TITLE: Monotone injection of an ordinal into $[0,1]$
QUESTION [5 upvotes]: This is related to my recent question and would provide a natural positive answer to Question 2. I am sure this must be known to experts.

Question: Is there a monotone injection $(\omega_1,<) \to ([0,1],<)$ ?

REPLY [6 votes]: There is a far reaching generalization of this due to Friedman and Shelah. Suppose $X$ is a Borel set in a Polish space and $<$ is a linear order of $X$ that is a Borel subset of $X\times X$. Then there is no order preserving map from $\omega_1$ into $(X,<)$.
The Friedman-Shelah result follows from a later structure theorem of Harrington and Shelah
who proved that for any such Borel linear order $(X,<)$ there is a Borel measurable order preserving
map into ${\bf R}^\alpha$ for some countable ordinal $\alpha$ where ${\bf R}^\alpha$ is ordered lexicographically.  The arguments for $[0,1]$ above can be generalized to show
that ${\bf R}^\alpha$ has no $\omega_1$-chains.