Re: The set of necessary FISONs

On 19.02.2025 19:58, Jim Burns wrote:

Not proven for the set {F} of FISONs,
which is not a FISON.

1) Induction covers all elements of an infinite inductive set.
F(1) ∈ F und F(n) ∈ F ==> F(n+1) ∈ F describes the infinite inductive set F of FISONs.
2) Subtraction all FISONs {1, 2, 3, ..., n} satisfying |ℕ \ {1, 2, 3, ..., n}| = ℵo from F leaves the empty set.
These two well-established arguments prove my case:
UF = ℕ  ==> Ø = ℕ.

 The sum of any two natural numbers
is a natural number.
 The "sum" of all the natural numbers,
for any reasonable definition of that,
will be larger than any natural number,
and not a natural number.
 

Like the product. Reason is the potential infinity of definable numbers.
Nevertheless: Zermelo proves the existence of an inductive *set* by induction.
Regards, WM