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

On 10/16/2024 12:55 PM, WM wrote:

On 16.10.2024 17:34, Jim Burns wrote:

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

On 16.10.2024 10:27, WM wrote:

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.

>
Maybe.

Proofs don't need 'maybe's.

Then not all natural numbers have been doubled.

No natural number is
the first to not.have a natural.number.double.

When doubling natural numbers (finite ordinals)
we obtain natural numbers.

>
Maybe.
>

When doubling all natural numbers
we obtain only natural numbers.

>
That is impossible.

There is no first natural number from which we obtain
(by doubling) anything not.a.natural.number.
The only set of natural numbers with no first
is the empty set.
There is no ▒▒▒▒▒ natural number from which we obtain
(by doubling) anything not.a.natural.number.

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

>
That is only possible in potential infinity.

𝕆ᶠⁱⁿ is what we mean by ℕ
They are "maybe" equal to precisely the extent
that 1+1 is "maybe" 2

But there the result is worthless.

Unlike, for example, non.well.ordered ordinals,
which you (WM) don't consider worthless.