Re: Replacement of Cardinality

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.
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.