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\begin{document}


\setbeamercovered{dynamic}

\title{Stable Homogeneous Trees}



\author{S.\ Jackson}



\institute{
Department of Mathematics\\
University of North Texas\\
%jackson@unt.edu
}






\date{
April 30, 2011\\
UNLV}


\begin{frame}
\titlepage
\end{frame}





\section{Introduction}

\begin{frame}
We consider the problem of getting the complete scale and Suslin cardinal analysis from 
just $\zf+\ad+\dc$. 
\mys


We recall the basic definitions and results.
\mys



$A\subseteq \ww$ is $\kappa$-Suslin if there is a tree $T$ on $\omega \times \kappa$ with 
\[ 
\begin{split}
A=p[T]& =\{ x \colon \exists f \in \kappa^\omega (x,f) \in [T]\}
\\ & = \{  x \colon \exists f \in \kappa^\omega \ \forall n\ (x \res n, f\res n) \in T\}
\end{split}
\]
\mys

We let $S(\kappa)$ be the collection of $\kappa$-Suslin sets, $S(<\kappa)=\bigcup_{\lambda<\kappa}
S(\lambda)$.
\mys



We say $\kappa$ is a {\color{red}Suslin cardinal} if $S(\kappa)-S(<\kappa)\neq \emptyset$.


\end{frame}





\begin{frame}
A {\color{red}semi-scale} on $A \subseteq \ww$ is a sequence of norms $\varphi_n \colon A \to \on$
such that if $x_i \in A$, $x_i \to x$, and $\varphi_n(x_i)$ are eventually equal to $\lambda_n$,
then $x \in A$.
\mys


A {\color{red}scale} on $A$ is a semi-scale which also satisfies 
$\varphi_n(x)\leq \lambda_n$.
\mys



\begin{fact}
For all cardinals $\kappa$, $A$ is $\kappa$-Suslin iff $A$ admits a scale into $\kappa$. 
\end{fact}



We may assume wlog that the norms of a (semi) scale are {\color{red}regular}, that is, they map
onto an ordinal. 
\end{frame}





\begin{frame}
If $A=p[T]$ is a Suslin representation for $A$, we get a semi-scale $\varphi^T_n$ on $A$ by:
$\varphi^T_n(x)=\ell_x(n)$, where $\ell_x$ is the {\color{red}left-most} branch 
of $T_x$. 
\mys


Conversely, given a semi-scale $\varphi_n$ on $A$, let 
$(s,\vec\alpha) \in T_\varphi$ iff $\exists x \sqsupseteq s\ \forall n <\lh(s)\ 
\varphi_n(x)=\alpha_n$. 
\mys


If $\varphi$ is a scale on $A$, then $(\varphi_0(x),\varphi_1(x),\dots)$ is the leftmost
branch of $T_\varphi$.
\mys



If $A=p[T]$ we can get a scale on $A$ by 
\[ \varphi_n(x)= \langle \ell_x(0), \ell_x(1),\dots,\ell_x(n-1)\rangle
\]
\end{frame}



\begin{frame}
We have the following facts.

\begin{fact}
For any tree $T$, we have: $T_{\varphi_T}\subseteq T$
and $[T_{\varphi_T}]=[T]$.

If $T=T_\varphi$
is the tree of a scale $\varphi_n$ then $T_{\varphi_T} =T$. 
\end{fact}



\begin{fact}
For any scale $\varphi_n$ on $A$ we have: $\varphi= \varphi_{T_\varphi}$.
\end{fact}



\begin{fact}
Both operations $T\mapsto T_{\varphi_T}$ and $\varphi\mapsto \varphi_{T_\varphi}$
are idempotent. [For any tree $T$ we have $\varphi_{T}= \varphi_{T_{\varphi_T}}$.]
\end{fact}
\end{frame}




\begin{frame}
Recall the definition of a $\bg$ scale.


\begin{defn}
A norm $\varphi\colon A \to \on$ is a $\bg$-norm if the following {\color{red}norm relations}
are in $\bg$:

\[
\begin{split}
x<^* y & \leftrightarrow x \in A \wedge (y \notin A \vee (y \in A \wedge \varphi(x)<\varphi(y))\\
x\leq^* y & \leftrightarrow x \in A \wedge (y \notin A \vee (y \in A \wedge \varphi(x)\leq\varphi(y))\
\end{split}
\]
\end{defn}

