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
> omega, and the number of all subsets always have bigger cardinality
> 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.
We can come back on ordinal later, but now I will focus the attention
on the cardinals, and prove indeed that 2^omega, or 2^N, or
equivalently the infinite cartesian product (of sets)
2X2X2X2X2X2X2X2X... , is NOT enumerable (and indeed vastly bigger that
the ordinal omega"^"omega.
You can look at the thread on the growing functions for a little more
on the ordinals. Actually my point was to remind people of the
difference between ordinal and cardinal, and, yes, they have different
addition, multiplication, etc.
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [EMAIL PROTECTED]
To unsubscribe from this group, send email to [EMAIL PROTECTED]
For more options, visit this group at