  We show that the existence of a first-order formula separating two monadic
second order formulas over countable ordinal words is decidable. This extends
the work of Henckell and Almeida on finite words, and of Place and Zeitoun on
$\omega$-words. For this, we develop the algebraic concept of monoid (resp.
$\omega$-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of
monoids (resp. $\omega$-semigroup, resp. ordinal monoid) that explicitly
includes a new operation capturing the loss of precision induced by first-order
indistinguishability. We also show the computability of FO-pointlike sets, and
the decidability of the covering problem for first-order logic on countable
ordinal words.
