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

On 9/13/2024 1:20 PM, WM wrote:

On 12.09.2024 22:54, Jim Burns wrote:

On 9/12/2024 1:48 PM, WM wrote:

On 12.09.2024 14:29, FromTheRafters wrote:

After serious thinking WM wrote :

But the endpoints of open intervals
will remain dark forever.

>
Open intervals simply don't contain endpoints,
dark or otherwise.

>
You simply don't know about them.

>
The boundary ∂A of set A is

>
not a result of understanding points.

If you decide ∂ᵂᴹA ≠ ∂A  then
whatever you say about ∂ᵂᴹA is not about ∂A

Fact is that the real axis is nothing but its points.

Yes.
ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] holds
points.between ratios of countable.to numbers.
for countable.to n′
⟨0,…,n′⟩ subsets are minimummed or empty.
k ∈ ⟨1,…,n′⟩ ⇒  ⟨0,…,n′-1⟩ ∋ j: j+1 = k
j ∈ ⟨0,…,n′⟩ ⇒  ⟨1,…,n′+1⟩ ∋ k: k = j+1

There is no gap.

Yes.
For each split S of ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]]
x is between S and (ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] \ S)

Every point has a next point but
next to defined points are dark points.

No.
Next.to.defined points are not.in ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]]

Next to every defined points there are ℵo dark points.
This configuration cannot be changed.

Whatever it is you are talking about,
we are talking about ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]]