In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, a Kurepa tree is a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that ar ...
(''T'', <) of height
ω1, each of whose levels is at most countable, and has at least
ℵ2 many branches. This concept was introduced by . 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Solovay showed in unpublished work that there are Kurepa trees in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
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It ...
. More precisely, the existence of Kurepa trees follows from the
diamond plus principle, which holds in the constructible universe. On the other hand, showed that if a
strongly inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of f ...
is
Lévy collapsed to ω
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
2 is inaccessible in the constructible universe.
A Kurepa tree with fewer than 2
ℵ1 branches is known as a
Jech–Kunen tree.
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 ordinal α<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an
ineffable cardinal, and conversely Jensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unless κ is ineffable.
Specializing a Kurepa tree
A Kurepa tree can be "killed" by
forcing the existence of a function whose value on any non-root node is an ordinal less than the rank of the node, such that whenever three nodes, one of which is a lower bound for the other two, are mapped to the same ordinal, then the three nodes are comparable. This can be done without
collapsing ℵ
1, and results in a tree with exactly ℵ
1 branches.
See also
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Aronszajn tree In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree o ...
*
Suslin tree In mathematics, a Suslin tree is a tree of height ω1 such that
every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin.
Every Suslin tree is an Aronszajn tree.
The existence of a Suslin tree is in ...
References
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Trees (set theory)
Independence results
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