The Suslin Hypothesis and Suslin algebras

Work in ZFC, so the axiom of choice is available as stated in The Axiom of Choice. The Suslin Hypothesis (SH) says that there is no Suslin line in the strong order-theoretic sense of Suslin lines in order language.

Let $B$ be a complete Boolean algebra as in Completeness, regular opens, and order continuity. It is atomless if every $0<b\in B$ has some $c$ with $0<c<b$. Regard $B^{+}=B\setminus \{0\}$ as a forcing order with stronger elements smaller. The algebra is ccc when $B^{+}$ is ccc in the sense of Compatibility, ccc and Knaster for posets, equivalently when every pairwise disjoint family of nonzero Boolean elements is countable.

The algebra $B$ is countably distributive when, for every double sequence $(b_{n,m}{)}_{n,m<\omega }$ in $B$,

${\bigwedge }_{n<\omega }{\bigvee }_{m<\omega }{b}_{n,m}={\bigvee }_{f\in {\omega }^{\omega }}{\bigwedge }_{n<\omega }{b}_{n,f(n)}\mathrm{.}$

A Suslin algebra is a complete, atomless, ccc, countably distributive Boolean algebra with $0\mathrm{\ne }1$. The last clause explicitly excludes the one-element algebra; atomlessness alone can be vacuous there. Empty joins and meets retain the complete-algebra conventions $\bigvee \mathrm{\varnothing }=0$ and $\bigwedge \mathrm{\varnothing }=1$. The displayed distributive law uses the nonempty index set $\omega$ in both coordinates, so it asserts no selection from an empty family.

The definition itself makes no choice. AC is declared because the equivalence and construction theorems on this page use simultaneous successor orders, maximal antichains, and countable enumerations.