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NNTP-Posting-Date: Mon, 30 Dec 2024 02:05:27 +0000
Subject: Re: Replacement of Cardinality (natural infinity)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 29 Dec 2024 18:05:40 -0800
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On 08/22/2024 07:04 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.
>
> Then, for that infinity caps the naturals as much as
> as infinity extends the naturals, is about how by
> various usual simple definitions, that the naturals
> are _compact_, the naturals make a space, and N the
> point-at-infinity is a one-point compactification of
> the space the naturals.
>
>
> So, really, some have that the naturals, always
> have these properties, being that they are compact,
> containing their compactification, and, that they
> are extra-ordinary, containing their order-type,
> either and both of those being infinite.
>
> Then, you'll welcome to aver that you've made
> no mention at all of infinity in your definition
> of finite, whole counting numbers.
>
> The natural numbers though are infinite, and infinite.
>