Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/19/2024 03:54 PM, Jim Burns wrote:

On 10/19/2024 2:19 PM, WM wrote:

On 19.10.2024 18:04, Jim Burns wrote:

On 10/19/2024 4:16 AM, WM wrote:

>

What you call a "set of finite ordinals" is
not a set
but a potentially infinite collection.

>
There is a general rule not open to further discussion:
Finite sets aren't potentially infinite collections.

>
Potentially infinite collections are
finite sets open to change.

>
That makes it easy.
>
No sets are open to change.
No sets are (your) potentially infinite,
No finite sets are potentially infinite.
>
The rule stands.
>

Proof:
If you double all your finite ordinals
you obtain only finite ordinals again,

>
Yes.
>

although the covered interval is
twice as large as the original interval
covered by "all" your finite ordinals.

>
No.
The least.upper.bound of finites is ω
The least.upper.bound of doubled finites is ω

>
Doubling halves the density and doubles the interval,
creating numbers which had not been doubled.
2n > n does not fail for any natural number.

>

The least.upper.bound of finites is ω

>
What ω is
is such that
k < ω  ⇔  k is a finite ordinal.
>
No k exists such that
k is a finite and k+1 > k is not a finite.
>
No k exists such that
k is an upper.bound of the finites.
>
ω is but anything prior to ω isn't
an upper.bound of the finites.
>
ω is the least.upper.bound of the finites.
>

The least.upper.bound of doubled finites is ω

>
A doubled finite is finite.
>
No k exists such that
2⋅k is a finite and 2⋅k+2 > 2⋅k is not a finite.
>
No k exists such that
2⋅k is an upper.bound of the doubled finites.
>
ω is but anything prior to ω isn't
an upper.bound of the doubled finites.
>
ω is the least.upper.bound of the doubled finites.
>
>

Isn't mathematics true?
You still haven't picked "anti or only" diagonal,
i.e. since they're together that's neither.
I mean, I see where you're going with that,
yet also I can assure you
it doesn't get all the way there.
Yet you trust Russell that's he's been to the mountain, ....
The omega is usually called a fixed-point besides
being a limit ordinal, also it's called a compactification
or one-point compactification of the integers for
the most usual sort of idea of a non-standard countable
model of integers with exactly one infinite member.
(This is not the usual Tipler's "omega-point", say,
which is not of so much relevant here though it
reflects convergence to a point at infinity as much
as it does mutual divergence, to infinity.)
Clearly it's not merely exactly a finite member, say, ....