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

On 12/31/24 12:17 PM, FromTheRafters wrote:

Richard Damon was thinking very hard :

On 12/30/24 4:38 PM, WM wrote:

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.

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

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Neither can you look at more numbers.

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Of course I can look at "more" numbers, as I can look at the one after where you stopped.
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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.

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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.

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

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But there is no Natural Number that isn't in a FISON, which is NOT what you have been saying.
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You don't seem to understand the difference between a set and its members.

 He has in the past explicitly stated such. Something like 'a set is nothing more than its elements' -- I don't think he has changed his tune.

Which is just a categorical error, showing that he doesn't know what he is talking about.
Yes, SOME of the properties of the set come from properties of its members, but just because we can add together two numbers, doesn't mean we can arbitrarily add together two sets of numbers, or even know what that would mean to do.