Re: The set of necessary FISONs

Am Wed, 26 Feb 2025 13:49:49 +0100 schrieb WM:

On 26.02.2025 13:22, joes wrote:

Am Wed, 26 Feb 2025 13:04:10 +0100 schrieb WM:

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.

They have, by the power of induction. (Infinity itself has been defined
separately.)

The power of induction proves ∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo

It does not prove |N \ U{F(n): n e U(F)}| = oo.

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?

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.

The *set* is not in the set.

No, why should it? But the set is proven to exist by induction.

Indeed, that you keep referring to removing |N| elements instead
of n e N is a mistake on your part. The existence has never been
disputed.

--
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.