Sujet : Re: Replacement of Cardinality
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.mathDate : 12. Aug 2024, 18:44:10
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <fcd3f5f1-fd6e-44ac-823d-fa567d5fb9ba@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11
User-Agent : Mozilla Thunderbird
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.
…