Suslin tree

In mathematics, a Suslin tree is a tree of height ω₁ such that every branch and every antichain is countable. They are named after Mikhail Yakovlevich Suslin.

Every Suslin tree is an Aronszajn tree.

The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by Kurepa (1935)) or a Suslin algebra. The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ₁) implies that there are no Suslin trees.

More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin tree is the same as a ω₁-Suslin tree. Jensen (1972) showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesis implies the existence of an ℵ₂-Suslin tree, is a longstanding open problem.

References

Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2
Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy.", Ann. Math. Logic, 4 (3): 229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729 erratum, ibid. 4 (1972), 443.
Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001
Kurepa, G. (1935), "Ensembles ordonnés et ramifiés", Publ. Math. Univ. Belgrade, 4: 1–138, JFM 61.0980.01, Zbl 0014.39401

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