\documentclass{beamer}
\usepackage{beamerthemesplit}
\usepackage[all]{xy}


\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[all]{xy}
\usepackage{eepic}
\usepackage{txfonts}



\theoremstyle{theorem}
\newtheorem*{thm}{Theorem}
\newtheorem*{lem}{Lemma}


\theoremstyle{definition}
%\newtheorem*{fact}{Fact}
\newtheorem*{defn}{Definition}
\newtheorem*{ques}{Question}
\newtheorem*{cor}{Corollary}
\newtheorem*{ex}{Example}
\newtheorem*{claim}{Claim}











\newcommand{\ww}{{\omega^\omega}}



\newcommand{\res}{\restriction}
\newcommand{\bg}{\boldsymbol{\Gamma}}
\newcommand{\bgd}{\check{\boldsymbol{\Gamma}}}
\newcommand{\bd}{\boldsymbol{\Delta}}
\newcommand{\bs}{\boldsymbol{\Sigma}}
\newcommand{\bp}{\boldsymbol{\Pi}}

\newcommand{\de}[2]{\boldsymbol{\delta}^{#1}_{#2}}


\newcommand{\on}{\text{On}}
\newcommand{\sP}{\mathcal{P}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}



\newcommand{\lh}{\text{lh}}

\newcommand{\ad}{\mathsf{AD}}
\newcommand{\dc}{\mathsf{DC}}
\newcommand{\zf}{\mathsf{ZF}}
\newcommand{\zfc}{\mathsf{ZFC}}

\newcommand{\cof}{\text{cof}}
\newcommand{\lr}{L(\mathbb{R})}
\newcommand{\sca}{\text{scale}}
\newcommand{\lex}{\text{lex}}

\newcommand{\mys}{\bigskip}
\newcommand{\hod}{\text{HOD}}
\newcommand{\gch}{\text{GCH}}
\newcommand{\ot}{\text{o.t.}}
\newcommand{\dom}{\text{dom}}

\newcommand{\sv}{\mathsf{v}}
\newcommand{\wli}{\mathbf{wlift}}
\newcommand{\sli}{\mathbf{slift}}

\newcommand{\sR}{\mathcal{R}}
\newcommand{\sA}{\mathcal{A}}

\newcommand{\ran}{\text{ran}}
\newcommand{\sbR}{\bar{\sR}}





\begin{document}


\setbeamercovered{dynamic}



\title{$\ad$, Large Cardinals, and Partition Properties}



\author{S.\ Jackson}




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






\date{{\color{red} ICWAC}\\
June, 2009, Bonn}






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



\section{Introduction}




\begin{frame}



We work in the base theory $\zf+\dc$. 
Most of the the time we will also assume $\ad$. 
\mys


{\color{red}General Question}:  Describe the exact large cardinal properties 
of all of the cardinals $\kappa<\Theta$ assuming $\ad$. 
\mys


The large cardinal properties must have versions that can be stated in 
a choice-less context. 
\mys


Some examples:  supercompact, measurable, Rowbottom, Jonsson. 
\mys



Strength can also be measured in terms of {\em partition properties}. 
\mys


{\color{red}General Question}: Describe the exact partition properties of all 
the cardinals below $\Theta$ assuming $\ad$. 
\end{frame}



\section{Large Cardinal Properties}

\begin{frame}



\frametitle{Large Cardinal Definitions}


\begin{defn}
$\kappa$ is {\em measurable} if there is a $\kappa$-complete ultrafilter on $\kappa$.
\end{defn}
\mys

From $\ad$ every ultrafilter on a set is countably additive, that is, is
a {\em measure}.
\mys


If $\kappa$ is measurable, then $\kappa$ has a normal measure. 
In general, $\kappa$ may have many normal measures. We say a measure $\mu$ on $\kappa$
is {\color{red} semi-normal} if $\mu(C)=1$ for every c.u.b.\ set $C \subseteq \kappa$. 


\end{frame}

\begin{frame}



If $\kappa$ is measurable, then $\kappa$ is regular. However, $\kappa$ need not be large.
\mys



Solovay showed $\omega_1$ is measurable. 
\mys


Moschovakis, Martin, Kunen
showed all the $\de 1 n$ are regular and then measurable. 
\end{frame}



\begin{frame}
There are other regular cardinals 
below the projective ordinals, and these were shown to be measurable as well. 
\mys


Using techniques of inner-model theory Steel showed:

\begin{thm}[Steel] ($\ad+V=\lr$) Every regular $\kappa <\Theta$ is measurable.
\end{thm}
\mys


The proof also showed  $\hod(x)$ satisfies the $\gch$ below $\Theta$. 
\mys

So,
any wellordered collection of subsets of $\kappa <\Theta$ has size $<\kappa^+$. 
\end{frame}



\subsection{Supercompactness}


\begin{frame}


\frametitle{Supercompactness}

\begin{defn}
$\kappa$ is $\lambda$-supercompact if there is a fine, normal measure
on $\sP_\kappa(\lambda)$.
\end{defn}
\mys

Solovay showed that $\omega_1$ is $<\Theta$ supercompact from $\ad_\R$. This was shown 
from $\ad$ (for $\leq \de 21$) by Harrington Kechris., and Woodin for $<\Theta$. 
\mys



Becker showed that $\omega_2$ is $\de 21$-supercompact from $\ad$. 
\end{frame}


\begin{frame}


We showed all the $\de 1n$ are $\de 21$-supercompact.
\mys


We generalized this too show:
\mys

\begin{thm}
All the regular $\kappa$ which are Suslin or the successor of a Suslin are $\de 21$-
supercompact. 
\end{thm}



\end{frame}





\begin{frame}
\begin{ques}
How supercompact are the other regular cardinals?
\end{ques}
\mys



In particular, how supercompact are the other regular cardinals below the projective ordinals?
\mys


Many questions about supercompactness measures remain.
\mys

A theorem of Woodin says that the supercompactness measures on $\sP_{\omega_1}(\lambda)$
are unique. This does not hold for $\kappa>\omega_1$. 
\mys


\begin{ques}
How many supercompactness measures are there on $\sP_\kappa(\lambda)$?
\end{ques}
\mys


For example, there are at least two distinct supercompactness measures on 
$\sP_{\de 13}(\de 14)$. 
\end{frame}








\subsection{Partition Properties}




\begin{frame}


\frametitle{Partition Properties}



Recall the Erd\H{o}s-Rado Partition notation.
\mys


\begin{defn}
$\kappa \rightarrow (\kappa)^\lambda$ if for every partition $\sP \colon 
(\kappa)^\lambda \to \{0,1\}$ of the increasing functions from $\lambda$ to 
$\kappa$ into two pieces, there is a homogeneous $H \subseteq \kappa$ of
size $\kappa$.
\end{defn}
\mys



