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

Am Mon, 18 Nov 2024 23:16:22 +0100 schrieb WM:

On 18.11.2024 22:58, FromTheRafters wrote:

on 11/18/2024, WM supposed :

On 18.11.2024 18:15, FromTheRafters wrote:

WM brought next idea :

>

|ℕ| - |ℕ| = 0 because if you subtract one element from ℕ then you
have no longer ℕ and therefore no longer |ℕ| describing it.

|N\{0}| = |N| = Aleph_0

If you remove one element from ℕ, then you have still ℵo but no longer
all elements of ℕ.

But you do have now a proper subset of the naturals the same size as
before.

It has one element less, hence the "size" ℵo is a very unsharp measure.

You can still say it is a subset, like Cantor did with "reality".

If |ℕ| describes the number of elements, then it has changed to |ℕ| -
1.

Minus one is not defined.

Subtracting an element is defined. |ℕ| - 1 is defined as the number of
elements minus 1.

And |N\{2}| = Aleph_0.

If you don't like |ℕ| then call this number the number of natural
numbers.

Why would I do that when it is the *SIZE* of the smallest infinite set.

The set of prime numbers is smaller.

There are infinitely many of them.

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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.