%Updated 3/01/00
%\undefine \eth
\input amstex
\NoBlackBoxes
\documentstyle{amsppt}
\settabs\+\hskip.1in&\hskip.55in& \cr
\def\psc#1#2{{\tabalign & #1 & \vtop{\hsize=4.325in\parindent 0pt
#2\strut }&\cr}\vskip2pt}
\def\ps{\hangindent .3in\hangafter 1}
\def\iput#1{}
\def\icopy#1{#1}
\font \sc cmcsc10
\font\Large=cmr8 scaled \magstep2
\magnification=\magstep1
\loadmsbm
\loadbold
\loadeusb
\def\ac#1{#1}
\pageheight{8.75truein}
\pagewidth{6.5truein}
\centerline{Additions to Part I: Numerical List of Forms}
\smallskip
\item{}{\bf [0 AR]} For all cardinals $m$ and $n$, if $2m=2n$,
then $m=n$. Sierpi\'nski \cite{1922}. (Brought to our attention
by W. Felscher.)
\medskip
\item{}{\bf [0 AS]} The Modified Ascoli Theorem. For any set $F$ of
continuous functions from $\Bbb R$ to $\Bbb R$, the following
conditions are equivalent:
\itemitem{(1)} Each sequence in $F$ has a subsequence that
converges continuously to some continuous function (not
necessarily in $F$).
\itemitem{(2)} (a) For each countable subset $G$ of $F$ and each
$x\in {\Bbb R}$, the set $G(x) = \{ g(x) : g\in G\}$ is bounded,
and
\itemitem{} (b) Each countable subset of $F$ is equicontinuous.\par
\ac{Rhineghost} \cite{2000} and note 10 \iput{Ascoli Theorem}
\medskip
Add a reference to Keremedis in form [1 DG].\newline
\item{}{\bf [1 DG]} Vector Space Kinna-Wagner Principle: For
every family $V = \{V_i : i \in K\}$ of non-trivial vector spaces
there is a family $F = \{F_i : i\in K\}$ such that for each
$i\in K$, $F_i$ is a non-empty, independent subset of $V_i$.
\ac{Keremedis} \cite{1999e} and note 127.
\medskip
\item{}{\bf [14 DA]} Keimel's Representation Theorem. Any hyperarchimedian
$l$-group can be imbedded into a Boolean product of a family of simple,
abelean $l$-groups. \ac{Gluschankof} \cite{1995} and note 151.
\iput{hyperarchimedian $l$-group}
\medskip
\item{}{\bf [14 DB]} If $G$ is an \icopy{achimedian $l$-group} with a weak
unit, then $G$ can be imbedded into $D(X)$ where $X$ is a locally compact
topological space and $D(X)$ is the set of continuous maps
$f: X\to \Bbb R \cup \{-\infty, \infty\}$ such that the inverse image
of $\Bbb R$ is dense in $X$. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DC]} \icopy{Clifford's Theorem}. An \icopy{abelean
$l$-group} is isomorphic to a subdirect product of totally ordered
abelian groups. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DD]} \icopy{Lorenzen's Theorem}. A representable
$l$-group is isomorphic to a subdirect product of totally ordered
groups. \ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DE]} \icopy{Holland's Theorem}. Any $l$-group is
isomorphic to a subdirect product of transitive $l$-groups.
\ac{Gluschankof} \cite{1995} and note 151.
\medskip
\item{}{\bf [14 DF]} Any $l$-group can be imbedded into the $l$-preserving
permutations of a totally ordered set. \ac{Gluschankof} \cite{1995} and
note 151.
\medskip
\item{}{\bf [14 DG]} In any hyperarchimedian $l$-group any proper
$l$-ideal can be extended to a prime one. \ac{Gluschankof} \cite{1989}
and note 151.
\medskip
\item{}{\bf [43 AI]} Every countably well-founded relation is
well-founded. (A relation $R$ is {\it (countably) well-founded}
if every (countable) set in the domain of $R$ has an $R$-minimal
element.) \ac{Diener} \cite{1994}.
\medskip
\item{}{\bf [94 Q]} The Classical Ascoli Theorem. For any set $F$ of
continuous functions from $\Bbb R$ to $\Bbb R$, the following
conditions are equivalent:
\itemitem{(1)} Each sequence in $F$ has a subsequence that
converges continuously to some continuous function (not
necessarily in $F$).
\itemitem{(2)} (a) For each $x\in {\Bbb R}$, the set $F(x) = \{
f(x) : f\in F\}$ is bounded, and
\itemitem{} (b) $F$ is equicontinuous.\par
\ac{Rhineghost} \cite{2000} and note 10 \iput{Ascoli Theorem}
\medskip
\item{}{\bf[94 R]} Weak Determinateness. If $A$ is a subset of
${\Bbb N}^{\Bbb N}$ with the property that\newline
\centerline{
$(\forall a\in A)(\forall x\in {\Bbb N}^{\Bbb N})
\left( x_n = a_n\hbox{ for } n=0 \hbox{ and } n \hbox{ odd } \to
x\in A\right)$}
\newline\iput{determinateness}
Then in the game $G(A)$ one of the two players has a winning
strategy. \ac{Rhineghost} \cite{2000} and note 153.
\noindent{\bf FORM 408.} If $\{f_i: i\in I\}$ is a family of functions
such that for each $i\in I$, $f_i\subseteq E\times W$, where $E$ and
$W$ are non-empty sets, and $\Cal B$ is a filter base on $I$ such that
\item {1.} For all $B\in\Cal B$ and all finite $F\subseteq E$ there
is an $i\in I$ such that $f_i$ is defined on $F$, and
\item {2.} For all $B \in\Cal B$ and all finite $F\subseteq E$ there
exist at most finitely many functions on $F$ which are restrictions
of the functions $f_i$ with $i\in I$,
\noindent
then there is a function $f$ with domain $E$ such that for each
finite $F\subseteq E$ and each $B\in\Cal B$ there is an $i\in I$
such that $f|F = f_i|F$. \ac{Felscher} \cite{1964}.
\medskip
\noindent{\bf FORM 409.} Suppose $(G,\Gamma)$ is a locally finite
graph (i.e. $G$ is a non-empty set and $\Gamma$ is a function from
$G$ to $\Cal P(G)$ such that for each $x\in G$, $\Gamma(x)$ and
$\Gamma^{-1}\{x\}$ are finite), $K$ is a finite set of integers, and
$T$ is a function mapping subsets of $K$ into subsets of $K$.
If for each finite subgraph $(A,\Gamma_A)$ there is a function $\phi_A$
such that for each $x\in A$, $\phi_A(x)\in T(\phi_A[\Gamma_A(x)])$, then
there is a function $\phi$ such that for all $x\in G$, $\phi(x)\in
T(\phi[\Gamma(x)])$. \ac{Foster} \cite{1964}.
\medskip
\noindent{\bf FORM 410.} RC (Reflexive Compactness): The closed
unit ball of a reflexive normed space is compact for the weak
topology. \ac{Delhomm\'e/Morillon} \cite{2000} and note 23.
\medskip
\noindent{\bf FORM 411.} RCuc (Reflexive Compactness for uniformly
convex Banach spaces): The closed unit ball of a uniformly convex
Banach space is compact for the weak topology.
\ac{Delhomm\'e/Morillon}
\cite{2000} and note 23.
\medskip
\noindent{\bf FORM 412.} RCh (Reflexive Compactness for Hilbert
spaces): The closed unit ball of a Hilbert space is compact for the
weak topology. \ac{Delhomm\'e/Morillon} \cite{2000} and note 23.
\bye