Re: The set of necessary FISONs

Am Sat, 15 Feb 2025 12:46:34 +0100 schrieb WM:

On 14.02.2025 18:43, joes wrote:

Am Fri, 14 Feb 2025 13:52:27 +0100 schrieb WM:

On 13.02.2025 17:38, Jim Burns wrote:

On 2/12/2025 5:19 AM, WM wrote:

>

Induction proves all elements and even the set (Zermelo, Z).

No. Not the set. Each member of the set.

All members of the set.

No, it does not include a member called „all members”.

Induction proves the existence of the inductive set of all its members.
Induction concerns the whole set.

No, you can not substitute a set into a sentence about naturals.

Set intensionality means same.membered sets are one set.
It does not mean sets are self.members.

All members are proved. This means none remains. Zermelo obtains from
that the existence of the infinite set Z.

Wrong. There is indeed a number, albeit not natural, of FISONs you can
NOT leave out: this number is infinite, and induction doesn’t reach it.

Read, learn, try to understand.

Are you saying there are infinite naturals?

What else you snipped:

The erroneous step is from „every finite number”
to „an infinite number”.

Induction proves an infinite number.

No, it does not prove a sentence to hold for infinite numbers
substituted into it, only finite ones, which there are infinitely many
of.

From „you can leave out any finite number of FISONs” it does not follow
that „you can leave out an infinite number of them”. There are no
infinite naturals, they are all finite.

Induction is valid without any infinite sets.

Induction creates infinite sets.
If i contruct x(1) and from x(n) follows x(n+1) then the set of all
x(n) is infinite.

Sure. But no x is.

Even though there are infinitely many!

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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.