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\title{An Easton Theorem for Level
by Level Equivalence
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, strongly
compact cardinal, strong cardinal,
level by level equivalence between strong
compactness and supercompactness, Easton theorem.}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{June 28, 2004}
\begin{document}
\maketitle
\begin{abstract}
We establish
an Easton theorem
for the least supercompact
cardinal that is consistent
with the level by level equivalence
between strong compactness and
supercompactness.
In both our ground model and the
model witnessing the conclusions
of our theorem, there are no
restrictions on the structure
of the class of
supercompact cardinals.
We also briefly indicate how our
methods of proof yield an
Easton theorem that is consistent
with the level by level equivalence
between strong compactness and
supercompactness in a universe
with a restricted number of large cardinals.
We conclude by posing some related
open questions.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s0}
In \cite{A01}, \cite{AH4},
and \cite{A03}, results concerning the
possible failures of GCH that are
consistent with the level by level
equivalence between strong compactness
and supercompactness were established.
%in universes with no restrictions on the size of
%the class of supercompact cardinals was begun.
We refer readers to these papers for
further details on the kinds of
theorems proven.
We note only that no theorem of
an Easton nature was established
in any of those articles.
By this we mean that no forcing extension
was constructed in which
level by level equivalence between
strong compactness and supercompactness
holds, and in which the size of the
power set of each member of a class
of regular cardinals is controlled
by a fixed ground model function.
The purpose of this paper is to
address the issue raised in
the preceding paragraph by proving an
Easton theorem that is consistent
with the level by level equivalence
between strong compactness and
supercompactness in a universe
in which the structure of the
class of supercompact cardinals
can be arbitrary.
At the end of the article,
we also briefly sketch how our methods
of proof yield another Easton theorem
that is consistent with the
level by level equivalence between
strong compactness and supercompactness
in a universe with a restricted
number of large cardinals.
Specifically,
%we begin by establishing the following.
%we will spend most of our time
%establishing the following theorem.
we establish the following theorem.
\begin{theorem}\label{t1}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds + $\K \neq
\emptyset$ is the class of
supercompact cardinals +
$\gk$ is the least
supercompact cardinal''.
Let $A = \{\gd \le \gk : \gd$ is
either a strong cardinal or the
regular limit of strong cardinals$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i1} $F(\gd) \in (\gd, \gd^*)$
is a cardinal, where
$\gd^*$ is the least strong
cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i2} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i3} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F(\gd)) = F(\gd) = \gd^{+}$ or
$j(F(\gd)) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\K$ is the class of
supercompact cardinals (so
$\gk$ is the least supercompact
cardinal) + Level by level equivlance
between strong compactness and
supercompactness holds +
For every $\gd \in A$,
$2^\gd = F(\gd)$''.
\end{theorem}
In both Theorem \ref{t1} and
the result to be sketched later,
it is unfortunately the case that
the Easton function $F$ is defined
only on a restricted set of
regular cardinals below the least
supercompact cardinal.
Furthermore, there are restrictions
placed on $F$'s range by clause (\ref{i3})
because of our methods of proof.
We will return to these issues in
the discussion given at the end
of this paper.
We note that there are many
natural functions satisfying
the conditions given in the
statement of Theorem \ref{t1}.
For instance,
if we let $B = \{\gd \in A : \gd$
is either a strong cardinal which
isn't a limit of strong cardinals
or a non-measurable limit of
strong cardinals$\}$ and
$C = A - B = \{\gd \le \gk : \gd$
is a measurable limit of strong
cardinals$\}$,
then
\[
F(\gd) = \left\{ \begin{array}{cc}
\gd^{+ 17} & \mbox{if
$\gd \in B$}\\
\gd^{++} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent and
\[
G(\gd) = \left\{ \begin{array}{lcl}
{\rm The \ least \ inaccessible
\ above \ } \gd & \mbox{if
$\gd \in B$}\\
\gd^{+} & \mbox{if
$\gd \in C$}
\end{array}
\right.
\]
\noindent are examples of such functions.
In fact, the Easton function can essentially
take on arbitrary values when $\gd \in B$,
subject to the restrictions given above in
clauses (\ref{i1}) and (\ref{i2}).
Before presenting the proofs
of our theorems, we briefly
state some preliminary information.
Our notation and terminology
will follow that given in \cite{A03}.
We do wish to mention a few
things explicitly, however.
When forcing, $q \ge p$ means
that $q$ is stronger than $p$.
A partial ordering $\FP$ is
$\gk$-directed closed for
$\gk$ a cardinal if
every directed set of conditions
of size less than $\gk$ has
an upper bound.
For $\gk$ a regular cardinal and
$\ga > \gk$ an ordinal,
${\rm Add}(\gk, \ga)$ is the
standard Cohen partial ordering
for adding $\ga$ Cohen subsets
of $\gk$.
