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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Thu, 22 Aug 2024 12:17:10 -0700
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On 8/21/2024 7:15 PM, Ross Finlayson wrote:
> On 08/21/2024 06:49 PM, Richard Damon wrote:
>> On 8/21/24 9:23 PM, Ross Finlayson wrote:
>>> On 08/21/2024 05:10 PM, Richard Damon wrote:
>>>> On 8/21/24 8:32 AM, WM wrote:
>>>>> Le 21/08/2024 à 13:32, Richard Damon a écrit :
>>>>>> On 8/21/24 6:44 AM, WM wrote:
>>>>>>> Le 20/08/2024 à 23:25, FromTheRafters a écrit :
>>>>>>>> WM explained :
>>>>>>>>> Le 20/08/2024 à 12:31, FromTheRafters a écrit :
>>>>>>>>>> on 8/19/2024, Richard Damon supposed :
>>>>>>>>>
>>>>>>>>>>> You can not derive a first number > 0 in any of the Number
>>>>>>>>>>> System that we have been talking about, Unit Fractions,
>>>>>>>>>>> Rationals or Reals, so you can't claim it to exist.
>>>>>>>>>>
>>>>>>>>>> Not in their natural ordering.
>>>>>>>>>
>>>>>>>>> Dark numbers have no discernible order. It is impossible to find
>>>>>>>>> the smallest unit fraction or the next one or the next one. It is
>>>>>>>>> only possible to prove that NUF(x) grows by 1 at every unit
>>>>>>>>> fraction. It starts from 0.
>>>>>>>
>>>>>>>> Normally, the unit fractions are listed in the sequence one over
>>>>>>>> one, one over two, one over three etcetera. There is a first but no
>>>>>>>> last. Now you have started from the wrong 'end'
>>>>>>>
>>>>>>> No, I have started from the other end. It exists at x > 0 because
>>>>>>> NUF(0) = 0.
>>>>>
>>>>>> But the other end doesn't "begin" with a first Natural Number Unit
>>>>>> fraction, if it has a beginning that will be a trans-finite number.
>>>>>
>>>>> No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all and
>>>>> only reciprocals of natural numbers.
>>>>>
>>>>> Regards, WM
>>>>>
>>>>>
>>>>
>>>> Can't be, because if it WAS 1/n, then 1/(n+1) would be before it, and
>>>> thus your claim is wrong. If 1/(n+1) wasn't smaller than 1/n, then we
>>>> just have that 1/n - 1/(n+1) wouldn't be > 0, so it can't be.
>>>>
>>>> BY DEFINITION, there is no "Highest" Natural Number, if n exists, so
>>>> does n+1, and your formula says you accept that n+1 exists, or you
>>>> couldn't use it.
>>>>
>>>> If you don't have that property, you don't have the Natural Numbers.
>>>>
>>>> PERIOD.
>>>>
>>>> DEFINITION.
>>>>
>>>> If you claim your mathematics say it can't be, then your mathematics
>>>> were just proven to not be abble to handle the unbounded set of the
>>>> Natural Numbers.
>>>>
>>>> Sorry, that is just the facts.
>>>
>>> Perhaps you'd like to learn about Conway's "Surreal Numbers",
>>> which make one for omega and further fill out a "non-Archimedean"
>>> field, that otherwise it's the same usual definition since Archimedes,
>>> in terms of field reals, the Archimedean field.
>>
>> I know about the surreal numbers. But that would just further confuse
>> the idiot. Just pointing out that his "dark numbers' can't be a part of
>> the Natural Numbers breaks his logic.
>>
>> Of course, since is logic can't even handle the simple unbounded Natural
>> Numbers, he has no hope in understanding the Surreal numbers.
>>
>>
>>>
>>> So, the usual idea of a "non-Archimedean field" is Conway's "sur-reals".
>>>
>>>
>>> Then, there's also a logical argument that if there are infinitely-many
>>> integers then that there are concomitantly infinitely-grand integers,
>>> that it belies the definition and makes it so that inductive inference
>>> by itself doesn't suffice, that "Eudoxus doesn't suffice", that
>>> "if there are infinite integers there are infinite integers"
>>> and so on, that logic automatically provides.
>>>
>>> So, you can find ways to make the points that there is a
>>> fixed-point, to the integers, that the integers _are_ compact,
>>> that there is an infinite member of otherwise the finite set,
>>> and these kinds of things, while at the same time the usual
>>> formalism's only use is that inductive inference never ends,
>>> abruptly.
>>>
>>>
>>
> 
> The thing is that let's say for example that WM is just another
> one of these linguistic chat-bots, just saying that he never really
> shows up having learned anything or changed, mostly, always just
> stuck at the same point.
[...]

The Dr. Trolls-a-Lot 9000 models?

lol ;^D