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

On 17.11.2024 17:59, FromTheRafters wrote:

WM pretended :

On 17.11.2024 12:38, FromTheRafters wrote:

WM presented the following explanation :

On 17.11.2024 12:01, FromTheRafters wrote:

WM was thinking very hard :

On 16.11.2024 22:33, Moebius wrote:
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For example "aleph_0 - aleph_0" is not defined.

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Small wonder. ℵo means only infinitely many: |ℕ|, |ℚ|, and many others.
|ℕ|-|ℕ| however is defined.

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No, it is not.

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If sets are invariable then ℕ \ ℕ is empty.
If |ℕ| concerns only the elements of ℕ, then |ℕ|-|ℕ|= 0.

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So, you're saying that if I take aleph_zero natural numbers and I remove the aleph_zero odd numbers from consideration in a new set, I will have a new emptyset instead of E?

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Try to understand. "aleph_0 - aleph_0" is not defined.

 Try to understand that |N| equals aleph_zero.

Of course. ℵo equals |ℕ|, equals |ℚ|, equals all countable sets. It is simply another name for infinitely "many". |ℕ| however is a fixed infinite number. Note that sets are invariable.
Regards, WM