If $\gk$ is inaccessible and
$\FP = \la \la \FP_\ga,
\dot \FQ_\ga \ra : \ga < \gk + 1 \ra$
is a reverse Easton iteration of length
$\gk + 1$
such that at stage $\ga$, a non-trivial 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 also wish to emphasize
what we mean when we say that
level by level equivalence between
strong compactness and supercompactness holds.
Specifically, we will say
that a model of ZFC witnesses
level by level equivalence between
strong compactness and supercompactness
iff for every measurable cardinal
$\gk$ and every regular $\gl > \gk$,
$\gk$ is $\gl$ strongly compact iff
$\gk$ is $\gl$ supercompact, except
if $\gk$ is a measurable limit
of cardinals $\gd$ which are $\gl$ strongly
compact.
Models containing supercompact
cardinals which also witness the
level by level equivalence between
strong compactness and supercompactness
and satisfy GCH
were first constructed in \cite{AS97a}.
Note that the exception in the
previous paragraph
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,
we will henceforth say that
$\gk$ is a witness
to the Menas exception at $\gl$.
If $\gk$ is measurable and
for every regular cardinal
$\gl > \gk$, $\gk$ is $\gl$
strongly compact iff $\gk$ is
$\gl$ supercompact, then we will
say that $\gk$ is a witness to
level by level equivalence between
strong compactness and supercompactness.
Finally, we mention that we are assuming
familiarity with the
large cardinal notions of measurability, strongness,
%superstrongness,
strong compactness, and supercompactness.
Interested readers may consult
\cite{J} or \cite{K}
%, \cite{K}, or \cite{SRK}
for further details.
%We note only that the cardinal
%$\gk$ is ${<}\gl$ supercompact if $\gk$ is $\gd$
%supercompact for every cardinal $\gd < \gl$.
\section{The Proof of Theorem \ref{t1}}\label{s1}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Let $V$, $A$, $F$, and $\gk$ be as in the
hypotheses for Theorem \ref{t1}.
Let $\FP$ be the usual reverse
Easton iteration of
length $\gk + 1$ which begins
by adding a Cohen subset of $\go$
and then adds, for every $\gd \in A$,
$F(\gd)$ many Cohen subsets of $\gd$.
We explicitly point out that the
definition of $\FP$ immediately
implies that the only non-trivial
stages of forcing occur at
members of $A$.
The standard Easton arguments
(see, e.g., \cite{J}) then show that
forcing with $\FP$ preserves all
cardinals and cofinalities and that
in $V^\FP$, $2^\gd = F(\gd)$ for
every $\gd \in A$.
%Note that since $\gk \in A$,
%$V^\FP \models ``2^\gk = \gk^{++}$''.
Further, because it is possible to write
$\FP = \FQ \ast \dot \FR$, where
$\card{\FQ} = \go$,
$\FQ$ is non-trivial, and
$\forces_{\FQ} ``\dot \FR$ is
$\ha_2$-directed closed'', by Hamkins'
gap forcing
results of \cite{H2} and \cite{H3},
any cardinal $\gd$ which is
$\gl$ supercompact in
$V^\FP$ had to have been $\gl$ supercompact
in $V$.
\begin{lemma}\label{l1}
$V^\FP \models ``\gk$ is the least
supercompact cardinal''.
\end{lemma}
\begin{proof}
By our remarks in the preceding
paragraph, since forcing with $\FP$
creates no new supercompact cardinals,
it suffices to show that
$V^\FP \models ``\gk$ is supercompact''.
To do this,
let $\gl > \gk^+ = 2^\gk$ be a regular cardinal,
and let
$j : V \to M$ be an elementary
embedding witnessing the
$\gl$ supercompactness of $\gk$
generated by a supercompact ultrafilter
over $P_\gk(\gl)$ with
$F(\gk) = j(F(\gk))$.
Let $\theta = F(\gk) = j(F(\gk))$ (so
$\theta$ is either $\gk^+$ or
$\gk^{++}$).
Note that since $V \models ``\gk$ is
the least supercompact cardinal'',
$M \models ``j(\gk) > \gk$ is the
least supercompact cardinal'', i.e.,
$M \models ``\gk$ isn't supercompact''.
By the remarks immediately following
Lemma 2.1 of \cite{AC2},
$M \models ``\gk$ is a regular
cardinal which is a limit of
strong cardinals''.
Also, as in Lemma 2.4 of \cite{AC2},
$M \models ``$No cardinal
$\gd \in (\gk, \gl]$ is strong'',
since otherwise, $\gk$ is supercompact
up to a strong cardinal and hence is
fully supercompact in $M$.
This means we can write
$j(\FP) = \FP_\gk \ast \dot
\add(\gk, \theta) \ast \dot \FQ
\ast \dot \add(j(\gk), j(\theta)) =
\FP \ast \dot \FQ
\ast \dot \add(j(\gk), j(\theta))$,
where the first
ordinal in the field of $\dot \FQ$
is above $\gl$.
