Re: The set of necessary FISONs

On 25.02.2025 11:37, joes wrote:

Am Tue, 25 Feb 2025 10:55:35 +0100 schrieb WM:

There are more FISONs than any natural number: infinitely many.

Yes potential infinity. There are more FISONs than any fixed natural
number, but the number of FISONs is a natural number

Immediate contradiction: there is no largest natural.

There is no largest FISON either. Nevertheless, the number of FISONs is a natural number, although not fixed.

 

because the sequence
{1}
{2, 1}
{3, 2, 1}
...
has no limit.

It does, actually, converge on N.

Not those which satisfy ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

 

Further induction produces no actual infinity.

WDYM "further"? Induction goes up to infinity.

Further I state that induction does not produce actual infinity.

 

Only all FISONs = natural numbers are the matter of my proof.
According to Zermelo they make up the set ℕ.

The set N is not a natural number

Of course. Nobody said so.

You pretend to remove the whole set.

I remove by induction all natural numbers. What of ℕ remains in your
opinion?

Huh?

That is not a natural number.

 

But Zermelo claims to produce the set ℕ. Is he wrong?
Hint: In fact he produces ℕ_def.

No, you are wrong.

You claimed yourself that induction produces only a finite number of
elements. Zermelo used induction.

I didn't. The *elements* are finite.

The elements n are finite and contain the number of elements {1, 2, 3, ..., n} which also are finite.
Regards, WM