Re: Replacement of Cardinality

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

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

Is there a gap?

∀ᵒʳᵈα: α=α
(which seems to mean) all ordinals are visibleᵂᴹ
(which seems to mean) there is no gap.

Zero is fixed and firm, existing
with certainty. The positive reals reach till zero
with no doubt.
There NUF = 0.
And because of
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
it cannot increase by more than
1 per point x.

x = 1/0  :⇔  0⋅x = 1
¬∃x ∈ R: x = 1/0