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

On 12/31/2024 09:17 AM, 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:

>

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.

>

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.

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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.

Sets are defined by their elements,
then though all the relations of sets
are defined by all the relations of sets,
in set theory, a theory with one relation, elt.
Then there's class/set distinction, that's
another fine point ignored by all inductive bots.