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

On 07.01.2025 21:08, FromTheRafters wrote:

WM was thinking very hard :

On 07.01.2025 15:05, FromTheRafters wrote:

WM wrote :

On 07.01.2025 10:51, FromTheRafters wrote:
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First several von Neumann ordinals

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v. Neumann was bright but not bright enough.
Why should we use the nomenclature of his disproven theory?

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Disproven?
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0     =     {}     =     ∅
1     =     {0}     =     {∅}
2     =     {0,1}     =     {∅,{∅}}
3     =     {0,1,2}     =     {∅,{∅},{∅,{∅}}}
4     =     {0,1,2,3}     =     {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
=====================================
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Notice they start at zero (emptyset)

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It would be more important to reach the full set.

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It does, in the infinite union.

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It does not because the infinite union cannot be larger than every unioned FISON.

 Just because you saya so?

No, because there is nothing that could increase the union above all numbers supplied by the FISONs.

 

Every unioned FISON is smaller than any definable fraction of the full set ℕ.

 So what?

What should cause a closer approximation?
Find a FISON that surpasses 1 % of ℕ. Fail.
Find a union of FISONs that surpasses 1 % of ℕ, i.e. which multiplied by 100 is |ℕ|. Fail. However that must be a finite union, because the union surpassing 20 % of ℕ must be larger. That cannot happen with an infinite union.
Regards, WM