Aronszajn tree

A $\kappa$-tree (http://planetmath.org/TreeSetTheoretic) $T$ for which for all 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.