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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 19 Feb 2025 21:23:45 -0500
Organization: i2pn2 (i2pn.org)
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On 2/19/25 11:52 AM, WM wrote:
> Am 18.02.2025 um 19:22 schrieb Jim Burns:
>> On 2/18/2025 10:22 AM, WM wrote:
> 
>> You (WM) have located a problem.
>> You try to work around it by not.mentioning it.
>> What you're not.mentioning is your assumption
>> that none of these sets are infinite.
> 
> Wrong. Induction has been invented for infinite sets.
> Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch 
> des folgenden ... Axioms. [Zermelo: Untersuchungen über die Grundlagen 
> der Mengenlehre I, S. 266]
> 
>> ⋃{F} = ℕ
> 
> Proof: If UF = ℕ is assumed, then F(1) can be omitted without changing 
> the union of the remainder. And if F(n) can be omitted without changing 
> this union, then also F(n+1) can be omitted without changing this union. 
> That makes the omitted FISONs the inductive collection of all FISONs and 
> proves the implication: If UF = ℕ, then { } = ℕ.
> 
> Regards, WM

But induction doesn't subtract elements.

All you have shown is that the no element in the set of all FISON is 
neeeded.

The problem is your "set" UF isn't being defined by a proper set theory, 
but just by Naive Set Theory.

Remember, "Induction" doesn't build a set, it is a test to see that a 
set contains the Natural Numbers (or their equivalent).

But that is just a fact that you don't understand, because your think 
"axioms" are proven.