Each $A_{\leq \alpha} =\{ x \in A \colon \varphi(x)\leq \alpha\}$ is in $\bd$. 
\mys


If $\bd$ is closed under $\wedge$, then each $A_\alpha=
\{ x \in A \colon \varphi(x)= \alpha\}$ is in $\bd$ as well. 
\end{frame}




\begin{frame}
\begin{defn}
$\varphi_n$ is a $\bg$-scale if all of the norm relation $<^*_n$, $\leq^*_n$ are in $\bg$.
\end{defn}


\begin{defn}
$\bg$ has the scale property, $\sca(\bg)$, if every $A \in \bg$ admits a $\bg$ scale.
\end{defn}
\mys


The scale property is the main structural representation for sets $A \subseteq \ww$ in
descriptive set theory.
\mys



If $\sca(\bg)$  and $\wedge \bd=\bd$, then $T_\varphi$ is a tree ($\varphi$ a $\bg$-scale on a $\bg$ set $A$) 
on an ordinal 
$\leq \delta(\bg)$ where:


\begin{defn}
$\delta(\bg)=$ the supremum of the lengths of the $\bd$ prewellorderings of $\ww$. 
\end{defn}
\end{frame}


\subsection{Homogeneous Trees}


\begin{frame}
\frametitle{Homogeneous Trees}


The notion of homogeneous tree was formulated independently by {\color{red}Kunen}
and {\color{red}Martin}, then isolated and 
given in its current form by {\color{red}Kechris}.
\mys


\begin{defn}
A {\color{red}measure} on a set $X$ is a countably additive ultrafilter on $X$.
\end{defn}
\mys


Recall that under $\ad$ every ultrafilter on a set $X$ is a measure.
\mys
\end{frame}



\begin{frame}
\begin{defn}
A tree $T$ is {\color{red}homogeneous} if there are measures $\{ \mu_s\colon s \in \omega^{<\omega}\}$
on $T_s$ satisfying:
\begin{itemize}
\item
If $s \sqsubseteq s'$ then $\mu_{s'}$ projects to $\mu_s$.
\item (homogeneity property)\\ 
For all $x \in \ww$ if $T_x$ is illfounded
then for any $A_{x \res n}$ with $\mu_{x\res n}(A_{x \res n})=1$, there is an 
$f$ such that $\forall n\ f\res n \in A_n$. 
\end{itemize}
\end{defn}
\mys


Note: This is equivalent to saying that the direct limit of the ultrapowers by the measures $\mu_{x \res n}$
is wellfounded.
\mys


Fact ($\zf$, Martin): If $A=p[T]$ where $T$ is homogeneous, then $A$ is determined.
\end{frame}


\begin{frame}
\frametitle{Weakly Homogeneous Trees}
\begin{defn}[Weakly Homogeneous, strong form]
$T$ is {\color{red}weakly homogeneous} if there are measures $\mu_{s,t}$ on $T_s$ defined for 
a set of $(s,t)$ in a tree, and sets $A_{s,t} \subseteq T_s$ satisfying:
\begin{itemize}
\item
$\mu_{s,t}(A_{s,t})=1$.
\item
If $s'$ immediately extends $s$ and $A_{s,t}$ is defined, 
then the $A_{s',t'}$ for $t'$ extending $t$ partition $T_{s'} \cap \{\vec \alpha 
\colon \pi_{s',s}(\vec \alpha) \in A_{s,t}\}$.
\item
If $(s',t') \sqsupseteq (s,t)$ then $\pi_{(s',t'),(s,t)}(\mu_{(s',t')})=\mu_{(s,t)}$. 
\item (homogeneity property)\\ 
For any $x , y \in \ww$, if $\mu_{x \res n, y\res n}$ is defined for all $n$ and 
$\exists f\ \forall n\ (x \res n, f \res n) \in A_{x\res n, y\res n}$, then 
for any $B_{x\res n, y \res n}$ with 
$\mu_{x \res n, y \res n}(B_{x \res n, y\res n})=1$ we have that 
$\exists g \ \forall n\ (x\res n, g\res n) \in B_{x \res n, y \res n}$.
\end{itemize}
\end{defn}
\end{frame}


\begin{frame}
\begin{defn}[Weakly Homogeneous, weak form]
There are measures $\mu_{s,t}$ with $\mu_{s,t}(T_s)=1$
satisfying:
\begin{itemize}
\item
If $(s',t') \sqsupseteq (s,t)$ then $\pi_{(s',t'),(s,t)}(\mu_{s',t'})=\mu_{s,t}$. 
\item
For any $x \in \ww$, if $T_x$ is illfounded then $\exists y \in \ww$ such that 
$\langle \mu_{x \res n, y\res n} \rangle$ has wellfounded direct limit: for any $\mu_{x \res n, y \res n}
(B_{x \res n, y \res n})=1$  there is a $g$ such that $\forall n\ (x \res n, g \res n) \in 
B_{x \res n, y\res n}$. 
\end{itemize}
\end{defn}


