I have a problem with this exercise, it says:
Given an ordered set $(S,\lt)$, an ordinal $\alpha$ embeds into $S$ if there exist an injective function $f:\alpha \rightarrow S$ such that $x \lt y \implies f(x) \lt f(y)$.
What is the minimum cardinality of a set $S$ such that every $\alpha \lt \omega_2$ embeds into $S$?
I am pretty sure that the answer is $\aleph_1$, and maybe a useful fact is that if a non-empty set satisfies $(S,\lt) \sim (S,\lt) \times \omega_1$ (where $(S,\lt) \times \omega_1$ is ordered with the lexicographical order and $\sim$ means there is an isomorphism of orders between the tho set, $i.e.$ a non decreasing bijection) then every $\alpha \lt \omega_2$ embeds into $S$ (it can be proven by transfinite recursion).
I just need to construct an example of a set $|S|=\aleph_1$ that satisfies the request of the problem, thank you in advance for your help.