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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Thu, 20 Feb 2025 07:29:36 -0500
Organization: i2pn2 (i2pn.org)
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On 2/20/25 4:44 AM, WM wrote:
> On 20.02.2025 03:23, Richard Damon wrote:
>> On 2/19/25 11:52 AM, WM wrote:
> 
>>> 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.
> 
>> But induction doesn't subtract elements.
> 
> By induction we can prove what FISONs are useless and can be subtracted, 
> namely all FISONs satisfying |ℕ \ {1, 2, 3, ..., n}| = ℵo.

No, you can prove that none of them are NECESSARY, but they can do the job.



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

No, YOU (and your logic) is not useful.

>>
>> The problem is your "set" UF isn't being defined by a proper set 
>> theory, but just by Naive Set Theory.
> 
> My set is defined by induction like the set of definable numbers or 
> FISONs is defined by Peano, Dedekind, Cantor, Zermelo, Schmidt, v. 
> Neumann, Lorenzen.

But sets aren't defined "by induction", so your set isn't defined.

Please re-read those theories and see how to actually construct a set.

Of course, if you think the axiom of infinity was proven by something, 
that shows you don't understand the basic meaning of the words, since 
axioms, BY DEFINITION, can not be proven.

> 
> Regards, WM
>