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\title{A Remark on the Tree Property in a
Choiceless Context
\thanks{2000 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal, tree property,
indestructibility, symmetric inner model.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
% express his gratitude to
Ralf Schindler for
helpful correspondence on the subject
matter of this paper which considerably
improved and clarified its presentation.}\\
% and CUNY Collaborative Incentive grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/apter\\
awapter@alum.mit.edu}
\date{November 29, 2009\\
(revised March 7, 2011)}
\begin{document}
\maketitle
\begin{abstract}
We show that
the consistency of the theory
``ZF + DC + Every successor cardinal is regular +
Every limit cardinal is singular + Every
successor cardinal satisfies the tree property''
follows from the consistency of a proper
class of supercompact cardinals.
This extends earlier results due to the author
showing that the
consistency of the theory ``ZF + $\neg{\rm AC}_\go$ +
Every successor cardinal is regular + Every
limit cardinal is singular + Every successor
cardinal satisfies the tree property'' follows
from hypotheses stronger in consistency
strength than a supercompact limit of
supercompact cardinals.
A lower bound in consistency strength is
provided by a result of Busche and Schindler,
who showed that the consistency of the theory
``ZF + Every successor cardinal is regular +
Every limit cardinal is singular + Every successor
cardinal satisfies the tree property'' implies
the consistency of ${\rm AD}^{L(R)}$.
%This extends, combines,
%and unifies in a choiceless context earlier
%results due to Mitchell, Abraham, Magidor
%and Shelah, and Cummings and Foreman.
\end{abstract}
\baselineskip=24pt
%\section{Introduction and Preliminaries}\label{s1}
We begin by
very briefly mentioning some of our conventions
and terminology. As usual, for
$\gk$ a regular cardinal, a {\em $\gk$-tree}
is a tree of height $\gk$, all of whose
levels have cardinality less than $\gk$.
$\gk$ satisfies the {\em tree property}
if every $\gk$-tree has a branch of length $\gk$.
A $\gk$-tree not satisfying the tree property
is called a {\em $\gk$-Aronszajn tree}.
For our purposes, all $\gk$-trees will be of
cardinality $\gk$ and will have base set
$\gk \times \gk$. This means that every
$\gk$-tree may be coded by a set of ordinals.
%Attempting to obtain the tree property at successor cardinals
The study of the tree property at successor cardinals is one which
has had a long and rich history in set theory.
Classically, Aronszajn (see \cite[Theorem 9.16,
pages 116--117]{J}) showed that
assuming the Axiom of Choice,
the tree property must fail at $\ha_1$, i.e.,
that $\ha_1$ carries an $\ha_1$-Aronszajn tree.
In his doctoral dissertation
\cite{Si}, Silver demonstrated that if the
tree property holds at a cardinal $\gk > \ha_1$,
then $\gk$ must be weakly compact in $L$.
In his doctoral dissertation \cite{Mi}, Mitchell
proved that the tree property at the successor of
a regular cardinal greater than $\ha_1$ can be forced
from a weakly compact cardinal.
Taken together, Silver's and Mitchell's results
consequently show that the tree property at the
successor of a regular cardinal greater than $\ha_1$
is equiconsistent with the existence of a
weakly compact cardinal.
Abraham \cite{Ab} demonstrated that relative to the
existence of a supercompact cardinal with a
weakly compact cardinal above it, it is consistent for
$2^{\ha_0} = \ha_2$ and for $\ha_2$ and $\ha_3$ both
to satisfy the tree property.
Shelah \cite{MS} showed that the successor
of a singular limit of strongly compact cardinals
satisfies the tree property, and
Magidor and Shelah together \cite{MS} proved that further,
relative to a huge cardinal with $\go$ many
supercompact cardinals above it, it is consistent for
SCH to hold at $\ha_\go$ and for
$\ha_{\go + 1}$ to satisfy the tree property.
