Suslin tree
In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most 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.
See also
[edit]- Glossary of set theory
- Kurepa tree
- List of statements independent of ZFC
- List of unsolved problems in set theory
- Suslin's problem
References
[edit]- 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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