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

On 1/20/25 7:26 AM, WM wrote:

On 19.01.2025 13:32, Richard Damon wrote:

On 1/19/25 5:43 AM, WM wrote:

 

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So, you admit that your FISONs are worthless?

 No. FISONs are useful for showing the difference between the potentially infinite collection of definable numbers and the set ℕ.

Which shows that there purpose is to LIE.

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Name any FISON that is required to produce ℕ by a union of FISONs.

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There is non, because you need almost ALL of the FISONS to do that,

 Gobbledegook.

No, just your stupidity.

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The set of required FISONs is well defined because for every FISON we can decide whether it is required in the union.

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But YOU can only do that for a finite number of them, not almost all of them.

 Wrong. Every set of ordinal numbers has a first one.

Right, but not necessarily a last one.
You don't seem to understand that difference.
FISONs always have a last element, as they are (by definition) finite, and finite sets have a first and last element.
INFINITE sets do not need that. The set of Natual Numbers does not have a highest member.

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Further, according to set theory, every well defined set of ordinal numbers has a first element. The FISONs F(n) must obey this theorem because they can be ordered by their greates ordinals n.
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You simply violate this fact.

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Where?

 By not giving a first one.

Because the set you ask for isn't a FISON, or a finite set of them.
All you are proving is your logic needs to handle infinite sets, but can't so it blows itself up.

 

The error is your handwaving claim that infinitely many FISONs are required. Infinitely many failures will not yield a success.
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And YOUR "handwaving" just shows your stupidity, and inability to understand your stupidity.
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I don't "Handwave" the claim of needing infinitely many FISONs,

 I can prove that no FISON is required. That is mathematics.

But you need SOMETHING, and without your FISONs you have nothing.
The problem is you need to establish the rules you are working under, but your Naive Mathematics can't be fully axiomized because it is based on an inconsistant footing.

 Regards, WM