Re: Replacement of Cardinality

On 8/12/2024 9:50 AM, WM wrote:

Le 11/08/2024 à 19:56, Jim Burns a écrit :

On 8/11/2024 8:29 AM, WM wrote:

Le 10/08/2024 à 19:28, Jim Burns a écrit :

What causes an exception:
nₓ ∈ ℕ without ⅟(nₓ+1) ?

>
The end of the positivee axis.

>

∀n ∈ ℕ: 1/n - 1/(n+1) > 0

>
∀n ∈ ℕ: 1/n > 1/(n+1) > 0

 If 1/(n+1) exists.

∀n ∈ ℕ: 1/n - 1/(n+1) > 0

asserts 1/(n+1) exists, '∀n ∈ ℕ'
along with asserting other things.

Each positive unit fraction is not
the first positive unit fraction.
>
What causes an exception: nₓ ∈ ℕ:
⅟nₓ > 0 without ⅟(nₓ+1) > 0 ?

>
The end of the positive axis.

There is no ⅟nₓ before the end of the positive axis
without ⅟(nₓ+1) before the end of the positive axis.
⎛ Assume otherwise.
⎜ Assume ⅟nₓ > 0 ≥ ⅟(nₓ+1)
⎜
⎜ nₓ⋅⅟nₓ = 1
⎜ ⅟nₓ > 0  ∧  1 > 0  ⇒  nₓ > 0
⎜ nₓ > 0  ⇒  nₓ+1 > 0
⎜ (nₓ+1)⋅⅟(nₓ+1) = 1
⎜ nₓ+1 > 0  ∧  1 > 0  ⇒  ⅟(nₓ+1) > 0
⎝ Contradiction.
There is no exception to not.being the lower end.