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

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.