Re: Replacement of Cardinality (natural infinity)

On 08/23/2024 03:02 PM, Ross Finlayson wrote:

On 08/22/2024 11:53 PM, Chris M. Thomasson wrote:

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:

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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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Humm... infinitely-grand? Kind of sounds like "really, _really_ hyper
large and grand, I swear!"... There is no largest natural number.
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There's an infinite, though.
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Then, 'big, biggest, and bigger' has models to fill its roles.
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Of course computer words are finite width
so you never have to worry about except
there being arbitrarily large bounds.
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Reading Boethius or something about the Trinity yet
about also for the technical and logical why in
Boethius it's like "the Sun, the Sun, the Sun", or
"the sword, the brand, the blade", here this sort
of relative then superlative and comparative, makes
for something like Infinity and something like Zero
and something like One that usual meager models of
the least often aren't themselves really _complete_
because there's a real complete _replete_.
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For example, the usual simplest notion of limit,
frames in the negative that something's
not-not-close enough, yet classically there's
motion and objects actually do part and meet.
So, a fuller completion of the infinite limit
and not just "a case for a limit existing" instead
"the case that the limit exists", here as what
was framed as "an inductive supertask" yet is
as simple as there is "a deductive inference",
has that the infinite-many are real in any matters
of real dynamics, with infinitely-many orders
of acceleration, so, there being a big-end and
a little-end of the resulting series, then
"it must be" that there's naturally infinity,
and here a "natural infinity".
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Despite the usual "I can't count that" then
it's like "well, obviously one must think it".
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Then, it's like that that's the only thing like
that that there is, extra-ordinary paleo-classical
super-standard natural infinity, mathematical infinity.
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It's not that it's a replacement for cardinality,
it's just that cardinality isn't a replacement for it.
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