Re: Contradiction of bijections as a measure for infinite sets

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=0 ∨ ∃⟦0,k⦆: ∀i ∈ ⟦0,k⦆: i⁺¹ ∈ ⦅0,k⟧
Visibleᵂᴹ or darkᵂᴹ,
if
∃⟦0,k⦆: ∀i ∈ ⟦0,k⦆: i⁺¹ ∈ ⦅0,k⟧
then
∃⟦0,k⁺¹⦆: ∀i ∈ ⟦0,k⁺¹⦆: i⁺¹ ∈ ⦅0,k⁺¹⟧
  ⟦0,k⁺¹⦆ = ⟦0,k⦆∪{k}
Visibleᵂᴹ or darkᵂᴹ,
if
k is a natural number,
then
then k⁺¹ is a natural number.
Visibleᵂᴹ or darkᵂᴹ,
the successor operation is closed in
the natural numbers.