We say $\kappa$ has the {\color{red} strong} partition property if $\kappa 
\rightarrow (\kappa)^\kappa$, and say $\kappa$ has the {\color{red} weak}
partition property if $\kappa \rightarrow (\kappa)^\lambda$ for all $\lambda<\kappa$.
\mys

In $\zfc$ no infinite exponent partition relations hold. So, $\zfc$ large cardinals properties
formulated using partition properties (e.g., Ramsey, Rowbottom, Jonsson) use exponent
$<\omega$. 
\mys
\end{frame}


\begin{frame}
We usually use a c.u.b.\ reformulation of the partition property. This involves the notion of
{\color{red} type} of a function.
\mys



\begin{defn}
We say $f \colon \alpha \to \on$ has {\color{red} uniform cofinality} $\omega$
if there is an $f' \colon \alpha \times \omega \to \on$ which is increasing 
in the second argument and $f(\beta)=\sup_n f'(\beta,n)$ for all $\beta <\alpha$.
\end{defn}
\end{frame}



\begin{frame}
\begin{defn}
$f \colon \alpha \to \on$ is of the {\color{red} correct type} if 
$f$ is increasing, everywhere discontinuous, and of uniform cofinality $\omega$.
\end{defn}
\mys

We can likewise define uniform cofinality $\omega_1$, $\omega_2$, etc.
\mys

More generally, for any $g \colon \alpha \to \on$ we can define $f$ having {\color{red}uniform
cofinality $g$}: there is an 
$$f' \colon \{ (\beta,\gamma) \colon \beta < \alpha, \gamma<g(\beta) \}$$
with $f(\beta)= \sup_\gamma f'(\beta,\gamma)$.
\mys

We use frequently the ``almost everywhere'' versions of these notions with respect
to some measure $\mu$ on $\dom(f)$.
\end{frame}



\begin{frame}
\begin{ex}
For functions $f \colon \omega_1 \to \omega_1$ there are two uniform cofinalities  almost everywhere with
respect to the normal measure on $\omega_1$. Namely, $f(\alpha)$ has uniform cofinality $\omega$ and
$f(\alpha)$ has uniform cofinality $\alpha$. 
\end{ex}
\mys




For functions $f \colon \lambda \to \kappa$ of type $g$, let $\lambda'=\ot(\dom(g))$.
\mys


The partition relation $\kappa \rightarrow (\kappa)^{\lambda'}$ 
induces a partition relation $\kappa \underset{g}{\longrightarrow} (\kappa)^\lambda$ of functions 
$f\colon \lambda \to \kappa $ of {\color{red}type $g$} with c.u.b.\ homogeneous sets
(increasing, discontinuous, and of uniform cofinality $g$).
\end{frame}






\begin{frame}
The simplest type is:

\begin{defn}
$f$ has the {\color{red}correct type} if $f$ is increasing, discontinuous, and of 
uniform cofinality $\omega$. 
\end{defn}




\begin{ex}
We have the strong partition relation on $\omega_1$ for functions 
of the correct type, and also for functions of type $g(\alpha)=\alpha$. 
\end{ex}


The first induces the $\omega$-cofinal normal measure on $\omega_2$, and the second the
$\omega_1$-cofinal normal measure on $\omega_2$. 
\end{frame}





\begin{frame}
An easy argument shows that 
$$
\kappa \rightarrow (\kappa)^{\lambda'} \Rightarrow \kappa \underset{g}{\longrightarrow}
(\kappa)^\lambda
$$
\mys



We  say $\kappa \underset{\text{c.u.b.}}{\longrightarrow} (\kappa)^\lambda$ 
if the c.u.b.\ version of the partition property holds for $f$ of the correct type.
\mys


On the other hand, $\kappa   \underset{\text{c.u.b.}}{\longrightarrow} (\kappa)^\lambda$ 
implies $\kappa \rightarrow (\kappa)^\lambda$. 
\mys


So, for $\lambda=\omega \cdot \lambda$, the usual version $\kappa 
\rightarrow (\kappa)^\lambda$ is equivalent to the c.u.b.\ version
$\kappa \underset{\text{c.u.b.}}{\longrightarrow} (\kappa)^\lambda$.
\end{frame} 



\begin{frame}
We officially adopt the c.u.b.\ versions of the partition properties. 
\mys


\begin{itemize}
\item
$\kappa \rightarrow (\kappa)^2$ implies (using c.u.b. version) that $\kappa$ is measurable.

\item
In fact, the $\omega$-c.u.b.\ filter is a normal measure.




\item
This implies $\kappa \rightarrow 
(\kappa)^{<\omega}$.


\item
So, $\kappa$ is measurable iff $\kappa \rightarrow (\kappa)^2$ 
iff $\kappa \rightarrow (\kappa)^{<\omega}$.
\end{itemize}
\mys





So, assuming $V=\lr$, this holds for all regular cardinals $\kappa<\Theta$.
\end{frame}



\subsection{Jonsson and Rowbottom}



\begin{frame}


\frametitle{Jonsson and Rowbottom Cardinals}
\begin{defn}
$\kappa$ is {\color{red}Jonsson} if for every $f \colon \kappa^{<\omega} \to 
\kappa$ there is an $A \subseteq \kappa$, $|A|=\kappa$ such that 
$f''(A^{<\omega}) \neq \kappa$. 
\end{defn}
\mys

Equivalent to the non-existence of a Jonsson algebra. 
\mys



\begin{defn}
$\kappa$ is {\color{red}Rowbottom} if for every $f \colon \kappa^{<\omega} 
\to \delta <\kappa$, there is an $A \subseteq \kappa$, $|A|=\kappa$, with 
$f''(A^{<\omega})$ countable.
\end{defn}
\end{frame}



\begin{frame}
$\zf$ facts.

\begin{itemize}
\item
Measurable $\Rightarrow$ Ramsey $\Rightarrow$ Rowbottom $\Rightarrow$ Jonsson.
\item
A singular Rowbottom cardinal must have cofinality $\omega$.
\item
Jonsson cardinals imply $0^\#$ exists.
\end{itemize}
\mys

Some $\zfc$ facts.

\begin{itemize}
\item
Jonsson and Rowbottom cardinals are equiconsistent.
\item
It is consistent that every Jonsson cardinal is Ramsey.
\item
(Tryba, Woodin)
If $\kappa$ is a regular Jonsson cardinal then every stationary
$S \subseteq \kappa$ reflects. In particular, successors of regulars are not
Jonsson.
\end{itemize}
\end{frame}




\begin{frame}
Using the cardinal structure given by $\ad$ we showed. 
\mys

\begin{thm}[J, L\"{o}we]
Assume $\ad$. then every cardinal below $\aleph_{\omega_1}$ is Jonsson.
\end{thm}
\mys

Woodin then showed using inner-model theory:
\mys


\begin{thm}[Woodin]
Assume $\ad +V=\lr$ (and so forth).  Then every $\kappa <\Theta$ is Jonsson.
\end{thm}


