Re: Replacement of Cardinality

Le 25/08/2024 à 12:21, Jim Burns a écrit :

On 8/24/2024 10:11 AM, WM wrote:

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

>
Yes.

 That's a definition of ω
 

Nevertheless
almost all natural numbers are bewteen ℕᵈᵉᶠ and ω.

 Something between ℕᵈᵉᶠ and ω is
a lower.than.lowest upper.bound of ℕᵈᵉᶠ

No. Dark numbers are not any bounds. Bounds are definable.

Something between ℕᵈᵉᶠ and ω
does not exist.

So the blind man argues.

Each non.empty set of ordinals holds a first.

Like each set of unit fractions.

 

Do the natnumbers reach till omega?

 

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

>
ω - 1 is the greates natural number.

 if
ω-1 and each j before ω-1 can be stepped.uo.to,
then
ω and each j before ω can be stepped.up.to.

Dark numbers cannot be stepped to.
Regards, WM