Sinapova \cite{Si1} improved
Magidor and Shelah's result of \cite{MS} by
constructing a model in which
SCH holds at $\ha_\go$ and
$\ha_{\go + 1}$ satisfies the tree property
using only $\go$ many supercompact cardinals.
Cummings and Foreman \cite{CF} demonstrated that
relative to the existence of $\go$ many
supercompact cardinals, it is consistent for
$2^{\ha_n} = \ha_{n + 2}$ for every $n < \go$
and for every $\ha_n$ for $1 < n < \go$ to
satisfy the tree property.
Schindler \cite{Sc} showed that if both
$\ha_2$ and $\ha_3$ satisfy the tree
property, then there is an inner model
with a strong cardinal.
Foreman, Magidor, and Schindler \cite{FMS} proved
that if $\ha_n$ has
the tree property for all $1 < n < \go$ and
$\ha_\go$ is a strong limit cardinal, then for
all $X \in H_{\ha_\go}$ and all $n < \go$,
$M^\sharp_n(X)$ exists. The work of
\cite{Sc} and \cite{FMS} therefore yields the
necessity of strong hypotheses for the
aforementioned results of Abraham and Cummings-Foreman.
Left open by all of these theorems, however, is
the following
\bigskip\setlength{\parindent}{0pt}
Question: Is it possible to combine and extend
these results by obtaining a model of ZFC in which
every successor cardinal greater than
$\ha_1$ satisfies the tree property?
\bigskip\setlength{\parindent}{1.5em}
Unfortunately, we are unable to answer the above
Question in a ZFC context.
The purpose of this note is to
extend an earlier result (which follows
from the main theorem of \cite{A85}) and
construct a model of ZF + DC in which
every successor cardinal satisfies the tree property.
%provide a complete answer in a choiceless context to our Question above.
Specifically, we prove the following theorem.
\begin{theorem}\label{t1}
Suppose $V \models ``$ZFC + There is a proper
class of supercompact cardinals''.
There is then a partial ordering $\FP \subseteq V$
and a symmetric inner model $N \subseteq V^\FP$ such that
$N \models ``$ZF + DC + Every successor cardinal
is regular + Every limit cardinal is singular +
Every successor cardinal satisfies the tree property''.
\end{theorem}
We take this opportunity to make several
remarks concerning Theorem \ref{t1}.
%As we have already mentioned, Theorem \ref{t1}
A corollary of
the main theorem of \cite{A85} is the construction,
using an almost huge cardinal,\footnote{The
statement of the main theorem of \cite{A85}
indicates the use of a 3 huge cardinal.
As was pointed out by Moti Gitik, and as was
mentioned at the end of \cite{A85}, an
almost huge cardinal suffices for the proof.}
of a model $N^*$ for the theory
``ZF + $\neg{\rm AC}_\go$ +
%Every successor cardinal is regular +
Every limit cardinal is singular + Every successor
cardinal is weakly compact''.\footnote{In the
context of the failure of the Axiom of Choice, the
cardinal $\gk$ is {\em weakly compact} if it
satisfies the partition relation $\gk \to (\gk)^2$.}
The work of \cite{A92} shows that the assumption
of the existence of an almost huge cardinal may be
weakened to a technical hypothesis which is, in
consistency strength, strictly in between a
supercompact limit of supercompact cardinals and
an almost huge cardinal.
Since the proof that a weakly compact cardinal
satisfies the tree property requires no
use of the Axiom of Choice (see, e.g., the proof
of \cite[Lemma 9.26(i), page 120]{J}),
$N^*$ is automatically a model for the theory
``ZF + $\neg{\rm AC}_\go$ +
%Every successor cardinal is regular +
Every limit cardinal is singular + Every successor
cardinal satisfies the tree property''.
Thus, the model $N$ of Theorem \ref{t1}
generalizes the model $N^*$ of \cite{A85} and
\cite{A92}, in the sense that $N$ satisfies a
weak fragment of the Axiom of Choice and is
constructed using a weaker hypothesis than that
found in either \cite{A85} or \cite{A92}.