We also believe we can show:


\begin{thm}[J, L\"{o}we]
Assume $\ad$. Every cardinal below $\aleph_{\omega_1}$ of cofinality 
$\omega$ is Rowbottom.
\end{thm}

\end{frame}






\begin{frame}
\begin{ques}
Assume $\ad$. Is every cardinal $\kappa< \Theta$  of cofinality $\omega$
Rowbottom?
\end{ques}
\mys


\begin{ques}
What partition properties do the various cardinals below $\aleph_{\omega_1}$
have assuming $\ad$? What about general cardinals below $\Theta$?
\end{ques}
\end{frame}



\subsection{AD Cardinal structure}

\begin{frame}


\frametitle{Review of cardinal structure}
We assume $\ad$ henceforth. 


\begin{defn}
$\de 1n$ is the supremum of the lengths of the $\bd^1_n$ prewellorderings of the reals.
\end{defn}





\begin{itemize}
\item
$\de 11=\omega_1$, $\de 13=\omega_{\omega+1}$, $\de 15=\omega_{\omega^{\omega^\omega} +1}$,
$\de 17=\omega_{\omega(5)+1}$, etc. ($\omega(n+1)=\omega^{\omega(n)}$). 
\item 
$\de 1{2n+2}=(\de 1{2n+1})^+$.
\item
$\de 1{2n+1}=(\lambda_{2n+1})^+$ where $\cof(\lambda_{2n+1})=\omega$.  The 
$\lambda_{2n+1}$, $\de 1{2n+1}$ are the Suslin cardinals below the projective ordinals. 
\item
The same pattern continues below $\aleph_{\omega_1}$. 
\end{itemize}
\end{frame}

\begin{frame}


\begin{itemize}
\item
$\de 1{2n+1}$ has the strong partition property.
\item
There are $2^{n+1}-1$ many regular cardinals between $\de 1{2n+1}$ and $\de 1{2n+3}$.
\item
All the successor cardinals between $\de 1{2n+1}$ and $\de 1{2n+3}$
have cofinality $> \de 1{2n+1}$.
\end{itemize}
\end{frame}





\begin{frame}
\frametitle{Picture of the Cardinal Structure}


\begin{picture}(300,200)(0,-150)

\put(0,0){\line(1,0){300}}

\put(10,-5){\line(0,1){10}}
\put(7,-12){$\omega$}


\put(20,-5){\line(0,1){10}}
\put(17,-12){{\color{red}$\de 11$}}


\put(35,-5){\line(0,1){10}}
\put(32,-12){\color{blue} $\omega_2$}


\put(50,-5){\line(0,1){10}}
\put(47,-12){$\omega_3$}

\put(65,-5){\line(0,1){10}}
\put(62,-12){$\omega_4$}

\put(85,-12){$\cdots$}


\put(105,-5){\line(0,1){10}}
\put(102,-12){$\lambda_3$}


\put(125,-5){\line(0,1){10}}
\put(122,-12){\color{red}$\de 13$}


\put(145,-5){\line(0,1){10}}
\put(142,-12){\color{blue}$\omega_{\omega+2}$}


\put(185,-5){\line(0,1){10}}
\put(182,-12){\color{blue}$\omega_{\omega\cdot 2 +1}$}



\put(225,-5){\line(0,1){10}}
\put(222,-12){\color{blue}$\omega_{\omega^\omega +1}$}



\put(265,-5){\line(0,1){10}}
\put(262,-12){$\lambda_5$}


\put(285,-5){\line(0,1){10}}
\put(282,-12){\color{red}$\de 15$}




\end{picture}


\end{frame}



\section{Below $\de 13$}


\begin{frame}


\frametitle{Below $\de 13$}

\begin{thm}
All the $\omega_n$ are Jonsson and $\omega_\omega$ is Rowbottom.
\end{thm}
\mys

We give the proofs using methods that will generalize (which we sketch later).
\mys


\begin{defn}
$W_1^m$ is the $m$-fold product of the $\omega$-cofinal, normal measure on 
$\omega_1$. 
\end{defn}
\mys


$W_1^m$ is the measure induced from the weak partition relation on $\de 11$
and the ordered set $\{ 1,2,\dots,m\}$
\mys
\end{frame}



\begin{frame}
\begin{fact}[Martin]
If $\kappa$ has the strong partition property and $\mu$ is a measure on $\kappa$,
then $j_\mu(\kappa)$ is a cardinal. 
\end{fact}
\mys


\begin{fact}
If $\kappa$ has the strong partition property and $\mu$ is a semi-normal measure on $\kappa$,
then $j_\mu(\kappa)$ is a regular cardinal. 
\end{fact}
\mys


\begin{lem}
$j_{W_1^m}(\omega_1)=\omega_{m+1}$.
\end{lem}
\end{frame}








\begin{frame}
Consider the algebra with single generator $\sv_1$ and operation 
$\oplus$. So the terms are 
$t=\underbrace{\sv_1 \oplus \cdots \oplus \sv_1}_m$
\mys


We assign $o(\sv_1)=1$, and $o(s\oplus t)=o(s)+o(t)$, so $o(t)=m$ above.
\mys


We have an ordering $<_t$ associated to the term $t$, namely the order type of $m$. 
\mys


Corresponding to this term we associate two measures.
\mys


So, $\ot(t)=$ order-type of $<_t= m$.
\mys

\end{frame}



\begin{frame}
${\color{red} \wli(t)}$ is the measure defined by the weak partition relation on 
$\de 11$ and $<_t$. So, $\wli(t)=W_1^m$. 
\mys




${\color{red} \sli(t)}$ is the measure on $\omega_{m+1}$ induced from the strong
partition relation on $\de 11$ and the measure $\wli(t)$.
\mys


The measure $\sli(t)$ depends on the way in which we identify 
$(\omega_1)^m$ with $\omega_1$ (there are $(m-1)!$ ways to do this).
\mys



It turns out not to matter, so we take reverse lexicographic order
(first by largest, then next largest, etc.).
\mys


The fact that the ordering doesn't matter is related to the ``global
embedding theorem.''
\end{frame}


\begin{frame}
\frametitle{Types of Functions on $(\omega_1)^n$}

\begin{fact}
If $f \colon (\omega_1)^n \to \on$, then there is a $W_1^n$ measure one set
restricted to which $f$ is ordered by:

$$
f(\alpha_1,\dots,\alpha_n) \leq f(\beta_1,\dots,\beta_n) \text{ iff }
(\alpha_{\pi(1)},\dots, \alpha_{\pi(j)}) 
\leq_{\lex}
(\beta_{\pi(1)},\dots, \beta_{\pi(j)})$$

where $\pi=(\pi(1),\dots,\pi(j))$ ($j \leq n$) is a partial permutation.
\end{fact}
\mys


 
If $f \colon (\omega_1)^n \to \omega_1$, then $\pi(1)$ is maximal. 
If $j=n$ we say $f$ depends on all its arguments.
\end{frame}

\begin{frame}
\begin{fact}
If $f \colon (\omega_1)^n \to \on$, then almost everywhere $f(\vec \alpha)$
either has uniform cofinality $\omega$ or has uniform cofinality $\alpha_{\pi(k)}$
for some $k \leq j$.
\end{fact}
\mys

