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)
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
3
|
|
|||
|
|
|
6
|
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 |
|||
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
1
|
Sorry, almost all pertinent answers (even with the exact reference of the Sierpinski article) were given in the article http://mathoverflow.net/questions/57597/universal-order-type ; so the only question I have left is "where did Cantor state it ?" |
|||
|
|
