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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 21 Jan 2025 18:41:16 -0500
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On 1/21/25 7:44 AM, WM wrote:
> On 21.01.2025 13:17, Richard Damon wrote:
>> On 1/21/25 6:45 AM, WM wrote:
>>> All finite initial segments of natural numbers, FISONs F(n) = {1, 2, 
>>> 3, ..., n} as well as their union are less than the set ℕ of natural 
>>> numbers.
>>>
>>> Proof: Assume UF(n) = ℕ. The small FISONs are not necessary. What is 
>>> the first necessary FISON? There is none! All can be dropped. But 
>>> according to Cantor's Theorem B, every non-empty set of different 
>>> numbers of the first and the second number class has a smallest 
>>> number, a minimum. This proves that the set of indices n of necessary 
>>> F(n), by not having a first element, is empty.
> 
>> Which is a proof of ANY, not ALL together,
> 
> It is a proof of not any. The proof that not all together are necessary 
> is this: U{F(1), F(2), F(3), ...} = U{F(2), F(3), F(4), ...}.
> 
> Regards, WM

which doesn't prove your claim about the Natural Numbers.

You keep on diverting to your strawmen when you get caught.

Yes, no finite set is infinite.

No union of a finite number of finite sets is infinite.

But this doesn't say that the infinite doesn't exist, and that we can't 
make the Natural Numbers from a union of an infinite set of FISONs.

And, because FISONs are finite, no less than an infinite number of them 
should be expected to be needed.

This doesn't mean we need ALL of them, just an infinite number of them.

Your claims are just proven to be your lies.