In mathematics, a Suslin tree is a

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

of height ω₁ such that every branch and every

antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its w ...

is at most

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

. They are named after

Mikhail Yakovlevich Suslin Mikhail Yakovlevich Suslin (russian: Михаи́л Я́ковлевич Су́слин; , November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fiel ...

. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...

of ZFC, and is equivalent to the existence of a Suslin line (shown by ) or a

Suslin algebra In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin. The existence of Suslin algebras is independe ...

. The

diamond principle In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe () and that implies the continuum hypothesis. Jensen extracted ...

, a consequence of

V=L The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructib ...

, implies that there is a Suslin tree, and

Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consi ...

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. showed that if

then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the

Generalized Continuum Hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent t ...

implies the existence of an ℵ₂-Suslin tree, is a longstanding open problem.

References

Thomas Jech Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years. Life He was educated at Charles University (his advisor was Petr Vopěnka) and from ...

, ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics,Springer, * erratum, ibid. 4 (1972), 443. * * Trees (set theory) Independence results {{settheory-stub

See also

References