Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively)

On 30.12.2024 17:08, Richard Damon wrote:

On 12/30/24 3:44 AM, WM wrote:

On 30.12.2024 01:36, Richard Damon wrote:

On 12/29/24 5:31 PM, WM wrote:

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My theorem: Every union of FISONs which stay below a certain threshold stays below that threshold.
Find a counterexample. Don't claim it but prove it. Fail.

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But what does that prove?

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That proves the existence of dark numbers.

 How?
 It shows that there are numbers you didn't look at, but you admit to not looking at all the numbers.

Neither can you look at more numbers.

 

The finite numbers are finite, and that is it.

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FISONs do not grow by unioning them. All FISONs and their unions stay below 1 % of ℕ.

Proof: Every FISON that is multiplied by 100 remains a FISON that can be multiplied by 100 without changing this property.
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If ℕ is an actually infinite set then it is not made by FISONs.

That is what I have been telling you for a long time.

 Who said it is made of FISONs, It could be said that it is the union of *ALL* of the infinite set of FISONs, but the union of an infinte set of things can be bigger than any member of that infinite set.

That is a belief shared by real fools only. Every union of FISONs which stay below a certain threshold stays below that threshold. All FISONs stay below 1 % of ℕ. Proof: Every FISON that is multiplied by 100 remains a FISON that can be multiplied by 100 without changing this property.
Regards, WM