Kurepa tree

Summary

In set theory, a Kurepa tree is a tree $(T, <)$ of height $\omega_1$, each of whose levels is countable, which has at least $\aleph_2$ many branches. This concept was introduced by (txt). The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that such trees do not exist) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe (Jech 1971).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, (txt) showed that if a strongly inaccessible cardinal is Lévy collapsed to $\omega_2$ 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.