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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 10 Jan 2025 22:38:51 +0100
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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