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What am I describing?

  1. May 12, 2008 #1
    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?
  2. jcsd
  3. May 13, 2008 #2
    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.
  4. May 13, 2008 #3
    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".
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