Suslin tree

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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.) They are named after Mikhail Yakovlevich Suslin.

Every Suslin tree is an Aronszajn tree.

The existence of a Suslin tree is logically 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.

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