Re: Replacement of Cardinality

On 8/19/2024 7:57 AM, WM wrote:

Le 18/08/2024 à 11:30, joes a écrit :

Am Sat, 17 Aug 2024 13:37:51 +0000 schrieb WM:

NUF(x) grows from 0 to more,
but at no point it grows by more than 1.

>
How does it even reach infinity then?

>
It passes through darkness.
There is no end visible.
That means growth without end.
We call it infinity.

Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ
⎜⎛ in the sense proposed by
⎜⎝ Paul Gustav Samuel Stäckelᵖᵍˢˢ (1862...1919)
⎜ iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S
Lemma 1.
⎛ No set B has both
⎜ finiteᵖᵍˢˢ order ⟨B,<⟩ and
⎝ infiniteᵖᵍˢˢ order ⟨B,◀⟩.
Lemma 2.
⎛ No set B, no element x, no order '◀' exists
⎜ such that
⎝ ⟨B,◀⟩ is finiteᵖᵍˢˢ and ⟨B∪{x},◀⟩ is infiniteᵖᵍˢˢ.