In addition, in the model $N$ of Theorem \ref{t1},
if $\gk$ is the successor of a limit cardinal,
then $\gk$ is not weakly compact. This is since
our construction will guarantee that
$V \subseteq N$, so by the way in which
$N$ has been built, there is a $\gk^+$ sequence of
subsets of $\gk$. By the proofs of
\cite[Lemmas 9.4 and 9.5, page 110]{J},
which require no use of the Axiom of Choice,
if there is a $\gk^+$ sequence of
subsets of $\gk$, then
$\gk$ is not weakly compact.
%$\gk \not\to (\gk)^2$.
This provides another contrast with the
model $N^*$ of \cite{A85} and \cite{A92},
in which {\em every} successor cardinal
is weakly compact.
Further, by the work of Busche and Schindler \cite{BuSc},
${\rm AD}^{L(R)}$ provides a lower bound in
consistency strength for the theory ``ZF +
Every successor cardinal is regular + Every limit
cardinal is singular + Every successor cardinal
satisfies the tree property''.
Finally, Theorem \ref{t1} illustrates how the situation
may differ when the Axiom of Choice is false.
In particular, the tree property for $\ha_1$ will
hold in the model $N$ witnessing the conclusions
of Theorem \ref{t1}. As we have already observed,
this is impossible when the full Axiom of Choice is true.
%\begin{pf}
Turning now to the proof of Theorem \ref{t1}, let
$V \models ``$ZFC + There is a proper class of
supercompact cardinals''. As in the proof of
\cite[Theorem 1]{A83}, we assume without loss
of generality that in $V$, each supercompact
cardinal $\gk$ is Laver indestructible \cite{L}
under $\gk$-directed closed forcing, and that there
is no inaccessible limit of supercompact cardinals
in $V$.\footnote{The
proof of \cite[Theorem 1]{A83} proceeds
by first forcing every member of
%a fixed class ${\cal K}^*$
the class of supercompact cardinals to be
Laver indestructible. However, since the
iteration used contains a {\em low gap} in Hamkins'
sense of \cite{H2} and \cite{H3}, by
Hamkins' Gap Forcing Theorem of \cite{H2}
and \cite{H3}, it creates no new supercompact
cardinals. This means that we may assume
without loss of generality that
all ground model supercompact cardinals
are indestructible, and that there
is no inaccessible limit of supercompact
cardinals in the ground model.}
Our model $N$ witnessing the conclusions of Theorem
\ref{t1} is the model $N$ of \cite[Theorem 1]{A83}.
In order to describe $N$ more precisely, let
${\cal K} = \{\go\} \cup \{\gk \mid \gk$ is
either a supercompact cardinal or the
successor of a limit of supercompact cardinals$\}$.
Assume that $\la \gk_i \mid i \in {\rm Ord} \ra$
enumerates ${\cal K}$ in increasing order. For each
$i \in {\rm Ord}$, let $\FP_i =
{\rm Coll}(\gk_i, {<}\gk_{i + 1})$, i.e.,
$\FP_i$ is the L\'evy collapse of all cardinals
in the open interval $(\gk_i, \gk_{i + 1})$ to $\gk_i$.
Let $\FP = \prod_{i \in {\rm Ord}} \FP_i$ be the
set support proper class product ordered componentwise,
and let $G$ be $V$-generic over $\FP$.
$V[G]$, being a model of AC, is not our desired
choiceless symmetric inner model $N$ witnessing the
conclusions of Theorem \ref{t1}. In order
to define $N$, we first note that by the Product
Lemma, for $i \in {\rm Ord}$, $G_i$, the projection
of $G$ onto $\FP_i$, is $V$-generic over $\FP_i$.
In addition, for each $\gd \in (\gk_i, \gk_{i + 1})$,
let $\FP_i \rest \gd = \{p \in \FP_i \mid
\dom(p_i) \subseteq \gk_i \times \gd\}$, and let
$G_i \rest \gd = \{p \in G_i \mid p \in \FP_i \rest \gd\}$.
