
$V \models ``\gd < \gk$ and $\gk$ is
the least supercompact cardinal''.
Consequently, regardless if
$\gd = \gk$ or $\gd < \gk$, to show that
$V^\FP = V^{\FP_\gg \ast \dot \FQ} \models
``\gd$ is $\gl$ supercompact'', it suffices
to show that $V^{\FP_\gg} \models ``\gd$
is $\gl$ supercompact''.
To do this, we first observe that if
$\gg < \gd$, then $\card{\FP_\gg} < \gd$,
so by the results of \cite{LS},
$V^{\FP_\gg} \models ``\gd$ is $\gl$ supercompact''.
We hence assume without loss of generality that $\gg = \gd$.
Let then $j : V \to M$ be an elementary embedding
witnessing the $\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over $P_\gd(\gl)$
such that $M \models ``\gd$ isn't $\gl$ supercompact''.
Since ${\rm cp}(j) = \gd$, $\gg = \gd$, $\gl > \gd$,
and GCH holds in $V$,
$M \models ``\gd$ has nontrivial Mitchell rank
and is a limit of strong cardinals''. Also,
$M \models ``$No cardinal $\eta \in (\gd, \gl]$
is strong'', because if not, then by closure,
$M \models ``\gd$ is $\ga$ supercompact for all
$\ga < \eta$, where $\eta$ is strong''.
As we have already observed, this means that
$M \models ``\gd$ is supercompact'', a
contradiction to the fact that
$M \models ``\gd$ isn't $\gl$ supercompact''.
It hence immediately follows that
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ'$,
where the first ordinal
on which $\dot \FQ'$ is forced to act nontrivially
%in the field of $\dot \FQ'$
is above $\gl$.
%A simplified form of the argument given in the proof of Lemma \ref{l1}
%(which doesn't require the construction of a master condition)
Once again, the usual diagonalization argument
(to which we referred in the proof of Lemma \ref{l1})
%(as given, e.g., in the construction of
%the generic object $G_1$ in \cite[Lemma 2.4]{AC2})
then applies and shows
that $j$ lifts in $V^{\FP_\gd} = V^{\FP_\gg}$ to
$j : V^{\FP_\gd} \to M^{j(\FP_\gd)}$, i.e.,
$V^{\FP_\gg} \models ``\gd$ is $\gl$ supercompact''.
This completes the proof of Lemma \ref{l7}.
\end{proof}
By writing $\FP = \FQ \ast \dot \FR$, where
$\FQ$ is nontrivial, $\card{\FQ} = \go$, and
$\forces_{\FQ} ``\dot \FR$ is $\go$-strategically closed'',
the same proof as presented in Lemma \ref{l3}
shows that
$V^\FP \models ``\K$ is the class of
supercompact cardinals''.
From this, it immediately follows that
$V^\FP \models ``\gk$ is the least supercompact cardinal''.
By using the factorization of $\FP$
just given and replacing a reference to
Lemma \ref{l1} with a reference to Lemma \ref{l7},
the same proof as found in Lemma \ref{l2} applies
and shows that $V^\FP \models ``$Level by level
equivalence between strong compactness and
supercompactness holds''.
Since standard arguments once again
show that $V^\FP \models {\rm GCH}$,
this completes the proof of Theorem \ref{tnew}.
\end{proof}
\section{Some Additional Comments and
Concluding Remarks}\label{s2}
As we have already mentioned,
in $L$ and higher inner models,
the weakly compact cardinals are
precisely the class of inaccessible cardinals admitting
stationary reflection. One may wonder whether this
phenomenon is also possible in the context of the
level by level equivalence between strong
compactness and supercompactness.
The methods previously discussed in fact
allow us to establish the following theorem,
which is a generalized version of Theorem \ref{t1}.
\begin{theorem}\label{tnewa}
Let
$V \models ``$ZFC + $\gk$ is
supercompact + No cardinal is
supercompact up to an inaccessible cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is
supercompact up to an inaccessible cardinal''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds.
