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Topic: A canonical form for small ordinals
Replies: 12   Last Post: Jan 19, 2014 2:35 PM

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Paul

Posts: 474
Registered: 7/12/10
Re: A canonical form for small ordinals
Posted: Jan 17, 2014 4:00 AM
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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?
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> >
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> > Thank You,
>
> >
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> > Paul Epstein
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> >
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>
>
>
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> Uh, gamma is not always "definable". Then it is to e_0
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> (else it would be).
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>
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> This lets the tiniest little infinity that is undefinable,
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> be uncountable.
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>
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> Still, a "canonical form up to e_0" eg, the limits of
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> induction, that would be very interesting as canonical
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> forms up to large ordinals are very highly organized and
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> structured. Even the typical products of all the vector
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> spaces around us are usually in products of omega:
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> exactly as they are of integer spaces.


Ross,

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.

fom,

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.

Paul Epstein



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