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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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 fom Posts: 1,968 Registered: 12/4/12
Re: A canonical form for small ordinals
Posted: Jan 16, 2014 9:07 PM

On 1/16/2014 7:23 PM, 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.
>
>

He should look in "Proof Theory" by Takeuti,
Chapter 2 Section 11.

Well-formedness rules:

1) 0 is an ordinal

2) Let mu and mu_1, mu_2, ..., mu_n be ordinals.
Then mu_1 + mu_2 + ... + mu_n is an ordinal
and omega^mu is an ordinal

3) Only objects obtained using Rule 1 and Rule 2
are ordinals.

Of course, when mu = 0, then omega^mu = 1. So,
that is the source of units.

Takeuti uses this definition of ordinal
and an infinitary sequent calculus to
present a version of Henkin's proof of
the consistency of Peano arithmetic by
transfinite induction.

This is not transfinite induction in
the usual sense. The inductive definition
of a polynomial given above plays its
role as a syntactic representation. As
a syntactic representation it can be
syntactically decomposed. This is the
sense of the proof-theoretic reduction
by which the consistency proof proceeds.

The form asked for by the original poster
is essential for the proof since any
finite iteration of trailing units permits a
finitary transformation in any given proof
tree for an arithmetical statement. All
of the possible polynomials are grouped into
classes of syntactic forms and those forms
are related to one another by syntactic
decompositions to reestablish a form
with the trailing units.

There may be some simpler discussion
of this form, but, I know it can be
found in Takeuti.

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