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

On 9/14/2024 12:24 PM, WM wrote:

On 13.09.2024 23:13, Jim Burns wrote:

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

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.

>

There is no gap.

>
Yes.
>

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⟩]]]]

>
Not in what you know about these sets,
but existing,
because:
⎛ WM: There is no gap.
⎝ JB: Yes

⎛ WM: Two points are next to each other
⎜ means that
⎝ no point is between them.
ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] includes ℚ[ℤ[ℕ[⟨0,…,n⟩]]]
No positive distance r > 0 exists such that
an r.length gap exists in ℚ[ℤ[ℕ[⟨0,…,n⟩]]] where
there aren't any q′ ∈ ℚ[ℤ[ℕ[⟨0,…,n⟩]]]
⎛ ¬∃r ∈ ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]]: 0 < r  ∧
⎜ ∃q,q″ ∈ ℚ[ℤ[ℕ[⟨0,…,n⟩]]]: q+r ≤ q″  ∧
⎝ ¬∃q′ ∈ ℚ[ℤ[ℕ[⟨0,…,n⟩]]]: q < q′ < q″
Two points are a distance r > 0 apart.
There is no rational.free gap between them.
There are rationals between them.
The two points are not next to each other.
There are no two points next to each other 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⟩]]]]

>
That's not enough.

There aren't any more,
not if we are discussing a line which
includes all rational points and
includes enough points more such that
functions continuous at each point
don't skip.over anywhere.
That's enough points for physics and
many other "common sense" purposes.
However, although each ⟨0,…,n⟩ is finite,
ℕ[⟨0,…,n⟩], ℤ[ℕ[⟨0,…,n⟩]], ℚ[ℤ[ℕ[⟨0,…,n⟩]]],
and ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]] are infinite.
Describing ℕ[⟨0,…,n⟩], ℤ[ℕ[⟨0,…,n⟩]],
ℚ[ℤ[ℕ[⟨0,…,n⟩]]], and ℝ[ℚ[ℤ[ℕ[⟨0,…,n⟩]]]]
as finite,
  for example, as their having
  each nonempty.subset two.ended,
leads to gibberish and
is an incorrect description.
Note, too, that
adding more points can't make them finite.