Standard arguments show that $G_i \rest \gd$ is
$V$-generic over $\FP_i \rest \gd$.
Next, for each ordinal $\ga$,
let ${\cal F}_\ga = \prod_{i < \ga}
(\gk_i, \gk_{i + 1})$ be the full support
product of the open intervals $(\gk_i, \gk_{i + 1})$.
Let ${\cal F} = \bigcup_{\ga \in {\rm Ord}} {\cal F}_\ga$.
For each $f \in {\cal F}$,
$f = \la \ga_i \mid i < \ga \ra$,
define $G \rest f = \prod_{i < \ga} (G_i \rest \ga_i)$.
Observe that $G \rest f$ is $V$-generic over
$\FP \rest f = \prod_{i < \ga} (\FP_i \rest \ga_i)$.
In other words, every $f$ is a set
sequence of ordinals each of whose elements
is a member of a unique open interval of the form
$(\gk_i, \gk_{i+ 1})$, and every $G_i \rest \ga_i$
collapses each cardinal in the open interval
$(\gk_i, \ga_i)$ to $\gk_{i}$.
$N$ can now be intuitively described as the
least model of ZF extending $V$ which contains,
for each $f \in {\cal F}$, the set $G \rest f$.
In order to define $N$ more formally, we let ${\cal L}_1$ be the
ramified sublanguage of the forcing language $\cal L$ with respect to
$\FP$ which contains symbols $\check v$ for each $v \in V$, a unary
predicate symbol $\check V$ (to be interpreted
$\check V(\check v) \iff v \in
V$, i.e., $\check V$ allows us to
determine members of the ground model),
%those members of $N$ which are present in the ground model),
and symbols $\dot G \rest f$
for each $f \in {\cal F}$.
$N$ is then defined as follows.
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gd = \bigcup_{\ga < \gd} N_\ga$ if $\gd$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}
By \cite[Lemmas 1.1 -- 1.5 and Lemma 1.7]{A83},
$N \models ``$ZF + DC + Every successor
cardinal is regular + Every limit cardinal
is singular''. Further, by
\cite[Lemmas 1.1 -- 1.5]{A83}, if
$N \models ``\gl$ is a successor cardinal'',
then $N \models ``\gl = \gk$ for some $\gk \in {\cal K}$''.
%some $\gk \in {\cal K}$'' or $N \models ``$
Our proof of Theorem \ref{t1} is therefore
completed by the following lemma.
\begin{lemma}\label{l1}
$N \models ``$Every successor cardinal satisfies
the tree property''.
\end{lemma}
\begin{proof}
Suppose $N \models ``\gk$ is a successor cardinal
and ${\mathfrak T}$ is a $\gk$-tree''.
As we have just observed, $\gk \in {\cal K}$.
Let $i_0$ be the unique ordinal such that $\gk = \gk_{i_0}$.
Since ${\mathfrak T}$ may be coded by a
set of ordinals, by \cite[Lemma 1.1]{A83}, we can
assume that ${\mathfrak T} \in V[G \rest f]$
for some $f \in {\cal F}$, $f = \la \ga_i \mid i < j \ra$.
Note that if we wish, by padding if necessary,
we may assume without loss of generality that
$j$ is arbitrarily large.
Write $f = \la \la \ga_i \mid i < i_0 \ra,
\la \ga_i \mid i_0 \le i < j \ra \ra =
\la f_1, f_2 \ra$.
Represent $G \rest f$ as $(G \rest f_1) \times
(G \rest f_2) = G_1 \times G_2$, where
$G_1 \times G_2$ is $V$-generic over the
partial ordering $\FP_1 \times \FP_2 =
(\FP \rest f_1) \times (\FP \rest f_2)$.
If $V \models ``\gk$ is supercompact'',
then because each
$V$-supercompact cardinal is indestructible and
$G_2$ is $V$-generic over $\FP_2$, a partial ordering
which is $\gk$-directed closed,
$V[G_2] \models ``\gk$ is supercompact''.
