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 {{harvtxt|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. {{harvtxt|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
References
{{refbegin}}
- Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, {{isbn|3-540-44085-2}}
- {{citation
|last=Jensen|first= R. Björn
|title=The fine structure of the constructible hierarchy.
|journal=Ann. Math. Logic|volume= 4 |year=1972|pages= 229–308
|doi=10.1016/0003-4843(72)90001-0
|mr=0309729
|issue=3 |doi-access=free}} erratum, ibid. 4 (1972), 443.
- {{citation | last=Kunen | first=Kenneth | authorlink=Kenneth Kunen | title=Set theory | zbl=1262.03001 | series=Studies in Logic | volume=34 | location=London | publisher=College Publications | isbn=978-1-84890-050-9 | year=2011 }}
- {{citation|last=Kurepa|first=G.|year=1935|title=Ensembles ordonnés et ramifiés|journal=Publ. Math. Univ. Belgrade |volume=4|pages=1–138|url=http://elibrary.matf.bg.ac.rs/handle/123456789/326 | jfm=61.0980.01 | zbl=0014.39401 }}
{{refend}}
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