---
tag: EW1NAD47PM
kind: concept
name: Kurepa tree
aliases: [kurepa-conjecture, kurepa-family, kurepa-hypothesis, kurepas-hypothesis]
sources:
  - wikipedia: en:8267820@r1345895149
    locator: Kurepa tree
    note: adopted
    mention: wikipedia://en/8267820
    sections: [Summary]
same_as:
  wikidata: Q5935117
  wikipedia: Kurepa tree
inventory: [G8TRED27VZ]
---

## 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 [[$\aleph_2$|0V2H5PGJ9B]] 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Kurepa originally conjectured that such trees do not exist) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe (Jech 1971).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 ω<sub>2</sub> is inaccessible in the constructible universe.