Re: The set of necessary FISONs

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

On 25.02.2025 10:17, joes wrote:

Am Tue, 25 Feb 2025 10:02:47 +0100 schrieb WM:

On 24.02.2025 19:01, joes wrote:

Am Mon, 24 Feb 2025 15:59:18 +0100 schrieb WM:

>

These axioms can be applied to show that all FISONs can be removed.

Nope. Any number of them can be removed. Indeed, removing all changes
the union, as you have admitted.

There are not more than any number.

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.

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

It does, actually, converge on N.

Further induction produces no actual infinity.

WDYM "further"? Induction goes up to 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?

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.

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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.