Thus, since $M^\gl \subseteq M$,
it is the case that
in both $V$ and $M$,
$\forces_{\FP_\gk \ast \dot
\add(\gk, \theta)} ``\dot \FQ$ is
$\gl^+$-directed closed'', i.e.,
$\forces_{\FP} ``\dot \FQ$ is
$\gl^+$-directed closed''.
Let $G$ be $V$-generic over
$\FP_\gk$, and let $H$ be
$V[G]$-generic over
$\add(\gk, \theta)$.
Standard arguments show that
$M[G][H]$ remains $\gl$ closed
with respect to $V[G][H]$.
Since
$M[G][H] \models ``|\FQ| = j(\gk)$'' and
GCH holds in both $V$ and $M$,
there are $2^{j(\gk)} = j(\gk^+)$
dense open subsets of $\FQ$
present in $M[G][H]$. However, since
$|j(\gk^+)| =
|\{f : f : P_\gk(\gl) \to \gk^+$
is a function$\}| = \gl^+$ by GCH,
we can use the usual diagonalization
arguments
(as given, e.g., in the construction
of the generic object $G_1$ in
Lemma 2.4 of \cite{AC2})
to construct in
$V[G][H]$ an $M[G][H]$-generic
object $H'$ over $\FQ$ and lift $j$ to
$j : V[G] \to M[G][H][H']$ in $V[G][H]$.
Note that $M[G][H][H']$ remains
$\gl$ closed with respect to
$V[G][H][H'] = V[G][H]$.
Then, as the number of dense open subsets
of $\add(j(\gk), j(\theta))$ in
$M[G][H][H']$ is either
$j(\gk^{++})$ (if $F(\gk) =
\gk^+$) or $j(\gk^{+++})$ (if
$F(\gk) = \gk^{++}$), which
by GCH and the fact that
$\gl \ge \gk^{++} \ge \theta$
has size $\gl^+$ in
$V[G][H]$, and as $\add(j(\gk), j(\theta))$
is $\gl^+$-directed closed in both
$M[G][H][H']$ and $V[G][H]$ and
$V[G][H] \models ``\card{j''H} \le \gl$'',
we can once again
use the standard diagonalization arguments
to construct in $V[G][H]$ an
$M[G][H][H']$-generic object $H''$ containing
a master condition for $j''H$.
We can now fully lift $j$ to
$j : V[G][H] \to M[G][H][H'][H'']$ in
$V[G][H]$, thereby showing that
$V[G][H] \models ``\gk$ is $\gl$ supercompact''.
Since $\gl$ was an arbitrary
regular cardinal at least
$\gk^{++}$, this completes the
proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
If $\gd$ is a non-trivial
stage of forcing and
$\gl > \gd$ is a regular
cardinal such that
$V \models
``\gd$ is $\gl$ supercompact'', then
$V^\FP \models ``\gd$ is
$\gl$ supercompact''.
\end{lemma}
\begin{proof}
Write
$\FP = \FP_{\gd + 1} \ast \dot \FQ$,
and let $\gz = {(\gd^*)}^V$.
Since $\gd < \gk$, it must be the
case that $\gl < \gz$.
This is because otherwise,
$\gd$ is supercompact in $V$
up to a strong cardinal, so
as mentioned in the first
paragraph of the proof of Lemma \ref{l1},
%$V \models ``\gd$ is ${<}\gz$
%supercompact and $\gz$ is strong'',
%so by the proof of Lemma 2.4 of \cite{AC2},
$V \models ``\gd$ is supercompact''.
This is
a contradiction to the fact that
$\gk$ is the least $V$-supercompact cardinal. As
$\forces_{\FP_{\gd + 1}} ``\dot \FQ$ is
$\gz$-directed closed and
$\gz$ is inaccessible'',
to show that
$V^\FP \models ``\gd$ is $\gl$ supercompact'',
it hence suffices to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$
supercompact''.
In analogy to Lemma \ref{l1}, let
$\theta = F(\gd)$, i.e.,
$\theta$ is either $\gd^+$ or
$\gd^{++}$.
Lemma 2.1 of \cite{AC2} states that
any cardinal $\gg$ which is both
strong and (at least) $2^\gg$
supercompact is a limit of strong cardinals.
Thus, since $\gd \in A$ and by GCH,
$\gl \ge 2^\gd$, in $V$,
$\gd$ is a limit of strong cardinals.
By the definition of
$\FP$ and $\FP_{\gd + 1}$,
because $F(\gd) = \theta$,
we may therefore infer that
$|\FP_{\gd + 1}| = \theta$.
Suppose now $\gl \ge \theta$.
By the definition of $F$,
we may choose $j : V \to M$
as an elementary embedding
witnessing the $\gl$ supercompactness
of $\gd$ such that
$j(F(\gd)) = F(\gd) = \theta$. Since
$V \models ``$No cardinal below $\gk$
is supercompact'',
$M \models ``$No cardinal below $j(\gk)$
is supercompact'', i.e., as
$M \models ``\gd < \gk \le j(\gk)$'',
$M \models ``\gd$ isn't supercompact''.
