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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 22 Jan 2025 07:10:32 -0500
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On 1/22/25 5:34 AM, WM wrote:
> On 22.01.2025 00:41, Richard Damon wrote:
>> 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), ...}.
> 
>> which doesn't prove your claim about the Natural Numbers.
> 
> It proves what I said: not all are required.

And no one said you needed to take the Union of ALL the FISONs, just ALL 
of an infinite set of FISONs.

> 
>> 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.
> 
> 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 FISONs,
> by not having a first element, is empty.

But the "set of necessary FISONs" is a set from the disproved Naive Set 
Theory, as you "logic" is just a Naive Mathematics with Naive Logic.

>>
>> And, because FISONs are finite, no less than an infinite number of 
>> them should be expected to be needed.
> 
> Infinitely many fail like infinitely many traiangles would fail.

But Infinitely many also succeed, and thus YOUR logic is what failed.

>>
>> This doesn't mean we need ALL of them, just an infinite number of them.
> 
> Contradicted by Cantor's theorem.

Nope, you can have an infinite set not needed and another infinite set 
needed.

Aleph_0 / 2 is still Aleph_0.

> 
> Regards, WM
>