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Re: Cantor's first proof,
Posted:
Nov 14, 2012 2:41 AM
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WM wrote:
"LudovicoVan" <ju...@diegidio.name> wrote:
>> > Consider the set of all positive rational numbers as existing.
>> > Fill into the vase all numbers of the interval (0, 1]. Then enumerate
>> > one of them by 1 and take it off the vase.
>> > Fill into the vase all numbers of the interval (1, 2]. Then enumerate
>> > one of them by 2 and take it off the vase.
>> > Fill into the vase all numbers of the interval (2, 3]. Then enumerate
>> > one of them by 3 and take it off the vase.
>> > Continue until all rational numbers have been enumerated. Then all
>> > have been taken off the vase. The remaining set of not enumerated
>> > rationals is empty.
>> > This result is nonsense from the mathematical standpoint. The limit
>> > cannot be empty. Hence, it is impossible to enumerate all rational
>> > numbers.
>> Sorry but there is no "hence" that I can see, just twisted reasoning.
> Don't you know how the rationals of the first interval between 0 and 1
> were enumerated by Cantor? 1/2 is enumerated by 1, 1/3 is enumerated
> by 2 and so on.
> 1) 1/2
> 2) 1/3
> 3) 1/4
> 4) 2/3
> ...
> Now we do the same, but with all positive rationals.
No. A map defined in your way would not be one-to-one
because all images 1, 2, 3. are already used by those rationals
taken off from the intervals (n-1,n].
Regards
Michael
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