Suslin tree

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.

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. harvtxt|Jensen|1972 showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ.

Martin's axiom MA(&alefsym;₁) implies that there are no Suslin trees.

References

*Thomas Jech, "Set Theory", 3rd millennium ed., 2003, Springer Monographs in Mathematics,Springer, ISBN 3-540-44085-2
*citation|id=MR|0309729
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 erratum, ibid. 4 (1972), 443.

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