Re: The set of necessary FISONs

On 01.02.2025 20:21, Jim Burns wrote:

On 2/1/2025 7:56 AM, WM wrote:

There is the assumption that
a set with U(F(n)) = ℕ exists.
Without changing the union
we can remove every element by induction.
No element remains.
The set does not exist.

 Each finiteᵒᵘʳ initial segment F(k) of ⋃{F(n)}
can grow¹ to another initial segment F(k+1)
which is also finiteᵒᵘʳ, and is larger than F(k),
and is not larger than ⋃{F(n)}
 {F(n}} holds each finiteᵒᵘʳ initial segment F(k)
⋃{F(n)} is larger than each F(k).

But all F(n) can be discarded without changing the union.
F(1) can be discarded. If F(n) can be discarded, then F(n+1) can be discarded.
Note: Mathematical induction is a method for proving that a statement P(n) is true for every natural number n that is, that the infinitely many cases P(0),P(1),P(2),P(3),... all hold. [Wikipedia]
Therefore if U(F(n)) = ℕ, then  { } = ℕ
Regards, WM