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Re: Matheology § 038
Posted:
Jun 14, 2012 11:33 PM
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PotatoSauce wrote:
> On Thursday, June 14, 2012 4:46:37 PM UTC-4, WM wrote:
>>
>> That proof is required but not given. Nevertheless Cantor "counts" the
>> rational numbers by equating
>> lim(card(q_1, q_2, q_3, ..., q_n)) = card(lim(q_1, q_2, q_3, ...,
>> q_n))
>
> No.
>
>>
>> card(q_1, q_2, q_3, ..., q_n) = n
>> lim n = aleph_0
>> lim(q_1, q_2, q_3, ..., q_n) = |Q
>
> No wonder you are confused. Your interpretation is completely wrong.
>
> There are no limits involved in defining a bijection between natural
> numbers and rational numbers.
It is Mückenheim's great discovery that there are limits involved. This is
accomplished by mixing up notions like
- finite, but of indeterminable size
- growing in some kind of process and, although always being finite,
surpassing any fixed finite size
- being infinite and ordered and having a maximal element
- being infinite and ordered without a maximal element
- being "potentially infinite"
- being "actually infinite"
in the most stupid ways.
Some of the above-mentioned notions have a precise mathematical meaning,
some may be given such meaning, and the last two seem to be of dubious
philosophical origin, maybe introduced by Aristotle in a not fully
satisfactory attempt to come to terms with Zeno's paradoxes; see
http://arxiv.org/abs/math/0604639
which would be a topic for discussion of much greater interest than
Mückenheim's completely braindead crap.
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