On 1/16/2014 2:31 PM, Paul wrote: > I saw it asserted without proof that all ordinals, alpha, less than epsilon_0, can be uniquely expressed as omega ^ beta * (gamma + 1), for some gamma and some beta such that omega ^ beta < alpha, where omega is the smallest infinite ordinal. This isn't clear to me. Could anyone help or give a reference? > > Thank You, > > Paul Epstein >
Uh, gamma is not always "definable". Then it is to e_0 (else it would be).
This lets the tiniest little infinity that is undefinable, be uncountable.
Still, a "canonical form up to e_0" eg, the limits of induction, that would be very interesting as canonical forms up to large ordinals are very highly organized and structured. Even the typical products of all the vector spaces around us are usually in products of omega: exactly as they are of integer spaces.