Suslin hypothesis

It is known that every separable dense complete linear order without endpoints is isomorphic to the real line. The question of whether "separable" can be replaced by the weaker "countable chain condition" turned out to be independent of ZFC.

The Suslin hypothesis, also called the Souslin hypothesis, is the assertion that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line.

Equivalently, it is the assertion that every tree of height ω₁ either has a branch of length ω₁ or an anti-chain of cardinality ω₁. The Generalized Suslin Hypothesis asserts that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.

The Suslin hypothesis is independent of ZFC, and is independent of both the Generalized Continuum Hypothesis and of the negation of the Continuum Hypothesis. However, Martin's Axiom, when combined with the negation of the Continuum Hypothesis, implies the Suslin Hypothesis. It is not known whether the Generalized Suslin Hypothesis is consistent with the Generalized Continuum Hypothesis; however, since the combination impies the negation of the square principle at a singular strong limit cardinal--in fact, at all singular cardinals and all regular successor cardinals--it implies that the axiom of determinacy holds in L(R) and is believed to imply the existence of an inner model with a superstrong cardinal.