Re: The set of necessary FISONs

On 1/22/25 10:04 AM, WM wrote:

On 22.01.2025 13:10, Richard Damon wrote:

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.
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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.

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Which is a proof of ANY, not ALL together,

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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), ...}.

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

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It proves what I said: not all are required.

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

 Can't you read?
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.
 

But There is no single UF(n) that equals N, because you can ony get there from the union of an infinite set of FISONs.
Your question abourt the "first" necessary fission is an invalid nquestion, as it is the same as asking for the highest Natural Number, which doesn't exist,
Your "logic" is just built on the presumption that you can ask illogical questions and assume an answer, that is the flaw of "Naive" Set Theory, and your "Naive" logic, and "Naive" mathematics.
Your ignorance thinks that the infinite is finite, just bigger than you thought of, which just shows your stupidity, and that you are too stupid to know your stupidity,

Regards, WM