Hence, as in Lemma \ref{l1},
$j(\FP_{\gd + 1}) = \FP_\gd \ast \dot
\add(\gd, \theta) \ast \dot \FQ
\ast \dot \add(j(\gd), j(\theta))$,
where the first ordinal in the field of
$\dot \FQ$ is above $\gl$.
The argument given in the proof of
Lemma \ref{l1} may therefore now
be used to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
Thus, we assume that
$\gl < \theta$. Since $\gl > \gd$, this means that
$\gl = \gd^+$.
If $j(F(\gd)) =
F(\gd) = \gd^+$, then the
argument given in the preceding
paragraph shows that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
We consequently assume that
$\gl = \gd^+$ and $j(F(\gd)) =
F(\gd) = \gd^{++}$. Under these circumstances,
the argument given on
pages 119--120 of \cite{AS97a},
pages 88--90 of \cite{A02},
pages 832--833 of \cite{AH4}, or
pages 591--592 of \cite{A03}
(which is originally due to Magidor
and is also found earlier in
\cite{JMMP}, \cite{JW}, and \cite{Ma2})
can be used to show that
$V^{\FP_{\gd + 1}} \models ``\gd$ is $\gl$ supercompact''.
For the convenience of readers, we give this
argument here as well.
Write
$\FP_{\gd + 1} = \FP_\gd \ast \dot \add(\gd, \gd^{++})$.
Let $G = G_0 \ast G_1$ be
$V$-generic over $\FP_{\gd + 1}$. Fix
$j : V \to M$ an elementary embedding
witnessing the $\gl = \gd^+$
supercompactness of $\gd$ generated
by a supercompact ultrafilter
${\cal U}$ over $P_\gd(\gl)$ such that
$j(F(\gd)) = F(\gd) = \gd^{++}$.
We then have
$j(\FP_{\gd + 1}) = \FP_{\gd} \ast \dot \add(\gd, \gd^{++})
\ast \dot \FQ \ast \dot
\add(j(\gd), j(\gd^{++}))$.
%where $\dot \FR_1$ is a term for
%$\add(j(\gd), j(\gd^{++}))$ as computed in
%$M^{\FP_{\gd + 1} \ast \dot \add(\gd, \gd^{++})
%\ast \dot \FR_0}$.
Therefore, by using the
standard diagonalization argument
mentioned in Lemma \ref{l1}, since
$M[G_0][G_1]$ remains $\gl$ closed
with respect to $V[G_0][G_1]$ and
$V \models {\rm GCH}$, it is possible
working in $V[G_0][G_1]$ to construct an
$M[G_0][G_1]$-generic object $G_2$ over
$\FQ$ and lift $j$ to
$j : V[G_0] \to M[G_0][G_1][G_2]$.
It is then as before the case that
$M[G_0][G_1][G_2]$ remains $\gl$
closed with respect to
$V[G_0][G_1]$.
We construct now in $V[G_0][G_1]$ an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gd), j(\gd^{++}))$.
For $\ga \in (\gd, \gd^{++})$ and
$p \in \add(\gd, \gd^{++})$, let
$p \rest \ga = \{\la \la \rho, \gs \ra, \eta \ra \in p :
\gs < \ga\}$ and
$G_1 \rest \ga = \{p \rest \ga : p \in G_1\}$. Clearly,
$V[G_0][G_1] \models ``|G_1 \rest \ga| \le \gd^+$
for all $\ga \in (\gd, \gd^{++})$''. Thus, since
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$ is
$j(\gd)$-directed closed and $j(\gd) > \gd^{++}$,
$q_\ga = \bigcup\{j(p) : p \in G_1 \rest \ga\}$ is
well-defined and is an element of
${\add(j(\gd), j(\gd^{++}))}^{M[G_0][G_1][G_2]}$.
Further, if
$\la \rho, \gs \ra \in \dom(q_\ga) -
\dom(\bigcup_{\gb < \ga} q_\gb)$
($\bigcup_{\gb < \ga} q_\gb$ is well-defined by closure),
then
$\gs \in [\bigcup_{\gb < \ga} j(\gb), j(\ga))$. To see this,
assume to the contrary that
$\gs < \bigcup_{\gb < \ga} j(\gb)$. Let
$\gb$ be minimal such that $\gs < j(\gb)$.
It must thus be the case that for some
$p \in G_1 \rest \ga$,
$\la \rho, \gs \ra \in \dom(j(p))$.
Since by elementarity and the definitions of
$G_1 \rest \gb$ and $G_1 \rest \ga$, for
$p \rest \gb = q \in G_1 \rest \gb$,
$j(q) = j(p) \rest j(\gb) = j(p \rest \gb)$,
it must be the case that
$\la \rho, \gs \ra \in \dom(j(q))$. This means
$\la \rho, \gs \ra \in \dom(q_\gb)$, a contradiction.
