Kurepa tree

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In set theory, a Kurepa tree is a tree (T, <) of height \omega_1, each of whose levels is at most countable, and has at least \aleph_2 many branches. This concept was introduced by Kurepa (1935). The existence of a Kurepa tree (known as the Kurepa hypothesis) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe. On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévy collapsed to \omega_2 then, in the resulting model, there are no Kurepa trees.

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  • Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2. 
  • Kurepa, G. (1935), "Ensembles ordonnés et ramifiés", Publ. math. Univ. Belgrade 4: 1–138 
  • Silver, Jack (1971), "The independence of Kurepa's conjecture and two-cardinal conjectures in model theory", Axiomatic Set Theory, Proc. Sympos. Pure Math. XIII, Providence, R.I.: Amer. Math. Soc., pp. 383–390, MR 0277379