In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
, 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, e.g., including only woody plants with secondary growth, only ...
(''T'', <) of height
ω1, each of whose levels is
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, 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
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
with the axioms of
ZFC:
Solovay showed in unpublished work that there are Kurepa trees in
Gödel's
constructible universe . 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 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
*
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
References
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Trees (set theory)
Independence results
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