Re: The set of necessary FISONs

Am Tue, 25 Feb 2025 15:18:08 +0100 schrieb WM:

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.

Impossible. The set of FISONs does not change.

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.

Those what? The naturals are not sequences.

Further induction produces no actual infinity.

WDYM "further"? Induction goes up to infinity.

Further 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.

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?

You cannot remove a non-natural number of elements, such as all of them.

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.

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

--
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.