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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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ross.finlayson@gmail.com

Posts: 1,216
Registered: 2/15/09
Re: A canonical form for small ordinals
Posted: Jan 18, 2014 6:51 PM
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On 1/18/2014 3:48 PM, Ross A. Finlayson wrote:
> On 1/17/2014 3:27 AM, quasi wrote:
>> quasi wrote:
>>> quasi wrote:
>>>> William Elliot wrote:
>>>>> 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 beta < alpha,
>>>>>> where omega is the smallest infinite ordinal. This isn't
>>>>>> clear to me. Could anyone help or give a reference?

>>>>
>>>> Note that alpha = 0 must be excluded since the ordinal 0 has
>>>> no such representation.
>>>>

>>>>> Cantor's normal form.
>>>>
>>>> Assuming 0 < alpha < epsilon_0, then as suggested by
>>>> William Elliot, Cantor's normal form clinches it.
>>>>
>>>> Proposition:
>>>>
>>>> For all ordinals alpha > 0 there are exist unique ordinals
>>>> beta and gamma such that (omega^beta)*(gamma + 1).

>>>
>>> I meant: such that alpha = (omega^beta)*(gamma + 1).
>>>

>>>> Existence:
>>>>
>>>> Let beta be the least exponent in the Cantor normal form for
>>>> alpha. Then the normal form factors as
>>>>
>>>> (omega^beta)*(gamma + 1)
>>>>
>>>> for some gamma.

>>
>> Clarifying remark:
>>
>> The least exponent beta may occur more than once in the normal
>> form, but since the normal form has only finitely many terms,
>> it will still be the case that once omega^beta is factored out
>> (on the left) the remaining factor will not be a limit ordinal,
>> and hence can be expressed in the form gamma + 1, for some
>> ordinal gamma.
>>

>>>> Uniqueness:
>>>>
>>>> Suppose alpha has two representations:
>>>>
>>>> alpha = (omega^beta1)*(gamma1 + 1)
>>>>
>>>> alpha = (omega^beta2)*(gamma2 + 1)
>>>>
>>> >From the uniqueness of the normal form for alpha, together with
>>>> the uniqueness of the normal form for gamma1 + 1 and gamma2 + 1,
>>>> it follows that beta1 = beta2, and based on that, the uniqueness
>>>> of the normal form for alpha then yields gamma1 = gamma2.
>>>>
>>>> This completes the proof of the proposition.
>>>>
>>>> Now suppose 0 < alpha < epsilon_0.
>>>>
>>>> Applying the proposition, there are unique ordinals beta and
>>>> gamma such that alpha = (omega^beta)*(gamma + 1).
>>>>
>>>> Then
>>>>
>>>> alpha = (omega^beta)*(gamma + 1)
>>>>
>>>> => omega^beta <= alpha
>>>>
>>>> But the above, together with alpha < epsilon_0, yields
>>>>
>>>> beta < alpha
>>>>
>>>> as required.
>>>>
>>>> Bottom line:
>>>>
>>>> Sometimes William Elliot is right on target, and this was
>>>> one of those times.

>>
>> quasi
>>

>
> And, then back to Von Neumann? This is up to "epsilon_zero",
> "epsilon_nought" as it were, changing the order of exponentiation
> in the form, here is then defined, if so "defined" in the
> different, anyways I don't care as it is still induction through
> definable ordinals.
>
> Then, quasi, I'd as rather it be more directly productive in the
> constructive, here for me that conclusion is still quite proper.
>
>
> Interesting stuff, that, building for example all products spaces,
> here with the general interest in a result like this.
>
> William, Elliott, quasi, or Dr. Quasi or whatever it is these days,
> please explain what you see as the significance of the result you
> have proven.
>
>


Rather that's "Elliot".




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