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

Am Tue, 19 Nov 2024 16:24:52 +0100 schrieb WM:

On 18.11.2024 23:40, FromTheRafters wrote:

WM wrote on 11/18/2024 :

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.

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.

Comparing the size of sets by bijection. Bijection of finite sets give
you a same number of elements, bijection of infinite sets give you same
size of set.

Why? Because only potential infinity is involved. True bijections pr5ove
equinumerosity.

What is a "true" bijection?

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.

Nope!

The number of ℕ \ {1} is 1 less than ℕ.

And what, pray tell, is Aleph_0 - 1 ?

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.

No, it is not.

It is, because 4 and 8 are missing.

It is a subset.

 > There is a bijection.
Only between numbers which have more successors than predecessors,

All of them do.

although it is claimed that no successors are remaining.

--
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.