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Topic: Infinity and all the rest
Replies: 15   Last Post: Oct 24, 2013 8:56 PM

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Posts: 18,076
Registered: 1/29/05
Re: Infinity and all the rest
Posted: Oct 24, 2013 12:08 PM
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On Thursday, 24 October 2013 16:09:01 UTC+2, Michael Klemm wrote:
> WM wrote: On Thursday, 24 October 2013 14:20:42 UTC+2, Michael Klemm wrote: >>
> M_1 = empty set, M_{n+1} = (M_n u (n-1, n]) - {q_n}.
> But why is a recursive definition of sets M_n a super task?

This is the common wording because the sets can easily be understood in that way. But in fact this kind of argument has no teporal features. It is simply a sequence of sets like 1 - 2 + 4 - 8 +- ... is a simple sequence of numbers.

> What is curious in repect of this quite common method?

It is curious that so many mathematicians believe that this sequence of sets could converge to the empty set. Obviously it does not. It is simple to show that the number of intervals (n-1, n] without any q_n divided by the number of intervals with at least one q_n for large n diverges towards infinity.

It is even simpler to show that never 10 % of all rationals are enumerated and never all the rationals of even the tiniest interval with positive measure are enumerated. It is strange that some people believe the enumeration up to every natural n would prove something about the possibility to enumerate all rationals.

Regards, WM

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