In the latter case there is a partial permutation $\pi'$
extending $\pi$ which determines the uniform cofinality. 
\mys


\begin{ex}
If $f \colon (\omega_1)^3 \to \omega_1$ has type $\pi=(3,1,2)$, and 
$f(\alpha_1,\alpha_2,\alpha_3)$ has uniform cofinality $\alpha_2$,
then $\pi'=(4,1,3,2)$.
\end{ex}
\mys



The possible types are described by $(\pi,\omega)$, $(\pi,\pi')$,
or $(\pi,s)$ (the latter for continuous functions).
\end{frame}


\begin{frame}


{\color{blue}Convention}: For $f \colon (\omega_1)^n \to \omega_1$, we
write the arguments to $f$ in any order. 
\mys

\begin{defn}
Let $f \colon (\omega_1 )^n \to \omega_1$ be of type $\pi=(n,i_2,\dots,i_n)$. 
For $1 \leq j \leq n$, we define the {\color{red}$j^{\text{th}}$ invariant}, $
f(j)$ (of type $\pi \res j$),
by:

$$
f(j)(\alpha_n, \alpha_{i_2},\dots, \alpha_{i_j})=
\sup \{ f(\alpha_n,\dots, \alpha_{i_n}) \colon \vec \alpha \cong \pi \}.
$$

\noindent
Also,
$$
f^s(\alpha_n,\alpha_{i_2},\dots,\alpha_{i_j})=\sup_{\beta <\alpha_{i_j}}
f(j)(\alpha_n,\alpha_{i_2},\dots, \alpha_{i_{j-1}},\beta)
$$
\end{defn}
\mys
\end{frame}



\begin{frame}
\begin{ex}
Suppose $f \colon (\omega_1)^4 \to \omega_1$ has type $\pi=(4,1,3,2)$. 
Then $$f(3)(\alpha,\beta,\gamma)= \sup_\eta f(\alpha,\eta, \beta, \gamma)$$
$$f(2)(\alpha,\beta)=\sup_{\eta_1<\eta_2} f(\alpha,\eta_1,\eta_2,\beta).$$
\end{ex}
\mys




Given two functions $f,g \colon (\omega_1)^n \to \omega_1$, there is a $W_1^n$
measure one set on which they are ordered as follows. 
\mys


Say $f$ has type $(\pi, \pi')$, $g$ has type $(\sigma,\sigma')$. Say $[f] < [g]$.
\end{frame}


\begin{frame}
Let $j_m \leq n$ be least such that $\pi' \res (j_m+1) \ncong \sigma' \res (j_m+1)$.
\mys


$j_m-1$ is largest $j$ such that $[f(j)]$ could equal $[g(j)]$ for functions of these types.
\mys



Then there is a $j\leq j_m$ such that (almost everywhere) 

$$
f(\alpha_1,\dots,\alpha_n) <g(\beta_1,\dots,\beta_m) \text{ iff }
(\alpha_n,\alpha_{i_2},\dots,\alpha_{i_j}) \leq_\lex 
(\beta_n,\beta_{i_2},\dots,\beta_{i_k}).
$$
\mys




Note that $\pi \res j \cong \sigma\res j$. Here $(n,i_2,\dots,i_j)=
\pi \res j=\sigma \res j$.
\end{frame}



\subsection{Level $2$-Trees}

\begin{frame}


This generalizes immediately  to finitely many functions.
\mys

An arrangement of finitely many functions can be described by a level-$2$
{\color{red} tree of uniform cofinalities} $\sR$. 
\mys


This is a function with domain a finite subtree of $\omega^{<\omega}$ 
satisfying:

\begin{itemize}
\item
$\sR(\emptyset)=(1)$ (the unique permutation of length $1$).
\item
$\sR(i_1,\dots,i_k)$ is either the symbol $\omega$, the symbol $s$,  or a permutation extending 
$\sR(i_1,\dots,i_{k-1})$. 
\item
If $\sR(\vec \imath)=\omega$ or $s$, then $\vec \imath\,$ is maximal in $\dom(\sR)$.
\item
If $\sR(i_1,\dots,i_k)=s$, then $i_k$ is least node extending $(i_1,\dots,i_{k-1})$.

\end{itemize}


\end{frame}


\begin{frame}
\frametitle{Picture of a Level-$2$ Tree}



\begin{picture}(300,200)(-150,-200)

\put(-2.5,-3){$\bullet$ $(1)$}


\put(0,0){\line(-1,-1){50}}
\put(0,0){\line(1,-1){50}}
\put(0,0){\line(0,-1){50}}




\put(-53,-53){$\bullet$}
\put(-2,-53){$\bullet$}
\put(47,-53){$\bullet$}



\put(-50,-50){\line(-2,-3){33}}
\put(-50,-50){\line(0,-1){50}}
\put(-50,-50){\line(2,-3){33}}
\put(-86,-103){$\bullet$}
\put(-53,-103){$\bullet$}
\put(-20,-103){$\bullet$}




\put(50,-50){\line(-2,-3){33}}
\put(50,-50){\line(0,-1){50}}
\put(50,-50){\line(2,-3){33}}

\put(14,-103){$\bullet$}
\put(47,-103){$\bullet$}
\put(80,-103){$\bullet$}




\put(-80,-55){$(2,1)$}
\put(10,-55){$\omega$}
\put(55,-55){$(2,1)$}



\put(-126,-108){$(3,2,1)$}
\put(-73,-98){\tiny $(3,1,2)$}
\put(-29,-113){\small $(3,2,1)$}



\put(-50,-100){\line(-1,-2){25}}
\put(-50,-100){\line(1,-2){25}}
\put(-78,-153){$\bullet$}
\put(-28,-153){$\bullet$}

\put(-98,-163){$(4,1,3,2)$}
\put(-28,-163){$\omega$}


\put(14,-113){$\omega$}
\put(35,-113){$\small (3,1,2)$}
\put(80,-113){$\omega$}


\end{picture}
\end{frame}

\begin{frame}
Given the level-$2$ tree $\sR$, we define an ordering $<_\sR$. 
\mys

$\dom(<_\sR)$ is the set of tuples $(\alpha_1,i_1,\dots, \alpha_k,i_k)$ 
where $(i_1,\dots,i_k)$ is a terminal node of $\dom(\sR)$ and 
$(\alpha_1,\dots,\alpha_k) \cong \sR(i_1,\dots,i_{k-1})$. 
\mys


We say $f \colon \dom(<_\sR) \to \omega_1$ is of type $\sR$ if $f$ is order-preserving,
discontinuous (except for $s$ case), 
and $f(\alpha_1,i_1,\dots, \alpha_k,i_k)$ has uniform cofinality as specified by 
$\sR(i_1,\dots,i_k)$. If $\sR(i_1,\dots,i_k)=s$, then 
$f(\alpha_1,i_1,\dots,\alpha_k,i_k)$ is the supremum of smaller values for limit
$\alpha_k$ and of uniform cofinality $\omega$ otherwise.
\mys

