Kurepa tree

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a Kurepa tree is a <a href="/facts/Tree_(set_theory)/5zkqfoz3">tree</a> (T, <) of height <a href="/facts/First_uncountable_ordinal/Quu6AMEy">ω1</a>, each of whose levels is <a href="/facts/Countable/Wj4DmWPv">countable</a>, and has at least <a href="/facts/Aleph_number/kwe9I4v5">ℵ2</a> many branches. This concept was introduced by <a href="/facts/%25C4%2590uro_Kurepa/AkJIcHJf">Kurepa</a> (1935). The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is <a href="/facts/Relative_consistency/IHkdnA9M">consistent</a> with the axioms of <a href="/facts/ZFC/UYIsPbXw">ZFC</a>: <a href="/facts/Robert_M._Solovay/75zxRfT4">Solovay</a> showed in unpublished work that there are Kurepa trees in <a href="/facts/Kurt_G%25C3%25B6del/P14DqKqK">Gödel</a>'s <a href="/facts/Constructible_universe/pxU0cmL3">constructible universe</a> (Jech 1971). More precisely, the existence of Kurepa trees follows from the <a href="/facts/Diamond_principle/e2sVSVjl">diamond plus principle</a>, which holds in the constructible universe. On the other hand, <a href="/facts/Jack_Silver/wzrk5XVR">Silver</a> (1971) showed that if a <a href="/facts/Strongly_inaccessible_cardinal/EUydsllM">strongly inaccessible cardinal</a> is <a href="/facts/List_of_forcing_notions/WKNzD87w">Lévy collapsed</a> 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.
A Kurepa tree with fewer than 2ℵ1 branches is known as a <a href="/facts/Jech%25E2%2580%2593Kunen_tree/35jFmE0H">Jech–Kunen tree</a>.
More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but at most |α| elements of each infinite level α<κ, and the Kurepa hypothesis for κ is the statement that there is a κ-Kurepa tree. Sometimes the tree is also assumed to be binary. The existence of a binary κ-Kurepa tree is equivalent to the existence of a Kurepa family: a set of more than κ subsets of κ such that their intersections with any infinite <a href="/facts/Ordinal_number/GelFLjJt">ordinal</a> α<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an <a href="/facts/Ineffable_cardinal/wtVdudkc">ineffable cardinal</a>, and conversely <a href="/facts/Ronald_Jensen/Q5PxW1Mr">Jensen</a> showed that in the constructible universe for any uncountable <a href="/facts/Regular_cardinal/wrGvPz2c">regular cardinal</a> κ there is a κ-Kurepa tree unless κ is ineffable.

Kurepa tree open-in-new

Kurepa tree