Re: Incompleteness of Cantor's enumeration of the rational numbers

Am Wed, 20 Nov 2024 17:51:19 +0100 schrieb WM:

On 20.11.2024 15:15, FromTheRafters wrote:

WM explained on 11/20/2024 :

set theory claims that all natural numbers can be counted to such that
no successors remain.

 
No it doesn't.

 
Even all rationals and algebraics.
"we get the epitome (ω) of all real algebraic numbers [...] and with
respect to this order we can talk about the nth algebraic number where
not a single one of this epitome has been forgotten"
"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"

You are once again lacking in precision: every natural is finite and thus
countable. The number of all of them together, the cardinality of the set,
however is infinite (yet still countable), is not contained in N (not a
member of the naturals).

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.