Re: how

On 6/5/2024 4:39 PM, WM wrote:

Le 05/06/2024 à 21:28, Jim Burns a écrit :

On 6/5/2024 6:43 AM, WM wrote:

Le 04/06/2024 à 23:31, Jim Burns a écrit :

On 6/4/2024 10:10 AM, WM wrote:

Le 04/06/2024 à 04:07, Jim Burns a écrit :

Assumption (2.) describes
objects in our familiar arithmetic.

>
That is true

>
Thank you.

>

Assumption 2.
ℕ⁺ holds all.and.only
numbers countable.to by.1 from.0

>
Of course.

>
Assumption 2 in detail.

>
does not contradict that
contrary to Cantor's claim
most natural numbers are uncountable,
although with n
also n^n^n is countable.

(2.)
For numbers i,j,k countable.to by.1 from.0
i before j before k  implies  i before k
exactly one is true of
'j before k', 'j after k', 'j equals k'
i < j < k  ⇒  i < k
j <≠≯ k  ∨  j ≮=≯ k  ∨  j ≮≠> k
| Assume ∀ᴺj ∃ᴺk≠j: j<k
|
| ¬∃ᴺj ¬∃ᴺk≠j: j<k
| ¬∃ᴺj ∀ᴺk≠j: ¬(j<k)
| ¬∃ᴺj ∀ᴺk≠j: k<j
|
|  s/j/i  s/k/j  s/i/k
| ¬∃ᴺk ∀ᴺj≠k: j<k
Therefore,
∀ᴺj ∃ᴺk≠j: j<k  implies  ¬∃ᴺk ∀ᴺj≠k: j<k
Define
darkᵂᴹ k  ⇔  ∀ᴺj≠k: j<k
¬∃ᴺk: darkᵂᴹ k
Darkᵂᴹ numbers do not exist in ℕ⁺