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.

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.