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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Thu, 9 Jan 2025 13:20:16 -0800
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On 1/8/2025 5:26 PM, Richard Damon wrote:
> On 1/8/25 5:33 PM, Chris M. Thomasson wrote:
>> On 1/8/2025 4:25 AM, Richard Damon wrote:
>>> 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.
>>
>> [...]
>>
>> WM must think that Aleph_0 is some really big natural number.
>>
>> ;^)
>>
> 
> well, he admitted that he doesn't think infinity exists, so those 
> numbers that he uses must be finite, just bigger than you can think about.

Seems so! It simply must be a side effect of hyper finite thought?