Suslin Tree

In set theory, a Suslin tree is a tree of uncountable height, specifically height aleph one, in which every branch and every level is countable, named after the Russian mathematician Mikhail Suslin. The existence of a Suslin tree is equivalent to a negative answer to Suslin's problem, which asks whether a certain characterization of the real number line by its order properties can be weakened in a particular way and still uniquely describe the real line, and whether such a tree exists cannot be decided by the standard axioms of set theory alone. Under the axiom of constructibility a Suslin tree does exist, while under an axiom called Martin's axiom together with the negation of the continuum hypothesis no Suslin tree exists, making the question of Suslin trees one of the classic examples of a mathematical statement that is independent of the standard axioms.