Since $M[G_0][G_1][G_2] \models ``$GCH holds
for all cardinals at or above $j(\gd)$'',
$M[G_0][G_1][G_2] \models ``\add(j(\gd),
j(\gd^{++}))$ is
$j(\gd^+)$-c.c$.$ and has
$j(\gd^{++})$ many maximal antichains''.
This means that if
${\cal A} \in M[G_0][G_1][G_2]$ is a
maximal antichain of $\add(j(\gd), j(\gd^{++}))$,
${\cal A} \subseteq \add(j(\gd), \gb)$ for some
$\gb \in (j(\gd), j(\gd^{++}))$. Thus, since GCH in $V$
and the
fact $j$ is generated by a supercompact ultrafilter over
$P_\gd(\gd^+)$ imply that
$V \models ``|j(\gd^{++})| = \gd^{++}$'', we can let
$\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra \in
V[G_0][G_1]$ be an enumeration of all of the
maximal antichains of $\add(j(\gd), j(\gd^{++}))$
present in
$M[G_0][G_1][G_2]$.
Working in $V[G_0][G_1]$, we define
now an increasing sequence
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$ of
elements of $\add(j(\gd), j(\gd^{++}))$ such that
$\forall \ga \in (\gd, \gd^{++}) [r_\ga \ge q_\ga$ and
$r_\ga \in \add(j(\gd), j(\ga))]$ and such that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
Assuming we have such a sequence,
$G_3 = \{p \in \add(j(\gd), j(\gd^{++})) :
\exists r \in \la r_\ga : \ga \in (\gd, \gd^{++}) \ra
[r \ge p]$ is an
$M[G_0][G_1][G_2]$-generic object over
$\add(j(\gd), j(\gd^{++}))$. To define
$\la r_\ga : \ga \in (\gd, \gd^{++}) \ra$, if
$\ga$ is a limit, we let
$r_\ga = \bigcup_{\gb \in (\gd, \ga)} r_\gb$.
By the facts
$\la r_\gb : \gb \in (\gd, \ga) \ra$
is (strictly) increasing and
$M[G_0][G_1][G_2]$ is
$\gd^+$ closed with respect to
$V[G_0][G_1]$, this definition is valid.
Assuming now $r_\ga$ has been defined and
we wish to define $r_{\ga + 1}$, let
$\la {\cal B}_\gb : \gb < \eta \le \gd^+ \ra$
be the subsequence of
$\la {\cal A}_\gb : \gb \le \ga + 1 \ra$
containing each antichain ${\cal A}$ such that
${\cal A} \subseteq \add(j(\gd), j(\ga + 1))$.
Since
$q_\ga, r_\ga \in \add(j(\gd), j(\ga))$,
$q_{\ga + 1} \in \add(j(\gd), j(\ga + 1))$, and
$j(\ga) < j(\ga + 1)$, the condition
$r_{\ga + 1}' = r_\ga \cup q_{\ga + 1}$ is
well-defined, since by our earlier observations,
any new elements of
$\dom(q_{\ga + 1})$ won't be present in either
$\dom(q_\ga)$ or $\dom(r_\ga)$.
We can thus, using the fact
$M[G_0][G_1][G_2]$ is closed under
$\gd^+$ sequences with respect to
$V[G_0][G_1]$, define by induction
an increasing sequence
$\la s_\gb : \gb < \eta \ra$ such that
$s_0 \ge r_{\ga + 1}'$,
$s_\rho = \bigcup_{\gb < \rho} s_\gb$ if
$\rho$ is a limit ordinal, and
$s_{\gb + 1} \ge s_\gb$ is such that
$s_{\gb + 1}$ extends some element of
${\cal B}_\gb$. The just mentioned
closure fact implies
$r_{\ga + 1} = \bigcup_{\gb < \eta} s_\gb$
is a well-defined condition.
In order to show that $G_3$ is
$M[G_0][G_1][G_2]$-generic over
$\add(j(\gd), j(\gd^{++}))$, we must show that
$\forall {\cal A} \in
\la {\cal A}_\ga : \ga \in (\gd, \gd^{++}) \ra
\exists \gb \in (\gd, \gd^{++})
\exists r \in {\cal A} [r_\gb \ge r]$.
To do this, we first note that
$\la j(\ga) : \ga < \gd^{++} \ra$ is
unbounded in $j(\gd^{++})$. To see this, if
$\gb < j(\gd^{++})$ is an ordinal, then for some
$f : P_\gd(\gd^+) \to M$ representing $\gb$,
we can assume that for $p \in P_\gd(\gd^+)$,
$f(p) < \gd^{++}$. Thus, by the regularity of
$\gd^{++}$ in $V$,
$\gb_0 = \bigcup_{p \in P_\gd(\gd^+)} f(p) <
\gd^{++}$, and $j(\gb_0) > \gb$.
This means by our earlier remarks that if
${\cal A} \in \la {\cal A}_\ga : \ga <
\gd^{++} \ra$, ${\cal A} = {\cal A}_\rho$,
then we can let
$\gb \in (\gd, \gd^{++})$ be such that
${\cal A} \subseteq \add(j(\gd), j(\gb))$.