Since by our assumptions on $V$,
$V \models ``\gk$ is not a limit of supercompact cardinals'',
it follows that $V \models ``\card{\FP_1} < \gk$''.
Therefore, by the
L\'evy-Solovay results \cite{LS}, $\gk$ remains
supercompact in $V[G_2 \times G_1] =
V[G_1 \times G_2]$. Because $\gk$ is supercompact in
$V[G_1 \times G_2]$, $\gk$ is weakly compact in this
model as well and hence satisfies the tree property in
$V[G_1 \times G_2]$.
This means that in $V[G_1 \times G_2]
\subseteq N$, there is a branch of length $\gk$
through ${\mathfrak T}$.
If, however,
$V \models ``\gk$ is a successor of a singular limit of
supercompact cardinals''\footnote{Note
that by our assumptions on $V$, {\em every}
limit of supercompact cardinals is singular.},
then because each
$V$-supercompact cardinal is indestructible and
$G_2$ is $V$-generic over a partial ordering
which is $\gk$-directed closed,
$V[G_2] \models ``\gk$ is a successor of a
singular limit of supercompact cardinals''.
As we have just observed, if
$V \models ``\gl < \gk$ is supercompact'', then
$V[G_1 \times G_2] \models ``\gl$ is supercompact''.
Thus, since each component partial ordering of $\FP_1$
is appropriately directed closed,
$\gk$ is in $V[G_2 \times G_1] = V[G_1 \times G_2]$
a successor of a singular limit of
supercompact cardinals. Hence, by Shelah's theorem of \cite{MS},
$\gk$ satisfies the tree property in $V[G_1 \times G_2]$.
This means that in $V[G_1 \times G_2]
\subseteq N$, there is once again a branch of length $\gk$
through ${\mathfrak T}$.
%partial ordering having cardinality less than $\gk$.
Therefore, in either situation,
$N \models ``\gk$ satisfies the tree property''.
This completes the proof of both
Lemma \ref{l1} and Theorem \ref{t1}.
\end{proof}
%\end{pf}
We remark that our methods of proof will
actually allow $\FP$ to be taken as a
countable support product. However,
for consistency with \cite{A83}, we use
set support in the definition of $\FP$.
Further, it is possible to begin the
construction of $N$ by symmetrically
collapsing $\gk_0$ to be $\ha_2$
instead of $\ha_1$. If this is done, then
the arguments of \cite{A83} in conjunction
with standard techniques show that
%$N$ will be a model for the theory
$N \models$ ``ZF + ${\rm DC}_{\go_1}$ + $2^{\ha_0} = \ha_1$ +
The tree property fails at $\ha_1$ +
Every successor cardinal is regular + Every
limit cardinal is singular + Every successor
cardinal greater than $\ha_1$ satisfies the tree property''.
In this way, $N$ is slightly more ``choice like'',
although by its construction,
the full Axiom of Choice of course still fails in $N$.
In addition, an alternate way of inferring that if
$V \models ``\gk$ is supercompact'', then
$N \models ``\gk$ satisfies the tree property'' is to
note that by \cite[Lemma 1.2]{A83},
$N \models ``\gk$ is weakly compact''.
As we have already observed, this immediately
implies that $N \models ``\gk$ satisfies the tree property''.
We end with two questions.
First, we ask if the gap in consistency strength
between the upper and lower bounds for the theory
``ZF + DC + Every successor cardinal is regular +
Every limit cardinal is singular + Every successor
cardinal satisfies the tree property'' provided
in this paper can be narrowed somewhat. Finally,
as was mentioned earlier, Theorem \ref{t1} does not
answer our Question posed above in a ZFC context. We
conclude by asking if a positive answer
to this question can be found, or even if
it is possible to construct a model
of ZFC (starting from any suitable large
cardinal hypotheses) combining the
results of \cite{CF}, \cite{MS}, and
\cite{Si1}, i.e.,
a model of ZFC in which every regular $\ha_i$
for $1 < i \le \go + 1$ satisfies the tree property.
