Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
De : mikko.levanto (at) *nospam* iki.fi (Mikko)
Groupes : sci.logicDate : 14. Dec 2024, 09:52:40
Autres entêtes
Organisation : -
Message-ID : <vjjh0o$3u5ec$1@dont-email.me>
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On 2024-12-12 22:06:58 +0000, WM said:
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:
The pairing is between TWO sets, not the members of a set with itself.
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.
TWO different sets, not the elements of a set and some of the elements of
that same set.
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.
The set of natural numbers, if there is any such set, is Dedekind-infinte:
the successor function is a bijection between the set of all natural
numbers and non-zero natural numbers.
-- Mikko
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