Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12.12.2024 18:48, Mikko wrote:

On 2024-12-11 14:04:30 +0000, WM said:
 

On 11.12.2024 01:25, Richard Damon wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 10.12.2024 13:19, Richard Damon wrote:
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The pairing is between TWO sets, not the members of a set with itself.

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The pairing is between the elements. Otherwise you could pair R and Q by
simply claiming it.
"The infinite sequence thus defined has the peculiar property to contain
the positive rational numbers completely, and each of them only once at
a determined place." [Cantor] Note the numbers, not the set.

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TWO different sets, not the elements of a set and some of the elements of
that same set.

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In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. [Wikipedia].

 Do you happen to know any set that is Dedekind-infinite?
 

No, there is no such set. This is proven by my black hats = numbers of the form 10n: For every interval [1, n] the relative covering is at most 1/10. And more than all intervals are not available to supply numbers of the form 10n.
Regards, WM