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In [[set theory]], a '''Kurepa tree''' is a [[tree (set theory)|tree]] (''T'', <) of height <math>\omega_1</math>, each of whose levels is at most countable, and has at least <math>\aleph_2</math> many branches. It was named after [[Yugoslavia|Yugoslav]] mathematician [[Đuro Kurepa]]. The existence of a Kurepa tree (known as the '''Kurepa hypothesis''') is consistent with the axioms of [[ZFC]]: As [[Robert M. Solovay|Solovay]] showed, there are Kurepa trees in [[Kurt Gödel|Gödel]]'s [[constructible universe]]. On the other hand, as [[Jack Silver|Silver]] proved in 1971, if a [[strongly inaccessible cardinal]] is [[List of forcing notions#Levy collapsing|Lévy collapsed]] to <math>\omega_2</math> then, in the resulting model, there are no Kurepa trees. | |||
== See also == | |||
* [[Aronszajn tree]] | |||
* [[Suslin tree]] | |||
== References == | |||
* {{cite book|author=Jech, Thomas|title=Set Theory|publisher=Springer-Verlag|year=2002|isbn=3-540-44085-2}} | |||
[[Category:Trees (set theory)]] | |||
{{settheory-stub}} | |||
Revision as of 11:59, 16 March 2013
In set theory, a Kurepa tree is a tree (T, <) of height , each of whose levels is at most countable, and has at least many branches. It was named after Yugoslav mathematician Đuro Kurepa. The existence of a Kurepa tree (known as the Kurepa hypothesis) is consistent with the axioms of ZFC: As Solovay showed, there are Kurepa trees in Gödel's constructible universe. On the other hand, as Silver proved in 1971, if a strongly inaccessible cardinal is Lévy collapsed to then, in the resulting model, there are no Kurepa trees.
See also
References
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