{\color{blue}Alternate Version}:
There is a countable set $\sM$ of measures such that whenever $T_x$ is illfounded then there is a 
sequence $\mu_1,\mu_2,\dots$ of measure in $\sM$ with $\mu_n(T_{x \res n})=1$ 
such that $\langle \mu_{x \res n}\rangle$ has wellfounded direct limit.
\end{frame}


\begin{frame}
\begin{fact}[Woodin] In $\zfc$ the strong and weak forms of the definition are equivalent. 
\end{fact}


\begin{fact}
The weak form of the definition is enough to make the Martin-Solovay construction work.
\end{fact}


\begin{fact}
The strong form of the definition s equivalent to saying that 
there is a homogeneous tree $\bar{T}$ on $\omega \times \omega \times \kappa$ which is 
isomorphic to $T$ (under a bijection of $\omega \times \kappa$ with $\kappa$).
\end{fact}
\mys


So, a set $A \subseteq \ww$ is weakly homogeneously Suslin iff it is the projection of 
a strongly homogeneously Suslin set.
\end{frame}


\begin{frame}
\frametitle{Martin-Solovay Tree}
Let $(T,\vec \mu)$ be a weakly homogeneous tree with $A=p[T]$. The {\color{red}Martin-Solovay tree}
$T'$ is a tree with \[\ww-A=p[T']\] defined as follows.
\mys



$(s,\vec \alpha) \in T'(T,\vec \mu,\lambda)$ iff $\exists f \colon T_s \to \lambda$ order-preserving
w.r.t.\ the BK ordering on $T_s$ such that $\forall i <\lh(s)$
\[ \alpha_i= [f^i]_{\mu_{s_i,t_i}}
\]
where $(s_i,t_i)$ is a reasonable enumeration of $\omega^{<\omega}\times \omega^{<\omega}$
and $f^i=f\res \{ \vec \alpha \colon |\vec \alpha|=i \wedge (s_i,\vec \alpha)\in T\}$
if $s_i \sqsubseteq s$, and $f=0$ otherwise.
\end{frame}

\section{Stable Homogeneous Trees}



\begin{frame}
\frametitle{Stable Homogeneous Trees}

\begin{ques}
When is the semi-scale from the Martin-Solovay tree $T'$ a scale?
\end{ques}
\mys

We make the following definition for weakly and strongly homogeneous trees. 
For simplicity we state them for homogeneous trees.
\mys

For $(T, \mu_s)$ a homogeneous tree, if $\vec A=\{ A_s \colon s \in \omega^{<\omega}\}$ 
with $A_s \subseteq T_s$, we let 

\[
T^{\vec A} =\{ (s,\vec \alpha) \in T \colon \forall i \leq |s|\ \vec
\alpha \res i \in A_{s \res i} \}.
\]

Let $T_x$, $T^{\vec A}$ denote the sections of $T$ and $T^{\vec A}$ at $x$.
\mys


Let $f_x$, $f^{\vec A}_x$ denote the rank functions on $T_x$, $T^{\vec A}_x$ 
(for $T_x$ wellfounded). Likewise for $f_{x \res n}$ and $f^{\vec A}_{x \res n}$.
\end{frame}


\begin{frame}
\begin{defn}
The homogeneous tree $(T,\mu_s)$ is {\color{red}stable} if there are measure one sets $A_s$
w.r.t.\ the $\mu_s$ such that for any $x \in B=\ww-A$ ($A=p[T]$) 
and any measure one sets $B_{x \res n}$ (w.r.t.\ $\mu_{x \res n}$) we have that
\[
[f^{\vec A}_{x,n} ]_{\mu_{x \res n}}   \leq  [f^{\vec B}_{x,n}]_{\mu_{x \res n}}
\]
for all $n$.
\end{defn}
\mys

\begin{thm}
If $(T,\mu_s)$ is a stable homogeneous tree, then the semi-scale $\varphi_n$ from the
Martin-Solovay tree $T'$ constructed from $T^{\vec A}$ is a scale. Here 
$\varphi_n(x)=[f^{\vec A}_x]_{\mu_{x \res n}}$.
\end{thm}
\end{frame}


\begin{frame}


{\color{blue}Proof:}
Let $x \in B=\ww-A$. So, $T^{\vec A}_x$ is wellfounded. 
\mys



