Re: Replacement of Cardinality

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

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.

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
Something between ℕᵈᵉᶠ and ω
does not exist.

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

Define
the natnumbers reach k
if and only if
ordinal k and each j before k except 0
can be stepped.up.to,
and 0 is before k,
and the natnumbers reach 0
Implicit:
Each non.empty set of ordinals holds a first.

k ∈ ω  ⇔  the natnumbers reach k
The natnumbers only reach elements of ℕᵈᵉᶠ.

>
No.

Missing:
Any other meaning of

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 naturak 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.
and
ω+1 and each j before ω+1 can be stepped.up.to.
and
ω+2 and each j before ω+2 can be stepped.up.to.
and
...
and
ω is merely another big natural in ℕᵈᵉᶠ
and
ω is not the least.upper.bound of ℕᵈᵉᶠ AKA ω
if
ω-1 and each j before ω-1 canNOT be stepped.uo.to,
then
ω-1 is an upper.bound of ℕᵈᵉᶠ
and
ω is not the least.upper.bound of ℕᵈᵉᶠ AKA ω
ω-1 implies ω isn't ω

Easy to understand by the smallest unit fraction.

About that smallest unit fraction...
I have some bad news.