\end{frame}

\begin{frame}
\begin{lem}
Suppose $f \colon \dom(<_\sR)\to \omega_1$ and there is a c.u.b.\ 
$C$ such that $f \res C$ is of type $\sR$ ($f \res C$ means all the $\alpha_j$
are in $C$). Then there is an $f'$ satisfying:

\begin{itemize}
\item
$f$ has type $\sR$.
\item
$f'=f$ almost everywhere (i.e., $f'\res D=f\res D$ for some c.u.b.\ $D$).
\item
$\ran(f') \subseteq \ran(f)$.
\end{itemize}
\end{lem}

\begin{proof}
Define $f'(\alpha_1,i_i,\dots,\alpha_k,i_k)=f(\ell_C(\alpha_1),i_1,\dots,
\ell_C(\alpha_k),i_k)$,

\noindent
where $\ell_C(\alpha)$ is the $\alpha^{\text{th}}$ element of $C$.
\mys

This $f'$ works.
\end{proof}
\end{frame}


\subsection{$\omega_n$ Jonsson}

\begin{frame}



\frametitle{$\omega_n$ is Jonsson}

Fix $F \colon (\omega_n)^{< \omega} \to \omega_n$.
\mys


Let $\sR_1, \sR_2,\dots$ enumerate the level-$2$ trees describing  an arrangement
of functions $f_1,\dots, f_m, f_{m+1} \colon (\omega_1)^{n-1} \to \omega_1$. 
We assume each $f_i$ is of type $(\pi,\omega)$ where $\pi=(n-1,1,2,\dots,n-2)$.
\mys


For each $i$, consider the partition $\sP_i$ of functions 
$f \colon \dom(<_{\sR_i}) \to \omega_1$ of type $\sR_i$ according to whether
$F([f_1],\dots,[f_m])\neq [f_{m+1}]$. 
\mys

Here $f$ induces the functions $f_1,\dots, f_{m+1}$. 
\end{frame}


\begin{frame}
\begin{claim}
On the homogeneous side of each $\sP_i$ the stated property holds.
\end{claim}


\begin{proof}
Suppose $C$ were homogeneous for the contrary side. Let $C'=$ limit points of $C$.
Fix $f \colon \dom(<_{\sR_i}) \to C'$ of type $\sR_i$. Replace $f$ by 
$g$ where $g(\alpha_1,i_1,\dots,\alpha_{n-1},i_{n-1})=f(\cdots)$ 
except when $\vec \imath$ correspond to $f_{m+1}$ in which case we set 
$$g(\alpha_1,i_1,\dots,\alpha_{n-1},i_{n-1})=
N_C(f(\alpha_1,i_1,\dots, \alpha_{n-1},i_{n-1}).$$


Here $N_C(\alpha)$ is the least element of $C$ greater than $\alpha$.
\mys


Then $g$ is also of type $\sR_i$, has range in $C$, and induces 
$f_1,\dots, f_m, g_{m+1}$ with $[g_{m+1}]\neq [f_{m+1}]$.
\end{proof}

\end{frame}

\begin{frame}
Let now $C_i$ be homogeneous for the stated side of $\sP_i$, and let $C=\bigcap_i C_i$.
\mys


Let $A= \{ \alpha <\omega_n \colon \alpha=[f], f\colon (\omega_1)^{n-1} \to C, 
f \text{ of type } (\pi,\omega)\}$.
\mys

Easily $A$ has size $\omega_n$.
Fix $g \colon (\omega_1)^{n-1} \to C$ of type $(\pi,\omega)$. 



\begin{claim}
$[g] \notin F''(A^{<\omega})$.
\end{claim}

\begin{proof}
Suppose $[g]=F([f_1],\dots,[f_m])$. Let $i$ be such that $f_1,\dots,f_m,f_{m+1}=g$
has type $\sR_i$ almost everywhere. 

By the lemma, get $f'_1,\dots,f'_m$, $f'_{m+1}$ in $C$ everywhere. This contradicts the
homogeneity of $C \subseteq C_i$.
\end{proof}
\end{frame}



\subsection{$\omega_\omega$ Rowbottom}

\begin{frame}


\frametitle{$\omega_\omega$ is Rowbottom}

Fix a function $F \colon (\omega_\omega)^{< \omega} \to \omega_n$ for some $n$. 
\mys



Let $\sR_1, \sR_2,\dots$ enumerate the possible level-$2$ trees giving the types of 
finitely many $f_1,\dots, f_k$ where $[f_j] < \omega_\omega$. 
\mys



For each $i$, let $\sbR_i$ denote the type induced by $\sR_i$ by restricting to 
$n-1$ invariants of the $f_j$.
\mys



This is equivalent to restricting the tree $\sR_i$ to nodes of height $\leq n-1$. 
\end{frame}



\begin{frame}
\begin{ex}
Say $n=3$, $k=3$ (i.e., $f_1$, $f_2$, $f_3$), and $\sR_i$  and $\sbR_i$ as shown.
Here $\sR_i$ corresponds to $f_1$, $f_2$, $f_3$ representing $\alpha_1, \alpha_2
<\omega_4$, $\alpha_3<\omega_2$. Also, $[f_1(1)]=[f_2(1)]<[f_3(1)]$ and 
$[f_1(2)]<[f_2(2)]$. 
\end{ex}


\begin{picture}(300,200)(-90,-180)
\setlength{\unitlength}{0.75pt}

\put(-2.5,-3){$\bullet$ $(1)$}


\put(0,0){\line(-1,-1){50}}
\put(0,0){\line(1,-1){50}}
\put(-53,-53){$\bullet$}
\put(47,-53){$\bullet$}
\put(-86,-55){$(2,1)$}
\put(55,-55){$(2,1)$}





\put(-50,-50){\line(-2,-3){33}}
\put(-50,-50){\line(2,-3){33}}
\put(-86,-103){$\bullet$}
\put(-20,-103){$\bullet$}
\put(-132,-108){$(3,1,2)$}
\put(-13,-108){$(3,2,1)$}






\put(50,-50){\line(0,-1){50}}
\put(47,-103){$\bullet$}
\put(47,-113){$\omega$}


\put(-83,-100){\line(0,-1){50}}
\put(-17,-100){\line(0,-1){50}}
\put(-86,-153){$\bullet$}
\put(-20,-153){$\bullet$}
\put(-111,-163){$(4,2,3,1)$}
\put(-44,-163){$(4,2,1,3)$}



\put(197.5,-3){$\bullet$ $(1)$}


\put(200,0){\line(-1,-1){50}}
\put(200,0){\line(1,-1){50}}
\put(147,-53){$\bullet$}
\put(247,-53){$\bullet$}
\put(114,-55){$(2,1)$}
\put(255,-55){$(2,1)$}





