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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, effectively)
Date: Tue, 31 Dec 2024 19:05:24 +0100
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On 31.12.2024 16:37, Richard Damon wrote:
> 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?
>>>>
>>>> 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.
> 
> Of course I can look at "more" numbers, as I can look at the one after 
> where you stopped.

I don't stop but know that every FISON is finite as its name says and 
covers at most 1 % of ℕ. If you don't agree find a counterexample.

>>>> 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.
> 
> But there is no Natural Number that isn't in a FISON, which is NOT what 
> you have been saying.

You cannot find a natural number that isn't in a FISON. But there must 
be more because FISONs cover less than 1 % of ℕ.

> The problem is your logic can't have "Every" FISON, because "every 
> FISON" is an infinite set of them,

Every mathematician knows that every FISON is finite as its name says 
and covers at most 1 % of ℕ.

Regardes, WM