Is there some sort of interesting way of organizing certain order types that aren't ordinals? When I say certain order types, some examples include but are not limited to:
- order types of dense linear orders
- order types of dense linear orders without endpoints
I'm really just looking for any interesting perspective on certain order types that aren't ordinals, (like in the above examples). How do they compare? Can we classify all of them? For example, all countable dense linear orders without endpoints are isomorphic to $\mathbb{Q}$. There are different dense linear orders without endpoints of size $\mathfrak{c}$. But is there an interesting way to classify and/or compare all the order types of dense linear orders without endpoints? Or is this a fruitless endeavor with little to be gleamed?
Motivation:
I've really enjoyed studying $\textbf{ON}$, and I wanted to know if anybody's tried to paint a really good picture of any specific order.