I have searched on line trying to find the point of ordinal numbers, and even though I can find many sites defining them and explaining their properties, none of them goes beyond the basic reason for their existance, as:
Ordinal numbers are the "labels" needed to arrange collections of objects in order. An ordinal number is used to describe the order type of a well-ordered set
Or something similar. But why does it feel like they are trying to see how "big" they can go, as in defining, $\omega2 $ , $\omega^2$, $\omega^\omega$, $\omega^{\omega^\omega}$, $\varepsilon_0$,$\varepsilon_\varepsilon$,$\omega_1$, etc but without pointing at any application of these large numbers.
Is there a real application, even if it's only to support other areas of mathematics, or is it just a display of big numbers ? Isn't $\omega_1$ big enough for everything we know ?