Three immodest statements shape this talk:

At this point, we have a pretty good understanding of the combinatorics of $\omega_1$.
- Justin Moore, 2012, Stella's Cafe, Ithaca, New York

They [the ordinals $\omega_n$] each have their own lives.
- Stevo Todorcevic, 2014, Mehak, Ithaca, New York.

I would formulate the basic problem of set-theoretic topology as follows: To determine which set-theoretic structures have a connection with the intuitively given material of elementary polyhedral topology and hence deserve to be considered as geometrical figures – even if very general ones.
- Paul Alexandroff, 1932, Elementary Concepts of Topology

Moore's statement is some measure at once of success and of failure, of how much we might hope to but do not know of $\omega_2, \omega_3, \omega_4, \dots$ - a situation calling either for explanation or rectification. Alexandroff's program, broadly conceived, offers some beginnings of both.

We'll define the term "ordinal" in this talk. By its end, we hope, $\omega_1$ will "mean something" to you, and/or you'll experience the above quotes as "haunting."