Re: The set of necessary FISONs

Am Tue, 18 Feb 2025 09:54:02 +0100 schrieb WM:

On 17.02.2025 20:59, Jim Burns wrote:

On 2/17/2025 2:27 PM, WM wrote:

 

Proofs by induction cover all FISONs.

 
Proofs by induction prove that some property A(k) describes each
element of some inductive

 
set.
Here this property is that FISON F(n) can be removed without changing
the premise U({F(1), F(2), F(3), ...} \ {F(1), F(2), ..., F(n)}) = ℕ.
(*)
 

That reasoning is silent about whether the _set_ (not its elements) has
A(k).

Therefore I gave you an example that you should be able to understand:
If every human has ended, then the human race has ended.
Analogously: If every FISON has been removed without changing the union,
then the set {F(1), F(2), F(39, ...} has been removed without changing
the union.

But the union has changed from a nonempty set to an empty one.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.