Re: how

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

On 6/3/2024 3:50 PM, WM wrote:

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

That is true because our familiar arithmetic is based upon potential infinity.
Only in finished infinity dark numbers are required.

For each n in ℕ⁺
n countable.to from.0
n+1 is countable.to from.n
n+1 is countable.to from.0 through.n
n+1 is in ℕ⁺
n is not larger than all numbers in ℕ⁺

And in particular all n have ℵo dark successors.

𝔼 is the subset of even numbers in ℕ⁺
𝕆 is the subset of odd numbers in ℕ⁺
 ⎛ There is no number in 𝔼 larger than
⎝ all numbers in 𝕆
 ⎛ There is no number in 𝕆 larger than
⎝ all numbers in 𝔼

Dark numbers do not unveil their mystery.
But from NUF we can prove that a first one must exist although we cannot find it.

Only one of the two complementary sets
can and must contain the first point.

 Why?
Responding "Logic" or "Mathematics" is dodging.

But it is true. Like Bob's survival.
Regards, WM