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

On 10.10.2024 22:47, Jim Burns wrote:

On 10/9/2024 11:39 AM, WM wrote:

On 09.10.2024 17:11, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

 

No.
When we *in actual infinity*
multiply all |ℕ|natural numbers by 2,
then we keep |ℕ| numbers
but only half of them are smaller than ω,
i.e., are natural numbers.
The other half is larger than ω.

 You (WM) are treating ω as though it is (our) finite.

No, I am treating the natnumbers as though all could be treated and ω as though it could be doubled.

>
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2} .

 Each γ≠0 preceding ω is (our) finite.

There are two alternatives: Either doubling creates natnumbers, then they are not among those doubled, then we have not doubles all. Or we have doubled all but then larger numbers have been created.

β < ω  ∧  γ < ω  ⇒  β+γ < ω
 ⎛ Assume a counterexample.
⎜ Assume
⎜ β < ω  ∧  γ < ω  ∧  β+γ ≥ ω

Dark numbers have no discernible individuality.

β < ω  ∧  γ < ω  ⇒  β×γ < ω
 ⎛ Assume a counterexample.
⎜ Assume
⎜ β < ω  ∧  γ < ω  ∧  β×γ ≥ ω

Dark numbers have no discernible individuality.
Regards, WM