Further, in $V^\FP$, the inaccessible
cardinals reflecting stationary
sets are precisely the weakly
compact cardinals.
Finally, every regular Jonsson
cardinal in $V^\FP$ is
weakly compact.
\end{theorem}
\begin{sketch}
Suppose
$V \models ``$ZFC + $\gk$ is
supercompact + No cardinal is
supercompact up to an inaccessible cardinal''.
Without loss of generality, as in the
proofs of Theorems \ref{t1} and \ref{tnew},
we assume in addition that
$V \models ``$GCH + Level by level equivalence
between strong compactness and supercompactness holds''.
For $\gd$ an inaccessible cardinal,
redefine $\FP(\gd)$ to be the partial
ordering for adding a non-reflecting
stationary set of ordinals of cofinality
$\go$ to $\gd$.
($\FP(\gd)$ is composed of
characteristic functions of subsets of
$\gd$ consisting of ordinals of
cofinality $\go$ which are non-stationary
at their supremum nor have any initial
segments which are stationary, ordered by
end-extension --- a more precise definition
may be found in \cite[Section 1]{AC2}.)
The partial ordering $\FP$ used in the proof
of Theorem \ref{tnewa} is the reverse Easton
iteration of length $\gk$ which begins
by forcing with $\add(\go, 1)$ and then is
trivial forcing, except at cardinals which
are in $V$ both non-Ramsey and inaccessible.
At such a cardinal $\gd$, we force with the
partial ordering $\FP(\gd)$.
If $V \models ``\gd < \gl$ are such that
$\gd$ is $\gl$ supercompact and
$\gl$ is regular''
and $j : V \to M$ is an elementary embedding
witnessing the $\gl$ supercompactness of
$\gd$ generated by a supercompact ultrafilter
over $P_\gd(\gl)$, then since $\gl$
must be below the least inaccessible above $\gd$,
$j(\FP_\gd) = \FP_\gd \ast \dot \FQ$, where the
first ordinal at which $\dot \FQ$ is forced to
act nontrivially is well above $\gl$.
The usual diagonalization argument therefore
once again applies and allows us to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
With $\K$ having $\gk$ as its only member,
the arguments of Lemmas \ref{l2} -- \ref{l6}
suitably modified then yield
that $V^\FP$ is as desired.
This completes the proof sketch of \break Theorem \ref{tnewa}.
\end{sketch}
Of course, the large cardinal structure
of both our ground model and generic extension
in Theorem \ref{tnewa} is severely limited.
One may wonder if this is indeed necessary.
The following theorem, told to us by James Cummings,
shows that some restrictions are required.
\begin{theorem}\label{tnewb}
{\bf (Folklore)} Suppose $\gk$ is a
regular limit of
cardinals $\gd$ which are $\gk$ strongly compact.
Suppose in addition that the
regular cardinals below $\gk$ are
non-stationary (e.g., if $\gk$ is the least
regular limit of cardinals $\gd$ which
are $\gk$ strongly compact).
Then $\gk$ admits stationary reflection.
%A non-Mahlo inaccessible limit of
%strongly compact cardinals $\gk$ admits
%stationary reflection.
\end{theorem}
\begin{proof}
Take $S \subseteq \gk$ as being stationary.
Define $f : S \to \gk$ by $f(\gd) = \cof(\gd)$.
Since $f(\gd) \le \gd$, by Fodor's Theorem,
$f$ is either the identity on a stationary
subset of $S$, or $f(\gd) = \ga$ for some
fixed cardinal $\ga$ and all $\gd$ in a
stationary subset of $S$. If the former holds,
then the regular cardinals must be a stationary
subset of $\gk$, contradictory to our
hypotheses. Thus, fix
$T \subseteq S$ stationary and $\ga$ such that
$f(\gd) = \ga$ for all $\gd \in T$.
