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

On 10.01.2025 21:06, joes wrote:

Am Fri, 10 Jan 2025 20:17:37 +0100 schrieb WM:

On 10.01.2025 19:28, joes wrote:

Am Fri, 10 Jan 2025 18:21:43 +0100 schrieb WM:

>

I have no expectations about cardinality. I know that for every finite
initial segment the even numbers are about half of the natural
numbers.
This does not change anywhere. It is true up to every natural number.

You wrongly expect this to hold in the infinite.

No, I expect it is true for all natural numbers, none of which is
infinite.

But it is true for every natural

Of course. Otherwise you would have to find a counterexample.

(if you formalise it correctly)!

Irrelevant.
∀n ∈ ℕ: |{1, 2, 3, ..., 2n}|/|{2, 4, 6, ..., 2n}| = 2.

That doesn't make it true for N and G.

I am not interested in these letters but only in all natural numbers. All natural numbers are twice as many as all even natural numbers. If your N and G denote all natural numbers and all even numbers, then 2 is true also for them.
Regards, WM