\put(150,-50){\line(-2,-3){33}}
\put(150,-50){\line(2,-3){33}}
\put(114,-103){$\bullet$}
\put(180,-103){$\bullet$}
\put(88,-113){$(3,1,2)$}
\put(167,-113){$(3,2,1)$}






\put(250,-50){\line(0,-1){50}}
\put(247,-103){$\bullet$}
\put(247,-113){$\omega$}

\end{picture}
\end{frame}

\begin{frame}

For fixed $\sR_i$ corresponding to $f_1,\dots, f_k$, 
a non-terminal node 
$p=(i_1,i_2,\dots, i_n)$, we define an $\sR_{i,p}$ extending
$\sR_i$. 
\mys


$\sR_{i,p}$ has a copy of the tree $\sR_i$ below the node  $p$ 
copied to below $p$ and put completely above the original copy. 
\mys






This corresponds to adding  new function $g_1,\dots, g_\ell$ of the same type as $f_1,\dot,f_\ell$ 
where we assume $f_1,\dots,f_\ell$ correspond to nodes below $p$. 
Note that $g_a(n-1)=f_b(n-1)$ for $`1 \leq a,b \leq \ell$. 
\end{frame}

\begin{frame}
\begin{ex}
If $j$ of the previous example corresponds to the node labeled $(3,2,1$) 
then we have the following.
\end{ex}


\begin{picture}(300,200)(-150,-180)
\setlength{\unitlength}{0.75pt}

\put(-2.5,-3){$\bullet$ $(1)$}


\put(0,0){\line(-1,-1){50}}
\put(0,0){\line(1,-1){50}}
\put(-53,-53){$\bullet$}
\put(47,-53){$\bullet$}
\put(-86,-55){$(2,1)$}
\put(55,-55){$(2,1)$}





\put(-50,-50){\line(-2,-3){33}}
\put(-50,-50){\line(2,-3){33}}
\put(-86,-103){$\bullet$}
\put(-20,-103){$\bullet$}
\put(-132,-108){$(3,1,2)$}
\put(-13,-103){$(3,2,1)$}






\put(50,-50){\line(0,-1){50}}
\put(47,-103){$\bullet$}
\put(47,-113){$\omega$}


\put(-83,-100){\line(0,-1){50}}
\put(-17,-100){\line(0,-1){50}}
\put(-17,-100){\color{red}\line(1,-1){50}}
\put(-86,-153){$\bullet$}
\put(-20,-153){$\bullet$}
\put(29,-153){\color{red}$\bullet$}
\put(-111,-163){$(4,2,3,1)$}
\put(-44,-163){$(4,2,1,3)$}




\end{picture}
\end{frame}

\begin{frame}


{\color{red}The Partition}: 
We partition $f$ of type $\sR_{i,p}$ according whether 
$$
F([f_1],\dots,[f_\ell],[f_{\ell+1}],\dots,[f_k])=
F([g_1],\dots,[g_\ell],[f_{\ell+1}],\dots,[f_k]).
$$
\mys


Here $f\colon  \dom(<_{\sR_{i,p}}) \to \omega_1$ induces $f_1,\dots, f_k$ and 
$g_1,\dots, g_\ell$.
\mys





\begin{claim}
On the homogeneous side of the partition the stated property holds.
\end{claim}
\end{frame}

\begin{frame}
The claim follows easily from a sliding argument and the following.
\mys


\begin{fact}
For any c.u.b.\ $C\subset \omega_1$ there is an $f \dom(<_{\sR_i}) \to C$ 
such that there are $\theta \geq \omega_{n}$  many $f_\alpha
\colon \dom(<_{\sR_i}) \to C$ such that all pairs $(f_\alpha, f_\beta)$
have the same $n-1$ invariants and are of type $\sR_{i,p}$.
\end{fact}


\begin{proof}
Given $\alpha=[h]$, $h \colon (\omega_1)^{n-1} \to \omega_1$, define 
$f_\alpha$ using 

\begin{equation*}
\begin{split}
& f_\alpha(\alpha_1,i_1,\dots, \alpha_{n-1},i_{n-1},\alpha_n,\dots,i_m)=\\ & \qquad
f(\alpha_1,i_1,\dots, \alpha_{n-1},i_{n-1},h(\alpha_2,\dots,\alpha_{n}),\dots,i_m).
\end{split}
\end{equation*}
\end{proof}




The argument here uses the special types $\pi=(n,1,2,3,\dots,n-1)$
we are considering, but the proof can be made to work for other types. 
\end{frame}

\begin{frame}
From the fact the claim follows since $F$ has range in $\omega_{n-1}<\theta$.
\mys



Let $C \subseteq \omega_1$ be homogeneous for all the partitions. 
\mys


It follows that for $\alpha_1=[f_1],\dots, \alpha_k=[f_k]$ with the $f$'s having range in $C$
of type $\pi$, the value
$F(\alpha_1,\dots,\alpha_k)$ depends only on the type $\sR_i$ and the $n-1$
invariants $f_j(n-1)$. 
\mys


Let $D \subseteq C$ be the set of closure points of $C$
\mys



Fix $f$ of type $\pi$ into $D$. Let $A \subseteq \omega_\omega$ be the set of 
$g$ of type $(m,1,2,\dots,m-1)$ for some $m \geq n$ with range in $C$. 
\mys


Then $|A|=\omega_\omega$ and $F \res A^{<\omega}$ takes only countably many values
(depends only on the type $\sR_i$).
\end{frame}


\section{Below $\de 15$}


\begin{frame}


\frametitle{Below $\de 15$}

Most of the arguments given above for the cardinals below $\de 13$ are general. 
\mys

We need a representation of the cardinals below $\de 15$ which will
allow us to use these arguments.
\mys



Such a notational system was described (and proved) below 
$\de 15$ by J.\ and {\color{red}Khafizov}, and later 
by  J. and {\color{red}L\"{o}we} for the general case $\de 1{2n+1}$. 
\mys

Recall that before we had the algebra generated by a single generator $\sv_1$.
This algebra $\sA_1$ has height $\omega$.
\mys


We now extend this algebra.
\end{frame}


\subsection{The General Algebra}

\begin{frame}


\frametitle{The General Algebra}

Have {\color{red}generators} $\sv_1, \sv_2, \sv_3,\dots, \sv_\beta,\dots$ and 
{\color{red}operations} 
$\oplus$, $\otimes$. 
\mys

Let $\sA_\alpha$ be the free associative, left-distributive algebra over
$\{ \sv_\beta \colon \beta < \alpha\}$. 
\mys
 
For the cardinals below $\de 15$ we only need the first $\omega$ many generators. 
\mys


