Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: A canonical form for small ordinals
Posted: Jan 17, 2014 4:00 AM

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.

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

Date Subject Author
1/16/14 Paul
1/16/14 ross.finlayson@gmail.com
1/16/14 fom
1/17/14 Paul
1/17/14 William Elliot
1/17/14 quasi
1/17/14 quasi
1/17/14 quasi
1/17/14 Paul
1/18/14 ross.finlayson@gmail.com
1/18/14 ross.finlayson@gmail.com
1/19/14 Paul
1/19/14 ross.finlayson@gmail.com