In mathematics, a Suslin tree is a tree of height ω1 such that
every branch and every antichain is at most 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.
Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslintree is an Aronszajn tree. More...
that the axiom of constructibility (V = L) implies the existence of a Suslintree. The diamond principle ◊ says that there exists a ◊-sequence, a family...
collapsing ℵ1, and results in a tree with exactly ℵ1 branches. Aronszajn treeSuslintree Jech, Thomas J. (1971), "Trees", Journal of Symbolic Logic, 36:...
Yakovlevich Suslin. The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslintrees or Suslin lines...
Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslintree? If ℵω is a strong limit cardinal, is 2 ℵ ω < ℵ ω 1 {\displaystyle 2^{\aleph...
In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More...
homogeneously Suslin if it is the projection of a homogeneous tree. S {\displaystyle S} is said to be κ {\displaystyle \kappa } -homogeneously Suslin if it is...
Tennenbaum (1971) in their construction of a model of set theory with no Suslintree. They also showed that iterated forcing can construct models where Martin's...
no ℵ2-Suslintrees. Laver proved that the perfect subtree version of the Halpern–Läuchli theorem holds for the product of infinitely many trees. This...
continuous image of a Borel set in a Polish space. A is a Suslin set, the image of the Suslin operation. There is a Polish space Y {\displaystyle Y} and...
Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive set theory Analytic...
A Suslin cardinal is a cardinal λ such that there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. 4. The Suslin hypothesis...
nonexistence of Suslin lines. Ronald Jensen proved that CH does not imply the existence of a Suslin line. Existence of Kurepa trees is independent of...
ben Isaac ha-Levi Jacob ben Judah Landau Samuel ben Natronai Alexander Suslin Jacob Weil Isaac ben Asher ha-Levi Simha of Speyer Isaac Asir HaTikvah England...
ben Isaac ha-Levi Jacob ben Judah Landau Samuel ben Natronai Alexander Suslin Jacob Weil Isaac ben Asher ha-Levi Simha of Speyer Isaac Asir HaTikvah England...
supports was introduced by Solovay and Tennenbaum to show the consistency of Suslin's hypothesis. Easton introduced another type of iterated forcing to determine...
prime ideal theorem Ultrafilter Ultrafilter lemma Tree (set theory) Tree (descriptive set theory) Suslin's problem Absorption law Prewellordering Stone duality...
gave the first published proof of the consistency of the existence of a Suslin line. With Karel Prikry, he introduced the notion of precipitous ideal....
uncountable subset of the Real numbers with the usual ordering. Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms...
Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg Cantor Paul...
scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases...
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