Since $\gk$ is a limit of cardinals
$\gd$ which are $\gk$ strongly compact,
let $\gk_0 \in (\ga, \gk)$ be such that
$\gk_0$ is $\gk$ strongly compact. Since $\ga < \gk_0 < \gk$,
$\gk_0$ is $\gk$ strongly compact, and $\gk$ is regular,
$\gk$ admits stationary reflection for
stationary subsets composed of ordinals
of cofinality $\ga$. Thus, there is some
$\gd < \gk$ for which $T \cap \gd$, and
hence $S \cap \gd$, is stationary.
This completes the proof of Theorem \ref{tnewb}.
\end{proof}
As mentioned in Section \ref{s0},
it is possible to augment the
results of Theorems \ref{t1},
\ref{tnew}, and \ref{tnewa} so
as to obtain $\diamondsuit_\delta$
for every successor and Mahlo cardinal
$\gd$\footnote{Shelah
has recently shown in \cite{Sh1a} that GCH implies
$\diamondsuit_\gd$ holds for every successor cardinal
greater than or equal to $\ha_2$.
(It is of course impossible for GCH to imply
that $\diamondsuit_{\ha_1}$ holds, since by
the results of \cite{DJ}, there is a model
containing no Souslin trees
in which $2^{\ha_0} = \ha_1$.)
It is unknown, however,
(see \cite[Question 0.5]{Sh1a}) if there is
an analogous ZFC theorem (with or without GCH)
when $\gd$ is inaccessible. Thus,
a supplemental forcing is necessary to obtain
$\diamondsuit_{\ha_1}$, and the present
state of knowledge seems to require that further
additional forcing be done in order to obtain
$\diamondsuit_\gd$ at every Mahlo cardinal $\gd$.}
and $\square_\delta$ for every
$\delta$ in a stationary subset of
the least supercompact cardinal.
To do this, by
\cite[Theorem 1]{A05}, we assume without loss
of generality that our ground model
$V$ not only satisfies GCH and the
level by level equivalence between
strong compactness and supercompactness,
but also is a model for $\diamondsuit_\delta$
for every regular uncountable cardinal $\delta$ and
$\square_\delta$ for every $\delta$ in a
certain stationary subset of
the least supercompact cardinal $\gk$.
We then force with the partial
ordering $\FP$ used in the
proofs of either Theorems \ref{t1},
\ref{tnew}, or \ref{tnewa} of this paper.
By our work above, the resulting model
consequently witnesses the conclusions
of any of these theorems. We therefore
have to show that
we may additionally infer the
remaining desired properties.
By our previous work, $\gk$ remains
the least supercompact cardinal in $V^\FP$.
To see that $\square_\delta$ holds
on a stationary subset of $\gk$,
we note that since forcing with (any version of)
$\FP$ preserves cardinals
and cofinalities, it easily follows
that each instance of $\square_\delta$
remains an instance of $\square_\delta$
in $V^\FP$.
%Let $\gk_0$ be the least supercompact cardinal in $V$
%(which, by Lemma \ref{l3}, is the least supercompact
%cardinal in $V^\FP$).
Because $\FP$ may be written as
$\FP_{\gk} \ast \dot \FQ$, where
$\FP_{\gk}$ satisfies $\gk$-c.c$.$ and
$\forces_{\FP_{\gk}} ``\dot \FQ$ is
$\gk$-strategically closed''\footnote{In what follows,
depending upon the exact definition of
$\FP$, $\dot \FQ$ may be a term for
trivial forcing.},
another application of
\cite[Exercise H2, page 247]{Ku1} tells us
that any stationary subset of
$\gk$ in $V$ remains stationary
in $V^\FP$.
This means that in $V^\FP$,
$\square_\delta$ holds on a stationary
subset of the least supercompact cardinal.
Then, since for any Mahlo cardinal
$\delta$, we may write $\FP$ as
$\FP_\delta \ast \dot \FQ$, where
$\card{\FP_\delta} \le \delta$ and
$\forces_{\FP_\delta} ``\dot \FQ$ is
(at least) ${\prec}\delta$-strategically closed'',
an application of Facts \ref{f1} and \ref{f2}
yields that $\diamondsuit_\delta$ is
preserved in $V^\FP$.
