What is X? Ordered Proper-Class-Sized Sequence

[Dragonfall]

This may be a little hand-wavy:

Let $$a$$ be an ordered, proper-class-sized sequence $$a=(a_0,a_1,...,a_{\omega},...,a_{\omega_2},...,a_{\omega_{\omega}},...)$$ where $$a_i, i\in\mathbb{O}rd$$ are, say, 0,...,9. So that if we look only at those $$a$$ whose expansion on $$a_{\omega}$$ onwards are 0, we'd get something like the real numbers.

We order these things lexicographically (or antilexicographically, whichever it is that the reals are ordered by, I can never remember). So let X be the class of these things. What is X?

 

[Dragonfall]

I just realized that X can't technically be a proper class since X contains proper classes. Formally we'll have to invoke some sort of type theory argument and call X a "second order class" or something. But that's just a technicality.

 

[Dragonfall]

I think what I mean is that "there is no class X such that every linearly ordered set is isomorphic to a subset of X".