By construction, for $\eta > \max(\gb, \rho)$,
there is some $r \in {\cal A}$ such that
$r_\eta \ge r$.
And, as any
$p \in \add(\gd, \gd^{++})$ is such that for some
$\ga \in (\gd, \gd^{++})$, $p = p \rest \ga$,
$G_3$ is such that if
$p \in G_1$, $j(p) \in G_3$.
Thus, working in $V[G_0][G_1]$,
we have shown that $j$ lifts to
$j : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$,
i.e.,
$V[G_0][G_1] \models ``\gd$ is $\gl = \gd^+$
supercompact''.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$V^\FP \models ``$Every measurable cardinal
$\gd$ either is a witness to level by
level equivalence between strong
compactness and supercompactness or is a
witness to the Menas exception at $\gl$
for some $\gl > \gd$'', i.e.,
$V^\FP \models ``$Level by level equivalence
between strong compactness and
supercompactness holds''.
\end{lemma}
\begin{proof}
Since $\FP$ may be defined
so that $\card{\FP} \le \gk^{++}$,
by the L\'evy-Solovay results \cite{LS},
Lemma \ref{l3} is true for any cardinal
$\gd > \gk$. By Lemma \ref{l1}, Lemma \ref{l3}
is true for $\gd = \gk$. It thus
suffices to show that Lemma \ref{l3} holds
for any cardinal $\gd < \gk$.
To establish this last fact,
we consider the following two cases.
\bigskip
\noindent Case 1: $\gd$ is a
non-trivial stage of forcing.
Suppose $\gl > \gd$ is such that
$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
Recall from the first paragraph of the
proof of Theorem \ref{t1} that $\FP$
may be written as
$\FQ \ast \dot \FR$, where
$\card{\FQ} = \go$,
$\FQ$ is non-trivial, and
$\forces_{\FQ} ``\dot \FR$ is
$\ha_2$-directed closed''.
Further, it is easily seen that
any subset $x$ of $\gd$ in $V^\FP$
has a ``nice'' name $\gt$
of size below $\gd$ in $V$, i.e.,
there is a set $y$ in $V$,
$\card{y} < \gd$, such that any
ordinal forced by a condition in
$\FP$ to be in $\gt$ is an element
of $y$.
Therefore, in the terminology of
\cite{H2} and \cite{H3}, $\FP$ is a
``mild forcing with respect to $\gd$
admitting a gap at $\ha_1$'', so by
the results of \cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
Note now that $\gd$ cannot be a witness in $V$
to the Menas exception at $\gl$, i.e.,
$\gd$ is not in $V$ a limit of cardinals
which are $\gl$ supercompact.
To see this, suppose to the contrary
that this is the case.
We may then find $\gg < \gd$ such that
$\gg$ is $\gl$ supercompact.
Note that as
$\gd$ is a non-trivial stage of forcing,
$\gd$ is either a strong cardinal in $V$
or a regular limit of strong cardinals in $V$.
Thus, in either situation,
it would have to be the case that
for some $V$-strong cardinal
$\gz \in (\gg, \gd]$, $\gg$ is
$\gz$ supercompact, which as
we observed in the first paragraph
of the proofs of Lemmas \ref{l1} and \ref{l2}
immediately implies that $\gg$ is
supercompact in $V$.
This is impossible, as $\gg < \gk$
and $\gk$ is the least $V$-supercompact cardinal.
Hence, by level by level equivalence
between strong compactness and
supercompactness in $V$,
$\gd$ is $\gl$ supercompact in $V$.
By the second paragraph of the
proof of Lemma \ref{l2}, this means
that $\gd$ is in $V$ a limit of
strong cardinals.
We may therefore apply Lemma \ref{l2}
to infer that
$V^\FP \models ``\gd$ is $\gl$ supercompact''.
\bigskip
\noindent Case 2: $\gd$ is a trivial
stage of forcing. As before, suppose
$\gl > \gd$ is such that
$V^\FP \models ``\gd$ is $\gl$ strongly compact''.
Let $S$ be the set of non-trivial stages
of forcing below $\gd$, with $\gg$ either
the largest member of $S$ (if it exists),
or the supremum of the members of $S$ otherwise.
In the former situation, it must be true that
$F(\gg) < \gd$, for if $F(\gg) \ge \gd$,
then by the definition of $\FP$,
$V^{\FP_{\gg + 1}} \models ``2^\gg \ge \gd$''.
Let $\gz = {(\gg^*)}^V$.
Thus, by writing
$\FP = \FP_{\gg + 1} \ast \dot \FP^{\gg + 1}$,
since the first ordinal in the field of
$\dot \FP^{\gg + 1}$
%must be an ordinal above $\gd$,
must be $\gz$ (which as $\gd$
is a trivial stage of forcing
must be above $\gd$), and since
$F(\gg) \in (\gg, \gz)$,
it must be true that
$\forces_{\FP_{\gg + 1}} ``\dot \FP^{\gg + 1}$
is $\gz$-directed closed and
$\gz$ is inaccessible''.