This means we are able to prove the
following theorems.
\begin{theorem}\label{t1a}
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''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds, as does
$\diamondsuit_\delta$ for every
successor and Mahlo cardinal
$\delta$ and $\square_\delta$ for every
$\delta$ in a stationary subset of the
least supercompact cardinal.
Further, in $V^\FP$, the Mahlo
cardinals reflecting stationary
sets are precisely the weakly
compact cardinals.
Finally, every regular Jonsson
cardinal in $V^\FP$ is weakly compact.
\end{theorem}
\begin{theorem}\label{t1aa}
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 +
$\gk$ is the least supercompact cardinal''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds,
as does
$\diamondsuit_\gd$ for every
successor and Mahlo cardinal
$\gd$ and $\square_\gd$ for every
$\gd$ in a stationary subset of $\gk$.
Further, in $V^\FP$, there is a stationary
subset of $\gk$
%of the least supercompact cardinal $\gk$
composed of non-weakly compact Mahlo cardinals which
reflect stationary sets.
\end{theorem}
\begin{theorem}\label{t1aaa}
Let
$V \models ``$ZFC + $\gk$ is
supercompact + No cardinal is
supercompact up to an inaccessible cardinal''.
There is then a partial ordering
$\FP \subseteq V$ such that
$V^\FP \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact
up to an inaccessible cardinal''.
In $V^\FP$,
level by level equivalence
between strong compactness and
supercompactness holds,
as does
$\diamondsuit_\gd$ for every
successor and Mahlo cardinal
$\gd$ and $\square_\gd$ for every
$\gd$ in a stationary subset of $\gk$.
Further, in $V^\FP$, the inaccessible
cardinals reflecting stationary
sets are precisely the weakly
compact cardinals.
Finally, every regular Jonsson
cardinal in $V^\FP$ is
weakly compact.
\end{theorem}
If desired, it is possible to augment
the models of Theorems \ref{t1a} -- \ref{t1aaa}
still further, so that they satisfy additional
instances of $\square$. For example, it is
shown in \cite{AC3} that there are weak forms of
$\square$ which may hold above every
supercompact cardinal in a universe in which
the level by level equivalence between
strong compactness and supercompactness
is also true. Our methods demonstrate that
these additional instances of this
weak version of $\square$
(we refer readers of this paper to \cite{AC3}
for the exact statement) may be assumed to
be present in
the models of Theorems \ref{t1a} -- \ref{t1aaa}.
Of course, our theorems leave many questions open.
We conclude our paper by posing a few of them.
For instance,
%in Theorems \ref{t1}, \ref{tnewa}, \ref{t1a}, or \ref{t1aaa},
in any of our models,
are the regular Jonsson cardinals precisely
the Ramsey cardinals, as is the
case in higher inner models?
Are the regular Jonsson cardinals
precisely the weakly compact cardinals?
Is there even a reasonably uniform
characterization of the regular Jonsson
cardinals?
Or, counter-intuitively,
%contradictory to what one might intuitively expect,
are the non-weakly compact Mahlo cardinals
admitting stationary reflection in Theorems \ref{tnew}
and \ref{t1aa} also Jonsson cardinals?
In light of the gap between the assumptions of
Theorems \ref{tnewa} and \ref{tnewb}, is it possible
to prove a generalization of Theorem \ref{tnewa}
for a universe with a richer large
cardinal structure?
Finally, is it possible to extend
Theorems \ref{tnew} and \ref{t1aa} so that
there exist non-weakly compact Mahlo cardinals
which reflect stationary sets above some
supercompact cardinal(s)?
By \cite[pages 69--70]{Ku1a},
the partial ordering
$\FQ_\gd$ used in the proofs of
Theorems \ref{tnew} and \ref{t1aa} isn't even
%forcing the existence of such Mahlo cardinals isn't even
$\go_1$-directed closed, so a positive answer to
this question would require the introduction
of a highly directed closed partial ordering
which forces the existence of
%this sort of Mahlo cardinal.
the desired kind of Mahlo cardinal.
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\end{document}