# Re: Bijections (was OM = SIGMA1)

From: Mirek Dobsicek <m.dobsicek.domain.name.hidden>
Date: Wed, 30 Jan 2008 13:43:15 +0100

Bruno Marchal wrote:
>
> Le 20-nov.-07, à 12:14, Torgny Tholerus a écrit :
>
>> Bruno Marchal skrev:
>>> To sum up; finite ordinal and finite cardinal coincide. Concerning
>>> infinite "number" there are much ordinals than cardinals. In between
>>> two different infinite cardinal, there will be an infinity of ordinal.
>>> We have already seen that omega, omega+1, ... omega+omega,
>>> omega+omega+1, ....3.omega, ... 4.omega .... ....omega.omega .....
>>> omega.omega.omega, .....omega^omega ..... are all different ordinals,
>>> but all have the same cardinality.
>>>
>> Was it not an error there? 2^omega is just the number of all subsets
>> of
>> omega, and the number of all subsets always have bigger cardinality
>> than
>> the set.
>
>
> Yes, that is true.
>
>
>
>> So omega^omega can not have the same cardinality as omega.
>
>
> But addition, multiplication, and thus exponentiation are not the same
> operation for ordinals and cardinals. I should have written
> omega"^"omega, or something like that. That is why I have written
> 3.omega instead of 3*omega.

Uu, reading about cardinals and ordinals on Wikipeadia did not helped me
at this point.

Could you please elaborate more on this? Of course, only relatively to
its importance towards CT ...

Cheers,
Mirek

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Received on Wed Jan 30 2008 - 07:43:34 PST

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