# Kurepa tree

In set theory, a Kurepa tree is a tree (T, <) of height $\omega_1$, each of whose levels is at most countable, and has at least $\aleph_2$ many branches. It was named after Yugoslav mathematician Đuro Kurepa. The existence of a Kurepa tree (known as the Kurepa hypothesis) is consistent with the axioms of ZFC: As Solovay showed, there are Kurepa trees in Gödel's constructible universe. On the other hand, as Silver proved in 1971, if a strongly inaccessible cardinal is Lévy collapsed to $\omega_2$ then, in the resulting model, there are no Kurepa trees.