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

On 10/16/2024 4:50 AM, WM wrote:

On 16.10.2024 10:27, WM wrote:

On 15.10.2024 02:00, Jim Burns wrote:

On 10/14/2024 4:03 PM, WM wrote:

On 14.10.2024 21:47, Jim Burns wrote:

On 10/14/2024 2:07 PM, WM wrote:

Try again
considering the darkness of most numbers.

>
⎛ A set S of ordinals holds first.S or is empty.

>
That is true for visible ordinals only.

That description is irrelevant to
the mathematics of actual infinity
which is what I pursue.

>
Perhaps
there is a perfectly reasonable explanation
why you (WM) can only pursue that
while sounding as though you don't know
what an ordinal is.
>
So, what is that perfectly reasonable explanation?

>
There is a general rule not open to further discussion:

Triangles have three corners.
⎛ A set S of ordinals holds first.S or is empty.
⎜
⎜ Ordinal ξ has successor ξ+1 = ξ∪{ξ}
⎜
⎜ _Finite_ non.0 ordinal k
⎜  has predecessor k-1:  (k-1)+1 = k  and
⎜ each prior non.0 ordinal j < k
⎝  has predecessor j-1:  (j-1)+1 = j

There is a general rule not open to further discussion:
When doubling natural numbers
we obtain natural numbers which
have not been doubled.

>
CORRECTION:
When doubling natural numbers
we obtain even numbers which have not been doubled.

The set S of ordinals which
are finite  and for which
their double is not finite
doesn't hold first.S = 𝔊
⎛ Proof:
⎝ not( countable.to 2⋅(𝔊-1) ∧ not.countable.to 2⋅𝔊 )
The set S of ordinals which
are finite  and for which
their double is not finite
is empty.
When doubling natural numbers (finite ordinals)
we obtain natural numbers.
When doubling all natural numbers
we obtain only natural numbers.

In potential infinity
we obtain more even natural numbers
than have been doubled.
In actual infinity
we double ℕ and obtain
neither ℕ or a subset of ℕ.

There is a general rule not open to further discussion:
The natural numbers ℕ equal the finite ordinals 𝕆ᶠⁱⁿ
Perhaps
there is a perfectly reasonable explanation
why you (WM) can only pursue your actual infinity
while sounding as though you don't know
what a natural number is.