number of linear orders

Question

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved for $\kappa=\aleph_0$ first, and then for uncountable $\kappa$, or for all $\kappa$ right away?

Answer 1

I hadn't noticed this question before, Martin Goldstern just posted it in MathOverflow. My answer is here.

In short, the result for $\kappa=\aleph_0$ is due to Cantor (at least $2^{\aleph_0}$) and to Bernstein and Hausdorff, in 1901, independently (at most $2^{\aleph_0}$).

The general result is due to Hausdorff, and appears in Über eine gewisse Art geordneter Mengen, Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Classe 53 (1901), 460-475. (I give additional details and references at the link above.)