What is X? Ordered Proper-Class-Sized Sequence

  • Context: Graduate 
  • Thread starter Thread starter Dragonfall
  • Start date Start date
Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
2 replies · 3K views
Dragonfall
Messages
1,023
Reaction score
5
This may be a little hand-wavy:

Let [tex]a[/tex] be an ordered, proper-class-sized sequence [tex]a=(a_0,a_1,...,a_{\omega},...,a_{\omega_2},...,a_{\omega_{\omega}},...)[/tex] where [tex]a_i, i\in\mathbb{O}rd[/tex] are, say, 0,...,9. So that if we look only at those [tex]a[/tex] whose expansion on [tex]a_{\omega}[/tex] 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?
 
Physics news on Phys.org
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.
 
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".