Re: The set of necessary FISONs

On 12.03.2025 19:18, Jim Burns wrote:

On 3/12/2025 9:04 AM, WM wrote:

⎛ Menge Z, welche die Nullmenge als Element enthält und
⎜ so beschaffen ist, daß jedem ihrer Elemente a
⎝ ein weiteres Element der Form {a} entspricht

 ⎛ set Z, which contains the zero set as an element and
⎜ is such that each of its elements a
⎝ corresponds to another element of the form {a}
 

That is just the induction.

 A definition answers "What is Z ?"
An axiom answers "Does Z exist?"

The axiom ensure the existence of an infinite set by induction.
Only that is of interest for comparison with my proof.

The axiom does not create an inductive set.

It ensures its existence.

Z₀ -- defined as the emptiest inductive set --
doesn't hold darkᵂᴹ {}  and
doesn't hold any visibleᵂᴹ x and darkᵂᴹ {x}

Of course not. Inductive sets do not contain dark numbers.

 {darkᵂᴹ} = Z₀\{visibleᵂᴹ} = {}

Nonsense. Dark is Cantor's ℕ \ Z₀.

 

my induction is ensured

 Your induction is not ensured by
your declaration that it's ensured.

It is ensured by using induction.

How is your induction ensured?

Here you can see it:

 

ℕ \ F(1) = ℵo,
and if
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n) = ℵo
then
ℕ \ F(1) \ F(2) \ F(3) \ ... \ F(n+1) = ℵo.

 

Regards, WM