Kurepa tree

In set theory, a Kurepa tree is a tree (T, <) of height ω₁, each of whose levels is at most countable, and has at least ℵ₂ many branches. This concept was introduced by Kurepa (1935). The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe (Jech 1971). More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévy collapsed to ω₂ 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.