Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : wolfgang.mueckenheim (at) *nospam* tha.de (WM)
Groupes : sci.logicDate : 12. Dec 2024, 23:06:58
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <vjfmq3$2upa9$3@dont-email.me>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
User-Agent : Mozilla Thunderbird
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
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