Re: Replacement of Cardinality (natural infinity)

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:

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No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all
and only reciprocals of natural numbers.

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Can't be, because if it WAS 1/n, then 1/(n+1) would be before it,

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That is tadopted from definable numbers. It is not true for all dark
numbers.
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Regards, WM

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But you claim the Natural Numbers, which define the whole infinite
sequence.
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Every Natural Number has, by its definition, a successor, so there is
not last one.
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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.
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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.

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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.
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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.
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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.
>
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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.
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Then, you'll welcome to aver that you've made
no mention at all of infinity in your definition
of finite, whole counting numbers.
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The natural numbers though are infinite, and infinite.
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