Aleph-one-like linear order ($|L|={\aleph }_1,|L_{<x}|\le {\aleph }_0$)

An aleph-one-like linear order has size ${\aleph }_1$ and every point has only countably many predecessors. Consequently every proper initial segment is countable, since it is bounded by any point outside it. This differs from requiring every nonempty interval to be uncountable.

For a subset $S$ of the nonzero countable limit ordinals, let $B_{\alpha }=1+\mathbb{Q}$ when $\alpha \in S$ and $B_{\alpha }=\mathbb{Q}$ otherwise. The order sum is an aleph-one-like linear order whose nonempty proper initial segments are $\mathbb{Q}$ or $\mathbb{Q}+1$. At the limit cut before block $\delta$, the complementary final segment has a least point exactly when $\delta \in S$. Club agreement of countable filtrations makes this condition invariant modulo a nonstationary difference. Disjoint stationary subsets of omega-one consequently supply $2^{{\aleph }_1}$ nonisomorphic orders.