Worked solution: Independence of Suslin's Hypothesis from ZFC (Jensen, Solovay–Tennenbaum, 1971)

Step 1 of 8: Suslin's question: can "separable" be weakened to "ccc"?

In plain words

The real line $\mathbb{R}$ has several familiar order properties: it is dense (between any two points there is another), complete (no gaps), has no first or last point, and it is separable — the rationals $\mathbb{Q}$ sit inside it as a countable set that gets arbitrarily close to every point. Cantor proved these four properties, taken together, uniquely characterize $\mathbb{R}$ up to relabeling.

Separability has an easy consequence: any collection of pairwise disjoint open intervals must be countable, since each one has to contain a distinct rational number. Suslin asked, in 1920, whether this weaker consequence alone — called the countable chain condition — could replace separability in Cantor's characterization, or whether some mysterious "impostor" line could satisfy it without secretly having a countable dense subset.

R \text{ dense, complete, no endpoints, ccc} \ \overset{?}{\Longrightarrow}\ R \cong \mathbb{R}

Detailed analysis

A totally ordered set $R$ is a linear continuum if it is dense, complete (every bounded set has a supremum), and has no least or greatest element. Cantor's characterization theorem says: if $R$ is a linear continuum that is also separable (has a countable dense subset), then $R$ is order-isomorphic to $\mathbb{R}$ .

Mikhail Suslin, in a paper published posthumously in 1920, asked whether separability in this theorem can be replaced by the strictly weaker countable chain condition (ccc): every collection of pairwise disjoint nonempty open intervals of $R$ is countable. Every separable linear continuum is automatically ccc (each interval must trap a distinct rational), but Suslin's question was whether the converse implication also holds.

A hypothetical linear continuum satisfying ccc but not separable — hence not order-isomorphic to $\mathbb{R}$ — is called a Suslin line, and Suslin's Hypothesis ( $\mathrm{SH}$ ) is the statement that no Suslin line exists. To attack this question, the next step translates it from continuous orders into the more combinatorially tractable language of trees.

Terms in this step

Separable order: A linearly ordered set that contains a countable subset getting arbitrarily close to every point, the way $\mathbb{Q}$ sits densely inside $\mathbb{R}$ .
Countable chain condition (ccc): An order (or more generally a poset) satisfies ccc if every family of pairwise disjoint nonempty open intervals (or pairwise incompatible elements) is countable.

Knowledge used in this step

The ZFC axioms