This means that
$V^{\FP_{\gg + 1} \ast \dot \FP^{\gg + 1}} =
V^\FP \models ``2^\gg \ge \gd$'', which
contradicts the $\gl$ strong compactness
of $\gd$ in $V^\FP$.
It is therefore also true by the definition
of $\FP$ that $\card{\FP_{\gg + 1}} < \gd$.
The factorization of $\FP$ just given
%and the factorization of $\FP$
%given in Case 1
consequently yields that
we once more have
in the terminology of \cite{H2} and
\cite{H3} that $\FP$ is a ``mild forcing
with respect to $\gd$ admitting a gap
below $\gd$''.
Hence, again by the results of
\cite{H2} and \cite{H3},
$V \models ``\gd$ is $\gl$ strongly compact''.
By the level by level equivalence between
strong compactness and supercompactness in $V$,
this means that $\gd$ is either $\gl$
supercompact in $V$ or is a witness
to the Menas exception at $\gl$ in $V$.
%i.e., $\gd$ is a limit in $V$ of cardinals which are
%$\gl$ supercompact.
Regardless of which of these situations holds,
it must be the case that there is some cardinal
$\gr \le \gd$ such that $\gr$ is $\gl$
supercompact in $V$.
As earlier, $\gl < \gz$, since otherwise,
$\gr$ is supercompact in $V$ up to a strong
cardinal and hence is fully supercompact in $V$,
which contradicts that $\gr \le \gd < \gk$
and $\gk$ is the least $V$-supercompact cardinal.
Therefore, since
$\forces_{\FP_{\gg + 1}} ``\dot \FP^{\gg + 1}$ is
$\gz$-directed closed and
$\gz$ is inaccessible'', an application of
the results of \cite{LS} tells us that in
both $V^{\FP_{\gg + 1}}$ and
$V^{\FP_{\gg + 1} \ast \dot \FP^{\gg + 1}} = V^\FP$,
$\gd$ is either $\gl$ supercompact or is
a witness to the Menas exception at $\gl$.
If $\gg = \sup(S)$,
then by the definition of $\FP$,
$\gg$ has to be singular. Hence,
it must be possible to write
$\FP = \FP_{\gg + 1} \ast \dot \FP^{\gg + 1}$, where
$\card{\FP_{\gg + 1}} < \gd$.
The analysis given in the preceding paragraph
thus once again applies to show that in $V^\FP$,
$\gd$ is either $\gl$ supercompact or is a
witness to the Menas exception at $\gl$.
Therefore, regardless if we are in Case 1 or
Case 2, if
$V^\FP \models ``\gd$ is $\gl$ strongly
compact'', $\gd$ is either $\gl$
supercompact in $V^\FP$ or is a witness
to the Menas exception at $\gl$ in
$V^\FP$.
This just means that in $V^\FP$,
level by level equivalence between
strong compactness and supercompactness holds.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$V^\FP \models ``\mathfrak K$ is
the class of supercompact cardinals''.
\end{lemma}
\begin{proof}
By Lemma \ref{l1},
$V^\FP \models ``\gk$ is the
least supercompact cardinal''.
Thus, to prove Lemma \ref{l4},
it suffices to show that in
$V^\FP$, the class of supercompact
cardinals above $\gk$ is the same
as in $V$. However, since as
we observed in the proof of
Lemma \ref{l3}, $\FP$ may be
defined so that
$\card{\FP} \le \gk^{++}$, this
follows by the results of \cite{LS}.
This completes the proof of Lemma \ref{l4}.
\end{proof}
Lemmas \ref{l1} - \ref{l4} complete the
proof of Theorem \ref{t1}.
\end{proof}
\section{Concluding Remarks}\label{s3}
We note that the methods of proof
for Theorem \ref{t1} will yield
another Easton theorem consistent
with the level by level equivalence
between strong compactness and
supercompactness in the context
of a universe with a restricted
number of large cardinals.
Specifically,
suppose we start with a
ground model $V$ in which $\gk$
is the only supercompact cardinal
and no pair of cardinals
$\gd < \gl$ is such that
$\gd$ is $\gl$ supercompact and
$\gl$ is inaccessible.
Suppose further that we redefine
$\gd^*$ as the least inaccessible
cardinal above $\gd$, set
$A = \{\gd \le \gk : \gd$ is
inaccessible$\}$, specify $F$
in a way analogous to the
definition given in the
statement of Theorem \ref{t1}, and
define $\FP$ as in the proof of
Theorem \ref{t1} using the set $A$.
