In set theory, a Kurepa tree is a tree of height , each of whose levels is countable, which has at least many branches. This concept was introduced by Kurepa. The existence of a Kurepa tree is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe. More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver showed that if a strongly inaccessible cardinal is Lévy collapsed to then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe.