Re: Contradiction of bijections as a measure for infinite sets

Am 09.04.2024 um 03:54 schrieb Jim Burns:

On 4/8/2024 9:55 AM, WM wrote:

Le 07/04/2024 à 21:47, Jim Burns a écrit :

 

The successor operation is closed in
the natural numbers.

>
For visible numbers only.

 Visibleᵂᴹ or darkᵂᴹ,
k is a natural number  :⟺

k e IN. (I hope that you don't mind that slight simplification).

Visibleᵂᴹ or darkᵂᴹ,
the successor operation is closed in
the natural numbers.

As is stated as (i) a Peano axiom (where we do not differentiate between visibleᵂᴹ and darkᵂᴹ natural numbers)
An e IN: s(n) e IN
or (ii) can be PROVED in set theory, where s(x) := x u {x} and IN := the intersection of all "successor sets". [Hence IN is a successor set itself, which means: 0 e IN and An e IN: s(n) e IN.]
Again this holds for ALL natural numbers (no matter if they are visibleᵂᴹ or darkᵂᴹ).