Let $x_i \in B$ with $x_i \to x$ and assume each $\varphi_n(x_i)$ is eventually equal to $\lambda_n$.
Say $\varphi_n(x_i)=\lambda_m$ for all $i \geq n$. Assume $x_i \res i=x \res i$.
\mys


For each $n$ let $f_n \colon T^{\vec A}_{x_n\res n}=T^{\vec A}_{x \res n} \to \on$ be such that 
\[
\forall i <n \ \lambda_i=[f_n^i]_{\mu_{x\res i}}
\]

where $f_n^i$ is the subfunction of $f_n$ induced by restriction.
\mys


Fix measure one sets $B_{x \res n}$ such that for all $i_1,i_2 \geq n$ we have 
\[
f_{m_1}^n\res B_{x \res n} =f_{m_2}^n \res B_{x \res n}.
\]

Then the $f_n$ restricted to the $\vec{B}$ define an order-preserving $f\colon T^{\vec B}_x \to \on$.
By stability we have $[f^{\vec A}_x]_{\mu_{x \res n}}=[f^{\vec B}_x]_{\mu_{x \res n}}$ for all $n$,
so $\varphi$ is actually a scale.
\mys

\end{frame}

\subsection{Main Theorem}


\begin{frame}
\frametitle{Main theorem}

\begin{thm}[$\zf+\ad+\dc$]
Every homogeneous tree  $(T, \{\mu_s\})$ on $\kappa < \Theta$ is stable.
\end{thm}
\mys

Proof uses ideas from the {\color{red}Martin-Woodin} proof that all trees are weakly homogeneous.
\mys



{\color{blue}sketch of proof:}
Let $\nu$ be a fine measure on $\sP_{\omega_1}(\cup_n \sP(\kappa^n))$. 
\mys

For $\sigma \in \sP_{\omega_1}(\cup_n \sP(\kappa^n))$, let $A^\sigma =\{ A^\sigma_s\}_s$
where $A^\sigma_s= \cap \{ A \in \sigma \colon \mu_s(A)=1\}$.
\mys

We show that for $\nu$ almost all $\sigma$ that the $A^\sigma_s$ stabilize $T$. We suppose not.
\end{frame}



\begin{frame}
Uniformly in $\sigma$ we define a tree $U^\sigma$ on $\omega \times \lambda^{<\omega}\sim 
\omega \times \lambda$ where $\lambda= \sup j_{\mu_s}(\kappa)$.
\mys


For $x \in B$ the tree $(U_\sigma)_x$ attempts to produce ordinals $[f^i_j]_{\mu_{x \res i}}$
which witness that the sets $A^\sigma$ have not yet attained the minimal ranking functions. 
\mys



They do this, roughly speaking, by describing embeddings of $T_x$ (on measure one sets) into 
proper initial segments of $T^{A^\sigma}_x$.
\end{frame}


\begin{frame}




We set $((s(0),\dots,s(n-1)),(\beta_0,\dots, \beta_{n-1})) \in U^\sigma$ iff:
\mys


1.) $\beta_0 \in \omega$.
\mys

2.) Each $\beta_i$ for $i>0$ codes a finite sequence of integers $t_i$ extending $s$ (roughly, a commitment
that later extensions of $s$ must follow $t_i$) along with a finite sequence of ordinals 
$(\beta^i_0,\dots, \beta^i_{|t_i|-1})$. 
\mys


3.) $s$ must be compatible with all $t_i$ for $i<|s|$.
\mys


4.) Let $f^i_j$ represent $\beta^i_j$ with respect to $\mu_{s \res i}$. Then there are 
$\mu_{s \res i}$ measure one sets restricted to which the maps 
$(\eta_0,\dots \eta_{i-1}) \mapsto (f^i_0(\vec \eta),\dots,f^i_{|t_i|-1})$
give an order-preserving from the BK ordering on $T_{s_n}$ to the BK ordering on 
$T^{A^\sigma}$.
\end{frame}


\begin{frame}
5.) For $i<\beta_0$, each $f^i_j$ is almost everywhere the identity function, and for 
$i =\beta_0$ we have that almost everywhere that $(f^i_0(\vec \eta), \dots, f^i_{|t_i|-1}(\vec \eta))$
is a properly less than $\vec \eta$ in $T_{|t_i|}$.
\mys


