Suslin tree

In mathematics, a Suslin tree is a tree of height ω₁ such that every branch and every antichain is at most countable. (An antichain is a set of elements such that any two are incomparable.) They are named after Mikhail Yakovlevich Suslin.

Every Suslin tree is an Aronszajn tree.

The existence of a Suslin tree is logically 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.

See also

• Kurepa tree
• Suslin's 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 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