Re: Replacement of Cardinality

Le 24/08/2024 à 00:04, Jim Burns a écrit :

On 8/22/2024 8:16 AM, WM wrote:

Le 21/08/2024 à 20:34, Moebius a écrit :

 

Yeah, it's like claiming:
"There is an end
(to the natural numbers), because at and above omega
there is no natural number.

>
Of course, but
omega is somewhat ghostly.
Do the natnumbers reach till omega?

 ω is an upper.bound of ℕᵈᵉᶠ.
Of all upper.bounds of ℕᵈᵉᶠ, the lowest is ω.

Yes. Nevertheless almost all natural numbers are bewteen ℕᵈᵉᶠ and ω.

 Each element of ℕᵈᵉᶠ is not upper.bound of ℕᵈᵉᶠ.
No upper.bound of ℕᵈᵉᶠ is in ℕᵈᵉᶠ

Correct.

 

Do the natnumbers reach till omega?

 Define
the natnumbers reach k  ⇔
(∀ᵒʳᵈj≤k:(∃ᵒʳᵈi:j=i∪{i} ⇐ j≠0) ∧ 0<k) ∨ 0=k
 k ∈ ω  ⇔  the natnumbers reach k
The natnumbers only reach elements of ℕᵈᵉᶠ.

No.

 ω, an upper.bound of ℕᵈᵉᶠ, is not.in ℕᵈᵉᶠ.
The natnumbers do not reach ω

ω - 1 is the greates naturak number.
Easy to understand by the smallest unit fraction.
Regards, WM