Suslin tree
Encyclopedia
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.)
Every Suslin tree is an Aronszajn tree
.
The existence of a Suslin tree is undecidable in ZFC, and is equivalent to the existence of a Suslin line. 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 ω+-Suslin tree. 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.
Tree (set theory)
In set theory, a tree is a partially ordered set In set theory, a tree is a partially ordered set (poset) In set theory, a tree is a partially ordered set (poset) (T, In set theory, a tree is a partially ordered set (poset) (T, ...
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.)
Every Suslin tree is an Aronszajn tree
Aronszajn tree
In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree...
.
The existence of a Suslin tree is undecidable in ZFC, and is equivalent to the existence of a Suslin line. The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
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 ω+-Suslin tree. 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.