Aronszajn treeAronszajn2002-07-27 16:58:012006-06-24 09:59:00DefinitionAronszajn tree$\kappa$-Aronszajn treetree property% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
%\PMlinkescapeword{theory}\PMlinkescapeword{tree}
\PMlinkescapeword{trees}
A $\kappa$-\PMlinkname{tree}{TreeSetTheoretic} $T$ for which $|T_\alpha|<\kappa$ for all $\alpha<\kappa$ and which has no cofinal branches is called a \emph{$\kappa$-Aronszajn tree}. If $\kappa=\omega_1$ then it is referred to simply as an Aronszajn tree.
If there are no $\kappa$-Aronszajn trees for some $\kappa$ then we say $\kappa$ has the \emph{tree property}. $\omega$ has the tree property, but no singular cardinal has the tree property.