Let $s$, $t$, etc.\ denote terms in this algebra. 
\mys


For example $t=\sv_1 \oplus (( \sv_1 \oplus \sv_2) \otimes \sv_2) \oplus (\sv_4 \otimes \sv_3)
\in \sA_5$. 
\end{frame}


\begin{frame}
Each term can be viewed as a tree, exactly as in the case of ordinal arithmetic.
\mys


\begin{ex}
For $t=\sv_1 \oplus (( \sv_1 \oplus \sv_2) \otimes \sv_2) \oplus (\sv_4 \otimes \sv_3)$
we have the tree:
\end{ex}


\begin{center}
\xymatrix{&&\bullet \ar[dll]\ar[d]\ar[drr]&& \\
\sv_1 && \sv_2 \ar[dl] \ar[dr] && \sv_3 \ar[d] \\
& \sv_1 && \sv_2 & \sv_4
}
\end{center}
\end{frame}


\begin{frame}
We inductively assign to each generator $\sv_\alpha$ an ordinal 
{\color{red} height} $o(\sv_\alpha)$ and  a {\color{red}measure} 
$g(\sv_\alpha)$ which lives on an {\color{red} order-type} $\ot(\sv_\alpha)$. 
\mys


We then extend these assignments to general terms $t \in \sA$:
$o(t) \in \on$ is the height of $t$, $g(t)$ is the {\color{red} germ}
of $t$ (a collection of measures), and an order-type $\ot(t)$.
\mys

Fix an ordering on the $n$-tuples, say the G\"{o}del ordering
(order first by largest element, then next largest, etc.). 
\mys

We define $o(t)$ first.
\end{frame}


\begin{frame}

\begin{defn}
We define $o(t)$ inductively through the following. 
\begin{itemize}
\item
$o(\sv_1)=1$
\item
$o(s \oplus t)=o(s)+o(t)$
\item
$o(s \otimes t)=o(s) \cdot o(t)$.
\item
$o(\sv_\alpha)= \sup \{ o(t) \colon t \in \sA_\alpha\}$
\end{itemize}
\end{defn}
\mys 

This gives:
$o(\sv_1)=1$, $o(\sv_2)=\omega$, $o(\sv_3)=\omega^\omega$,
$o(\sv_4)=\omega^{\omega^2}, \dots, o(\sv_n)=\omega^{\omega^{n-2}}$.
\mys



For $\alpha \geq \omega$, $o(\sv_\alpha)=\omega^{\omega^\alpha}$.
\end{frame}

\begin{frame}
\begin{defn}
$S_1^n$ is the measure on $\omega_{n+1}$ induced by the strong partition relation on 
$\omega_1$, functions $f \colon (\omega_1)^n \to \omega_1$
of type $\pi=(n,1,2,\dots,n-1)$ and the measure $W_1^n$ on $(\omega_1)^n$.
\end{defn}
\mys



Note that $S_1^1$ is the $\omega$-cofinal normal measure on $\omega_2$.
Also, $S_1^n=\sli(\sv_1 \cdot n)$ (using $(n,1,2,\dots,n-1)$
in identifying $(\omega_1)^n$ with $\omega_1$). 
\end{frame}

\begin{frame}


\begin{defn}
We define $g(\sv_n)$ for $n <\omega$ as follows.

\begin{itemize}
\item
 $g(\sv_1)=$ the principal measure on $1$ (a single point). $\ot(\sv_1)=1$.
\item
$g(\sv_2)=W_1^1$. $\ot(\sv_2)=\omega_1$. 
\item
$g(\sv_3)=S_1^1$. $\ot(\sv_3)=\omega_2$. 
\item
$g(\sv_n)=S_1^{n-2}$. $\ot(\sv_n)=\omega_{n-1}$. 
\end{itemize}
\end{defn}
\end{frame}


\begin{frame}

For $t \in \sA$, let $T_t$ be the finite tree with nodes labeled by generators
corresponding to $t$. Then $g(t)$ is the collection of measures 
$\{ \nu^{i_1,\dots,i_k} \}$ for $(i_1,\dots,i_k) \in T_t$. 
\mys

Let $<_t$ be lexicographic ordering on tuples 
$(i_1,\beta_1,\dots,i_k,\beta_k)$ where $(i_1,\dots,i_k) \in \dom(T_t)$
and $\beta_j <\ot(g(\sv^{\vec \imath}))$. 
\mys

Then $\ot(t)$ is the length of $<_t$. This is obtained from the 
$\ot(\sv^{\vec \imath})$ using $+$, $\cdot$ as in the case of $o(t)$.
\mys


\begin{ex}
For $t=\sv_1 \oplus (( \sv_1 \oplus \sv_2) \otimes \sv_2) \oplus (\sv_4 \otimes \sv_3)$
we have $\ot(t)=1+(1+\omega) \cdot \omega +\omega^{\omega^2} \cdot \omega^\omega=
\omega^{\omega^2} \cdot \omega^\omega$.
\end{ex}
\end{frame}

\begin{frame}
\begin{defn}
{\color{red}$\wli(t)$} is the measure on $\de 13$ induced by the weak partition relation on 
$\de 13$, function $f \colon \dom(<_t) \to \de 13$ continuous (uniform cofinality
$\omega$ at successors), and the measures $g(t)$.
\end{defn}
\mys

$\sli(t)$ is naturally a measure on tuples $\gamma^{\vec \imath}$, $\vec \imath
\in T_t$. 
\mys

\begin{defn}
{\color{red}$\sli(t)$} is the measure on $j_\mu(\de 13)$, $\mu=\sli(t)$, 
induced by the strong partition relation on $\de 13$, functions 
$F \colon \de 13 \to \de 13$ of the correct type, and the measure 
$\sli(t)$ on $\de 13$.
\end{defn}

As before, $\sli(t)$ depends on the bijection between $(\de 13)^n$ and $\de 13$ used.
\end{frame}


\begin{frame}
\begin{thm}
For any $t \in \sA_\omega$ and $\mu=\wli(t)$, $j_\mu(\de 13)=
\aleph_{\omega +o(t)+1}$.
\end{thm}
\mys


So, we represent the successor cardinals $\de 13< \kappa< \de 15$ as 
$\kappa=j_\wli(t)(\de 13)$ for $t \in \sA_\omega$. 
\mys

Recall the successor cardinals $\de 11 <\kappa < \de 13$ were represented as 
$k=j_\wli(y)(\de 11)$ for $t \in \sA_1$. 

