What is the cardinality of $X=\{ \alpha \text{ ordinal }| \alpha\text{ is the order type of } A \subset \mathbb R\}$?

Question

I found out that $\aleph_1$ is an upper bound. I can show that any well ordered subset of $ \mathbb R$ is countable, so its order type is an element of $\omega_1$. I'd like to show that $|X|= \aleph_1$, but I don't know how to do it. Some hint?

Answer 1

Hint: Try proving that any countable ordinal $\alpha$ embeds in $\mathbb{R}$ by induction on $\alpha$. You'll make use of the countability assumption at limit stages, in that any countable limit ordinal has countable cofinality.

Alternatively, you can prove that any countable totally ordered set $X$ at all embeds in $\mathbb{R}$, by enumerating the elements of $X$ and defining an embedding $f:X\to\mathbb{R}$ on one element at a time.