%if we redefine
%$\gd^*$ as the least inaccessible
%cardinal above $\gd$, then
Easy modifications of the proofs
of Lemmas \ref{l1} - \ref{l3}
obtained roughly speaking by
replacing the word ``strong''
with the word ``inaccessible''
and simplifying the relevant arguments
(the details of which we leave to the readers
of this paper) will then give the
following result.
\begin{theorem}\label{t1a}
Let
$V \models ``$ZFC + GCH +
Level by level equivalence
between strong compactness
and supercompactness holds +
$\gk$ is supercompact +
For no pair of cardinals
$\gd < \gl$ is it the case that
$\gd$ is $\gl$ supercompact and
$\gl$ is inaccessible''.
Let $A = \{\gd \le \gk : \gd$ is
inaccessible$\} =
\{\gd : \gd$ is inaccessible$\}$.
Suppose that $F : A \to \gk$,
$F \in V$
is a function with the
following properties.
\begin{enumerate}
\item\label{i3a} $F(\gd) \in (\gd, \gd^*)$
is a cardinal.
%where $\gd^*$ is the least inaccessible cardinal above $\gd$.
%\item If $\gd < \gg$, then
%$F(\gd) < F(\gg)$.
\item\label{i4} ${\rm cof}(F(\gd)) > \gd$.
\item\label{i5} If $\gd \in A$ is
$\gl$ supercompact for $\gl > \gd$,
then there is an elementary embedding
$j : V \to M$ witnessing the
$\gl$ supercompactness of $\gd$
generated by a supercompact ultrafilter over
$P_\gd(\gl)$
such that either
$j(F(\gd)) = F(\gd) = \gd^{+}$ or
$j(F(\gd)) = F(\gd) = \gd^{++}$.
\end{enumerate}
\noindent There is then a partial ordering
$\FP \in V$ such that
$V^\FP \models ``$ZFC +
$\gk$ is supercompact +
For no pair of cardinals
$\gd < \gl$ is it the case that
$\gd$ is $\gl$ supercompact and
$\gl$ is inaccessible +
Level by level equivlance
between strong compactness and
supercompactness holds +
For every inaccessible cardinal,
$2^\gd = F(\gd)$''.
\end{theorem}
%We are, however, most interested
%in establishing results along
%the line of Theorem \ref{t1}
%when there are no restrictions
%on the structure of the class
%of supercompact cardinals
%(as is the case with Theorem \ref{t1}).
We are, however, interested in
establishing additional results along
the lines of Theorems \ref{t1} and \ref{t1a}.
We therefore conclude this paper by asking what
other sorts of Easton theorems are
consistent with the level by level
equivalence between strong
compactness and supercompactness.
In particular, is it possible for
the domain of the $F$ of Theorems
\ref{t1} and \ref{t1a}
to be all regular cardinals
at or below $\gk$, as in Easton's
original result?
Is it possible for the domain of
$F$ to include regular cardinals
above $\gk$?
In general, which regular cardinals
may be included in $F$'s domain, and
which cardinals may be included in
$F$'s range?
Note that these questions are
(non-trivially) valid in
any universe containing supercompact
cardinals, regardless if there are
any restrictions on the number of
large cardinals present.
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\end{document}
\begin{graveyard}
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}
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.
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 standard interval notation.
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 an Easton support iteration of length $\gk$
so that at stage $\ga$, a non-trivial 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.
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),
then player II has a strategy which
ensures the game can always be continued.
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 $f \in V$.
A result which will be used in the proof
of Theorem \ref{t1} is
a corollary of Theorems 3 and 31
and Corollary 14 of
Hamkins' paper \cite{H5}.
This theorem is a generalization of
Hamkins' Gap Forcing Theorem and
Corollary 16 of
\cite{H2} and \cite{H3}
(and we refer readers to \cite{H2},
\cite{H3}, and \cite{H5} for
further details).
We therefore state the theorem
we will be using now, along
with some associated terminology.
%quoting freely from \cite{H2} and \cite{H3}.
Suppose $\FP$ is a partial ordering
which can be written as
$\FQ \ast \dot \FR$, where
$|\FQ| \le \gd$,
$\FQ$ is non-trivial, and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H5},
$\FP$ {\rm admits a closure point at $\gd$}.
In Hamkins' terminology of \cite{H2}
and \cite{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}, \cite{H3}, \cite{H5},
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 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}
A result which will be key in the proof
of Theorem \ref{t1} is Hamkins'
Gap Forcing Theorem of \cite{H2} and \cite{H3}.
We therefore state this theorem 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
$|\FQ| < \gd$ and
$\forces_\FQ ``\dot \FR$ is
$\gd$-strategically closed''.
In Hamkins' terminology of
\cite{H2} and \cite{H3},
$\FP$ admits a gap at $\gd$.
Also, as in the terminology of
\cite{H2} and \cite{H3} (and elsewhere),
an embedding
$j : \ov V \to \ov M$ is
amenable to $\ov V$ when
$j \rest A \in \ov V$ for any
$A \in \ov V$.
The Gap Forcing Theorem is then
the following.
\begin{theorem}\label{t2}
{\bf (Hamkins' Gap Forcing Theorem)}
Suppose that $V[G]$ is a forcing
extension obtained by forcing 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 $V$.
\end{theorem}
\end{graveyard}