Re: Cantor's "proof"

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 09/25/04 

Date: Fri, 24 Sep 2004 22:14:19 -0600

In article <opseuaogk43uk9lu@cs81133.pp.htv.fi>,
 Keckman <keckman@welho.com> wrote:

> On Thu, 23 Sep 2004 11:39:33 -0600, Virgil
> <ITSnetNOTcom#virgil@COMCAST.com> wrote:
>
> The more you create numbers that are not in list(n) the more you put them
> to next list.

Where there are more numbers missing than there are listed. This is true
for every list! As long as each list of reals is countable (an image of
N under some function), there are more real numbers missing from the
list than in it.

> > A possible solution to the endless sequence of nuts who do not
> > understand Cantor's diagonal proof might be to show that Cantor's
> > method, slightly modified, creates at least as many numbers not in a
> > given list there are in the list.
> >
> >
> > Start as usual with the standard non-member of the list, say A_0,
> > differing from the nth member of the list, say L_n, in the nth decimal
> > place, in one of the usual ways to avoid the dual representation problem
> > of a sequence terminating in infinitely many zeros of infinitely many
> > nines.
> >
> > Then consider the sequence C_(m+n), 0 < n, in which C_(m+n) differs
> > C_m at decimal place m, 0 <= m < n, and from L_n at decimal place
> > (m+n).
> >
> > We now have as many non-members as members of the given list.
> >
> >
> >
> >
> >
> > But we can carry this even further, for every function f:N -> N on the
> > naturals that is strictly monotone increasing (so that f(n+1) > f(n) for
> > all n in N) we can now construct a D_(m+n) to differ from all previous
> > D_x's and such that D_(m+n) differs from L_n at position f(n).
> >
> > Since, by other methods, it can be shown that the set of strictly
> > monotone functions on N is uncountable, we have now "constructed"
> > uncountably many non-members of the original list.
> >
> > Thus we show that there are MORE non-listed than listed for ANY list.

Relevant Pages

  • Re: i still havent get it
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  • Re: i still havent get it
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  • Re: Cantors "proof"
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  • Re: Cantors second proof
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  • Re: Cantors diagonal proof wrong?
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