Re: Bijections (was OM = SIGMA1)

```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
```
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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