Re: The set of necessary FISONs

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.
 Regards, WM

Which is a proof of ANY, not ALL together,
Your logic just can't handle that infinite set, so comes up with illogical answers.
Your logic is based on the proven incorrect Naive Set Theory, not any of the modern set theories that fixed it.