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NNTP-Posting-Date: Tue, 31 Dec 2024 20:04:49 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, effectively)
Newsgroups: sci.math
References: <vg7cp8$9jka$1@dont-email.me> <vk23m7$31l8v$1@dont-email.me>
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 31 Dec 2024 12:04:51 -0800
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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?
>>>>>
>>>>> 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.
>>
>>>>
>>>>>> The finite numbers are finite, and that is it.
>>>>>
>>>>> FISONs do not grow by unioning them. All FISONs and their unions
>>>>> stay below 1 % of ℕ.
>>>
>>>>> Proof: Every FISON that is multiplied by 100 remains a FISON that
>>>>> can be multiplied by 100 without changing this property.
>>>>>
>>>>> 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 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.

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.