\end{frame}

\begin{frame}
This representation has other applications. For example, we can easily read off 
the cofinalities of the successor cardinals. 
\mys



\begin{thm} \label{cofcomp}
Suppose $\de 13< \aleph_{\alpha+1} < \de 15$. Let $\alpha=\omega^{\beta_1}+ \dots +
\omega^{\beta_n}$, where $\omega^\omega> \beta_1  \geq  \dots \geq
\beta_n$, be the normal form for $\alpha$. Then:
\begin{itemize}
\item
If $\beta_n=0$, then $\cof(\kappa)=\de 14=\aleph_{\omega+2}$.
\item
If $\beta_n>0$, and is a successor ordinal, then $\cof(\kappa)
=\aleph_{\omega \cdot 2+1}$.
\item
If $\beta_n >0$ and is a limit ordinal, then $\cof(\kappa)=
\aleph_{\omega^\omega +1}$.
\end{itemize}
\end{thm}
\end{frame}

\subsection{Rowbottom Example}

\begin{frame}



\frametitle{A Rowbottom Example}
The proofs of Jonsson and Rowbottom are now similar to the arguments
given earlier 
below $\de 13$. 
\mys


\begin{ex}
We sketch the proof that $\kappa=\aleph_{\omega^{\omega \cdot 2}}$ is Rowbottom. 
\end{ex}
\mys

$\aleph_{\omega^{\omega\cdot 2}}=\sup_n \aleph_{\omega^\omega \cdot \omega^n}$,
so we consider the terms $$t_n= \sv_3 \otimes \underbrace{\sv_2 \otimes \cdots
\otimes \sv_2}_n.$$
\mys




Analysis of types of functions and the sliding lemmas are proved as before.

\end{frame}


\begin{frame}
If $f \colon \dom(<_{t_n}) \to \de 13$, then we define the 
invariants $f(0)$, $f(1), \dots, f(n)=f$ (note: we regard $f$ as being from $(\omega_1)^n$
to $\de 13$ having uniform cofinality $\omega_2$). 
\mys

$f$ represents $(\alpha_0,\alpha_1,\dots, \alpha_n)$, where 
$\alpha_0=\sup(f)$, $\alpha_1=[f(1)]$, etc. 
\mys

We identify $\vec \alpha$ with an ordinal by ordering first by $\alpha_0$,
then say by $\alpha_1$, etc. 
\mys



We consider $F \colon \de 13  \to \de 13$  of the correct type. We define the invariants 
$F(1),\dots, F(n+1)=F$. 
\mys

$$F(j)(\alpha_0,\dots, \alpha_{j-1})=\sup_{\alpha_j,\dots,\alpha_n} F(\alpha_0,\dots,\alpha_n).$$
\end{frame}

\begin{frame}





We sketch a proof of the key point:
\mys

\begin{lem}
Let $C \subseteq \de 13$ be c.u.b., and $\delta < \aleph_{\omega^{\omega \cdot 2}}$. 
Let $n < \omega$. Then there is an $F \colon \dom(\mu_n) \to C$  
such that there are $\theta \geq \delta$
many $[F']$ for $F' \colon \dom(\mu_{m})\to C$ with $F'(n+1)=F$.
\end{lem}
\end{frame}

\begin{frame}

\begin{proof}
Consider the ordering $<'$ on tuples $(\alpha_0,\dots,\alpha_n,\beta)$
where $\vec \alpha \in \dom(\mu_n)$ and $\beta < h(\alpha_0)$ for some fixed function $h
\colon \de 13 \to \de 13$. 
\mys

Choose $m>n$ and $h$ so that the function $G(\alpha_0,\dots, \alpha_m)=h(\alpha_0)$
satisfies $[G]_{\mu_m} > \delta$. 






Fix $F \colon \dom(<') \to C'$ of uniform cofinality $\omega_2$. . 
\mys


For $\gamma=[H]_{\mu_m}<[G]_{\mu_m}$, set 

$$
F_\gamma(\alpha_0,\dots,\alpha_n)= F'(\alpha_0,\dots,\alpha_n, H(\alpha_0,\dots,\alpha_m)).
$$







\end{proof}
\end{frame}


\subsection{A Cardinal Computation}






\begin{frame}

\frametitle{A Cardinal Computation}

\begin{ex}
We consider the cardinal $\kappa=\aleph_{\omega^{\omega \cdot 2} +\omega^3+1}$.
We identify a description giving this cardinal and the corresponding term in the algebra
$A_\omega$. We sketch a proof that these actually represent this cardinal.
\end{ex}
\mys


Let $t= (\sv_3 \otimes \sv_3) \oplus(\sv_2\oplus \sv_2 \oplus \sv_2)$. 
\mys

So, $o(t)=\omega^{\omega\cdot 2}+ \omega\cdot 3$. 
\mys

Also, $\ot(t)= \omega_2 \cdot \omega_2+ \omega_1 \cdot 3$.
\end{frame}

\begin{frame}

We define a {\color{red} description} relative to the measure sequence 
$$W_3, S_1^4, W_1^3, S_1^5, W_1^2.$$ 


We denote elements of these measure spaces by: $f$, $h_1$, $(\eta_1,\eta_2,\eta_3)$, 
$h_2$, $(\eta_4,\eta_5)= f, \vec h$. 
\mys



$d$ is the object which defines the ordinal in the iterated ultrapower 
assigning to $f$, $h_1, \dots$, the ordinal $(d; f; h_1, (\eta_1,\eta_2,\eta_3),
h_2, (\eta_4,\eta_5)) <\omega_3$ which is represented w.r.t.\ $W_1^2$ by 
$(\alpha,\beta) \mapsto (d;f;\vec h;(\alpha,\beta))$ where:



$$d(f;\vec h;\alpha,\beta)=h_1^s(\eta_2, \eta_3, h_3^s(2)(\eta_5,\alpha),\beta).$$
\end{frame}

\begin{frame}
Let $\mu=\sli(t)$. So, $\mu$ is a measure on $\de 13$. 
\mys

Let $\kappa_1= j_\mu(\de 13)$. So, $\kappa_1 \geq \kappa$. 
\mys


Let $\kappa_2= (d;f;\vec h)$ be the ordinal represented by the description. From the 
description analysis, $\kappa_2 \leq \kappa$. 
\mys

To finish, it suffices to show that $\kappa_1 \leq \kappa_2$. 
It then follows that 
$\kappa=j_\mu(\de 13)=(d;f;\vec h)$.
\end{frame}


\begin{frame}
We define an embedding $\pi \colon j_\mu(\de 13) \to (d;W_3; S_1^4,W_1^3,S_1^5,W_1^2)$. 
For $F \colon \de 13 \to \de 13$ representing $[F]_\mu$, let $\pi([F])$ be represented in the iterated ultrapower
by:

$$
\pi([F])(f;\vec h)=F([\theta]),$$

where $\theta \colon <_t \to \de13$ is defined as follows. 
\mys

For $\delta_1<\delta_2 <\delta_3<\omega_1$, let $$\theta(\delta_1,\delta_2,\delta_3)= 
f([(\alpha,\beta) \mapsto h_1(\eta_2,\eta_3,h_3(\eta_4,\delta_1,\delta_2,\delta_3), \beta)]).$$


For $\rho_1=[\ell_1]<[\ell_2]=\rho_2 <\omega_2$, let $$\theta(\rho_1,\rho_2)= 
f([(\alpha,\beta)\mapsto h_1(\eta_1, \ell_1(\alpha),\ell_2(\alpha),\beta)]).$$ 










\end{frame}


















\end{document}