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

On 14.12.2024 09:52, Mikko wrote:

On 2024-12-12 22:06:58 +0000, WM said:

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,

If ℕ is a set, i.e. if it is complete such that all numbers can be used for indexing sequences or in other mappings, then it can also be exhausted such that no element remains. Then the sequence of intersections of endsegments
E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content. Then, by the law
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
the content must become finite.

is Dedekind-infinte:
the successor function is a bijection between the set of all natural
numbers and non-zero natural numbers.

This "bijection" appears possible but it is not. This is better demonstrated by the "bijection" between the sets ℕ = {1, 2, 3, ...} and D = {10n | n ∈ ℕ}. It is contradicted because for every interval (0, n] the relative covering is not more than 1/10, and there are no further numbers 10n beyond all natural numbers n. The sequence 1/10, 1/10, 1/10, ... has limit 1/10.
Regards, WM