• Source: 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

Kata Kunci Pencarian:

• Daftar masalah matematika yang belum terpecahkan
• Suslin tree
• Suslin's problem
• Aronszajn tree
• Tree (set theory)
• Diamond principle
• Mikhail Suslin
• Kurepa tree
• List of unsolved problems in mathematics
• Suslin algebra
• Suslin representation