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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Wed, 8 Jan 2025 07:25:38 -0500
Organization: i2pn2 (i2pn.org)
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On 1/8/25 4:04 AM, WM wrote:
> On 08.01.2025 00:30, Richard Damon wrote:
>> On 1/7/25 7:51 AM, WM wrote:
> 
>>>> No, ZF doesn't have as an axiom that the set of Natural Numbers exist.
>>>
>>> AoI: There exists an infinite set S.
>>
>> Which isn't that the NATURAL NUMBERS are an infinite set.
> 
> The infinite set has been designed by Zermelo according to Dedekind's 
> definition of the natural numbers, as Zermelo noted. https:// 
> gdz.sub.uni-goettingen.de/id/PPN235181684_0065? 
> tify=%7B%22pages%22%3A%5B276%5D%2C%22pan%22%3A%7B%22x%22%3A0.461%2C%22y%22%3A1.103%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.884%7D

So? it doesn't mean that ZF has made it an axiom that the set of Natural 
Numbers exist, he has made his Axiom of Infinity to be designed so that 
the existance of the Natural Numbers can be derived from it. You confuse 
cause from effect.

IT is good to know where you are trying to go, or it can be hard to get 
there.

IF you claim that the axiom of infinity is NOT valid, then why do you 
keep on using the results of it in your logic? One of your problems is 
you have ADMITTED that you "logic" isn't axiomized (since you admit you 
can't provide a set of actual axioms to define it) and thus you admit 
that your "logic" isn't actually LOGIC. Your "Theorem" can't be actually 
a Theorem, as you don't have any axioms on which to prove it.

> 
>>>>> Every union of FISONs {1, 2, 3, ..., n} which stay below this 
>>>>> threshold stays below this threshold too.
>>>>
>>>> But not the union of *EVERY* FISON, the FULL INFINITE set of them. 
>>>
>>> All are below 1 %.
>>
>> No, 
> 
> Show one FISON that is larger than 1 %.

There isn't one, but that doesn't matter,

Show me a Natural Number that is bigger than Aleph_0 / 100?

It doesn't exist, because Aleph_0 is infinite, and an infinite number 
divided by ANY finite value is still that infinite value, and thus there 
is no finite value greater than that.

This doesn't give you your "dark numbers" as "non-defined finite 
numbers", but shows that your logic is just broken.

>>
>> So, all you prove is that a finite subset of an infinite set doesn't 
>> cover all of the infinite set.
>>
>> Your logic can't handle *EVERY* FISON at once.
> 
> Show one FISON that is larger than 1 %.

Doesn't exist, and doesn't need to.

Again, you are just proving your stupidity.

The creation of the infinite set of N isn't based on having an infinite 
member in it, but on having an infinite number of members, that build 
something having properties different from any of its individual member.

Your "logic" can't handle the difference between the set itself, and the 
individual members, and that is what make it blow itself up into 
smithereens on its inconsistancies.

> 
> Regards, WM
> 
> 
>>
>>>
>>>> SOething your "logic" can't handle,
>>>
>>> Luckily.
>>
>> So, you think you are "lucky" to be ignorant?
>>
>> I guess that just shows how your logic works.
>>
>>>
>>> Regards, WM
>>>
>>
>