The Suslin hypothesis is independent of ZFC, in both directions

A Suslin line is a dense complete linear order without endpoints that satisfies the countable chain condition (every family of pairwise disjoint nonempty open intervals is countable) but is not separable. The Suslin hypothesis (SH) says that no Suslin line exists; equivalently, that every such order satisfying the ccc is order-isomorphic to $\mathbb{R}$.

Kurepa (1935): three equivalent forms. Over ZFC, a Suslin line exists if and only if a Suslin tree exists (a tree of height $\omega_1$ with no uncountable chain and no uncountable antichain), if and only if a Suslin algebra exists (a Boolean algebra that is complete, atomless, countably distributive and satisfies the countable chain condition).

The following are relative to the consistency of ZFC.

(a) Con(ZFC + SH). Solovay and Tennenbaum (1971) force SH by iterating ccc forcings that kill Suslin trees. The clean statement of what they proved is: MA + (not CH) implies SH, together with the consistency of MA + (not CH) recorded in Martin's Axiom ‡.

(b) Con(ZFC + not SH). Jech (1967) and Tennenbaum (1968) force a Suslin tree into existence. Jensen (1972) then proved the sharper result that the diamond principle implies a Suslin tree exists, and diamond holds in the constructible universe, so a Suslin line exists in $L$.

(c) The consequence for the ccc. If a Suslin line exists then the countable chain condition is not productive: there is a ccc partial order whose square is not ccc. So "a product of ccc spaces is ccc" is not a ZFC theorem, while under MA + (not CH) it is.

• Not proved in this library. None of (a), (b), (c) is proved here, and neither trees of height $\omega_1$ nor the diamond principle is defined here.

• What would prove it. For (a), iterated ccc forcing as in Martin's Axiom ‡. For (b), the fine structure of $L$ far enough to derive diamond, plus the tree construction from diamond. For (c), the combinatorics of Suslin trees. All lie in a forcing and inner-model track this library does not contain.

• Why it matters here. Suslin's 1920 question is the natural sequel to the order-theoretic characterisation of $\mathbb{R}$: the classical theorem says a dense complete separable linear order without endpoints is order-isomorphic to $\mathbb{R}$, and the question is whether "separable" can be weakened to "ccc". The answer is that ZFC does not decide it, so the characterisation of the real line cannot be improved in that direction by any argument in this library. It is also the reason the library must not use "ccc implies separable" as a step anywhere. The ZFC-provable substitute is available and needs no independence at all: the Cantor cube $\{0,1\}^{\kappa}$ for $\kappa$ larger than the continuum is ccc and not separable, which settles the question for topological spaces even though it says nothing about linear orders (Well-order and well-ordered set ↗, Cardinal (initial ordinal) and cardinality ↗, $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874) ↗).

• Conditional discipline. (a) and (b) are relative consistency statements; the implications inside them, MA + (not CH) implies SH, and diamond implies not SH, are ordinary ZFC theorems. Nothing here asserts that a Suslin line exists or that one does not.