Size of Totally Ordered Set with Countable Predecessors

Question

Assume Choice. Let $S$ be a set, and $\trianglelefteq$ be a total order on $S$. If for all $s \in S$, the set $\{t:t\trianglelefteq s\}$ is countable, what are the possible cardinalities of $S$? Obviously, any cardinality $\le \omega_1$ has such a total ordering.

Answer 1

There are no other possibilities. Let $\kappa$ be the cofinality of the order; clearly $\kappa\le\omega_1$. But then if $\langle x_\xi:\xi<\kappa\rangle$ is a cofinal sequence, and $S_\xi=\{x\in S:x\trianglelefteq x_\xi\}$ for $\xi<\kappa$, we have

$$|S|=\left|\bigcup_{\xi<\kappa}S_\xi\right|\le\kappa\cdot\omega\le\omega_1\;.$$