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.

See also

• Glossary of set theory
• Kurepa tree
• List of statements independent of ZFC
• List of unsolved problems in set theory
• Suslin's problem

References

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer,
• erratum, ibid. 4 (1972), 443.