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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality (natural infinity)
Date: Thu, 22 Aug 2024 23:53:08 -0700
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On 8/22/2024 7:04 PM, Ross Finlayson wrote:
> On 08/22/2024 06:23 PM, Richard Damon wrote:
>> On 8/22/24 8:19 AM, WM wrote:
>>> Le 22/08/2024 à 02:10, Richard Damon a écrit :
>>>> On 8/21/24 8:32 AM, WM wrote:
>>>
>>>>> No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all
>>>>> and only reciprocals of natural numbers.
>>>
>>>> Can't be, because if it WAS 1/n, then 1/(n+1) would be before it,
>>>
>>> That is tadopted from definable numbers. It is not true for all dark
>>> numbers.
>>>
>>> Regards, WM
>>
>> But you claim the Natural Numbers, which define the whole infinite
>> sequence.
>>
>> Every Natural Number has, by its definition, a successor, so there is
>> not last one.
>>
>> And, by your own definitions, if you can use the number individually,
>> which you did for 1/n and thus n, you can use "normal mathematics" on
>> it, that that says that if n exists, so does n+1 as we have a definition
>> for that number, thus there is not last definable number.
>>
>> Yes, if 1/n was a "dark number" we might not be able to find the n+1 in
>> the dark numbers, but none of those are Natural Numbers, but must be
>> some beyond-finite set of numbers.
> 
> One might imagine that the definition of "natural"
> numbers is exactly insofar as what exist, "natural"
> in the sense of being an entire model of integers.
> Then, "whole" numbers are usually the word for
> integers, the counting integer or whole numbers,
> that "natural" integers, for example, often include
> zero, then as with regards to whether they include
> infinity, or not.
> 
> So, some have for something like extra-ordinary sets,
> that N = N+, that is to say, being "merely infinite" is a
> big enough ordinal that it contains itself, and that
> that's automatically "natural" because there's not
> even anything that can be done about it, it arises
> from naive and thus natural quantification over
> the elements, there is no rule number one barring it,
> so, "naturals" might have infinitely-grand members.

Humm... infinitely-grand? Kind of sounds like "really, _really_ hyper 
large and grand, I swear!"... There is no largest natural number.