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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 26 Jan 2025 17:51:18 -0500
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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

Thinking a bit about this, the "Set of Necessary FISONs" will be empty, 
becuase no particular FISON is needed, you just need an infinite set of 
them.

A simple proof of this is that we can build up the set at least two 
different ways using two infinite sets with no members in common.

The first set it the set of all ODD FISONs, i.e. FISONs whoes highest 
member is an odd number.

There is no Natural Number not covered by this union, as for every 
Natural Number, either it is itself odd, and thus part of its own FISON, 
or the number one greater than itself (which does exist) will be odd, 
and this number will exist in that FISON, and thus in the union of them.

We can also do that with the set of even FISONs.

For a FISON to be in the set of "Necessary" it would need to be in EVERY 
set that meets the requriment, but since no set does, there are no 
"necessary" FISONs.

Just like there are no "necessaery" Numbers to have the sum of the set 
to be zero. We can generate the sum of zero many different ways, and 
have no one necessary value for the set.

The one requirement we can see is that the set of FISONs being unioned 
together must be infinite, as any finite number of finite set unioned 
together results in a finite set, so can't cover an infinite set.

And, it seems, as long as you DO have an infinite set of FISONs that you 
are unioning together, you will cover the full set of Natural Numbers.

Note, the fact that on one set is "necessary" doesn't mean what you try 
to make it mean, but that is because you logic just doesn't understand 
that nature of the infinite, and you are too stupid to understand that 
limitation in your logic, which makes your brain just a giant black hole 
that no intelegence can get out of.