Suslin's problem

Independent of the axiomsFoundations of mathematics

Statement

Let $(L, <)$ be a non-empty, dense, complete linear order without endpoints such that every collection of pairwise disjoint non-empty open intervals in $L$ is countable (the countable chain condition, ccc). Must $(L, <)$ be order-isomorphic to the real line $(\mathbb{R}, <)$ ?

Thomas Jech (1967) and Stanley Tennenbaum (1968) constructed forcing models containing Suslin lines (refuting Suslin's Hypothesis), Ronald Jensen (1968) proved that Suslin lines exist in Gödel's constructible universe $L$ via the diamond principle $\diamondsuit$ , and Robert Solovay and Tennenbaum (1971) invented iterated forcing to prove that Suslin's Hypothesis holds under $\text{MA} + \neg\text{CH}$ .

In 1972 Ronald Jensen proved that the Generalized Continuum Hypothesis (GCH) is also compatible with Suslin's Hypothesis by introducing morasses and fine-structure forcing, showing that CH alone does not decide the existence of Suslin trees.

References

Robert M. Solovay, Stanley Tennenbaum (1971). Iterated Cohen extensions and Souslin's problem · DOI:10.2307/1970745
Ronald B. Jensen (1972). The fine structure of the constructible hierarchy · DOI:10.1016/0003-4843(72)90001-0
Thomas Jech (2003). Set Theory

History of the problem

1920Mikhail Suslin's problem appears in the first volume of Fundamenta Mathematicae, asking whether every complete dense unbounded linear order satisfying the countable chain condition is separable.
1943Edwin W. Miller proves that a counterexample to Suslin's problem (a Suslin line) exists if and only if there exists a Suslin tree: a tree of height $\omega_1$ in which every branch and every antichain is countable.
1967Thomas Jech and, independently in 1968, Stanley Tennenbaum use forcing to construct models of ZFC in which a Suslin tree exists, proving that Suslin's Hypothesis cannot be proved from ZFC.
1968Ronald Jensen extracts the combinatorial principle $\diamondsuit$ from Gödel's constructible universe $L$ and shows that $V = L$ implies the existence of a Suslin tree.Kurt Gödel
1971Robert Solovay and Stanley Tennenbaum develop iterated Cohen extensions to destroy all Suslin trees without collapsing $\omega_1$ , proving that Suslin's Hypothesis is consistent with ZFC.Paul Cohen