# Aronszajn tree

A $\kappa$-tree (http://planetmath.org/TreeSetTheoretic) $T$ for which $|T_{\alpha}|<\kappa$ for all $\alpha<\kappa$ and which has no cofinal branches is called a $\kappa$-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 tree property. $\omega$ has the tree property, but no singular cardinal has the tree property.

Title Aronszajn tree AronszajnTree 2013-03-22 12:52:34 2013-03-22 12:52:34 Henry (455) Henry (455) 10 Henry (455) Definition msc 03E05 msc 05C05 TreeSetTheoretic Antichain SuslinTree WeaklyCompactCardinalsAndTheTreeProperty Aronszajn tree $\kappa$-Aronszajn tree tree property