Le 21-nov.-07, à 08:49, Torgny Tholerus a écrit :

>  meekerdb skrev:Torgny Tholerus wrote:
>>> An ultrafinitist comment to this:
>>> ======
>>> You can add this complementary sequence to the end of the list.  That
>>> will make you have a list with this complementary sequence included.
>>> But then you can make a new complementary sequence, that is not
>>> inluded.  But you can then add this new sequence to the end of the
>>> extended list, and then you have a bijection with this new sequence
>>> also.  And if you try to make another new sequence, I will add that
>>> sequence too, and this I will do an infinite number of times.  So you
>>> will not be able to prove that there is no bijection...
>>> ======
>>> What is wrong with this conclusion?
>> You'd have to insert the new sequence in the beginning, as there is no
>> "end of the list".
>  Why can't you add something to the end of the list?  In an earlier 
> message Bruno wrote:
>  "Now omega+1 is the set of all ordinal strictly lesser than omega+1, 
> with the convention above. This gives {0, 1, 2, 3, ... omega} = {0, 1, 
> 2, 3, 4, ....{0, 1, 2, 3, 4, ....}}."
>  In this sentence he added omega to the end of the list of natural 
> numbers...

Adding something to the end or to the middle or to the beginning of an 
infinite  list, does not change the cardinality of that list. And in 
Cantor proof, we are interested only in the cardinality notion.

Adding something to the beginning or to the end of a infinite ORDERED 
list, well, it does not change the cardinal of the set involved, but it 
obviosuly produce different order on those sets, and this can give 
different ordinal, which denote type of order (isomorphic order).

The ordered set {0, 1, 2, 3, ...} has the same cardinality that the 
ordered set {1, 2, 3, 4, ... 0} (where by definition 0 is bigger than 
all natural numbers). But they both denote different ordinal, omega, 
and omega+1 respectively. Note that {1, 0, 2, 3, 4, ...} is a different 
order than {0, 1, 2, 3, ...}, but both order here are isomorphic, and 
correspond to the same ordinal (omega).

That is why 1+omega = omega, and omega+1 is different from omega. 
Adding one object in front of a list does not change the type of the 
order. Adding an element at the end of an infinite list does change the 
type of the order. {0, 1, 2, 3, ...} has no bigger element, but {1, 2, 
3, ... 0} has a bigger element. So, you cannot by simple relabelling of 
the elements get the same type of order (and thus they correspond to 
different ordinals).

OK?  (this stuff will not be used for Church Thesis, unless we go very 
far ...later).



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