Re: Replacement of Cardinality

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.
So, here it's not unlike just a usual premise of the counter-intuitive,
that there is exhaustion, in terms, besides the unbounded, in terms.
Then, what happens is the constant flailing makes a combination of
a sort of straw-man, and a soft-ball, that the straw-man is that
anyone who makes or even refers to the same point is tarred with
the same brush, of that being a useless repetitive crank, and as
well it soft-balls the given "blessed" argument otherwise, showing
how that over so many times it's defended, from a weak attack,
that it must be so very strong, the "soft-ball" vis-a-vis the
"hard-ball", where the soft-ball is not a fair challenge.
Then, besides, for example, that "the definition", of natural
integers, has that for example "Russell's paradox" is applied
to otherwise finite sets and results _always_ whatever's the
"infinite" so resulting _also in it_, "the definition", that
"the definition" that all the members are finite is _false_,
has that then as an _axiom_, that's given, that all members
are finite is an _axiom_, then has of course that there's
another model of the integers where that's only a fragment
and another where that's only an extension, and that there
are models of integers that do _not_ meet "the definition",
and particularly if they do at all it's an _axiom_.
So, this makes Russell up to be a real flake and hypocrite,
and having lied to you, or confirmed your biases or given
an easy out from otherwise the _required_ larger, fuller dialectic.
So, you want to know this so that later people who fully do,
will not have so plainly hood-winked you about it.
Anyways there are definitely models of _natural_ integers
and there's nothing wrong with them and all the facts according
to at least one _law of large numbers_, can be so demonstrated,
that have infinitely-grand among their infinitely-many elements.
And, it's a result of logic that there are _not_ "standard models"
of natural integers, at all, that "standard" meaning Archimedean
pretty much are _only_ either fragments or extensions.
So, one of the cheapest things in the world these days,
is to set up a billion chat-bots full of non-sense,
and hit them with the straw-man soft-ball, fallacy.
Anyways, looking around the Wiki, for example, for
"non-standard analysis", "sur-real numbers", and so
on, and a lot of other concepts with regards to
"non-standard" and "infinite", "infinitary" numbers,
you'll find that even in the past few years that
what was entirely one-sided as that these inquitive
and even required concepts like exhaustion versus
the unbounded, were just "wrong" or "not even wrong",
that those are setup themselves to be just like so.
So, it's better not to be a hypocrite _now_,
so you won't be just wrong, _then_.