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\end{document}
Relative to the existence of a proper class
of supercompact cardinals, we establish
the consistency of the theory
``ZF + DC + Every successor cardinal is regular +
Every limit cardinal is singular + Every
successor cardinal satisfies the tree property''.
This extends, combines,
and unifies in a choiceless context earlier
results due to Mitchell, Abraham, Magidor
and Shelah, and Cummings and Foreman.
\begin{lemma}\label{l1}
$N \models ``$Every successor cardinal satisfies
the tree property''.
\end{lemma}
\begin{proof}
Suppose $N \models ``\gk$ is a successor cardinal
and ${\mathfrak T}$ is a $\gk$-tree''.
As we have just observed, $\gk \in {\cal K}$.
Let $i_0$ be the unique ordinal such that $\gk = \gk_{i_0}$.
Since ${\mathfrak T}$ may be coded by a
set of ordinals, by \cite[Lemma 1.1]{A83}, we can
assume that ${\mathfrak T} \in V[G \rest f]$
for some $f \in {\cal F}$, $f = \la \ga_i \mid i < j \ra$.
Note that if we wish, by padding if necessary,
we may assume without loss of generality that
$j$ is arbitrarily large.
Write $f = \la \la \ga_i \mid i < i_0 \ra,
\la \ga_i \mid i_0 \le i < j \ra \ra =
\la f_1, f_2 \ra$.
Represent $G \rest f$ as $(G \rest f_1) \times
(G \rest f_2) = G_1 \times G_2$, where
$G_1 \times G_2$ is $V$-generic over the
partial ordering $\FP_1 \times \FP_2 =
(\FP \rest f_1) \times (\FP \rest f_2)$.
If $V \models ``\gk$ is supercompact'',
then by \cite[Lemma 1.2]{A83},
$N \models ``\gk$ is weakly compact''.
As we have already observed, this means that
$N \models ``\gk$ satisfies the tree property,
i.e., there is a branch of length $\gk$
through ${\mathfrak T}$''.
If, however,
$V \models ``\gk$ is a successor of a singular limit of
supercompact cardinals''\footnote{Note
that by our assumptions on $V$, {\em every}
limit of supercompact cardinals is singular.},
then because each
$V$-supercompact cardinal is indestructible and
$G_2$ is $V$-generic over a partial ordering
which is $\gk$-directed closed,
$V[G_2] \models ``\gk$ is a successor of a
singular limit of supercompact cardinals''.
If $V \models ``\gl < \gk$ is supercompact'',
then as each
$V$-supercompact cardinal is indestructible and
$G_2$ is $V$-generic over $\FP_2$, a partial ordering
which is $\gk$-directed closed,
$V[G_2] \models ``\gl$ is supercompact''.
Since by our assumptions on $V$,
$V \models ``\gl$ is not a limit of supercompact cardinals'',
it follows that $V \models ``\card{\FP_1} < \gl$''.
Therefore, by the
L\'evy-Solovay results \cite{LS}, $\gl$ remains
supercompact in $V[G_2 \times G_1] =
V[G_1 \times G_2]$.
%As we have just observed, if
%$V \models ``\gl < \gk$ is supercompact'', then
%$V[G_1 \times G_2] \models ``\gl$ is supercompact''.
Thus, since each component partial ordering of $\FP_1$
is appropriately directed closed,
$\gk$ is in $V[G_2 \times G_1] = V[G_1 \times G_2]$
a successor of a singular limit of
supercompact cardinals. Hence, by Shelah's theorem of \cite{MS},
$\gk$ satisfies the tree property in $V[G_1 \times G_2]$.
This means that in $V[G_1 \times G_2]
\subseteq N$, there is once again a branch of length $\gk$
through ${\mathfrak T}$.
%partial ordering having cardinality less than $\gk$.
Therefore, in either situation,
$N \models ``\gk$ satisfies the tree property''.
This completes the proof of both
Lemma \ref{l1} and Theorem \ref{t1}.
\end{proof}
%\end{pf}