Re: The set of necessary FISONs

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.

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