6.) We weave in the Martin-Solovay tree into $U_\sigma$ as well. Say at every even level of 
$U_\sigma$ we put in ordinals from the Martin-Solovay tree (so a branch through $(U_\sigma)_x$
also gives a branch through $(T')_x$ and so proves $x \in B$).
\end{frame}



\begin{frame}
For any $x$,  $(U^\sigma)_x$ is illfounded iff $x \in B$ and there are measure one sets 
$B_{x \res n} \subseteq A^\sigma_{x \res n}$ (with respect to $\mu_{x \res n}$) such that 
for some $m$, $[f^{\vec B}_x]_{\mu_{x \res m}} < [f^{A^\sigma}_x]_{\mu_{x \res m}}$.
\mys


So, for $\nu$ almost all $\sigma$ there is a leftmost branch $(x^\sigma, {\vec \beta}^\sigma)$
through $U^\sigma$. By countable additivity we may fix $x=x^\sigma$ on a measure one set. 
\mys


Fix any $\sigma$ where 
$x^\sigma=x$ and also $\sigma$ contains measure one sets $C_{x \res n}$ (w.r.t.\ the 
$\mu_{x \res n}$) which attain the minimal values for $[f^{\vec C}_x]_{\mu_{x \res n}}$
(which we can do by countable choice).
\mys

The branch $\vec \beta^\sigma$ 
gives a proper embedding from $T^{\vec D}_x$ into $T^{A^\sigma}_x\subseteq T^{\vec C}_x$
for some measure one sets $D_{x \res n}$. This contradicts the choice of the $C_{x \res n}$. 



\end{frame}


\subsection{Application}
\begin{frame}
\frametitle{Application}

We use the above result to complete the Scale/Suslin cardinal analysis developed by the 
Cabal, assuming $\zf+\ad+\dc$. Reader can consult chapter 3 of Handbook article
``Structural Consequences of Determinacy.''
\mys


Assume $\kappa$ is a limit Suslin cardinal with $\bg=\S(\kappa)$ closed under quantifiers. 
(Type IV case). 
\mys

Let $\Lambda$ be the selfdual {\color{red}Martin pointclass} 
which constructs the nest Suslin cardinal.
Martin defined this class, Steel gave another characterization. 
\mys



Let $\lambda=o(\Lambda)=$next Suslin cardinal after $\kappa$. 
\mys

\end{frame}



\begin{frame}
Let $\Sigma^\lambda_0=\bigcup_\omega \Lambda$. Define $\Pi^\lambda_n$, $\Sigma^\lambda_n$
as usual.
\mys

Previous results showed $\sca(\bs^\lambda_{2n})$, $\sca(\bp^\lambda_{2n+1})$
for $n \geq 1$ with corresponding Suslin cardinals 

\[
\delta_1=\delta(\bp^\lambda_1)=\lambda^+, \hspace{10pt} \lambda_3=(\delta_3)^-, 
\hspace{10pt} \delta_3=\delta(\bp^\lambda_3),\dots
\]


Previous arguments did nor give that $\Lambda$ was a self-justifying system, nor the scale property
at $\bs^\lambda_0$, $\bp^\lambda_1$
\mys


\begin{thm}
$\Lambda$ is self-justifying and $\sca(\bs^\lambda_0)$, $\sca(\bp^\lambda_1)$.
\end{thm}
\end{frame}



\begin{frame}

{\color{blue}Proof:}
Let $A$ be $\bg$-complete and write $A=p[T]$ with $T$ a tree on $\omega \times \kappa$. 
By Martin-Steel we may assume $T$ is homogeneous. By theorem we may assume $T$
has been stabilized. 
\mys


For $x \in B=\ww-A$, let $\varphi_n(x)=[f_x]_{\mu_{x \res n}}$, where $f_x \colon 
T_x \to \on$ is the ranking function. 
\mys


A direct computation shows $\varphi_n$ is a $\Lambda$-norm. The $\{ \varphi_n\}$
form a scale since $T$ is stable. 
\mys



This shows $B$ admits a scale all of whose norms are in $\Lambda$. It follows that 
this also holds for all sets in $\Lambda$. The other results easily follow.














\end{frame}










\end{document}
