Aronszajn Tree

In set theory, an Aronszajn tree is a tree of uncountable height that has no uncountable branch and no uncountable level; every Suslin tree, for example, is an Aronszajn tree. More generally, for a cardinal number kappa, a kappa-Aronszajn tree is a tree of height kappa in which every level has size less than kappa and every branch has height less than kappa, so that the ordinary Aronszajn trees are exactly the aleph-one-Aronszajn trees. They are named for Nachman Aronszajn, who constructed one in 1934, a construction later described by Kurepa in 1935. A cardinal for which no kappa-Aronszajn tree exists is said to have the tree property.