Re: The set of necessary FISONs

On 2/6/25 3:54 AM, WM wrote:

On 06.02.2025 01:46, Jim Burns wrote:

On 2/5/2025 2:18 PM, WM wrote:

 

The axiom of induction:
∀P( P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)))
>
P(1): U(F(n) \ F(1)) = ℕ.
>
P(k): U(F(n) \ {F(1), F(2), ..., F(k)}) = ℕ
==>
P(k+1): U(F(n) \ {F(1), F(2), ..., F(k+1)}) = ℕ.

>
A description, not a magic spell.
>
...which could also be written...

 Why should we use two different writings?

>

I can remove all FISONs from U(F(n))
without changing the claimed union.

>
What you (WM) mean by 'remove all...without changing..."
is that, by induction,

 it is proved that all F(n) can be subtracted like by induction all natural numbers can be subtracted from ℕ:
 {1} can be subtracted because it is an element of in ℕ.
If {n} has been subtracted, then {n+1} can be subtracted because it is an element of in ℕ.
 By the axiom of induction the result is the empty set.
 Regards, WM

Which just means that no one FISON is needed to build the set of Natural Numbers, as there are always ones bigger.
You just don't understand what you are saying, because your mind is just broken by your naive logic.