Re: The set of necessary FISONs

On 26.02.2025 12:53, joes wrote:

Am Wed, 26 Feb 2025 12:40:13 +0100 schrieb WM:

On 25.02.2025 16:31, joes wrote:

 

The set of FISONs does not change.

It is the set of definaed natural numbers. It depends on the systemlike
the set of known prime numbers.

Nope. There are inf. many ("defined") FISONs.

Defining is a human activity. Never an infinity will be defined.

 

I state that induction does not produce actual infinity.

Whatever. Induction does not include infinity, although it does prove
infinitely many formulas containing finite numbers, but not the one
formula saying "all". It is not that strong.

Then consider only the natural numbers which are subject to induction.

Yes, please do that, and not the number Infinity ("all naturals").

The number of naturals is not itself a natural number. So by "removing
every natural" you are removing a non-natural number. Get this?

Of course: Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden, seinem wesentlichen Inhalte von Herrn Dedekind herrührenden Axioms. ... Der Bereich enthält mindestens eine Menge Z, welche die Nullmenge als Element enthält und so beschaffen ist, daß jedem ihrer Elemente a ein weiteres Element der Form {a} entspricht ... Die Menge Z_0 enthält die Elemente 0, {0}, {{0}}, usw. und möge als "Zahlenreihe" bezeichnet werden, ...  Sie bildet das einfachste Beispiel einer "abzählbar unendlichen" Menge. [E. Zermelo: Untersuchungen über die Grundlagen

 

The other way around: the number of finite elements is infinite.
Anyway, no infinite element is produced.

Right. Zermelo produces all finite elements, no infinite element. So do
I. That is the set.

And the number of these elements is not finite and not an element.

He ascertains the set.
Regards, WM