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

On 9/27/2024 2:54 PM, WM wrote:

On 25.09.2024 20:40, Jim Burns wrote:

On 9/25/2024 11:51 AM, WM wrote:

That means NUF increases by 1
at every point occupied by a unit fraction.

>
There are numbers (cardinalities) which increase by 1
and other numbers (cardinalities), which
don't increase by 1.

>
No.

You invoke _axiom.1_

Every countable set is countable,
i.e., it increases one by one.

Axiom.1 _describes_
what you are currently discussing.
Axiom.1 means
⎛ If
⎜ the set of unit fraction can't increase by 1
⎜ then
⎝ we aren't discussing that set.
Axiom.1 does not mean
⎛ If
⎜ we are discussing the set of unit fractions
⎜ then
⎝ that set can increase by 1
We know axioms are true
by applying them only where they are true.
It does not work in the opposite direction,
not the case that sets _become_ finite
because there is an Axiom of Finity.
What you want is
to tell us we've been wrong about sets.
Axioms can't do that.
A new axiom moves the discussion.
The (now two) discussions talk past each other.

For each positive point x
for each number (cardinality) k which can increase by 1
there are more.than.k unit.fractions between 0 and x

>
That is a misinterpretation of
the law valid for small numbers.

| "If the law supposes that," said Mr. Bumble,
| squeezing his hat emphatically in both hands,
| "the law is a ass — a idiot."
|
-- Charles Dickens, "Oliver Twist"
Describe by axiom the unit fractions.
Go one axiom further and
describe √2 and the unit fractions as one more --
and you will be wrong.
Calling it a law only means the law is a ass.
It doesn't mean you and the "law" are right.
√2 → ⅟1
⅟1 → ⅟2
⅟2 → ⅟3
⅟3 → ⅟4
...
Descriptions.
⎛
⎜ A unit fraction is
⎜ reciprocal to a naturalnumber/-0
⎜
⎜ A natural.number.set≠{} is minimummed.
⎜ A natural.number≠0 is predecessored.
⎝ A natural number is successored.

For each positive point x
the number (cardinality) of
unit.fractions between 0 and x
is not
any number (cardinality) which increases by 1
Instead, it is
a number (cardinality) which doesn't increase by 1.

>
For every x NUF increases by not more than 1.

For every x>0 and x′>0
NUF increases by not more and not less than 0.
⅟⌈1+⅟x⌉ → ⅟⌈1+⅟x′⌉
⅟⌈2+⅟x⌉ → ⅟⌈2+⅟x′⌉
⅟⌈3+⅟x⌉ → ⅟⌈3+⅟x′⌉
⅟⌈4+⅟x⌉ → ⅟⌈4+⅟x′⌉
...

Each positive point is undercut by
some finite.unit.fraction.

>
Repetition
is apparently what you (JB) think mathematics is!

Sometimes.
🎜 Aleph.naught bottles of beer on the wall,
🎝 Aleph.naught bottles of beer.
🎜 Take one down, pass it around,
🎝 Aleph.naught bottles of beer on the wall.
🎜 Aleph.naught bottles of beer on the wall,
🎝 Aleph.naught bottles of beer.
🎜 Take one down, pass it around,
🎝 Aleph.naught bottles of beer on the wall.
...

Each positive point is
undercut by some finite.unit.fraction.

⎛ Assume otherwise.
⎜ Assume
⎜ δ>0 is not undercut by
⎜ some finite.unit.fraction.
⎜ 0 < δ  ∧  ¬(⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< δ)
⎜
⎜ β = greatest.lower.bound ⅟ℕᵈᵉᶠ
⎜ α < β < γ  ⇒
⎜ ¬(⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< α)  ∧  ⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< γ
⎜
⎜ ¬(⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< δ)
⎜ 0 < δ ≤ β
⎜
⎜ 0 < β < 2⋅β
⎜ ⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< 2⋅β
⎜ 0 < β ≤ ⅟k < 2⋅β
⎜ 0 < ¼⋅⅟k < ½⋅β < β ≤ ⅟k < 2⋅β
⎜ ⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< ½⋅β
⎜
⎜ However,
⎜ 0 < ½⋅β < β
⎜ ¬(⅟ℕᵈᵉᶠ ₑₓᵢₛₜₛ< ½⋅β)
⎝ Contradiction.
Therefore,
δ>0 is not.undercut by
  some finite.unit.fraction.

Prove that
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
is wrong or agree.

Prove that
∀n ∈ ℕ: 1/n > 1/(n+1) > 0
is wrong or agree that
all unit.fractions are not.first.