It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum hypothesis, it is possible to construct a similar "universal order" of cardinal $\aleph\_1$. I have two questions : 1) Where did Cantor prove his theorem 2) Can CH be weakened in the previous result ? For reference, Sierpinski construct an ordering on binary sequences indexed by countable ordinals (up to some ordinal $<\omega\_1$), having the desired property than any countable Dedekind cut is separated by some element ; this is actually the same order than the one on surreal numbers born before day $\omega\_1$ (as easily shown by the Gondor construction) 



Cantor theorem on orders
Concerning question 2: 


a. It is consistent that $2^{\aleph\_0}= \aleph\_2$ and there is a universal order of size $\aleph\_1$.


b. It is consistent that $2^{\aleph\_0}= \aleph\_2$ and there is no universal 
order of size $\aleph\_1$.


See Kojman+Shelah, JSL. [preprint](http://www.cs.bgu.ac.il/~kojman/UnivOrd.pdf)


