On Friday, January 17, 2014 1:23:06 AM UTC, Ross A. Finlayson wrote: > 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.
Sorry, I got the statement wrong. Beta < alpha is what I should have said, not that omega ^ beta < alpha. The unproved lemma is: All ordinals, alpha, such that alpha is less than epsilon_0, can be uniquely expressed as omega ^ beta * (gamma + 1), for some gamma and some beta such that beta < alpha, where omega is the smallest infinite ordinal.
So this is indeed a canonical form for epsilon_0 (or e_0 in Ross's terminology). However, it would be nice to see a proof or a reference.
You did give a reference and thank you very much for your post. However, I don't have access to a library, or an unlimited book budget, so free e-references would be ideal.
If I am able to consult Takeuti, I will give more substantial feedback.
Thanks to Ross and fom, and thanks in advance to any other helpers on this question.